A geometrical practise, named Pantometria diuided into three bookes, longimetra, planimetra, and stereometria, containing rules manifolde for mensuration of all lines, superficies and solides: with sundry straunge conclusions both by instrument and without, and also by perspectiue glasses, to set forth the true description or exact plat of an whole region: framed by Leonard Digges gentleman, lately finished by Thomas Digges his sonne. Who hathe also thereunto adioyned a mathematicall treatise of the fiue regulare Platonicall bodies, and their Metamorphosis or transformation into fiue other equilater vniforme solides Geometricall, of his owne inuention, hitherto not mentioned of by any geometricians.

A GEOMETRICAL Practise, named PANTOMETRIA, diuided into three Bookes, Longimetra, Planimetra and Stereometria, containing Rules manifolde for mensuration of all lines, Superficies and Solides: with sundry straunge conclusions both by instrument and without, and also by Perspectiue glasses, to set forth the true description or exact plat of an whole Region: framed by Leonard Digges Gentleman, lately finished by Thomas Digges his sonne.

Who hathe also thereunto adioyned a Mathematicall treatise of the fiue regulare Platonicall bodies, and their Metamorphosis or transformation into fiue other equilater vnifoorme solides Geometricall, of his owne inuention, hi∣therto not mentioned of by any Geometricians.

[illustration]

Imprinted at London by Henrie Bynneman ANNO. 1571.

To the right honorable my singular good Lorde Sir Nicolas Bacon Knight, Lord keper of the great seale of England.

CAlling to memo∣rie right honourable, and my singular good Lorde, the great fauour your Lordship bare my father in his life time, and the conference it pleased your ho∣nor to vse vvith him touching the Sciences Mathematicall, especially in Geometrical mē∣surations, perusing also of late certaine volumes that he in his youthe time long sithens had compiled in the English tongue, among other I found this Geo∣metricall practise, vvhich my father (if God had spared him life) minded to haue presented your Honoure vvithall, but vntimely Death preuenting his determination, I thought it my part to ac∣complish the same, asvvell for the satisfaction of his desire, as al∣so to shevve my selfe not vnmindfull of so many good turnes as your honor from time to time most aboundantly hath bestovv∣ed on me, hauing therfore supplied such partes of this treatise as

vvere left obscure or vnperfect, adioyning therunto a Discourse Geometricall of the fiue regulare or Platonicall bodyes, contai∣ning sundry Theoricall and practicall propositions of the mani∣folde proportions arising by mutuall conference of these solides Inscription, Circumscription or Transformation, and novv at the last fully finished the same, I am bolde to exhibite and dedicate it to your honor, as an eternall memoriall of your Lordshippes great fauoure tovvardes the furtherance of learning, and a pub∣like testimonie of my bounden duetie: hoping your honor vvill rather respect the good vvill vvherevvith it is presented, than the vvorthinesse of the present, not agreeable I confesse to the ex∣cellent knovvledge vvhervvith your Lordship is indued, euen in the very fountains thēselues vvhence these conclusions as springs or braunches are deriued. And yet suche as I nothing dout your honor vvill both accept in good parte, and also at vacant leysure from affaires of more importance, delite your selfe vvithall, the rather for that it containeth sundry suche nevve inuented Theo∣remes, and other straunge conclusions, as no Geometers haue hi∣therto, in any language published. Wherby your Lordship shall not only incourage me heereafter to attempt greater matters, but also as it vvere vvith a soueraigne medicine preuent the poy∣soned infection of enuious backbiting tongues: for as the veritie of these experimentes and rules shall neuer be impugned, being so firmely grounded, garded, and defended vvith Geometricall demonstration, against vvhose puissance no subtile Sophistrie or craftie coloured arguments can preuaile, so think I there is none so impudently malitious, as vvill or dare reproue them for vaine or improfitable, vvhen they shall perceiue your Lordshippe, (vvhose learned iudgement, grauitie and vvisedome is sufficient∣ly knovven to the vvorld) doth allovv and accepte them as fra∣grante floures selecte and gathered out of the pleasant gardynes Mathematicall, meete to delite any noble, free, or vvell disposed minde, and profitable frutes seruing most commodiouslye to sundry necessary vses in a publike vveale, and suche as shall ther∣by

receyue pleasure or commoditie, muste of duetie yelde con∣digne thankes vnto your Lordshippe, vnder vvhose protection and patronage I haue not feared to sende abroade (as a vvan∣dring pilgrime) this Orphane and fatherlesse child, the vvhich as I perceiue of your Honour fauourably accepted, so meane I, God sparing life, to imploy no small portion of this my shorte and transitorie time in storing our natiue tongue vvith Mathema∣ticall demonstrations, and some suche other rare experiments and practicall conclusions, as no forrayne Realme hath hytherto been, I suppose, partaker of. In the meane time I leaue longer to detayne your honour vvith my rude and homely tale, from more seriouse and vvaygh∣tie affayres, committing youre Lordshippe to the tuition of the Almightie, vvho graunte you a long healthfull honorable life, accom∣panyed vvith perfecte felicitie,

Your Honours moste bounden Thomas Digges.

The Preface to the Reader.

ALthough Geometrie for the certayntie thereof haue suche a priuiledge as fewe other Sciences, béeing so fortified with Demonstration that no Precepte or Rule thereof for the veritie can be reproued, yet least some, either of igno∣raunce or malice shoulde af∣firme it vnprofitable, not ser∣uing to any necessarie vse in a publike Weale, I thought good somewhat to say thereof, and firste the sentence of Plato written at the entrie into his Schole commeth to my remembraunce, 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉. therby excluding al such as were ignoraunt of Geome∣trie as vnable or vnméete to attayn higher secrets or misteries of Philosophie. Aristotle also intreating of Morall Philoso∣phie, in the fifth Booke of his Ethikes, with geometricall Fi∣gures most beautifully paynteth out Iustice, discerning and seuerally comparing sundrie partes thereof with Geometri∣call and Arithmeticall proportions. But his Interpreters, whose works are yet extante, for want of skill in these mat∣ters, haue so blemished, darkned and defaced his meaning, that scarcely any resemblance thereof shall to the Reader appeare.

In sundrie other his works also of naturall Philosophie, as the Physikes, Meteores, de Coelo & Mundo. &c. ye shal finde sun∣drye Demonstrations, that without Geometrie may not possi∣bly be vnderstanded. And to leue Philosophie, how necessarie it is to attayne exacte knowledge in Astronomie, Musike, Per∣spectiue, Cosmographie and Nauigation, with many other Sciences and faculties, who so meanely trauayleth therein shall soone finde. But to omitte talking of Geometrie in ge∣nerall, and to speake priuately of this Treatize, I thinke good first to open the order and effect thereof, and then somewhat to saye of the applyances. Like as euery body geometricall is in∣uironed with Superficies, and euery Superficies inclosed with lines, so that it consisteth of these thrée, Longitude, Latitude, and Profunditie, and without consideration of these thrée no solide may be measured: so is this Treatise also diuided into thrée Bookes. The first entreating of lines sheweth sundry meanes to measure all maner lengthes, heightes, distances, and pro∣fundities. The seconde termed Planimetra settes foorth diuerse meanes and rules to measure manyfolde superficies, playne Conuex and Concaue, whether they be compounde with straighte or circulare lines, or mixte of bothe. In the thirde na∣med Stereometria, is sette out the exacte mensuration of sundry solides, replenished with a number of rules and preceptes, ga∣thered out of Euclide, Archimedes and Appolonius Pergeus his Conykes, wherein the Reader shall not a little delighte him∣selfe with the finenesse and subtiltie of their inuentions, espe∣cially if he indeuor himselfe to search out the reason, cause and demonstration of them. And nowe somewhat to speake of the commoditie of these conclusions, as the skilfull in Architecture can applye the Stereometria to serue his turne in preordinance and forecasting both of the charges, quantities and proportion of all parcels necessarily appertayning to any kinde of buyl∣dings: so Planimetra maye serue for disposing all manner grounde plattes of Cities, Townes, Fortes, Castles, Pallaces

or other edifices. The Marshall of the fielde shall also moste spéedely thereby appoynt place conuenient for his Campe, di∣stributing euery parte thereof according to the number of his men, horse, cariage. &c. Also in suruaying, parting and diuiding of Lands and woods, it is most requisite aswell for exact as spée∣dye dispatch therin, and therof we haue notable recorde in Hi∣stories how much this science auayled the Egiptians when as by the inundation of Nylus their whole countrey was so drou∣ned that with slime of the water all their Bounds and marks were defaced: Yet certayne wise men ayded with knowledge in this science, founde out and distributed to euery man his owne. The other parte named Longimetra the ingeniouse practizioner wil apply to Topographie, fortification, conducting of mines vnder the earth, and shooting of greate ordinance. So that as there is no kinde of man, of what vocation or degrée so euer he be, but shall finde matter both to exercise his witte and diuersly to pleasure him selfe, so surely for a gentleman especi∣ally that professech the warres, aswell for discoueries made by sea, as fortification, placing of Campes, & conducting of Armies on the lande, how necessarie it is to be able exactly to describe the true plattes, symetrie and porportion of Fortes, campes, townes and countryes, coastes & harboroughes, I thinke there are none so vnskilfull, but will confesse these Geometricall mensurations most requisite. But if any there be so rude, igno∣raunte and vnlearned, or so much blinded with selfe liking, that can not be content to acknowledge any thing requisite in a per∣fite souldier that is wanting in them selues (for suche surely they are that moste arrogantly will maintayne this fonde opi∣nion) let them but regard and marke the renoumed Captaine Alexander the great, who had this knowledge in such high esti∣mation, that he seldome or neuer in his manifoulde conquests wold attempt any thing, whether it wer fort, toune or coūtrey, but first he would haue the exacte Topographie therof, and ther∣vpon

inuente, deuise and after execute his warlyke Polycies. On the contrarie side, for negligence of this knowledge we may reade how Aniball the most worthie and famouse Soul∣dier that euer hath béen (all circumstances duely wayed) was by that paynefull, graue and right valiaunt Captayne Quintus Fabius Maximus ledde and trayned into a fielde inclosed with stepe hilles and déepe riuers, where this noble Consull had so inuironed his Hoste by fortifying of two mountaynes, that he with all his Armie were in maruaylouse ieopardie to haue pe∣rished for famine, had not the craftie witte of Aniball inuented a present pollicie to escape suche imminente calamitie. Cyrus also that greate Kinge and mightie Monarche, was he not through ignoraunce of Topographie, euen in pursuing of victo∣rie intrapped and discomfited with al his power by the Scithian quéene Thomiris at the riuer of Oaxis? I passe ouer innume∣rable examples of all ages, manyfestly declaring the greate aduauntage or disaduauntage a Captayne may receyue, aswell inuading as defending by the strong or weake situation of pla∣ces, and by the foreknowledge or ignoraunce of them: neither is there any liberall or frée minde, whether he be of profession Warlike or Ciuile, that will not take great delight and plea∣sure to sée how by Arte a man may measure the distances of places remote and farre a sunder, approching nighe none of them, and that aswell, yea and farre more exactly than if with Corde or pole he should paynfully passe them ouer. Herevpon did the Poetes fayne Atlas of suche huge and mightie perso∣nage, sustayning and vpholding the earth and mightie Masse of heauenly spheres, for that this man notwithstanding he were imprisoned in a mortall carkasse, and thereby detayned in this moste inferior and vilest portion of the vniuersall world, far∣dest distante from that passing pleasant and beautifull frame of celestiall Orbes, yet his diuine minde ayded with this sci∣ence of Geometricall mensurations, founde out the quanti∣ties, distances, courses, and straunge intricate miraculouse

motions of these resplendent heauenly Globes of Sunne, Moone, Planets and Starres fixed, leauing the rules and pre∣ceptes thereof to his posteritie. Archimedes also (as some sup∣pose) with a glasse framed by reuolution of a section Parabo∣licall, fired the Romane nauie in the sea comming to the siege of Syracusa. But to leaue these celestiall causes and things doone of antiquitie long ago, my father by his continual payn∣full practises, assisted with demonstrations Mathematicall, was able, and sundrie times hath by proportional Glasses duely situate in conuenient angles, not onely discouered things farre off, read letters, numbred péeces of money with the very coyne and superscription thereof, cast by some of his freends of purpose vppon Downes in open fieldes, but also seuen myles of declared what hath béen doon at that instante in priuate pla∣ces: He hath also sundrie times by the Sunne beames fired Powder, and dischargde Ordinaunce halfe a myle and more distante, whiche things I am the boulder to reporte, for that there are yet liuing diuerse (of these his dooings) Oculati testes, and many other matters farre more straunge and rare which I omitte as impertinente to this place. But for inuention of these conclusions, I haue hearde him say, nothing euer helped him so muche as the exquisite knowledge he had by continuall practise attained in Geometrical mensuratiōs. And for science in great Ordinance especially to shoote exactly at Randons (a qualitie not vnméete for a Gentleman) without rules Geo∣metrical, and perfect skill in these mensurations, he shall ne∣uer know any thing: thus haue I partely declared the plea∣sure and commoditie that any well disposed minde may reape by these thrée bookes of my fathers. But somewhat to say con∣cerning the last Treatize of the fiue regulare bodies and their Metamorphosis adioyned by my selfe, for that I knowe some, yea many shall there be that will not spare to say, as they haue done by others, it is a fonde toy, a meare curiouse trifle, ser∣uing to no vse or commoditie: surely I meane not greatly to

labour to attayne the good opinion of suche▪ hauing learned howe the disprayse of the ignoraunt and vitiouse is no lesse cōmendation than the prayse of the wise and vertuouse, béeing a precedent infallible from the beginning and alway conty∣nuing that ignoraunce enuieth knowledge, and vice vertue. Euery mā disliketh or extolleth things accordingly as they are agreable or repugnant to his appetite, and rather would I en∣ioyning the company of Euclide, Archimedes, Appollonius Per∣geus, and other geometers writing of the like toyes & curiosities be condemned, than in publishing matter agreable to suche mens phantasie and conceypt, be of them highly commended, and of the learned laught at or contemned, but howsoeuer Epicurus, Midas, or their like, giuen onely to lucre and world∣lye pleasure liste to thinke thereof, I nothing mistruste of suche as couet the vnderstanding of matters harde and diffi∣cile, desiring the knowledge of things somewhat passing the reache and capacitie of the common sorte, wherein onely the nature of man surmounteth beastly kinde, or by proofe and assay in cases of like difficultie able to iudge them selues, this Treatize shall not be mislyked but thankfully receyued, and for the reste perswasions are but vayne, for as no words can adde stomacke or make the cowarde valiaunt, so surely suche two footed Moules and Todes whom destinie and nature hath ordayned to craule within the earth, and suck vpon the muck, maye not possibly by any vehement exhortation be reduced or moued to taste or sauour any whitte of vertue, science, or any suche celestiall influence, my hope is if any fault escape, as in suche long and intricate tediouse calculations of irrationall numbers maye happen to be some, the discrete and modest Reader will rather of curtesie amende it, than with enuiouse cauillation, vngratefullye requite my paynefull trauayles, whereby I shall be prouoked not onely to publishe the demon∣strations of these and many moe straunge and rare Mathe∣maticall Theoremes, hytherto hidden and not knowen to the

worlde, but also to imprinte sundry other volumes of my fa∣thers, which he long sithence compiled in the Englishe tong, desiring rather with playne and profitable conclusions to store his natiue language and benefite his Coun∣trey men, than by publishing in the Latin rare and curiouse demonstrations, to purchase fame among straungers.

Elementes of Geometrie, or Diffinitions. (Book 1)

A Poynt I call whiche cannot be diuided, whose parte is nothing.

A Lyne is a length without breadth or thicknesse, whose ex∣tremities are two poyntes.

THe shortest drawen betwéene two Poyntes is a streight line, the contrary are crooked lines.

[illustration]

A Superficies is that hath length and breadth onely, béeing bounded or determined with lines.

A Playne Superficies is that whiche lieth equally and euen∣ly betwéene his lines or boundes.

[illustration]

A Playne Angle is the inclination of two lines lying in one playne Superficies, concurring or méeting in a poynt.

IF those lines that containe the Angle be straight, it is called a right lined Angle, and those two lines his containing sides, but if a third straight line be drawne crosse the former from one to the other, that shall be called the subtending side.

[illustration]

OF straight lined angles there are thrée kindes, the Ortho∣gonall, the Obtuse, and the Acute Angle.

WHen any right line falleth Perpendicularly vpon an o∣ther, that is to say, making the Angles on either side e∣quall, eche of those Angles is an Orthogonall or right Angle, and that falling line a Perpendicular.

[illustration]

BAC the right angle contained of the Perpendicular, and one part of the ground line equall to BAD the right angle contained of the Perpendicular, and the other portion of the grounde line, and therfore both Orthogonall.

The Brode or Obtuse Angle is greater than the Orthogonal.

[illustration]

The Acute or sharpe, is lesser than the right angle.

[illustration]

A Figure is comprehended within limites and bounds, whe∣ther it be one or many.

A Circle is a plaine figure, determined with one line, which is called a Circumference, in whose mids there is a point named his Centre. From the which all right lines drawne to the circumference are equall.

[illustration]

A Semicircle or halfe Circle, doth conteine both the Dime∣tient and Centre of his circle, with the precise halfe of his circumference.

A Right line drawne through the Centre vnto the Circum∣ference of both sides, is named his Diameter or Dimeti∣ent, the halfe of it is called his Semidiameter.

[illustration]

ALl Straight lines besides the Diameter in any Circle pulled from one part of the Circumference to the other, be called cordes.

THe portion of the Circumference from that corde compre∣hended, is named an Aroke.

A Touch line is that toucheth a circle in a Pointe.

[illustration]

[illustration]

EMong Right lined figures, suche as haue onely thrée si∣des are Triangles, whereof there be sundrie sortes bearing seuerall names, according to the diuersitie of their sides and Angles.

IF the Triangles thrée sides be euery of them of like length, it is called an Equilater Triangle.

ISoscheles is such a Triangle as hath onely two sides like, the thirde being vnequall, and that is the Base.

Schalenum hath thrée vnequall sides.

[illustration]

A Rightangled Triangle is suche a one as hath one Righte Angle.

AN Obtusiangle Triangle hath one obtuse angle, and is called Ambligonium.

Oxigonium hath all acute or sharpe Angles.

[illustration]

THere be also foure sided Figures called Quadrangles, whose Opposite sides and angles are equall, suche are na∣med Paralelogrammes, whereof there are but foure sortes.

IF all the sides be equall, and al the angles right, than is that Paralelogramme called a square.

IF one side containing the right Angle, be longer than the o∣ther containing side, then is that figure called a Rectangle.

IF all the sides be equall, and no angle aright, then is it called Rhombus.

BUt if it haue only the Opposite sides equall, and the other that containe an Angle vnequall, it shall be named Rhom∣boides.

[illustration]

All other quadrangles are Trapezia.

POlygona, are such Figures as haue moe than foure sides, whose angles if they be all like and equall, they are termed Equiangle Polygona.

[illustration]

AL other plain Superficies, whether they be enuironed with straight or crooked lines, shalbe named irregulare figures.

WHen two right lines drawne in one plaine Superficies, are so equedistantly placed, that though they were infi∣nitely extended on either side, yet would neuer méete nor con∣curre, they shall be called Paralleles.

[illustration]

A Quadrant is the fourth part of a Circle, included with two Semidiameters commonly diuided in .90. portions, which partes are named grades or degrées.

The first Chapter. Hovve Perpendiculares vppon any straight line are ereared,

ADmit AB were the line to be crossed, 〈◊〉〈◊〉 that ye desired a Perpendiculare or plumbe line in C, open your compasse, put the one foote in C, make of either side the line one pricke, D.E. Nowe extend the compasse to the widenesse of bothe, or shorter, putting the one foote in D and the other immouea∣ble, making an arcke ouer and vnder C. This done discretely, remoue the cōpasse frō that Centre to E (remaining so opened) there fixe one foote, with the other crosse the arke afore made aboue and beneath C, where make two points, or these letters FG. Then take a Ruler and lay him vpon bothe the poyntes crossing the centre C. Thus drawe your plumbe or squire line FCG. In like manner any line may be deuided in halfe, or cir∣cle in four equall partes. Sée the example on the side folowing▪

[illustration]

The second Chapter. Hovve perpendiculare or hanging lines are dravven from a point assigned to any right line lying in the same playne Superficies▪

SUppose C the point from whence I wold let fal a Perpendiculare to the streight line A.B. open therefore your cumpasse of suche wydnesse, that placing the one foote in C. ye may reache with the other beyond the line A. B. and drawing an Arke, note where it ma∣keth

[illustration]

intersections with the same line, whiche in this Figure are marked with the letters D. and E. then placing one foote of your cumpasse in E draw an arcke directlye vnder C. and then fixing one foot agayne of your compasse

in D. crosse the former arcke in F, finally drawe the streight line CF. for that is a perpendicular to the line AB.

PIthagoras inuention might here take

[illustration]

place, who did finde these numbers 3.4. and 5. or like ioyned measures to make a right angle.

The third Chapter. From any point assigned to extende a Parallele to any other right line lying in the same Superficies.

FRom the point assigned let fall a perpendicular to that line, and from some other pointe in that line ereare a perpendiculare, as ye were in the last Chapters taught, then opening youre com∣passe to the length of the perpendiculare let fall from the point assigned, measure out the like length in the perpendiculare ereared, beginning from the ground line: then laying a ruler to the point assigned, and the ende of that length drawe a streight line, for that shall bée a Paralelle to the other.

Example.

ADmit A the point assigned, BC

[illustration]

the line AD. the perpendicular let fall from A, CF. the perpendiculare ereared from C. my compasse opened to the wydnesse of AD, I set one fote in C, cutting CF, with the other

in E, then applying my ruler, I drawe the line A E, which is a Pa∣rallele to B.

The 4. Chapter. To diuide any limited right line into as many equall partes as ye list.

FOrasmuch as hereafter in drawing of plattes and mensuration of ground by instrument, the diuision of right lines into manye equall partes is requi∣red, I thought good to gyue instruction thereof before I entreate of those matters: ye shall therefore vpon either end or limite of the diuisible line, ereare a Perpendiculare, the one vpward the other dounwarde, and opening your compasse at aduenture, measure out so many partes in either perpendicu∣lare, as you would make diuisions in your line, and drawing right lines frō the pointes in one perpendiculare to the points in the other, beginning from the first of the one, to the laste of the other, ye shall diuide the line gyuen, into so manye equall partes as there be diuisions in your perpendiculares.

Example.

ADmit the line AB whiche I would diuide into 7 equall porti∣ons, I ereare vppon A and B, the perpendiculars AC, BD, as you may behold in the Figure: and opening my compasse at ad∣uenture, I measure oute 7 partes ending at EF. then drawing lines from the diuisions of the one to the diuisions in the other (beginning from the last in one perpendicular, to the firste in the other) you may

beholde in the Figure the line

[illustration]

A B, parted into 7 equall por∣tions, in this manner maye you proceede infinitely to diuide it in∣to as many portions as you list.

The 5. Chapter Hovv equall angles are made.

FIxe one foote of your compasse vppon the con∣course or méeting of those two right lines that contayne the angle whose like or equall you would make, and opening youre compasse at pleasure, describe an arke cutting the two con∣tayning sides of the angle: then draw an other right line & pla∣cing one foote of the compasse (remayning immouable) there∣on, with the other describe an Arcke rysing from that laste drawen line, then resorte to your angle, and open youre com∣passe to the wydnesse or distance of the two intersections made by the arcke in the two contayning sides, and transporting the same distance to your second arcke, sette one foote of the com∣passe at the beginning thereof, I meane where it ryseth from the line, with the other cutte the laste described Arcke, then laying youre ruler to that intersection and to the centre of the arcke, drawe a right line till it concurre with the other: thus haue you a new angle equall to the former.

Example.

SVppose BAC, the angle whose like or equall I desire, DE, the arcke drawen with one foote of my compasse, whyle the other re∣mayned in A, the compasse im∣mouable,

[illustration]

I set one foote in the line FG, drawing with the other the arke KI. This done, I open my compasse to the di∣stance of DE, and placing one foote in K, with the other I crosse the arcke in I, finally, laying the ruler to I and F, I draw the line FH. and thus haue I made the angle HFG. equall to the firste angle BAC.

The 6. Chapter. To make a triangle equall to any other right lined triangle assigned.

FIrste, as yée were taught in the laste proposition make an angle equall to some one angle in that triangle, it forceth not which of them it be. Then extende oute those streight lines that contayne this angle till they be of equall length with the contayning sides of his corresponding angle in the triangle. This done, couple y^e endes of those two right lines togither with a thirde, and so haue you framed a Triangle equall to the former.

Example.

THe triangle assigned is ABC, to his angle at B by the former probleme I frame an equal or like, conteined with the lines DEF, extending ED and EF, till

[illustration]

they become of equall lengthe with AB and BC, whiche I may easely do, by extending my compasse first to AB, BC, and after transferring those di∣stances or lengths to DE and EF. Finally applying my ru∣ler to the ends or limits of those lines, I drawe the subtending side DF, and thus haue I framed a new triangle DEF, equall to the other ABC.

I Thinke it not amisse before I entreate of Geometricall mensurations, to premise certaine Theoremes whereby the ingenious may redily conceiue the grounde, reason, and de∣monstration of suche rules as shall ensue.

The first Theoreme.

ENy tvvo right lines crossing one an other, make the contrary or verticall angles equall.

The seconde Theoreme.

IF any right line fall vpon tvvo Parallele right lines, it maketh the outvvarde angle on the one, equall to the invvarde angle on the other, and the tvvo invvarde opposite angles on contrarye sides of the falling line also equall.

The thirde Theoreme.

IF any side of a triangle be produced, the outvvarde angle is equall to the tvvo invvarde opposite angles, and all three an∣gles of any triangle ioyned togither, are equall vnto tvvo right angles.

The fourth Theoreme.

IN equiangle triangles, al their sides are proportional asvvel such as conteyne the equall angles, as also their subtendente sides.

The fifth Theoreme.

IF any foure quantities be proportionall, the firste multiplied in the fourth, produceth a quantitie equall to that vvhich is made by multiplication of the seconde in the thirde.

The sixth Theoreme.

THE visible beames falling on playne conuese or concaue glasses, are reflected in equall angles.

The seuenth Theoreme.

IN right angled triangles the square of the side subtending the right angle, is equall to bothe the squares of his conteyning sides.

The eight Theoreme.

AL Parallelogrammes are double to the triangles that are de∣scribed vpon their bases their altitudes being equall.

The ninth Theoreme.

AL like or equiangle Figures retayne double the proportion of their correspondent sides.

The tenth Theoreme.

IF from any angle of a triangle to his subtendent side, a perpen∣diculare descende the square of that subtendente side, or basis added to one of the conteyning sides square, surmounteth the square of the third side, by a rightangled Parallelogramme con∣teyned of the vvhole base and double that his portion, vvhiche lyeth betvveene the perpendiculare and the laste named thirde sides subtendente angle.

The .7. Chapter. The description of the Quadrant Geometricall.

[illustration]

FIrst ye must make a common simple large quadrante thus, with your compasse drawe an Arke or Circumference, that

may be more, or at the least sufficient for a quadrant, then put both the féete of your compasse in that arke, making two Prickes. Now the distaunce of these two poyntes diuided in two equall parts, adding one portion to the aforesaid circum∣ference or distance, sheweth a precise quadrant, ye ought then to pull of eche side from the centre a line to the vttermoste poynts, which be the extremes of your quadrant. Again draw a line from your centre A to the middes of the quadrants cir∣cumference C, and if ye liste ye may diuide that quadrant into 90. grades thus: First in 3. partes, then euery thirde in 3. so haue ye 9. portions: Now euery of them in 2. riseth .18. Then eche in 5. equall partes maketh .90. degrées.

Of the Scale.

FOr the Scale ye shall draw from either side of your qua∣drant a right hanging line (as is declared) touching the middle line in one poynt, so haue ye the sides of your scale eche to be deuided in 12.60.100.1000 poynts all marked from the centre A, the mo the more commodious. Forget not to haue two equall fine plates of brasse persed in the middes (for your sightes) and placed on the side AD, as ye sée EF, with a line and plummet falling out of the centre A. I call the scale in this quadrant the two sides within diuided in certayne portions or partes. And those .12. partes next to your sights I name poynts of right shadow: thother side of the scale portions or poynts of contrary shadow: better it were, yea and more for the purpose if eche side had .60.100. or, 1000. diuisions.

The .8. Chapter. The vse of the Scale, shewing perpendiculare or directe heightes by their shadowes.

[illustration]

COnuey the lefte side of your quadrant Geometricall to∣warde the Sunne, the threade and Plummet hauing their frée course mouing it vp or downe, vntill bothe your sightes haue receyued the sunne beames. Then yf your thread be founde in the twelfth parte, shadowes of all things (béeing perpendiculare eleuated) be equall with their bo∣dies, yf the plummet with the thread be perceiued cutting the parts next to the sightes, which I call right shadowes, then euery thing direct is more than his shadow. By that proportion which 12. excéedeth the partes where the thread was founde: yf it fall on the firste parte of righte, take the shadow 12. times to make the heigthe, yf it chaunce on the second portion sixe times, on the third foure times, on the fourth thrise, in the fifth twise and two fiftes of the shadow, in the sixte poynte twise, in the seuenth once, and fiue seuenths of the shadow, in the eight portion once and a halfe, in the ninth once and the thirde part, in the tenthe once and the fifte parte: in the eleuenthe poynte ye shal take

the shadow once and the eleuenth parte of that shadowe: or in fewe wordes, multiplye the length of the shadowe by 12. and the producte di∣uide by the partes in whiche you founde the threade, your quotient she∣weth the heigth: but and if it be in the partes of contrarye-shadowe, augment the lengthe of the shadowe with the partes declared by the Plummet, and the encrease diuide by 12. so commeth the altitude also. Ensample, in the figure that goeth before it is playne to be perceyued. When the thread falleth on 12 portions, the shadowe is equal with the thing it selfe. In 6 of right it is but halfe, in sixe of contrarie it is twice the heigth: So to conclude, ye may sée as the side in righte excéedeth the partes, so doth the altitude or bodie the shadowe, and contrary in con∣trary shadowe, beholde your figure how the threade cutteth 6 partes of contrary shadowe in the Quadrant next to the right hande, the shadowe BC then being 210 foote, multiplie (as I haue sayde) the length of the shadowe 210 feete, with 6 the partes cut by the threade, encreaseth 1260, that diuided by twelue, riseth 105: the altitude of suche a bodie whiche had a shadowe then 210 féete. Thus of all such like.

The .9. Chapter. Of Vigetius concerning heigthes.

[illustration]

HE affirmeth by any certaine measure directely standyng (whose shadow is knowne) the heigth of any other thing the shadowe then measured not to be hid, performed by the rule of proportion. Example.

Suppose the shadowe of any thing 210 feete. Now say 20 the shadow of a thing knowne giueth 10, what shall 210, riseth 105: the heigth.

The .10. Chapter. VVithout shadovve or any supputation by your quadrant geometricall to take heigthes approchable.

[illustration]

LIfte vp ingeniously your quadrant exactly made towarde the thing to be measured, looking diligentlye through both the sightes, going backe or forwarde as occasion is giuen vntill ye see the toppe, so that your fine or subtile thread fall iustly vpon the twelfth poynte. Now if you measure the distaunce from you to the base (which base here I call the poynt directly vnder the toppe, then haue ye the altitude of the highest summitie to the right poynt or base in heigthe equall with your stan∣ding, adioyning vnto it the heigth of your eye downewarde. Ensample, The lyne and plummet in the figure afore falleth precisely in the twelfthe portion, the space then being from you to the base, whiche is from A to C 15. foote. To this ye must adde the heigth of your eye (here ymagined 5. foote) so haue ye 20 féete, the true altitude from A to B. As the length of the corde from the eye to C sheweth the measure to be layde backe: so doth the touche of the line and plummet in C, declare where ye muste beginne to lay the measure backe.

The .11. Chapter. VVith the ayde of tvvo places to search out improcheable heightes.

[illustration]

SEeke two stations going hither and thither, yea towarde or from the thing ye intende to measure, so that in the one place the thread may fall in 12, the other station in 6. pointes of right shadow, then if yée double the distance of both places, the summitie shal appeare from that parte of the thing measured whiche is equall in heigth with your eye, or if your standing be euen with the base, ioyning to that doubled distance the heigth of your eye, ye haue the whole altitude from the grounde, &c. If the one roome cause the thread to fal in 12. the other in 8. of right shadow, then triple the space, so haue ye the heigth also. Or the one in 12. the other in 9. of right shadow, then quadruplate the distance: yea the one vnder 12. that other 6. of contrarie shadowe, then the space betwéene both stati∣ons is equall with that you measure, euer vnderstanding from youre eye vpwarde. Or if the plummet be enforced to fal vnder 6 points of the contrary shadowe, the other vnder 4 partes of the same, or in 4 and 3 of contrary, in al these the distance of the place is equall with the altitude. So then in measuring the space betwéene the two places ye haue gotten the heigth from your eye vp, putting vnto it (as I haue sayd) the length from your sight dounwarde, the iust altitude of the whole appeareth: the Base being euen with your standing. Example, This Figure decla∣reth the falling of the thread vnder 6 of right, and 12. Also vnder 12 and 6 of contrarie, by doubling the space betwéene the two firste places, the altitude appeareth. In like manner the distance from the middle stan∣ding to the last, bringeth the heigth from your sight.

Howe lengthes in heigth are knowen.

I Would not haue you ignorant héere how to knowe lengths which be in heigth not easie to come vnto, first by your art afore mentioned get eyther heigth, subduce the lesse height out of the more, of force youre desi∣red length remayneth. Or thus, let the plummet fall vnder 12 portions, marke your place, go in toward the thing (the threade as it was) vntill ye sée the base of that length, so the distance betwéene the two standings is vndoubtedly the lengthe: héere néedeth no Example.

The 12. Chapter. Hovv by your Quadrant vvith calculation speedely to knovv all heightes accessible.

YOur Quadrant as tofore is sayd handsomly eleuated against or toward the thing to be measured, percey∣uing thorow the sights not more than the top. Marke well the diuisions of pointes touched in your scale, if they be of right shadow, multiplie the distance from you to the base by 12, and diuide by the partes of your Scale which your thread made manifest. But and if they bée of contrarie shadow, worke contrarely: that is augment by the parts, and make particion by 12, remembring euer to adde the heigth of your eye dounward to your Quotient, so haue ye your desire: the Base being equall with your standing.

Example.

Admit the thread with the plummet note 6 partes of contrarie, as ye may see in this Figure: the distāce frō the base A to your stāding B 115 foote, mul∣tiplie that by 6, so haue ye 690, the which diuided by 12, yeldeth 57 ½ foote: to this adioyne fiue foote (being the heigth of your sight to the grounde) conclude the Altitude 62 ½ foote.

Heere yee shall note that in mensuration of heightes with this in∣strument, it shall be requisite first to finde what parte of the Altitude is leuell with your eye, which you may thus doo:

CAuse the plummet and thread to fall vpon the side line of your qua∣drant where the degrées beginne, and then searching thorough the sightes, that part which you can espie of youre Altitude, (the plummet hanging vppon the foresaid laterall line) is leuell with youre eye, the heigth whereof from the base compared with the altitude of youre eye, discouereth the vnequalitie or difference of the ground: that is to saye, how much higher or lower the base of the thing to be measured, is than the ground at your station which difference as ye shall sée cause added or detracted from your heights found as is before declared, yeldeth most exactely the true altitude, and thus shall you be assured neuer to erre, how vneuen or vnleuell so euer the ground bée.

[illustration]

The .13. Chapter. To get inaccessible heightes by supputation (vvith the helpe of tvvo places) supposing either side of the Scale diuided, 100 partes.

IF your thread in the first station fal vpon 50 points of contra∣rie, with those diuide 100, so haue ye 2. In y^e other place (going right back or forward no way declyning) admit it note 25 of contrarie, now 100 diuided with 25 riseth 4, withdrawe 2 from 4, 2 is left your diuidēt, mete the space betwéene both standings, and di∣uide that by 2, youre diuisor, so haue ye the heigth from the eye vppe. Note, if the difference of the Quotient be 1, the space betwéene the stan∣dings shalbe equall with the desired heigth, adding youre stature. If 2, the space is double to the altitude, if 3, thréefolde, &c.

Or thus worke: Reduce the partes of contrarie shadow vnto portions of right, and then doo as you would with pointes of right: that reduction is made thus, multiplie 100 in him selfe, so haue ye 10000, the which di∣uided by euery parte of contrarie shadowe, so shall they be as pointes of right shadowe. And if you haue made two stations, pull the lesse Quoti∣ent from the great, the reste waighe as you haue ben instructed. No end hath the Geometer in finding true measures, many I might say infinite mo wayes heightes are found, by any two equall things orthogenally ioyned, by staffe, corde, squire, triangle glasse, &c. as bréefly followeth.

By any two things of one length ioyned thus in right Angle, Altitudes are found.

THe ende C applied to your eye, goe backewarde or forwarde as you shall sée cause, till you can espie the toppe and base of youre Altitude, by the extremes AB: so doing the distance

[illustration]

betwéene youre foote and the base, is equall to y^e heigth, without adioyning the altitude of your eye, whiche in all the reste before shewed is required, only héere ye shall take héede so to couple AB and CD, that in be∣holding the altitude, your line AB may de∣pende perpendiculare or equedistant to the heigth. In like sorte may you mete the di∣stance of any two things in sight, and that exactely, if you vse discretion in placing AB, that it be alway parallele to the line measured.

The 14. Chapter. Heightes are ingeniously searched out by a staffe.

IF any staf be erected, the measurer vpō his backe beholding y^e top of the thing, the distāce of the eie from the foot of the thing sheweth the heigth. Or thus receyue my mynd more largely, prepare a right staffe diuided in 12 or mo equal parts, y^t done, set it right vp a certeyn distāce (as ye list) from the heigth which you wil measure. Now go right frō that staf some space at pleasure: laying your eye to the ground equal with the base of the thing to be measured, mo∣uing back or drawing néer to that staffe, vntil ye may rightly and plain∣ly

sée the very summitie or vpmost parte of the thing to be measured, by the top of your staffe, which performed make a marke wher your eie had his place: Now measure the distance or space from the staffe to your eye with the staffe it selfe, and note what proportion the staffe hath to the distance, the same shall youre heigthe haue to the lengthe from your eie to the base of that Altitude. Ensample. The staffe CD (in this figure) and the distance CE are equall, therefore affirme the heigth AB to be equall yea so long as the distance betwéen your eie and the base of that required heigth which is AE, if otherwise according to the pro∣portion afore mentioned, ye may by the rule (called the golden precept,) bring the iust heigth thus, méete the ground betwéene your eie & the staffe suppose it 12 foote, then the distance from your eie to the base 200 foote, your staffe 5 foote, say of 12 commeth 5, what shall come of 200, so haue ye 83 féete and ⅓ your exact heigth.

The .15. Chapter. Ye may also heereby redely measure all lengthes standing in heigth as the altitude of any Tovver standing on a hill, or the length of a steeple aboue the battlements, or the distance be∣tvveene story and story in edefices, your selfe standing on the grounde.

FOr Example I would measure the distance betwéene B and G, suppose the line visuall EG cut your staffe (which I would wishe diuided in 12 partes) vppon the fourth part from the toppe. Then woorke by the rule of proportion: saying 20 parts the distance betwéene the eie and the staffe geueth 200 foote, the distance of the tower, what yeldeth 4 parts, thus shall you finde the fourth number proportionall 40 foot, which is the exact length of BG. And thus may you measure only by a staffe di∣uided in 12 equall partes (without any other instrument) any altitude, how so euer it be situate.

Beholde the Figure on the other side.

[illustration]

The like is brought to passe by the Squire, beholde the Figure.

ALwayes DC will retaine the same proportion to DE, that BF doth to AB, so that BF measured, you may by the golden rule attayne the heigth of AB, or contrary if you knowe the Altitude AB, you may proportionally learne the longitude FB: Pleasanter to practize is this than the former and moste exact for Altitudes. But where as some do vse it as well for longitudes and distances, makyng the staffe a side common of

[illustration]

[illustration]

the greater and lesser triangles: though demonstration beare it, I allow not of it, the angle made with the squire and staffe, groweth so acute, and vnsensible, that great errour ensueth the least mistakyng.

The .16. Chapter. By a Glasse heighthes may be pleasantly practized and founde on this vvise.

CAst a glasse on the grounde, so it may lye equall, yea euen in heigth with the base of the thing to be measured, your eie on the glasse fixed, go from it vntil ye sée no more than the very top of that thing of which ye require the heigth. Then let a Plummet with a line come from your eie to the grounde, marke the fall of that Plummet, loke what proportion the distance beareth (from the saide Plummet to the Centre of the glasse) compared to the length of the corde, the same shall the space betwéene the glasse and the base of the heigth, haue to the altitude desired.

[illustration]

THe precept of proportion may be as well héere vsed as afore, and so to attaine heightes. Beholde the figure, as the distance DC from the fall of the Plummet to the centre of the glasse, is equall with the line ED, falling from the sight E so the length CE from the glasse to the base of the tower, swarueth not from the desired Altitude AB.

The .17. Chapter. To measure the deepenesse of any vvell by the Quadrant Geometricall.

NOwe to returne to the first instrument Geometricall and so to end, the bredth or Diameter knowne, set your Qua∣drant on the end of your Well in the very toppe, then lift vp or put downe this Quadrant vntill ye see the bottome on the contrary side, marke the poyntes cut, looke what proportion the partes haue to the whole side, the same shal

the Diameter haue to the length or déepenesse. Therfore by the number of the pointes forme diuide 12, so the Quotient vttereth howe often the 〈◊〉〈◊〉 of the Well maketh the depth, or multiply the breadth by 12, the 〈◊〉〈◊〉 diuide by the pointes found, the depth also appereth. Ensample. In 〈◊〉〈◊〉 figures folowing there be 3 pointes of the Quadrant cut &c. 3 in 1•• is contained foure times, so is the Diameter in the length or profundi∣tie. In like manner of proportion (as is declared) ye may gather the lēgth of things ascending some high Turret. Farther note whatsoeuer I haue sayd of the Quadrant appertaining to heightes know that to be spoken of the square Geometricall, which héere shalbe put forthe for lengthes only, one reason one ground serueth them both. As easie also is it to know how much water, I meane how many quartes, gallons, or other measure are cōtained in these Welles, or in any other Regular excauate body, which shall plainely appéere in the last kinde of Geometrie in place due.

[illustration]

By this instrument also ye may knovve from any spring or foun∣taine, vvhether the vvater may be conueyed by Pipes or o∣thervvise, to any other place or places hovv farre distant so e∣uer they be, yea though there be many hilles and valleis be∣tvvene.

I Would not haue you ignorant that the nature of water is such, as by pipes it may be rered aboue the fountaine hee, and caried ouer hilles or mountains how high so euer they be, so that the vent or end where the water must issue out of the Pipe be inferior to the Fountaine whence it is de∣riued: neither néede you care though there be many valleis in the way, for experience teacheth that the lower your water falleth, the more fréely it will runne, and the more pure and holesome it shall be. Al∣wayes it behoueth you to haue consideration of the Fountaine whence it procéedeth, and the Sestarne, Well, or other place whereinto it falleth▪ that this be lower than the other. It is also to be wayed how this differēce of highnesse and lownesse is to be accompted, some suppose that all places lying in a straight leuel line from the spring hed, are of one heigth, which opinion is erronious, because the water (being an heauy body as the earth is) presseth and tendeth alway to the center. And in all his courses (being not violently forced contrary to his nature) moueth downeward, or at the least vniformly and equodistantly frō that centre. Wherby it is manifest that no spring can of his own nature run in leuel or right line frō his hed, for this equidistant course to the centre, is an arke or portion of a circle. But euery leuel right line (considering it is a contingent or touch line) is caryed aboue the circumference, and the farther it is extended, the farther distāte it is frō the Centre, so that either the water must make his course vpward vtterly contrary to his nature, or else it shall decline frō this le∣uel right line. In déede the globe of the earth & water being so great, any smal Arke or portion of their Circumference will not sensibly differ frō a straight line. And yet in conueying of waters any great distance, very ex∣perience wil bewray an error. The meane therfore to attaine perfection héerin, is to finde the difference betwéene this straight and Circular line, wherof héereafter I shal intreat more at large: only héere will I opē how without any error sensible, (sufficient for any Mechanicall operation) by the aid of sundry stations ye shall accomplish this purpose.

First it behoueth ye diligently to marke at the fountaine head the Su∣perficies of the water, and if the ground so serue that ye may place your eie euen therwith, take your Quadrant, and turning your selfe towarde the place whether you meane to conuay this water, (if it be within sight) espie the same through the sightes of your Quadrant, meting diligently the fall of the Plummet, which if it cutte any of the Degrées, ye may con∣clude

it is not possible for the water naturally to runne thither. But if your Sestourne or place be not to be séene at the Spring head, then es∣pie some other marke through the sight of your Quadrant on either side towarde your place, alwayes causing the thréede and Plummet to fall directly vppon the laterall line where your Degrées beginne, then re∣mouing and situating your eie at the poynte or marke which you laste es∣pyed, finde throughe your sightes a newe marke, causing your Plummet and thread to fall vppon the aforesayd laterall line, and thus procéede from station to station, till you come to the sight of your last place, then if your Plummet and thread cutte the Degrées, ye may conclude as a∣fore▪ but if the grounde at the Spring head be suche, as you cannot con∣ueniently place your eie as I haue sayde, then let fall from your eie, or some place of equall heigth with your eie, a string and Plummet to the brimme of the water, measuring the length thereof. If at your laste sta∣tion your Plummet and thread hanging as I haue tofore sayde, your vi∣suall line passing throughe the sightes of your Quadrant fall aboue the Sestourne where this water should issue out, erreare a Pole or suche like thing to the heigth or length that the string was at the Fountaine head: and if your visuall line reache higher than the toppe or summitie of that Pole also, ye may conclude that this water may be deriued thi∣ther. And if from that parte of the Pole your visuall line cutteth, ye a∣bate the Perpendiculer from your eie to the water at the fountain head, and for euery Mile trauailing 4 inches, the poynte where you leaue is exactly leuell with the Superficies of the water, and so highe it may be brought, and not aboue. It behoueth ye also to take order that your sta∣tions be not aboue 200 or 300 pace at the moste a sunder, otherwise error sensible may ensue. Ye may also (if néede require) at euery station erecte an high Pole, and so may you passe ouer both mountaines and valleys, alwayes noting at euery station, what portion of the Pole your line vi∣sual dothe cutte, reseruing them to be added or subtracted as you sée cause, at your last station. Superfluous, yea rather teadious should it be to vse moe woordes in so plaine a matter, the ingenious Practisioner will finde sundrye wayes to healpe himselfe as occasion requires by sight of the grounde. &c.

The .18. Chapter. To get the length or distance of any place or marke in sight, be it neuer so farre, and that vvithout instrument.

AMong many practises I finde sixe wayes principally be had in estimatiō, the first ensuing without any instru¦ment, other requiring aide of instruments, whose name compositions and vse folow as séemeth méete. Although in measuring of lengthes after the mindes of many wri¦ters the auoyding of hilles, & in few woords most plaine hath bin desired, least great errors should ensue, héere such things shal not be required: Only it shalbe néedfull at the time of your measuring to haue ground at libertie on the one side. This commoditie had the ground leuel or otherwise woorke thus, at the beginning of your length set vp a staf•• or marke which may be séene a farre of, then go from it Orthogonally •• squirewise of which side ye will 200 foote, or as ye list, the more ground•• the better, put vp there a staffe also: Now conuay your selfe to the first staffe or mark going back frō it 300 foote more or lesse at your pleasure, set vp there the third staffe, so that the first marke or staffe and it agrée•• in a right line from your sight to the farthest point of your length by the iudgement of your eie. Nowe go sidewise from thence as afore in a right angle vntill the second marke offer it selfe aright betwéene the extréem•• part of your length and sight, there put vp the fourth staffe. All this per∣formed, séeke out the distance betwéene the first staffe and the second that name your first distance, then the length betwéene the first & third staffe, call that the second distance. Againe the space betwéene the thirde and the fourth staffe is the third distance, subduce the first distance from the third, so remaineth your Diuisor, then multiply the third distance by the second, & the product diuide by your diuident or diuisor, the Quotient she∣weth the true length from the third staffe to the fortresse or marke desi∣red. For more plainnesse beholde the Figure.

[illustration]

Example.

HEre this letter A representeth the fortresse Castle or marke which is the ex∣treame or fardest part of lēgth to be measured, B is your first staf, C the secōd staffe, differing from B the first orthogonally 100 foote, D the third staffe being distant back frō the first in a right line with the marke 133 foote. E is the fourth staffe running sydewise orthogonally or in a squire from the third, vntill the far∣dest part of your length or marke is perceyued in a right line with the second staf, being distant from D the thirde staffe 120 foote. Nowe by subtraction subduce 100 from 120, there remayneth your diuisor 20. Then multiplie 133 with 120, so ryseth 15960, which diuided by 20 commeth 798 foote the true distance betwene D and A.

But forasmuch as this conclusion is to be done without instrument, and here orthogonall motion sidewise is required, it shall be requisite also to declare howe an orthogonal or right angle is vpō the sodaine to be made, ye shal therfore (according to Pythagoras inuention mentioned among the diffinitions at the beginning of this Booke) take 3 staues, cordes or such like, making the one 4 such partes as the o∣ther is 3 and the third 5: This done conioyne their endes togither and the angle subtended of the longest staffe is a right, whiche firste placed at B, and after at D, directing one of his comprehending sides to A, the other shall guyde you to C and E, or if you desire with more expedition to dispatche and not tarry the pro∣portioning of the cordes or such like to this Pythagoricall rule, take any 3 staues, stickes or threades, and conioyne them, making a triangle, it forceth not of what forme or fashion it bee: then placing one angle thereof at B, turning one side to A, direct your selfe sydewise by the other, alway••s remēbring to place the same an∣gle at D, and departing sydewise agayn in like manner, in all the reste do as before is declared. Thus vsing any meane diligence, ye shall most exactely measure any distance.

The 19. Chapter. VVith halbardes, pikes or staues hauing no other instrumentes, you may measure the distance betvvene any tvvo markes lying in a right line from you, not approching any of them.

YOu shal first (as was declared in the last Chapter) prepare an angle with ioyning any 3 staues or such like togither, which you must (at your standing) place in such sorte that one of the sides cōtayning the Angle, may lye directly toward the marke: thē setting vp a staffe, pike or

other marke there, departe sidewise as the other side of your angle shall direct you, so farre as you list, the more ground the better. And there set vp your seconde staffe or marke, then goe directly backe from your firste staffe (alwayes kéeping it exactly betwéene youre sight and the markes) as many score againe or pike length as ye liste, setting vp a third staffe. This done, you shall place the same angle you vsed at youre firste staffe, nowe agayne at your third staffe, in all pointes as it was before: The one side of the angle lying directly toward the first staf, the other side will show you whither you shall go to place your fourth staffe, for passing on still in a right line with that syde of your angle, you shall at the last finde the second staffe iustly situate betwéene you and the fardest marke, there set vp the fourth staffe, then remoue your angle agayne to the second staf, and placing there as before the one side euen with the first staffe passe on in a right line with the other till you come directly betwéene your néerest marke and the fourth staffe, there pitche vp the fift. Now shall you mea∣sure how many pace, halberd or pike length is betweene youre firste and second staffe, deducting that from the distance betwéene the third & fourth, and this remayne you shall reserue for a diuisor. Then multiplie the di∣stance betwéene the seconde and fifte staffe in the distance betwéene the third and fourth staffe, the producte diuided by your reserued diuisor, yéel∣deth in the quotient the true distance betwene these two markes.

Example.

AB the two markes whose distance I would measure, C my standing place where I set vp my first staffe, I my triangle made of three stauos, halberdes, billes, or any such like things, KLM, the angle whiche I will now vse in this practize N, first placed at C, secondly at D, thirdly at E, at C and D, the situation is all one, but at E, it somewhat differeth as you may beholde in this figure, which I wold haue you note lest happely ye be deceyued in your practize, the first staffe C, the second E, the third staffe D, F the 4, G the fift staffe, CE the distance betweene the first and third deducted from DF, the distance between the second and fourth there remayneth HF your diuisor, which measured, I admit 50 halberd lengths. The distance betwene GE 3•• halberde lenghtes, the space betwene DF 100 pike length. Now, 100 multiplied in 30 produceth 3000, which diuided by 50 leueth in the quotient 60, I conclude therfore the distance betweene AB 60 pike lengthes. This one thing is specially to be noted, that whatsoeuer you mete the space GE withall, whether it be halberd, bill, arrow or staffe that ye vse the same in measu∣ring HF, and as for DF, it forceth not what you measure it withall, marry what

soeuer it be, your quotient shall beare the same denomination: precisenes is to be vsed in placing of your triangle, and in measuring EG and HF, otherwise error may ensue, especially if DF be but a small distance, and the angle at B very sharpe, there needeth in this matter no furder admonition, small practize will resolue all doubtes.

[illustration]

The .20. Chapter. To measure the distance betvveene any tvvo markes hovvsoeuer they be situate, thoughe there be riuers or suche like impedimentes be∣tvveene you, as ye cannot approche nighe any of them, and that vvithout instrument also.

YOure angle as before hath ben sayde, prepared of any thrée sta∣ues, you shall fyrst at pleasure set vp one staffe, and applying thervnto your angle in suche sort, that the one conteyning side lye directly to one of your markes (which here for distinction I will call the first) go backwarde to and fro till you fynde your seconde marke precisely couered with your staffe, noting what parte of the side subtending the angle is cut by your lyne visuall, and there make a fyne notche, or marke vpon that subtending staffe, whiche done you shall go sidewise from the first erreared staffe, as the other containing side of your triangle will direct you so farre as ye list, and then set vp your seconde staffe, yet passe on from thence in a right lyne as many foote pace or other measure, as you will, setting vp againe the thirde staffe, now at your se∣conde staffe situate your triangle in all respects as it was at the first, and passe on from thence in a right line with that containing side of your an∣gle that riseth from your staues, and cōmeth somwhat toward the marke going so farre till you espie your selfe iustly betwéene youre thirde staffe and the first marke, there set vp the fourth staffe, then resort to your an∣gle againe, and standing behinde that second staffe, note whether a right line from the angle to that notche (before made on the subtendent staffe or side of the triangle) will directe you, for that way precisely shall you go on till you come in a right line with the second marke and third staffe erecting there the fift staffe. This done you must measure the distance be∣twéene youre seconde and thirde staffe, reseruing that for a diuisor, then multiply the distance betwéene your fyrste and thirde staffe in the di∣stance betwéene the fourth and fyfte staffe, the producte diuided by your reserued diuisor, yeldeth in the quotient the true distaunce betwéene the two markes.

Example.

AB the two markes whose distance I would mete C, my first staffe, I my triangle made of 3 staues placed thereat, as you may see in the figure directing with the one conteyning side to A, the first marke, and with the other to D and E my seconde and thirde staues, H is the fine notche or mark vpon the side sub∣tendent to my angle, where the line visual from C to the seconde marke B passeth, my triangle I situate now at D as it was before at C, the one conteyning side lying euen with the errered staues, the other directeth to my forth staffe F, placed in aright line with E, the thirde staffe, and A the firste marke. Agayne my line visuall proceeding from D to H the subtill notche in the subtendente side

of the angle, extendeth to my fifte staffe G, situate exactlye betweene E the thirde staffe, and B the other marke: This done, I measure the distaunce be∣tweene my seconde and thirde staffe, finding it 20 foote, likewise betweene the fourth and fifth staffe 72 foote, finally betweene the firste and thirde staffe 65. pase, so that according to the rule before giuen, multiplying 65 in 72, I haue 4680 whiche diuided by 20 yeldeth in the quotient 234, so many pase is there betwene A and B. I haue not here set out the figure in iust proportion, answering to these numbers, for that is not requisite, but in suche fourme as may best open and make manyfeste the situation of the staues and triangle, wherein consisteth all the difficultie of this practise.

[illustration]

A proofe or demonstration Geometrical of the former conclusion.

TO auoyde tediouse recitalles of the premisses, I suppose the figure in all re∣spects made as it was before: the proposition that I will here demonstrate shall be, that DE retayneth the same proportion to EC, that GF doth to AB,

whiche admitted or assumed as a Lemma, the rule before giuen of multiplying FG in EC, and diuiding by DE, to produce AB may two wayes be approued geometrically by the 16 proposition of the 6 boke of Euclides elements, and arith∣metically by the 19 proposition of the 7 boke of the same elements, wherin I mind

[illustration]

not to vse mo words, considering it is nothing els but the calculation by the rule of three, whose demonstration hath been handled by diuerse, and well knowen to any, meanely in these matters trauayled. But how these DE and EC become to be proportionall with GF and AB hath not ben by any hitherto proued, this Lemma therefore or proposition I minde to demonstrate. Firste it is apparante that DF is equidistante to AC by the 2•• propositon of the firste of Euclides elements, the words are these.

〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉. And may thus be englished, if any right line falling or passing through two other right lines, making the outwarde angle equal to the inwarde opposite on the same side, or the two inwarde angles ioyned togither equall vnto two right angles, those two right lines are parallele: but here the line EC passing through the lines AC, DF maketh the outward angle FDE equal to the inward opposite on the same side ACD by supposition, bicause they were bothe made with one angle of the triangle, I may therfore conclude by this Theoreme, that DF is equidistante to CA, and farther inferre by the seconde Theoreme of the sixth booke, that AF to FF hath the same proportion that CD hath to DE, the proposition is this. 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉

〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉.

Yf a parallele line be drawen to any side of a triangle it shal proportionallye cut the two other sides, and if two sides of a triangle be proportionally diuided the line that coupleth those diuisions shal be parallele to the other side. It is manyfest by the first part of this theoreme that DF being as it was before proued parallele to AC one side of the triangle ACE that it doth proportionally diuide the two other sides CE, EA, in the poynts FD the like shall be proued of DG, for seeing the right line CE falleth on the two right lines DG, CB making the outwarde angle HDE equal to the inwarde and opposite angle HCD on the same side of the line CE which in the construction of the figure was supposed, it must needes follow by the 28 propositiō of the first booke of Euclide tofore recited, that DG is parallele to CB, and forasmuch as in the triangle BCE, DG is drawen parallele to the one side CB, it shal by the seconde proposition of the 6 booke of Euclide (be∣fore also recited) deuide the two other sides CE, EB proportionally, so that BG shall retayne the same proportion to GE that CD dothe to DE, and so conse∣quently the same that AF doth to FE, as it is playne by the eleuenth theoreme of the fifth booke of Euclide: his words be these.

〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉. The sence thereof is this, that any two proportions beeing equall or agreable to any one are also equal betwene themselues, as here first it was proued that AF to FE bare the same pro∣portion that CD to DE, and now that BG to GE retaineth the same proportion that CD to DE: therfore by the theoreme last recited AF to FE and BG to GE, shall be proportional: so haue you now two sides of the triangle AEB pro∣portionall with two sides of the triangle FEG, and the angles conteyned of those sides equal. Therfore by the 6 theoreme of the 6 booke those two triangles are e∣quiangle, the words of that theoreme are these.

〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉.

Yf two triangles haue an angle in the one equal to one angle in the other, and a∣bout those equal angles the sides proportional, those two triangles shall be equi∣angle, and those angles, equal whose subtendent sides are proportional, Euclide also in his 4 proposition of the same booke sayth thus.

〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉.

In equiangle triangles aswell the contayning as the subtending sides of equall angles are proportionall, I may therfore affirme (seeing the triangles ABE, EFG are equiangle) that AB hath the same proportion to FG, that AE hath to FE, but before it was proued that AF and FE were proportionall to CD and DE, conioynedly therefore AE to FE, shall retayne the same proportion that CE doth to DE, by the 18 proposition in the fifte Booke of Euclide, saying thus: 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉.

If magnitudes disioynedly or seperatly be proportionall, conioynedly or com∣pounded, they shall also bee proportionall, wherevppon I may finally inferre by the 11 proposition of Euclides fifte Booke tofore recited, that AB to GF retayneth the same proportiō that CE doth to DE, bicause they both obserue the same pro∣portion that AE doth to EF. And thus to conclude, it is manifest that AB, FG, CE, DE, are 4 quantities proportionall: whereof three knowen, the fourth AB by the rule of proportion is to bee found. Very like vnto this is the Demon∣stration of the other, and by the selfesame propositions to be proued, and therefore superfluous to vse mo wordes.

The .21. Chapter Hovv ye may most pleasantly and exactely vvith a playne glasse from an highe cliffe, measure the distance of any shippe or shippes on the sea as follovveth.

THE best kinde of glasse for this purpose is of stéele finely pullished, so that the Superficies thereof be smoothe, ney∣ther conuexe nor concaue, but flatte and playne as may bée possible. This glasse it behoueth yée to hange vp aboue the toppe of the cliffe with the pullished side dounwarde eque∣distant to the Horizon wherein you must vse great diligence, for if there happen any error in the situation thereof, great inconuenience maye fol∣lowe in your mensurations. This done, let a plumbeline fall from the centre of your glasse to the Superficies or grounde platte on the toppe of the cliffe: (which ground platte) also you must vse some diligence in the choyse thereof that it be as leuell and playne as ye can finde any, but if it be not altogither euen or exacte leuell, ye shall supplie that wante as I

shall hereafter shewe you: but to returne to the purpose, your Glasse thus situate, turning your face towarde the shippe or other marke on the sea, whose distance ye desire, goe backwarde, alway hauing your eye fi∣xed on the glasse till suche time as ye can sée the shippe, or rather the very hull next to the water therein, that done let an other plumbe line fall from your eye to the ground, then circumspectly measure both the length of these plumbe lines, and also the distance betwéene this plumbe lyne and the other that fell from the centre of the glasse, this done ye shall de∣ducte the lengthe of the perpendiculare from your eye, out of the lengthe of the other perpendiculare from your glasse, and the difference reserue for a diuisor, then multiplie the distance of the two perpendiculare lines, by the heigth of the cliffe, I meane from the water vpward to the glasse, and the producte diuide by your reserued diuisor, the quotient will shewe you the exacte distance to the shippe or marke. But yf your grounde be not leuel, ye shall by your quadrant searche the difference or inequalitie therof, & if it be lower at the glasse than at the viewing station, you shall deduct the difference from your diuisor, but if contrariwise, the difference shal be added to the diuisor, & then shal ye work as I haue before declared.

[illustration]

Example.

SVppose a shippe on the sea C, whose distaunce I desire to knowe standing on the cliffe GFE, your glasse A equidistantly eleuate, AB the perpendicu∣lare lyne 72 inches, E your viewing station, DE the altitude of your eye 69 in∣ches, BE the distance betweene your two perpendiculare lines 20 foote, FG the heigth of the cliffe from the sea 96 pase. Now by deductings 69 inches the alti∣tude of your eye from 72 the lengthe of AB, the plumbe line from the glasse, there remayneth 3 your Diuisor; yf the grounde be leuell, otherwise ye muste adde or detracte the difference or vnequalitie according to the rules to fore gi∣uen, but here supposing your grounde leuell your diuisor remayneth 3. then re∣solue 20 foote into inches, and multiplye the same in the altitude of the cliffe vp to the glasse 98, so haue ye 23520, whiche diuided by 3 yeldeth in the quotient 7840, whiche is the exacte distaunce of that parte of the shippe whiche ye did es∣pye from the centre of your glasse.

YOu may on this manner from an highe hill or mountayne, hauing any playne or leuell grounde on the toppe, not onely measure the distance of any marke that ye can sée, but also set foorth the true platte and proportion of an whole Countrey, with all the Townes, Coastes, Harboronghes. &c.

For yf you mene circularely about your glasse, alway when you espy any marke, setting vp a staffe, writing thervppon the name of the place ye sée, whether it be village, porte, roade, or such lyke, ye shall in the end situate as it were the whole countrey in due proportion vpon your plat¦fourme, so that measuring the distaunce of euery staffe set vp from the middle lyne perpendiculare falling from the glasse, and the distaunce likewise of euery staffe from other, ye may (working by the golden rule) finde out the exacte distaunce of euery towne, village, porte, roade or suche lyke from your platfourme, and also how farre euery one is di∣stante off from other. Thus muche I thought good to open concerning the effects of a playne Glasse, very pleasant to practise, yea most exactlye seruing for the description of a playne champion countrey. But maruey∣louse are the conclusions that may be perfourmed by glasses concaue and conuex of circulare and parabolicall fourmes, vsing for multiplication of beames sometime the ayde of glasses transparent, whiche by fraction should vnite or dissipate the images or figures presented by the reflection

of other. By these kinde of glasses or rather frames of them, placed in due angles, ye may not onely set out the proportion of an whole region, yea represent before your eye the liuely ymage of euery towne, vil∣lage, &c. and that in as little or great space or place as ye will prescribe, but also augment and dilate any parcell thereof, so that whereas at the firste apparance an whole towne shall present it selfe so small and com∣pacte together that ye shall not discerne any difference of streates, ye may by applycation of glasses in due proportion cause any peculiare house, or roume thereof dilate and shew it selfe in as ample fourme as the whole towne firste appeared, so that ye shall discerne any trifle, or reade any letter lying there open, especially if the sonne beames may come vnto it, as playnly as if you wer corporally present, although it be distante from you as farre as eye can discrye: But ef these conclusions I minde not here more to intreate, hauing at large in a volume by it selfe opened the miraculous effectes of perspectiue glasses. And that not onely in matters of discouerie, but also by the sunne beames to fire, pou∣der, or any other combustible matter, whiche Archimedes is recorded to haue done at Syracusa in Sicilie, when the Romane Nauy approched that Towne. Some haue fondly surmised he did it with a portion of a section Parabolical artificially made to reflect and vnite the sonne beames a great distance of, and for the construction of this glasse take great pay∣nes with highe curiositie to write large and many intricate demonstra∣tions, but it is a méere fansie and vtterly impossible, with any one glasse whatsoeuer it be to fire any thing, onely one thousand pase off, no though it were a 100 foote ouer, marry true it is, the Parabola for his small di∣stance, most perfectly doth vnite beames, and most vehemently burneth of all other reflecting glasses. But how by applycation of mo glasses to extende this vnitie or concourse of beames in his full force, yea to aug∣ment and multiply the same, that the farder it is caried the more violent∣ly it shall pearse and burne. Hoc opus hic labor est, wherein God sparing lyfe, and the tyme with oportunitie seruing, I minde to imparte with my countrey men some suche secretes, as hath I suppose in this our age ben reuealed to very fewe, no lesse seruing for the securitie and defence of our naturall countrey, than surely to be meruailed at of straungers.

The .22. Chapter. The making of an Instrument named the Geometricall square.

[illustration]

YE shall prepare a fine playne Plate, or a cleane foure square planed boarde, yea, for want of them, foure equal, smooth, and wel tryed rulers, of what length, bredth or thicknesse ye lyst, the longer the better, yet in my fantasie to auoyde painefull cariage, it is most commodious that euery of them be but a foote and an halfe in length, one inche in breadth, a quarter or more thicke, ye muste

ioyne them by the helpe of some Artificers squirewise, vpon those rulers plate or boorde iustly ioyned, drawe foure lines perpendiculare or squire the one to the other: Now diuide the two fardest sides from the centre, eache hauing 1200 portions at the last, marke all from your centre, forget not to haue an index, not with commune sightes, but thus, let the néerest be a thinne plate halfe an ynche broade, and 3 ynches in heigth, and in the middes a fine slytte, the second and fardest from the centre of that length, a streigth pynne with a little knobbe in the toppe. These sightes must be iustly set vpon the line fiduciall of your Index. This Index I would wishe also marked with 2000 suche diuisions as the scale side hath 1200, it hath place in the cētre, and there made to tarry, so that with ease it may be tur∣ned from the first to any pointe. The exacte handling of this instrument with most comlynesse framed, I commit to the diligent maker. For more instruction, behold the Figure, ye may commodiously describe a quadrant in your square the making of which is declared.

The .23. Chapter. You may redely hereby vvithout Arithmetike mete the distance of any marke.

THis instrument handsomely placed vppon his staffe or o∣therwyse, laye the line fiduciall of your index vppon the be∣ginning of the degrées in your Quadrant, and turne youre whole instrument (the index not moued) till ye may espye through the sightes your marke then remoue the index to the contrary side of the quadrāt, placing the line fiducial on the side line where the degrées ende, and looking through the sightes, ye shall espie a marke si∣dewise, some certein number of scores, the more the better. This done, set vp a staffe where the centre of your instrument was, and placing it again at the marke last espyed, set your index on the beginning of the degrées, mouing your whole instrument, till you finde through the sightes the ••••af at the first statiō, then remoue your index (the quadrant keeping his place) till ye may agayne espie through the sights your marke, which done, note the degrées cut by the line fiducial: and then worke thus, vpon some euen smothe superficies whether it be boord plate or paper: draw first a streight line, & opening your compasse to some small distance, cal that space a score

and make so many such diuisions in your line as there is scores between•• your stations. Then ereare vpon the one end of your line a perpēdicular, and fixing the one foote of your cōpasse at the other end, opening it to what wydnesse ye thinke good, draw an arcke rising from the same line that re∣presenteth your stationarie distance, and diuiding it into degrées (as you were taught in making the quadrant) extende from the centre to the nū∣ber of grades cut by your line fiduciall a right line, till it concurre with the perpendiculare before ereared. Then sée how ofte that space (whiche re∣presented the score in diuiding your stationarie distance) is conteyned in the perpendiculare, so many score is the marke off from your first station, and by diuiding the Hypothenusal line, you may in like māner finde the di∣stance from the second station.

Example.

A the first station, C the second, D the marke, AC foure score pase: degree•• of the quadrant cut at the second station, 71 ½: H the vnitie or measure represen∣ting one score, EF 4 partes, GF 12, GE 12 ⅔, or neere there about. Thus may you conclude the distance of the marke from the first station 12 score pace. The Hypoth••nusal line or distāce of the marke frō the secōd statiō 12 score & 13 pace.

[illustration]

The .24. Chapter. Hovv vvith your square Geometricall to tell any length so farre as ye may see by supputation.

YOu shall disagrée frō those writers that haue declared the vse of the square in the Latin tong, other than this which I shall now open vnto you. So order your Geometricall square, that al sides may be of like heigth from the groūd, to auoyde grasse, mole hilles and such other impediments, precisenes in this ordering is not so requisit, call to remē∣••rance that your square hath two principall lines, one squire vnto the o∣••her running from the cētre of either side to the beginning of your points: ••et your index▪ yea the line fiduciall vpon one of those lines, that side lying ••long with the index towardes the marke. Nowe, the extreame parte of ••our length perceyued through your sightes, turns the index (the square ••ot moued) to the other principall line, squire to the firste, looking agayne ••hrough the sightes, and noting some marke a good distance from you the ••ore ground the surer: This done and a staffe pitched vp where the cen∣••re of your instrument stood, conuey that instrument to the second marke, ••urning it and your index to the place where you first were, the index be∣••ng in the principall line as afore, euen so soone as ye can espie your first ••tation through the sights, remoue that index vntil you may sée y^e extreame ••art of your length, your sight receyuing it diligently note the points tou∣••hed: Nowe if the index fall on the left side of your scale, I meane y^e side which falleth perpendicular to that side of the square issuing from the cen∣••re, wheron your index was first placed, then must ye multiplie the space betwéene the first and second place by these partes cut, and diuide by 1200, the quotient is your desire. But if the index fall on the right side of the scale, then shall you worke contrary, multiplying the space betwene your stations in 1200, and diuiding by the partes cut, or ye may reduce y^e partes of the right side, to partes proportionall of the left, and worke with them according to the first rule thus: Diuide the square of 1200 by the partes cut in the right side of your scale, the quotient is the parts proportionall, which encreased by the distance of your stations, making partitiō by 1200, the quotient is the true distance of the marke from your first station.

Example.

A Is the place to be measured, B, the marke where I first disposed myne instru∣ment, frō it I go orthogonally to C the index cutting there 400 in the right side of youre square, the distance betweene B and C, I haue supposed 80 pace, where∣fore multiplie 1200 by 80, so there commeth 96000, whiche diuided by 400, de∣clareth vnto mee 240 pace, the true length frome B to A. Or by diuiding 1440000 the square of 1200, with 400 the partes cutte, you shall produce

in the quotient 3600, your proportionall partes founde by the rule of reduction, which augmented in 80, yeldeth 288000, and that diuided by 1200, bringeth in the quotient 240, which is the longitude AB, agreeing with the former operati∣on. This manner of reduction, I would wishe you diligently to note, for it shall hereafter in diuerse conclusions be vsed.

[illustration]

The .25. Chapter. Hovv to mete any line Hypothenusall as the distance from your eye standing in a valley to the top of an hill or highe turret. &c.

FIrst, if the hill or turret be stéepe vp, so that the foote be visible lying perpēdicularly vnder the top, ye shal first measure the di∣stance of the base, either by this instrumēt or otherwise as was before declared: and also the heigth of the top or summitie of y^e same hill or fower: which done, ye shall square aswel the longitude as the altitude, ioyning togither the productes, the roote quadrat of the whole nū∣ber, is the desired distance or line Hypothenusal: you may in this manner (approching nighe any town of warre) tell the iuste length of the scaling laders that shall reache from the brym of the ditch or edge of the counter∣scarfe, to the top of the wal or curtein, by adding the square of y^e ditches la∣titude, to the heigth of the curteyne aboue the leuel of y^e outwarde banck, for the roote of the producte will be the true length of the scaling ladder.

[illustration]

A the toppe of the hill, B the foote, C my station or the place of mine eie, A B 60 pace, CB 200 pace, the square of 60 is 3600, the square of 200 is 40000, these two ioyned together make 43600, whose Quadrate roote being about 208 pace 3 foote is the Hypothenusall line AC. Likewise AB the breadthe of the Ditche being 30 foote, and BC the altitude of the curtaine 20 foote, there two squares added together bring forthe 1300, whose Quadrate roote being 36 foote very nighe, is the length of the scaling ladder AC. But if the base of your moun∣taine be not visible, then ereare vp your Geometricall square, the index placed (as was before declared) towarde the toppe of the Hill A, and remouing the Index (your square standing immoueable) espie your second station Orthogonally at D where ye must place the Centre of your Instrument, and so situate your square againe, that you may beholde bothe your station and the mountaine toppe without stirring of the square, only remouing the Index: in all the rest doe as is before al∣ready sufficiently declared, beholde the Figure, there needeth no other Example. The last Chapiter well vnderstode, openeth this most plainly.

[illustration]

To measure the distance betvvene any tvvo markes that lie in one right line from your eye.

YOu may resolue this by the former, measuring howe farre ei∣ther is Distant from your selfe, and then deducte the one from the other, Or thus an other waye, the side of your Geometricall square directed towardes them, departe Orthogonally (as is tofore declared) 100 or 200 paces as ye list, the more the better, then place your Instrument againe, turning the side towarde your first station, re∣mouing the index to either of the markes, noting what partes at eyther

place the index dothe cut of the scale. And if the index at bothe times fall on the left side, deduct the lesse from the greater, with the number remaining, augment the distance betwéene your stations, and diuide by the whole side of the Scale, your Quotient is the distance. If the index at either time fall on the right side, then must you by the rule afore geuen, reduce them into partes proportionall, or if at one time the index fall on the left, at an other time on the right, then shall you only reduce the partes cut on the righte side, which done, deduct as before is sayd the lesser from the greater, and with the remainder multiply your distance stationary, the product diuided by 1200 yeldeth how farre one marke is beyond the other.

Example.

Admit AB the markes in a right line from C your first station, D the seconde station Orthogonally situate from C, where your square being placed, sup∣pose your Index first cut 800 partes on the left side, and after 900 partes on the right, these 900 of right ye must reduce, diuiding the square of 1200 by 900, as was taught in the former Chapiter: so will your Quotient amount to 1600, from which if ye withdrawe 800 the partes cut on the left side, there will remaine 800 which multiplied in 200 pace the distāce stationary CD, there amoūteth 160000 This diuided by 1200 yeldeth in the Quotient 133 ⅓ the distances therfore of AB your markes is 133 pace, one foote 8 inches.

The .26. Chapter. To measure the distance betvveene any tvvo markes lying in one plaine leuell ground vvith your eie or station hovve so euer they be situate vvithout supputation.

MEasure by the rules tofore giuen, how far either marke is of from you, then placing the index vpon the side of the square, turn your instrument til you can espie throughe the sights one of your markes, the square so remaining stedy, moue your index toward the other mark, & when you haue found that mark also through your sights, note what degrées of the Quadrant, the line fiducial cutteth,

this done, ye shall vppon some plaine borde, plate, or suche like, drawe a straight line, then open your Compasse some meane widenesse, and fixing one foote at the beginning of the line, wyth the other make an intersection. Nowe if the markes be many Miles off, you may terme that portion a Mile, or if the distance of the markes be small, a skore, but if they be verye néere at hande, this little line shall represent a pace only, héerein you muste vse discretion, respecting the distance of the markes, and so proportioning your line which is the vnitie of the woorke, that your Plate or borde may receiue the rest of your operations: then procéede with your compasse, ma∣king so many Diuisions in your drawne line, as there are Miles, scores or paces in the distance of one of your markes from your standing, this fini∣shed, open your Compasse at pleasure, fixing the one foote at the end of your line, with the other draw a Circumference or Arke, and this Arke you shall diuide into Degrées, as was taught before in the making of the Quadrant, beginning at your drawne line, and so passing on, till you come to suche number of Degrées, as was cutte by the line Fiduciall, then laying a Ru∣ler to the Center, applying it to that ende of the Arke, drawe an other straight line, and your Compasse againe opened to the length of your little line, (which I terme the vnitie of your woorke) begin at the forsayd Cen∣tre, making againe so many Diuisions in that latter line, as there was Miles, scores or paces in the distance of the other marke from your stāding: And if it fall out that in those Distances there be any odde score, pace, or féet, ye may diuide one of those litle lines or vnities of your work, into mo partes accordingly, and so sette forthe proportionally the exacte Distances of the two marks according to the measure first founde: this doon, ye shall couple together the endes of the two straight lines with an other straight line, finally opening your compasse to the length of the vnitie, beginning at the one end of this last drawn line, measure how many of these vnities is therin contained, for so many Miles, scores or pace (according to the De∣nominatiō of the vnitie) may ye say there is certainly betwéene those two marks. But if at the ende of this latter mesuring, there be any portion left lesse than the vnitie, you must as I haue tofore said by Diuision of the vni∣tie search out what portion it is. For more plainnesse behold the Example.

Example.

Admit I would mesure the distance betwene AB two churches, my self stan∣ding on a hill at C, first I suppose CA 10 myle ½, CB 13 ¼, the arke of the qua¦drant

[illustration]

[illustration]

cut by the lyne fiduciall .30. degrees, then resorting to some plaine boorde such lyke, I drawe the right line EF: My vnitie or myle I make D, and ope∣••••ng my compasse to that measure, I tell on 13 ¼, making an ende at F, then ope∣••••ng my compasse at pleasure, I make the arke HI, one foote of my compasse ••••stned in E, and beginnyng at H, I number toward I 30 grades, ending at K: ••••en draw I the lyne EK, foorth to G. which with my compasse extended to the ••••ngth of D, I diuide into 10 parts, and ½, finishing at G, and with a right line ••nioyning GF, I measure againe how many vnities is therin conteyned, I fynde ••⅔ or therabout, herefore I conclude betweene these two churches 6 ⅔ myles, this ••ude of measuring is good for the vnlearned, but suche as haue Arithmetike, o∣••er rules moste exact shall ensue.

The .27. Chapter. The composition of the instrument called Theodelitus.

IT is but a circle diuided in 360 grades or degrées, or a semicir•••• parted in 180 portions, and euery of those diuisiōs in 3 or rather 6 smaller partes, to it ye may adde the double scale, whose sin∣gle composition is mentioned. The sides of that scale diuided in 〈◊〉〈◊〉 60, or 100 parts. The index of that instrument with the sightes &c. are

not vnlike to that whiche the square hath: In his backe prepare a vice 〈◊〉〈◊〉 to be fastned in the top of some staffe if it be a circle as here: let you•• instrument be so large that from the centre to the degrées may be a 〈◊〉〈◊〉 in length, more if ye list, so shall you not erre in your practizes, the ba•••• side must be plaine and smooth to draw circles and lynes vppon, as shal 〈◊〉〈◊〉 declared: for a farther declaration of that I haue sayde, beholde thys fy••gure folowing.

[illustration]

The .28. Chapter.

To searche the beste proportion or simetrie of many places vvith the true distance approchyng neere none of them by the instrument named Theodelitus.

THis instrumēt vpon his staffe or otherwise in the field pla∣ced the index béeing in his diameter, let it direct your sight to some one place ••hich ye wil mesure. Truly in my fanta¦sy it wer more cōmodious if his dimetiēt or diameter wer first laid in a straight line, bringing the sight to y^e vttermost place toward the left hand, so conueying your index to euery 〈◊〉〈◊〉 or marke on the right syde, noting diligently the angle or angles of 〈◊〉〈◊〉 vpon some state stone or table prepared, which angles here I call grades or degrées from the dimetient apparantly cut by the lyne 〈◊〉〈◊〉 whilest he is broughte to euery marke. This performed, resorte to backe side of your instrument where necessitie requireth a circle or a ••••••icircle to be made, deuided exactly in 360, or if it be an halfe circle in degrées or portions, euen as your Theodelitus here is, from whose 〈◊〉〈◊〉 must finely drawe those angles of position noted before in your 〈◊〉〈◊〉 taken by youre instrument, so that after youre purpose hadde, they 〈◊〉〈◊〉 be cleane put out, then pull the index the instrumente vnmoued to∣••••••d the right hand, at pleasure obseruing through the fights some marke 〈◊〉〈◊〉 yardes from you or lesse as yée lyste. There shall bée youre seconde 〈◊〉〈◊〉, notyng vppon youre slate the angle of position from the dimetient 〈◊〉〈◊〉 lyne fiduciall, directing to the seconde place or marke, whyche ••word muste drawe in the back side from the Centre at large euen as you 〈◊〉〈◊〉 it in your instrument, then conuey your Theodelitus from thence to second marke or standyng place, causyng the diameter iustely to note 〈◊〉〈◊〉 first abiding. And here euen as tofore ye must search Angles of 〈◊〉〈◊〉 agayne, and marke them in the table or slate, which doon, resorte to the 〈◊〉〈◊〉, and vpon the last angle being the line directing to the seconde stan∣••••••g place, draw a circle as far or néere to the other as ye lyst: Or a 〈◊〉〈◊〉 duided in 180 degrées. Whose diuisyons muste take theyr begyn∣••ge at the lyne whyche is Diameter of the semicircle. Nowe drawe 〈◊〉〈◊〉 visuall lynes or angles of position laste taken by your instrument ••arge, and see where the lines méete, or a lyke toucheth his like. So 〈◊〉〈◊〉 you the due proportions, for the distaunce yée shall woorke thus, 〈◊〉〈◊〉 the lyne that goeth from the centre of the one circle or semicircle to

the other, in as many portions as ye thinke mete, or rather in so many 〈◊〉〈◊〉 fynd certaine measure, and by those parts diuide the lines betwixt eue••ry place of which ye require the length. Then multiply the portions that are betwéene any two sections or places in the distance of your two stati••ons, which I imagined here .300 pace, and make partition by those part•• that are betwixt the two centres, so haue ye the true length or distance 〈◊〉〈◊〉 two suche places. In like maner ye must do of the reste.

[illustration]

Example.

WHerby all thing may better appeare, ABC are the markes in the fielde to be measured, D the first abyding or standing place, wher ye shall set the cen∣tre of your instrument, his diameter lying directly agaynste A the firste marke EFGH the foure visuall lines running by the angles of position of the instrumēt vnto all the markes: the first noting no degre or portion: the second 20, the third 40, the fourth 90 degrees of the instrument, which directeth to the seconde station M, where ye shal now set the centre of your instrument, the diameter lying right agaynst your first abyding. here the lines visuall IKL running to the markes: out new angles of position agayne. The first noteth 55 degrees, the seconde 74, the third 85 grades. Now, if ye marke diligently where these lines crosse the other, ther is the true proportion of suche places, from those sections or crossings draw right lines as appeareth by the Figure. Now to get the distāce, ye shall seeke out the space betweene the two stations DM: being 300 pase or yardes, notwithstanding it is diuided but into 18 partes: also betwene the markes or places A and B, are cōtay∣ned 11 such partes. Seeing that I am ignorant what number of yardes be contay∣ned in those 11 portions, I am compelled to worke by the rule of proportion thus, 18 bringeth 300 yardes, what shall 11 bring mee? your quotient sheweth 183 and 2/6 that is ⅓, which maketh a foote: so betweene A and B are contayned 183 pases and a foote. Thus of all other aswell of DA, DB, DC, MC, MB, MA, as of CB, CA, the Reader must not bee ignorant that euen as I haue supposed by this figure, the instrument in the field placed and all things perfor∣med by it with all diligence marked vppon a slate or such like, so ought it to be ex∣actly drawen on the backeside of your instrument, eauen as the Figure afore de∣clareth.

Certes most excellent and farre passing all other is this kinde of mea∣suring, requiring great exercise, the distāce of one place had I am brought into the knowledge of many with the beste proportion.

The .29. Chapter The construction of an instrument Topographicall seruing most commodiously for all manner mensurations.

HAuing alreadie plainly declared the making of the Quadrant Geometricall with his scale therein contayned, whose vse is chéefly for altitudes and profundities: the composition also of the square and planisphere or circle named Theodelitus, for measuring lenghtes, breadthes and distances. Yt may séeme superfluous more to write of these matters, yet to finishe this treatise, I thinke it not amisse to shew how you may ioyne these thrée in one, whereby you shall frame an instrument of such perfection, that no māner altitude, latitude, longitude, or profunditie can offer it selfe, howsoeuer it be situate, which you may not both readely and most exactly measure. You shall therfore first prepare some large foure square pullished plate of Latin, wherein you may describe your Geometrical square, his sides diuided in 1200 parts at the lest, with index and sightes as was before shewed: describing also within the same square the Planisphere or circle called Theodelitus, then must you vppon an other fine pullished plate, drawe your Quadrant, or rather a semicircle diuided iustly into 180 grades, and within the same a double scale: euery side contayning at the leste an 120 partes, finally, fix∣ing on the dimetient thereof two sightes perpendicularly reared, and e∣quedistantly persed, so as the line visuall may passe parallele to that dia∣meter. You haue a double Quadrant Geometricall with a double scale, whiche you muste by the ayde of some skilfull Artificer, so place ouer the other plate wherein youre square Geometricall and Theodelitus was de∣scribed, that his centre maye exactly reste in a Perpendicular line from the centre of the planisphere or circle named Theodelitus his circū∣ference depending dounwarde. And this double Quadrant or semicircle, must in such sorte be connexed to the Perpendiculare erected from the centre of the planisphere, and alhidada at the foote thereof, that what way so euer the Diameter with sightes be turned, the Alhidada maye alway remayne exactly vnderneath it, directing bothe to one verticall circle or poincte of the Horison: this perpendiculare wherevnto the semicircle s centre is fastened, ought also to be marked with 200 partes equall to the diuisions of the scale beginning at the centre, so procéeding dounward til you come to the end of those 200 portions: more I néede not say of this instrument, considering the construction, if euery parte hath ben seueral∣ly de••lared sufficiently before, for the placing and ••onioyning of them, be∣hold the Figures.

[illustration]

IKLH the square Geometricall, MN his index with sightes, GEFO Theodelitus, GF his Alhidada er index with sightes AB the line perpen∣diculare from B dounward noted with 200 partes, equall to the diuisions of the scale, DRC the semicircle hauing on his Diameter two sightes fixed as was tofore declared. This is also to be noted, that the double scale is compound of two Geometricall squares, the one seruing for altitudes, the other for profundities. The square which the line perpendicular cut∣teth when the Diameter is directed to any markes lying lower than your station, I call the scale of profundities, the other shall for distinction be na∣med the scale of altitudes.

[illustration]

THis semicircle ought so to be placed that the centre B hang directly o∣uer the centre A and that the diameter DC with his sightes maye be moued vp and downe, and also sidewise whither you list, alwayes cary∣ing GF about directly vnder it. You must also prepare a staffe pyked at the ende, to pitche on the ground with a flat plate on the toppe to set this instrument vpon. It is also requisite that within Theodelitus you haue a néedle or fly so rectified, that being brought to his due place the crosse dia∣meters of the Planisphere may demonstrate the foure principall quarters of the Horizon, East, Weste, North and Southe: And this may you do by drawing a right line making an angle (with that one diameter of youre instrument representing the meridiane) equall to the variation of the cō∣passe in your region: which in England is 11 ¼ grades or néere therabout, and may be redely obserued in all places sundrie wayes. But thereof I mind not here to entreate, forasmuch as it appertayneth to Cosmographie, & nauigatiō, wherof I haue cōpiled a treatise by it self, touching y^e fabrica∣tiō this may suffise. Nowe for y^e vse great héede must you take in pitching

of the staffe wherevpon this instrument is placed, that it stande perpen∣dicularely, whiche by a lyne and plummet ye may trye, also when the ••nstrument is placed thereon, ye shall by a lyne and plummet fixed on ••he centre of the semicircle discerne whither it be rightly situate: for yf the thread and plummet hanging at libertie fall close by the perpendicu∣••are, then is it well: otherwise ye muste moue the staffe to and fro tyll ye finde it so. This done, it behoueth you also to set this instrument on your staffe, that the néedle haue his due place, so as the semidimetientes of Theodelitus may directe vnto the foure cardines or quarters of the Horizon, then what soeuer marke you espye, whose distance, altitude or profunditie you desire, turne the dimetient of the semicircle to and fro, ••p or downe, till through the sightes thereon fixed you haue espied it, ••lway the circle or Theodelitus remayning immoueable: finally you ••hall note bothe what degrées the Alhidada cutteth of the circle, and the perpendiculare of the semicircle, and also what partes of the perpendicu∣••are is intercepted with the scale, these numbers thus founde, you shall ••iuersly vse as shall hereafter be declared. This is farder to be noted that ••he double scale is compounde of two geometricall squares, the one ser∣uing for altitudes, the other for profundities. The square which the lyne perpendicular cutteth when the diameter is directed to any marke lying ••ower than your station, I call the scale of profundities, the other shall for distinction be named the scale of altitudes.

The .30. Chapter. By this instrument to knovve hovv many myles or pase any Shippe is distante from you, your selfe standing vpon an highe cliffe or plat forme by the sea coaste.

YOur Topographicall instrument equedistantly situate to the Hori∣zon, (as was before declared) turne the diameter of the semicircle towards the ship, and when you haue espyed through the sights the wall or lowest part of the hull next the water, note exactly what parte of the scale is touched with the lyne perpendiculare, thē measure the height of the cliffe, or rather the centre of your semicircles altitude aboue the sea, & multiply the same in the hole syde of your scale, diuiding by the par∣tes touches of the perpendiculare lyne in your scale, the quotient is the distance of the ship from the basis or foote of the cliffe lying perpendicu∣••arely vnder the centre of your instrument, but if you multiply the parts

of the perpendiculare intercepted with the Scale in the cliffes altitu••e before measured, and diuide by the partes of the scale cutte, he quoti••••t will shew the lyne Hypothenu••al, or distance of that parte of the shippe which your lyne visuall touched from your eye, or adioyning the square of the longitude firste founde to the square of the altitude, the roote qua∣drat of the product is also the true length of the line visuall.

Example.

[illustration]

ADmit I stande on the cliffe A, and see the shippe B lying at rode in the sea, I desire to know how farre of she is from me, mine instrument conue∣niently placed at C (as is tofore declared) I turne my dimetient of my semicir∣cle towarde the shippe, mouing it vp and downe tyll I espye through the sightes

••he wall of the hull next the sea, and therewithall I finde my perpendiculare cut∣••••ng the thirde parte of my Scale, then measure I the altitude of the cliffe aboue ••he sea EF, finding it 59 pase, wherevnto I adde one for the altitude of my in∣••••rument DC, so haue I 60 pase, the heigth of the semicircle aboue the sea, whiche ••ultiplyed in 120 the side of the scale, there amounteth 7200, whiche diuided by 3 ••eldeth in the quotient 2400 pase, that is two miles, 3 furlongs, 25 pases, the lon∣••itude HB. Now square 2400 pase, se haue you 5760000, wherevnto yf you ad∣••••yne the square of HD the product will amount to 5763600, whose quadrat ••••ote is the longitude of your line visuall, represented by DB, beeing not fully 〈◊〉〈◊〉 foote more than HB.

THis conclusion serueth most commodiously for all suche as shall haue committed to their charge any platfourme with ordinaunce, for here∣••y you may exactly at the firste viewe, tell the distance of any shippe or ••arke, so that hauing a table of Randons made, mounting your péeces ••ccordingly, no vessel can passe by your platfourme (though it be without ••oynte blancke) but you may with your ordinaunce at the first bouge hir ••nd neuer bestow vayne shotte. This instrument serueth also no lesse ••ptly to be situate on the rampire or mount within any towne of warre, whence you desire with shotte to beat the enemie aloofe of, before he shall ••pproche nighe. But to giue full instruction for shooting great ordinance, ••nd for the variation of Randons, hauing respecte to the length of the ••eece, waighte of the Bullet, force of powder, proportion of the concaue Cylinders, and distance of the marke, it woulde require a long discourse: I haue thereof in a booke by it selfe very largely, and I dare be boulde ••o say, sufficiently entreated: for conferring and conioyning Geometri∣••all demonstration, with my long continued painefull practises, I haue ••t the laste reduced that moste irregulare course and circuite of the Bul∣••et (framed and compounded of violent and naturall motions) within ••he bounds of numbers and arithmeticall rules, which whether it be dif∣••iculte or no, they onely knowe that haue or shal attempt to do the lyke. This surely I will say, that as the ignorant in Geometricall and Arith∣meticall proportions, shall neuer attayne perfection, though he turmoyle ••n powder and shot all the days of his life: so the Geometer, how excel∣••ent so euer he be, lening only to discourse of reason, without practise (yea

and that sundry wayes made) shall fall into manyfolde errours, or ine••∣tricable La••erinthes. Among many that I haue read concerning that matter, I note one Nicholas Tartalea an Italian, who surely for his singu∣lare inuention and perfect knowledge in Geometricall demonstrations, few or none in our time or many ages before may be compared with him: and yet handling this Argument, he hath erred euen in the princi∣pall, and as I might tearme them the veriest trifles: I meane touching the vttermost Randon and circuite of the Bullet, which he affirmeth to be made of a circular and right line: others haue supposed it to be compact of many right lines, making seueral angles proportionally to the seueral mounts of the péece, or hollow Cylinders, and many lyke opinions and manyfest errours haue diuerse wel séene in Geometrie for want of expe∣rience admitted and mainteyned. But of these in due place. Now to this purpose.

The 31. Chapter. To knovv hovv muche higher or lovver any marke is than the leuell from your eye, although there be suche impediments betvvene, that you can neither approche nigh vnto it, nor see the base.

HEre shall you vse great diligence in the placing of this in∣strument, that it be situate precisely equedistant to the Hori∣zon, which done you shall turne the Diameter of your moue∣able semicircle to and fro, till you can through the sightes es∣pye the marke, noting therwithall the poyntes or partes cut in the scale and perpendiculare, then measure the distance of the same marke from your eye, as you were taught before, which multiplied in the parts of the scale, yf you diuide by the partes of the perpendiculare, the quotient is the difference howe muche the marke is higher or lower than your eye. Also yf you adde the square of the partes cut in your Scale, to 14400. reseruing the producte for a Diuisor, and multiplye the Square of the distaunce Hypothenusall in the Square of the partes cutte in the Scale, diuiding the ofcome by the reserued Diuisor, the roote quadrates of the quotient is the foresaide difference or vnequalitie of leuelles. And thus may ye also finde how muche any one marke is higher or lower than an other, although they be far a sunder, & either of thē remote from

you by comparing their altitudes founde in this sorte together.

[illustration]

Admit B, my station where I place myne instrument A the mark whose alti∣tude I desire aboue the leuell of myne eye, though I maye not by reason of sun∣dry impedimentes approche nyghe vnto it, nor see the base. Fyrste I measure the Hypothenusall lyne AB, by the preceptes tofore giuen, whiche I suppose 500 pace. then perceyuing thorough the sightes of my semicircle the marke A, I fynde 10 partes in my scale of altitudes, intercepted with the perpendicular lyne, the square thereof ioygned to 14400 produceth 14500 my diuisoure. Then doo I multiplie the square of 500 in 100, the square of the partes cut in the Scale, so haue I 2••••00000, whiche diuided by 14500, the diuisor be∣fore reserued, your 〈◊〉〈◊〉 quadrate roote will he very nighe 41 pace 31 inches, and this is the true difference or vnequalitie of leuell betweene the marke A, and the centre of your semicircle B, so that if a well bee soonke of suche deapthe that the bottome thereof were lower than A 41 pace 31 inches, as I admitte the lyne AC, then maye you certainely affirme, that C the bottome of that well is leuell with B, and yet may you not thereby, inferre that from

a fountayne heade, lyinge of equall heygth wyth B, you may naturally d••rya•• water to C, for the leuell of waters is circulare, as I haue before in the booke de∣clared: And heere I thynke it not amisse to gyue you a precepte howe to fynde the diuersitie of these leuelles, wherby yee may exactly resolue sundry questions perteyning to water woorkes, wherein dyuers haue greately erred, obseruyng n••t this difference.

The .32. Chapter. To fynde the difference betvvene the streyght and cir∣cular or true vvater leuell from a fountaine at any place appoynted.

THis difference is nothyng else but the lengthe of a pen∣dicular lyne falling from the leuell right line of a foun∣tayne to the water leuell of the same, as in the figure followynge A is youre station, D the fountayn whence I woulde conuey water to the forte B standyng on a hill, C a poynte by imagination conceyued directely vnder B within the earthe in a leuell ryght lyne from the fountayne D, or rather the Superficies of the water therein contey∣ned. E the water leuell, that is to saye, the hyghest poyncte that any water will naturally runne at, béeinge conueyed by pype from D. The lyne CE is thys difference of leuelles whyche you shall thus at∣tayne. Fyrste it behooueth you to gette the distaunce of the fountayne from the place whyther you woulde conueye youre water, whyche distaunce you shall multiplie by it selfe, addynge the offcome to the square of the earthes semidimetiente, and oute of the producte ex∣tracte the roote Quadrate, from whych roote if you withdraw the fore∣sayde Semidiameter, the remaynder is youre desyred difference or lyue CE.

Example.

[illustration]

Admitte the distaunce DC 10 myles, the semidiameter of the earth, 5011 Italyan myles, euery myle conteyning 1000 pace geometrical, the pace being 5 foot: the square of this semidiameter is this nūber of pace 25110121000000. Likewise 10 myles the distaunce squared yeldeth 100000000 paces, this added to the square of the earthes semidiameter produceth 25110221000000 paces: Nowe if from the roote quadrate thereof ye subtracte the somidiameter, there wyll remayne 9 pace, 4 foote, and 11 ynches: so muche you maye assuredly say, that the water leuell E is vnder the other leuell at C. Nowe if you woulde knowe standyng at A by the fountayne not approchyng nyghe the Castle howe deepe it were requisite to sinke a well there to receyue this water you may thus doo, firste measure the lyne BC, that is to saye, howe hyghe the grounde platte of the Castell is aboue the leuell right lyne of the fountayne D, for this you are taughte howe to doo before, then searche oute the difference betweene the straighte and water leuell of the same fountayne by the rule giuen in the last chapter, these twoo ioygned together, doo produce the profunditie BE,

that is howe many pace foote and inches you shall synke a welle at the Castell to receyue water from that fountayne. Herein there neede no example, the pre∣misses well vnderstand, this conclusion is manyfest.

The .33. Chapter. You may also by this instrument meete the distance betvveene shippes on the sea, or other markes on the lande, hovv so euer they be situate, vvith the aid of Arithmeticall supputation.

FIrste measure the distance of eyther shippe from your sta∣tion, howe soeuer it bée, whyche you may doo by sundry meanes before declared, then takynge off the perpendi∣cular and semicircle, and fixing the Index of the square Geometricall vpon his due angle fyrst place it on the side of the square, whence the ••••••isions of the Scale begin to bée numbred, and turne the whole plate (••he Index kéepyng his place) till you can espy one of the shippes through ••he sightes. That doone moue it towarde the other shyppe, whiche when you can also espye, note the partes of the Scale touched with the lyne ••••ducial, and also the partes of the index, cutte wyth the syde of the scale, then shall you thus woorke. Multiplie the partes of the Scale by the distaunce of the shyppe whiche you laste espyed through your sightes, and diuide by the partes of the index, the quotient note, for it must serue you to double vse, fyrste square it and square also the distance last viewed through the sightes deduct one of these squares from the other, the roote square of the remaynder ye must compare wyth the distaunce of the fyrste shyppe, I meane that whiche at the fyrste your index lying on the syde of your square Geometricall yée espyed, detractyng the lesser from the greater, the remaynder yée shall agayne square, and adde it to the square of your reserued quotien•• DE, the roote quadrate of the producte is the exacte distaunce betwéene the two shippes.

Example.

[illustration]

I stand in the Castle A, the two shippes, whose distance I require B and C, AC measured as is tofore declared 2000 pace, AB 2500, the partes of the scale tou∣ched with the fiduciall line of the Index 50, the partes of the Index cut with the scale 130, B the ship first viewed, while the Index rested on the side of the square Geometricall, C the shippe last espied, when the scale was cut with the line fidu∣tial of the index, I multiply therfore according to the rule aboue geuē, 50 in 2000, so haue ye 100000, which diuided by 130 yeldeth 769 3/13 whose square detracted from the square of 2000, leueth 3408284 4/169, whose Quadrate roote is 1846 2/13 whiche deducted from 2500 the distance of the latter shippe from the Castle, there remai∣neth 653 31/13, whose square added to the square of 769 3/13, the Quotient tofore reser¦ued, produceth a number, whose roote is neere vnto 1009 ½, these two shippes there∣fore ye may conclude 1009 ½ pace a sunder.

The .34. Chapter. To dravv a platte of any coast or countrey, containing the true pro∣portion and Symetrie thereof, in suche sort that you may readely tell hovve farre any place is distant from other, and that vvithout Arithmeticke.

YOu shall ascende on some highe Tower, Hill, Cliffe, or o∣ther place, from whence you may commodiously behold on euery parte the whole Countrey rounde about adiacent in your Horrizon, there set vp your Instrument Topographi∣call on his staffe, and in suche sorte place it by ayde of the néedle, that the four Semidiameters may lie East, West, Northe and Southe, euery one answering his like quarter of the heauen, then turne the Diameter of your Semicircle, to euery Towne, Uillage, Hauen, Rode, or suche like, espying through the sightes, the middle or most notable marke in euery of them, noting therewithall in some Table by it selfe the Degrées cut by the Alhidada in the Circle, which I call the Angles of Position, and so make you a table of your first station. Then searche out your eye, viewing round about, some other loftie place, from whence you may behold again all these places, for that shal be your second station: and turning therunto the Diameter of your Semicircle, note also what parts of the Circle is touched with the line fiduciall of the Alhidada: This done, si∣tuate your Topographical instrument, in all respectes as was before sayd, and turning the Diameter of your Semicircle, espying through the sights, all suche markes as you sawe before, note againe the Degrées cut, or An∣gles of position, wryting the name of euery place, and his Degrées by it, so haue you an other table of your second station: with these tables you shall resorte to some plaine smoothe Superficies of borde, parchement, paper, or suche like, and thereon describe a large Circle, diuiding it as you were before taught, into 360 partes, like to the Circle in your instrument. Then from the Centre thereof to euery Degrée noted in your first table extende straight lines, wryting vppon euery of them the name of his place, and vp∣pon that line that representeth your second station fixe the one foote of your Compasse, opening the other at pleasure, drawe an other large Circle, di∣uiding it into 360 Grades, and from the Centre thereof, extend right lines to euery Degrée noted in your seconde Table, wryting as before vppon e∣uery of them the names of their places or markes, whereof they are the sight Angles. Finally you shall note diligently the concourse or crossing of euery two like lines, making thereon a Starre or suche like marke, with the name of the place correspondent. Nowe if you desire to knowe howe farre euery of these townes, villages. &c. are distant from other, you shall thus doe, measure the Distance betwéene your stations by instrument or otherwise, as you haue bin before taught, and diuide the right line betwéen

the Centres of your Circles into so many equal portiōs, as there are miles, Furlongs, or Scores betwéene your stations, then opening your compasse ••o one of those partes, you may measure from place to place, alway affir∣ming so many Miles, Furlongs or Scores (according to the Denominati∣on of that one parte whereunto you open your Compasse) to be betwéene place and place, as you finde by measuring there are of those partes. Some consideration you must haue in placing the Centre of your second Circle, so conueniently distant from the other, that the concourse or méeting of semblable positionall lines, may be within the Compasse of your Pa∣per. &c.

Example.

There is a sea coast hauing sundry harboroughes, townes, villages, castels and suche like situate thereon, whose platte in due proportion I require, with the exact distances of euery place from other, hauing therefore elected a loftie seate, from whence I may beholde all these places, (mine instrument situate as is declared) remouing the index to the fardest being a Castell standing in the mouthe of an Hauen, hauing receiued it throughe my sightes, the line fiduciall of mine Index, cutteth 30 Grades, thence remouing it to the next, being a village or fisher towne, the Index cutteth 50 degrees, againe at the next being a Citie, it cutteth 75 grades at the fourth being a great Baye I note bothe the entrances, at the Esterne side of the Baye it cutteth 75 Degrees, at the Westerne 100 Degrees, finally at the far∣dest place Westward, being a fort within the land, the Index doth cut 130 grades. Thus haue I the Table of my first station, as foloweth.

The Table of my first Station.

The Castle

30 Degrées

The Uillage,

50 Degrées

The Citye,

75 Degrées

The Esterne head of the Baye,

95 Grades

The Westerne head of the Baye,

97 ½ Grades

The Fort within the lande.

130 Grades

This done, I beholde an other highe Hill, from whence I may in like manner view all these places, & turning my Index therunto, I find the line Fiduciall lying vpon 180 degrées. Then carying my instrument thither, &

placing it in all pointes there, as it was at the first station, I turn my index againe to the first marke or castle, finding it to cut 15 grades, at the second 25 degrées, at the third 40 grades, and so to the rest as appéereth in this Ta∣ble folowing. And as I haue done of these few, so might I how many nota∣ble markes so euer there were, wryting their names seuerally with their Degrées noted, as ensueth.

The Table of my second Station.

The Castle

15 Degrées

The Uillage,

25 Degrées

The Citye,

40 Degrées

The Esterne head of the Baye,

50 Grades

The Westerne head of the Baye,

55 Grades

The Fort within the lande.

80 Grades

[illustration]

With these Tables repaire to a plaine smoothe Superficies, drawing therin a faire Circle as you sée deuided in 360 partes, and from his Centre A, I pul out right lines to euery Grade noted in my first table▪ AC the line 〈2+ pages missing〉〈2+ pages missing〉

••e castle, AD of the village, AE of the citie, and so foorth of the rest, ending 〈◊〉〈◊〉 the line AB, cutting 180 degrees in my circle, then making B a centre I 〈◊〉〈◊〉 a circle diuided as the other at A, and from his centre pull out streight 〈◊〉〈◊〉 to all those grades that were noted in my seconde table: nowe marking the ••••••course of semblable lines, that is to say, where the line of the castle issuing frō A 〈◊〉〈◊〉 with the line of the Castle proceeding from B, I note it with a starre as 〈◊〉〈◊〉 may beholde at the letter C, and thus prosecuting the like in the rest, always 〈◊〉〈◊〉 a starre or like marke vpon the concourse of correspondent right lines (all 〈◊〉〈◊〉 intersections not regarded) I haue finally situated al these places in due pro∣••••••tion, noting them with these letters CDEFGH: as in the figure you may see. Now to lerne the distance betweene euery of them, I diuide the space betweene B into 5 parts, for so many miles by mensuration I finde betwene my two 〈◊〉〈◊〉, then drawing right lines from C to D, from D to E, and so from euery one 〈◊〉〈◊〉, opening my compasse to one of these partes, I measure how many times it ••onteyned in euery of these right lines: wherby I conclude the number of miles betweene euery of them, as appeareth in the table folowing.

The Table.

The distance of euery marke from the stations.

• C 5 myles, 1 furlong.
• D 5 myles, ½ furlongs.
• F 5 myles, 3 furlongs.
• G 6 myles, 1 furlong.
• H 6 myles, 3 furlongs.
• C 9 myles, 6 furlongs.
• D 9 myles, 1 furlong.
• E 8 myles, 3 furlongs.
• F 7 myles.
• G 7 myles, 2 furlongs.
• H 5 myles.

The distance of euery marke from other.

• CD 1 myle 6 furlongs.
• CE 4 myles ½ furlongs.
• CF 5 myles 5 furlongs.
• CG 6 myles 1 ½ furlongs.
• CH 9 myles.
• DE 2 myles 3 furlongs.
• DF 4 myles 1 furlongs.
• DG 4 myles 4 furlongs.
• DH 7 myles 4 furlongs.
• EF 1 myle 7 ½ furlongs.
• EG 2 myles 2 ½ furlongs.
• EH 5 myles 5 furlongs.
• FG 5 furlongs.
• FH 3 myles 6 furlongs.
• GH 3 myles 4 furlongs.

Thus passing or chaunging your stations, you may make seueral plattes, conteyning the true proportion and distances of townes, villa∣ges, portes, rodes, hilles, riuers, and all other notable places through •••• whole Realme, but then how to reduce them all into one fayre carde or mappe you shall be taught hereafter.

The .35. Chapter. Hovv to reduce many plattes into one, and to make a fayre carde or mappe of an vvhole prouince or region, and that in as large or small roome as you vvil assigne, vvithout any arithmeticall calculation.

I Minde not here to set foorth the maner how to situate pla∣ces in their due longitude and latitude, neither howe to furnishe your mappe with Meridians, Paralleles, Zones Climates, and other circles correspondent to the heauenly sphere, for that apperteyneth to Cosmographic, onely in this place shall you learne Chorographically to make a carde, whereby you may redily know the true distance and situation of places one from an other, hauing therefore as is tofore declared (by sun∣dry plattes made in euery seuerall Horizon or Kenning) founde out the true distance of euery notable place from other, you shall make one en∣tier table of all, conteyning the number of myles, furlongs &c. betwéene place and place, beginning at the East, and so procéeding on till you come to the farthest Westwarde. This table thus finished you shal vpon your Parchement paper or other playne whatsoeuer it be, fitte for this pur∣pose, draw one streight line directly through the middle thereof, crossing the same with an other perpendicularly, againe as you wer taught at the beginning of this booke, then write at the endes of these lines the foure principall quarters, East, West, North and South. Nowe it behoueth you to consider (conferring your plattes togither) how farre distante the most Easterne place is from the Westerne, and likewise the Northerne from the farthest Southward, that you may accordingly so proportion your mile, as all these places fall within the compasse of your carde: now by the one side of your superficies, draw a streight line of competente length, then opening your compasse to the widenesse of that measure which ye will call your myle, marke out 20 of them in your last drawen

straight line, which you may garnish with other Parallels, diuiding eue∣•••• myle into his furlongs, this shall be called your scale, Now must you ••••eyne gesse (as néere as you can by comparing your plats) which is the ••iddlemost place of this countrey that you describe, and the same shall ••••u situate vpon the intersection of the two former drawē lines, making ••••ere a starre, and writing the name thereof, whether it be citie, village, ••••stle or suche like, then searche out in the same plat what other notable ••••ace or marke lyeth east, west, north, or south from it. And if you espye ••••y resort to your table prepared as I haue tofore said, searching out the ••••stance betwéene these places, and to so many miles or furlongs extend ••••ur compasse in the scale last made, then kéeping your compasse immo∣••••able set one foote thereof vpon the intersection or middle place, exten∣••••ng the other to that quarter wherin you found the place situate, make ••n intersectiō with the streight line, and there likewise make a starre or ••••her fine marke, writing the name of the place: But if you finde no no∣••••ble marke lying precisely east, west, north, or south from that foresaide ••iddle place, then take some other what you list in your plat, & note both ••••e distance & also the angle that it maketh, with the middlemost already marked in your carde, that is to say (if it lye not in one of these principal ••uarters from it) how many degrées it declineth, and describing vpon ••he intersection a circle diuided into grades, pull out from the centre a ••ight line to the like number of degrées that you found in your plat: final∣••y opening your compasse to the like number of miles in your scale that ••ou perceiue in the table to be betwéene those places, fixing one foote in ••he intersection with the other, cut this last drawen line, and at y^e section make a marke, writing thereby his name, thus haue you two places. Now for all the reste one rule shall suffise, whatsoeuer it be that you wil next marke, search out in your table his distance from bothe these alredy ••oted, and opening your compasse to the like number of miles in your scale, placing the one foote in either of the places alredy described, make an arke with the other, and where those arkes crosse there is the situa∣tion of the thirde place. And thus may you procéede to all the reste, onely taking héede that hauing opened your compasse to the number of myles you fixe the foote in his correspondent place, and so shall you (passing on from one to an other) exactly situate them all, as by the example folo∣wing shall more playnly appeare.

Example.

[illustration]

BDCE the Parchment, paper or other plaine whereon I would describe the countrey whereof I haue alreadie taken the plat or plattes, as was before taught, BC and DE the two crosse diameters, making intersection at A, I the line which I tearme a myle, HG the scale conteyning 20 of them, this myle if I list to make a large carde should be diuided into furlongs, yea and euery furlong into partes, and therein you muste vse consideration, appoynting the lengthe of your myle according as you desire to haue your charte great or small. Now it behoueth you to resorte to your plattes, searching out as nigh as you can by estimation the

••iddlemost place. And that in my laste platte (which heere I mynde to vse for ••xample sake) was my first station, heere placed at A, then finde I directly ••este from thence 5 myles my seconde station. Extending therefore my Com∣••asse to so manye in the Scale, and setting the one foote in A, with the o∣••••er, I make a section in AC at K, and that is my seconde station. Nowe will I place the Castle which I finde in my platte or table made by it, ••••stant from my firste station 5 ⅛ myles, from the seconde 9 ¾ myles. Firste there∣••••re I extende my Compasse in the scale to 5 ⅛ myles, and fixing the one foote in 〈◊〉〈◊〉, with the other I drawe an arcke, then opening my compasse to 9 ¾ partes in ••••e scale, and setting one foote in K, I crosse with the other the arcke tofore ••••awen in L, in like manner opening my compasse agayne to 5 ½ in the Scale, ••hich is the number of myles betweene the Village, and my first station. The one ••••ote in A, I drawe an arke, then extending the compasse agayne to 2 myles lac∣••••ng ¼ parte, the distance betweene the Castle and Village, I crosse the foresaide ••••cke in M, there is the Village. In this manner you shall proceede to the reste, ••aking fine prickes or other little markes for the true places of them, then ble∣••ishing all the arkes, circles, and other lines, which I suppose drawen with black ••••ad or such like, that you maye easely put oute or rase awaye, you shall finally ••autifie it with ymages and Figures, as you thinke most agreable and fitte to ••••presse and represente the patterne, I meane the country it selfe that you de∣••••ribe, as by drawing a Castle at L, a Village at M, a Citie at N, a Har∣••••roughe at OP, and so of the reste. Euen in like manner may you describe any Towne, Forte, Campe, or Pallace, and set out the true ground platte of all man∣••er Edifices, altering your Scale, and in steade of myles, diuiding it into scores, ••ace, foote, and such like small measures.

A note for sea Cardes.

YOU may also if you will diuide the circle at the myddes of your map into 32 partes, pulling out straight lines fro the cētre to the vttermost boūds of the charte representing y^e 32 windes, and vpon the same cētre de∣scribe an other circle so large as youre mappe can conteyne, the circumfe∣rence thereof will be parted into 32 equall portions with the foresayde lines representing y^e windes. Now, if vpō euery of these intersectiōs as a centre, you describe a circle diuiding euery of their circumferences into 32 winds, extēding frō their cētres streight lines thoroughout y^e whole map as before you shall make a sea carde wherin you may by the former rules

Coastes, Harboroughes, Rockes, Sandes, Shelues, Chanels, Rodes, with their soundings and depthe of ancorage, &c. But of these matters héere I meane not particularly to discourse, referring the more ample de∣claration of them to an other treatize of Cosmographie and nauigation, wherein I shall not only discouer the manifold errors that Mariners fall into by vsing their commune Chartes and rules, but also sette foorth true exacte and easie precepts for them, with sundrie rare conclusions hitherto not practized of any.

A note for Mines.

MOst commodiously also serueth this instrument to conducte Mynes vnder the earth, for noting the Angles of position in the Planis∣phere or Theodelitus, and also Angles of altitude or profunditie in the se∣micircle or scales appropriate therevnto, measuring the distances from Angle to Angle, you may make by the former preceptes moste certeine plattes of your iorneis, and thereby alwayes knowe vnder what place you are, and which way to directe your Myne to approche any other place you liste. Whereof an infinite number of strange and necessarie questi∣ons might be moued. It were an easie matter in such sorte to make great volumes. But who so well vnderstandeth the premisses, (I meane not by reading them onely, but by practizing also) shall not néede farther in∣struction in these or like cases. I might enlarge this booke with sundrie instrumentes, and many mo wayes and rules to measure, but as the pre∣misses are of them selues sufficient, so the diligent practizioner (searching out the reason and demonstration of them) shall be able of him selfe to in∣uent manifolde meanes to resolue the like or other stranger questions, whereof infinite may be proponed.

The ende of the fyrst Booke.

The second kynde of Geome∣trie called Planimetra. (Book 2)

HAVING accomplished the first part called Longimetra, con∣tayning sundrie rules to measure lēgthes, breadthes, heigths, depthes, and distances: I thinke it meete novv to proceede to the seconde kynde named Planimetra, vvherein ye shall haue rules for the mensuration of all manner playne Figures, to knovv their contente superficiall. And forasmuch as there is no Superficies, but is enuironed vvith lines either streight or curue. And all Figures com∣prehended vvith streight lines may be resolued into Triangles: It seemeth most meete, first to teache the measuring of Triangles as follovveth.

The fyrst Chapter. Of Triangles.

AS there are thrée kind of Angles, so is there also thrée kind of Triangles: the first is called Orthogonium, hauing one of his angles a right: the other Ambligonium, contayning one obtuse angle: the third kinde is called Oxigonium, whose thrée angles are all acute. Of right angled Triangles also there are two kindes, for eyther it hath two equall sides, and then is it called Isoscheles, or thrée vnequall, and that is Scalenum.

The right angled Isoscheles is thus measured.

MUltiplie one of the equall sides in it selfe, the halfe of the producte is the Area or superficiall contente.

Example.

ADmit ABC an Isoscheles right angled triangle, ha∣uing

[illustration]

the sides AB and BC that contayne the righte angle B equall, eyther of them being 10, I multiplie 10 in it selfe, thereof ariseth 100, the halfe 50, is the Area of that Isoscheles right angled triangle.

A right angled Scalenum you shall thus mete.

MEasure the longitude of two sides contayning the right angle, aug∣menting the one by the other, the halfe thereof is that triangular su∣perficiall contente.

Example.

Suppose the right angled Scalenum ABC, his

[illustration]

two sides that contayne the right angle AB 10, BC 16, the one augmented by the other, yeeldeth 160, whose halfe being 80, is that triangulare Area.

In euery of these right angled Triangles you shall not neede to measure more than two of their sides, for any two knowen by supputation the third may be found, whereof I shall giue you rules with examples.

The second Chapter. Any tvvo sides of right angled triangles knovven by calcula∣tion to finde the thirde.

EVCLIDE in the 47 propositiō of his first booke of Elemēts, hath proued by demonstration, that y^e squares of the two cō∣tayning sides ioyned togither, are equall to the square of y^e Hypothenusa, or third side subtending y^e right angle: wherby we may redely (any two sides being giuē) find y^e third thus:

〈◊〉〈◊〉 the two sides comprehending the right Angle be knowne, adde their ••••uares together, and the Radix Quadrate of the Product, is the Hypothe∣••••sa, but if you knowe the length of that Hypothenusa, and one other side, 〈◊〉〈◊〉 shall subtract from the square of that Hypothenusa, the square of that ••••her geuen side, and the roote Quadrate of the remainder is the third side ••••sired.

Example.

In the Triangle ABC, AB is 6, and

[illustration]

C ••, their squares ioyned make 100, ••hose roote is 10, and that is the Hypothe∣••usa. In the Triangle DEF, the Hypothe∣••usa is ••3 DE, 5. I desire the lengthe of EF, 〈◊〉〈◊〉 the square of DE, subtracted from 169 〈◊〉〈◊〉 square of the Hypothenusa, leaueth 144, ••hose Quadrate roote 12, is the side EF.

Of Obtuse angled Triangles, there are also two sortes, Isoscheles ••nd Scalenum.

The .3. Chapter. Ambligonium Isoscheles is thus measured.

YOu must first finde out the Perpendicular line, from the Ob∣tuse Angle to the contrary side, by deducting the square of halfe the side subtending the Obtuse angle, from the Square of one whole side containing the same angle, the Roote Quadrate of ••he Product is the Perpendicular, which multiplied in the halfe of the for∣••ayd subtending side, produceth the Area.

Example.

ABC the Isoscheles Ambligonium, AB

[illustration]

••0, BC 16, 10 squared is 100, 8 squared yel∣deth 64, which deducted from 100, leaueth ••6, whose Quadrate roote is 6, the Per∣pendicular AD: which multiplied by 8, pro∣duceth 48, and that is the Area of this Obtusiangle Isoscheles.

Ambligonium Scalenum, you shall thus measure.

FIrst you must search out his Perpendicular line in this manner, square euery side, then adde the square of the side subtending the Obtuse angle,

to one of the other squares, from the offcome abating the thirde square, the half of the remainder diuide by the forsaid subtending side, the square of the Quotient you shal deduct from that square which before you did adde to the square of the subtending side, and from the remaine, extract the roote Qua∣drate, for that is the line Perpendicular, which multiplied by half the fore∣named subtending side, will produce the content Superficiall of y^e triangle▪

Example.

ABD the Triangle, AB 20, AD 34, DB 42, the square of AB 400, of AD 1156, of BD 1764, 1764 added to 400 maketh 2164 from which subtracting 1156, there re∣maines 1008, the halfe beeing

[illustration]

504, diuided by 42, the side subtending the Obtuse angle, yeldeth in the Quotient 12, so much is the line BC, which squa¦red maketh 144, that subtra∣cted frō 400, leaueth 256, whose roote Quadrate being 16, is the Perpendicular AC. Now mul¦tiply 16 in 21, the halfe of the forsaid subtending side BD, so haue you 336 the super∣ficies of that Ambligonium Scalenum.

The 4. Chapter. Of Acutiangle Triangles called Oxigonia, there are three kindes.

FOr either the thrée sides are equall, and then is it an Equilate•• Triangle, or two sides only equal, which is Isoscheles, or al thrée vnequall, and that is a Scalenum.

Of Triangles Equilater.

You shall multiply the square of the side in it selfe, and the offcome in 3, the Product diuide by 16 the roote Quadrate of the Quotient is the Area, or multiply the Cube of halfe the Equiangle Triangles side in the Semi∣perimetrie of the Triangle, the Quadrate roote of the Product is the fore∣said Area also.

Example.

Admit the Triangles side 6, the squared square is 1296, which augmented by

•• yeldeth 3888, and that diuided by 16, bringeth in the Quotient▪ 243, whose

[illustration]

roote Quadrate beeing 15, and betweene 1/•• and 1••/31, is the superficiall content of that E∣quilater Triangle. The same number is founde by the seconde rule, for the Cube of 3 is 27, which multiplied in 9 the Semiperime∣try yeldeth 243, whose roote Quadrate is the Area of that Triangle agreeing with the former supputation.

Of Isoschele Acutiangle Triangles.

••Rom the square of one of the equall sides subtract the square of halfe the base or vnequall side, the Quadrate roote of the remainder multiplied in ••••lfe the base, produceth the content superficiall.

Example.

AB and AC the equall sides of the Isoschele Trian∣••••••,

[illustration]

either of them 6, the base BC 4, the halfe thereof 2, ••••ose square deducted from the square of 6, leaueth 32, ••••ose roote multiplied in 2, bringeth the roote of 128, which ••••ery nigh 11 7/2•• the Area of that Triangle.

Of Oxigonium Scalenum.

To finde the Area of Acutiangle triangles that haue side equall to other, it behoueth you to searche out 〈◊〉〈◊〉 line Perpendicular falling from one of the angles the contrary side, and multiply the same in halfe the base or side wheron ••alleth, the Producte is your desire. But to get the line Perpendiculare, 〈◊〉〈◊〉 shall thus worke: square euery side, then adde the square of the base, to 〈◊〉〈◊〉 square of one side, deducting fro the Producte the third square, halfe of 〈◊〉〈◊〉 Remainder diuided by the base, the Quotient you shall againe square 〈◊〉〈◊〉 deduct this number from that square which you adioyned to the square your base, the root Quadrate of the remainder is the line perpendicular.

Example.

〈◊〉〈◊〉 Acutiangle triangle Scalenū ABC, AB 13, BC 14, CA 15, the square of AB 169,

of BC 196, of CA 225, 169 added to 196, pr••∣duceth

[illustration]

365, from which 225 detracted, leaueth 140, whose medietie being 70, diuided by 14, yel∣deth 5 the line BD, the square thereof abated from the square of AB, leaueth 144, his root•• Quadrate is 12, the Perpendiculare AD▪ which augmented by 7 the medietie of the base, bringeth 84, the Area or Superficiall content of that Triangle.

Finally I thincke it not amisse to geue you one Rule vniuersal to mea∣sure all manner Triangles of what sorte so euer they be, and that without any regarde of the Perpendicular.

The .5. Chapter. A rule generall to measure all manner Triangles according to their plaine.

ADde all the sides of that Triangle together, taking halfe of the number which surmounteth. Now out of this halfe, you must pull by Subtraction euery side by it selfe, noting diligently the differences, that is, how much euery side dif∣fereth from that halfe, multiply the differences the one in the other, and the Product in the thirde, with that which riseth augment the halfe aboue mentioned, then séeke of that summe the Quadrate roote, so haue you the true content Superficial of that Triangle, the example shal ensue, wherby al that I haue said, shal the better appéere.

Admit the Triangle whose sides you haue measured ABC, of whome the

[illustration]

left side AB i•• 12, BC 16, AC 20, these sides ioy∣ned together, make 48, whose halfe is 24, from whome BC 16, differeth 8, AB 12, AC 4, so the differences are 8, 12, 4. Nowe multiply 8 with 12, riseth 96, the which augmented with 4, commeth 384, that multiplied in 24, the halfe if 48, sur∣mounteth 9216, of this summe the Radix is 96, which is the verye content of this Triangle.

Of Quadrangles (that is to saye plaine figures, hauing foure Angles and foure sides,) there are fiue sortes, as appéereth in the Diffinitions, the Square, the Rectangle, Rhombus, Rhomboides, and Trapezia.

The .6. Chapter. Of Squares.

FOr the square you shall onely measure

[illustration]

one side, multiplying the same in it self, so haue you the Area or content super∣ficiall.

Example.

I finde AB the one side of the square ABCD 10, whiche augmented by it selfe riseth 100 the Area of that square.

Of rectangles or right angled Paralelogrammes.

In right angled paralelogrammes ye muste measure the two vnequal sides, multiplying the one in the other, the product is the content superfi∣ciall of the figure.

Example.

I suppose the two sides AB, AC of that righte

[illustration]

angled Paralelogramme. ABCD the one 45, the other 110, these two multiplyed togither yelde 4950 the Area.

Of Rhombus and Rhomboides.

THese two figures haue one rule, it behoueth you to measure one side, and the perpendicular fallyng from one of the opposite angles to the same side, these multiplyed the one in the other produceth the Area.

Example.

Admitte ABCD the Rhombus, whose Area I desire: I measure the ••••de BD finding it 20, then muste I measure also the lengthe of a lyne perpen∣dicular

fallyng from A vppon BD,

[illustration]

which I suppose here 16, these two mul¦tiplied the one in the other bring 320, the superficiall content of the Rhom∣bus. Likewise in the Rhomboides sup∣pose I finde by mensuration the side BD 42, the lyne perpendicularly fal∣lyng from A vpon BD 16, multiplie these nūbers, I produce 672, the Area of that Romboides ABCD.

But bicause it may séeme somewhat difficult to get the length of those perpendiculars, bicause it is vncertain on what point of BD the perpen∣dicular line shall fall, I thinke good to prescribe you a rule how you may exactly and redily get all such lines perpendicularly fallyng, and that not only in these figures, but also in triangles, whiche shall be no small ease and discharge of laborsom trauayling, when you shall measure great fiel∣des, or champion playnes, &c.

The .7. Chapter. For measuring of lines perpendicular.

YOu shall prepare a right angle by conioyning thrée staues proportioned according to these nūbers, 3 4 and 5, as you were taught in the former boke, the one of the staues con∣teyning the right angle you must place directly vpon that side of your playn figure that subtendeth the angle from whence youre perpendiculare shoulde fall, marking ther∣withall whither the other conteining side directeth, if it lye euen with the foresayd opposite angle, then is the right line betwéene your station, & that opposite angle, the perpendicular. But if it directe not iustely to the an∣gle, you shall moue to and fro in that subtendent syde, till you fynde it so, then by the preceptes giuen in the fornier booke, you may sundry wayes measure the distaunce of the angle opposite from youre station, whereby you are brought in knowledge of this perpendiculars length.

Admit ABCD the paralelogramme, or ABC the triangle, whose per∣••••ndiculares falling from A to the contrary side I desire to measure, but because 〈◊〉〈◊〉 know not to what point in BC, or BD, these perpendicular lynes from A will 〈◊〉〈◊〉, I take my right angle made of the three staues GEF, placing EF in the

[illustration]

〈◊〉〈◊〉 BC, and BD, passing to and 〈◊〉〈◊〉 in those lynes till I find EG 〈◊〉〈◊〉 directly with the angle A, then 〈◊〉〈◊〉 I that a straight line from A, to 〈◊〉〈◊〉 station or angle E, is the perpen∣••••cular, whose length I may mea∣••••re either by Quadratum geome∣••••icū, or otherwise, without instru∣••ent, as in the first booke is decla∣••••d. Or thus deducting the square 〈◊〉〈◊〉 EB, the distance of my station ••o the ende or angle of eyther field 〈◊〉〈◊〉 B, from the square of the line A••, rising from that subtendent side 〈◊〉〈◊〉 the opposite angle, the roote qua∣••rate of the remaynder shall be the 〈◊〉〈◊〉 perpendicular, whiche multi∣••lyed in BD, yeldeth the Area of that paralelogramme: or in halfe BC, so ••aue ye the superficiall content of the triangle ABC.

The .8. Chapter. To measure Trapezia.

YOu shall measure the length of a diagonall or crosse line exten∣ded to opposite angles through the Superficies of the Trapezium and likewise the length of the two perpendicular lines, falling from the other angles vppon the sayd crosse lyne as you were ••aught in the last chapter, then adde those two perpendiculars together, ••ultipling the halfe of the product in that diagonall line, so haue yée the Area of that Trapezium.

Example.

ABCD the Trapezium, CB the diagonall or crosse lyne extnded fro

the angle C, to BE and F, the two poyntes where the perpendiculars shall 〈◊〉〈◊〉

[illustration]

from the other angles A, D, vppon the crosse line CB, these points I find with my right angled triangle, as was taughte in the laste Chapiter. Now to attein the Area, I measure the length of BC 20, AE 8, FD 6: 6 and 8 ioyned make 14, the half is 7, multiplied in 20, the line diagonal produceth 140, and that is the con∣tent superficiall of that Trapezium.

The .9. Chapter. Rules to measure all equiangle superficies hovve many sides soeuer they haue.

FIrste you must get the centre of youre figure, then from it pull a perpendicular line to the middes of some syde, sée how many perches or other measures it conteyneth, adde all the sides toge∣ther, multiplying halfe the summe in the perpendicular or han∣••ing line, so haue ye your purpose.

Example.

[illustration]

Imagine this figure BCDEFG and euery side of length 12, the centre A founde, drawe a lyne perpen∣dicular from it to the middle of the side BC, this line being 10 ⅖ multiplyed in 36 the halfe number of the sydes, bryngeth 374 ⅖ the superficiall contente of that figure.

An other Example.

FGH is here a Superficies beeing 5 square,

[illustration]

euery syde 10, Nowe from youre centre to the myddes of the syde pull the perpendicular lyne whych is 6 22/25 thys multiplyed in 25, the semipe∣rimetrie of that Fygure, yeldeth 172, and that is the Area of this pentagonall superficies.

A note to finde the centres of those equiangle Figures.

THe centre is found drawing lines from one angle to the contrarie, or from the middle of some side being odde to the opposite angle, the cut∣ting or concourse of those streight lines sheweth the place of the Centre: and thus as is declared, you maye readely measure all equiangle figures, what capacitie or number of sides soeuer they bée of.

The .10. Chapter. To measure the Superficiall content of any rightlined Figure of vvhat forme so euer it bee,

THe best rule I can perscibe, is to resolue it into Triangles, by drawing or ymagening lines from Angle to Angle, and so measure euery Triangle seuerally: Finally, adding all the productes togyther, ye shall haue the Area or contente Superficiall of that whole Figure, which althoughe it be of it selfe (the premisses well vnderstande) playne ynoughe, yet to auoyde all doubtes, I shall adioyne one example.

Admit ABCDE an irregular

[illustration]

pentagonum, whose side AB is 20, BC 30, CD 16, DE 24, EA 18 this figure you maye diuide in three triāgles, by drawing the two lines BE BD, thus by the rules tofore giuen, I measure first the Area of the tri∣angle ABE, whiche I finde 144, likewise BED, 312, the thirde tri∣angle BDC 240: these three nūbers ioyned togither, produce 696 which is the true Area or contente Superficiall of that irregular Superficies.

The .11. Chapter. A readie mean to find the content superficial of any great field, or cham∣pion playne, hovv irregular of forme or fashion soeuer it bee, vvithoute painfull trauayling about it, onely by measuring one side.

YE shall, as I haue taught in Longimetra (either with Theodelitus or your Topographicall instrument) draw vpon some leuell, smothe playne superficies, the exacte platte and symetry of that field, then either by your in∣strument, or otherwise: measure howe many rodde or perche, is contayned in some one side of that playne, so many equall diuisions shall you make, opening youre compasse accordingly in his correspondent side of your platte. This done, note well the fashion of the field, and drawing lines from angle to angle, or otherwise, diuide it into triāgles, or other regulare figures, as you shall sée cause: then opening your compasse to one of those diuisions, you maye therewith measure euery side, and line, in that figure as exactely as with corde, or pole, ye should paynfully pase it ouer, whereby with the ayde of these former preceptes, you shall seuerally measure euery triangle or o∣ther regular figure, and ioyning togither their contentes the resulting summe is your desire.

Example.

Suppose GHIKL that irregu∣gular

[illustration]

pentagonum, the true platte of some great field or playne made by one of the instruments Geometrical, as is before in the first booke declared and that by mensuration I finde the side GL 120 perches in length, ope∣ning therfore my cōpasse according∣ly, I diuide GL into 120 equall por∣tions, and proceding on to euery side I finde GH 90, HI 100, IK 150, KL 80, then beholding the forme thereof, I see it maye aptely bee parted in three Triangles by lines drawen from I, 60 G and L. In like manner I searche howe many such Diuisions there are in them, finding either of them 170 perches in length. Finally, by the rule generall of Triangles, I searche the Area of euery

Triangle, finding the fyrst that is IHG 360, IGL 780, IKL 600, all three ••••yned togyther, maketh 1740, so manye perches you maye conclude the Area of ••hat Figure, which reduced 10 Acres diuiding by 160 bringeth 10 Acres 3 ½ R••odes.

A note for Wooddes.

THIS way shall you moste spéedely and exactly measure all manner Woddelande whiche otherwise is very harde and tediouse preciselye ••o doe, forasmuche as ye can not at one viewe beholde euery parte there∣of, nor measure such Diagonall lines passing thoroughe it in sundrie ••laces, as were requisite in irregular Figures, but with the ayde of your ••nstrument Topographicall encompassing it rounde aboute, alwaye no∣••ing the Angles of Position at euery station, you maye firste make an exacte platte thereof, and after measure as you were instructed in the for∣mer Chapter, the Superficiall capacitie.

The .12. Chapter. Hovve you maye from an highe Hil, or Cliffe, measure hovv manye Acres, Roodes, or Perches, is contayned in any Fielde, Parke, VVood, or other playne Superficies, in the countrie rounde aboute you, not approching nighe them.

CALl to remembraunce howe you were taught in the first Booke by the Instrument Topographicall to set foorth the true platte of an whole Countrie, and euery parte there∣of, whiche, for asmuche as it is there at large sette out, it were héere superfluous to recite agayne, admitting therefore, (by the Arte there taughte) an exacte platte forme of the Fielde, Woodde, or other playne made, ye shall with your•• Compasse diuide some one side thereof into 40, 60, or 100 equall partes, as

you liste, and kéeping youre Compasse immouable, measure all suche o∣ther lines Perpendiculares, &c. as shall séeme requisite to attayne the Area thereof, and by the former preceptes, diuiding it into Triangles, Rightangled Parallelogrammes, or other regular figures, ye shall mea∣sure the contentes superficiall thereof, that is to say, how manye of those small squares, whereof euery little diuision was a side, is contayned in that superficies or platforme. This done, ye must also with youre square geometricall or other instrument from the hill or cliffe, measure y^e length of that side in the fielde, that the first diuided side in your platte did repre∣sente, I meane how many rodde or perche it is long then square aswell the number of perches in the side of the fielde, as also the number of diui∣sions in his corresponding side of youre platte, the number procéeding of the perches squared, ye shall multiplie in the Superficiall contente of youre platte tofore founde, and the Producte diuide by the square of the Diuisions in the syde of youre platte, the Quotient will be the number of perches, whiche diuided by 160, and the remayne by 40, the Quoti∣entes will shewe the Acres and roodes contayned in that fielde, parke, or woodde, you measure.

Example.

ADmitte ABCDEF the platte or proportionall patterne of a parke which from some Hill or Cliffe a farre off I haue drawen by the ayde of myne Instrument Topographicall, as was declared in the firste Booke, and for that it is a Figure of many sides, I searche howe it maye beste be resolued into regular Figures, whiche (as you maye perceyue) is readely done withdrawing the two lines, AC and FD, which partes the whole Figure in two Triangles and one right angled Parallelogramme, then opening my Compasse to some small di∣stance, I diuide some one side (for Example the side AF) into 40 partes, and keeping my Compasse immouable, I measure howe manye of those Diuisions are contayned in AC and FD, finding eyther of them 60. Now, to measure the Triangles ABC, DEF, I drawe Perpendiculares from B too AC, and from E to FD, finding BG 20, and HE 16, Nowe 40 multiplied in 60, bringeth 2400, the right angled Parallelogramme AD, 20 the Per∣pendicular in 30 halfe the Base yeeldeth 600, the Triangle ABC, 8 halfe the other perpendiculare, in 60, the base ariseth 480, the triangle DEF, these three

ioyned togither, produce 3480, the whole contente of ABCDEF, now as was

[illustration]

taught in Longimetra, with my square geo∣metricall, I must measure from the hill or cliffe the length of that side of the parke represented here by AF, which I admitte founde as is before sayde 356 rodde or perche, whose square beeing 126736 aug∣mēted in 2400, the content of my plat brin∣geth 304166400, this diuided by 1600, the square of the side AF yeldeth 190104, and this diuided by 160 produceth in the quoti∣ente 1188, and the remayne is 24. I con∣clude therefore that there is in that parke 1188 acres, and 24 perches.

In this sorte with small exercise vsing industrie, in making your plattes you may moste exactely and speedely surueye an whole countrey, with all his pastours, medowes, marshes, woddes, and euery perticulare inclosure, whereof although I might propound an infinite number of examples, and filt many leaues with varietie of rules, yet considering the premis∣ses to the ingeniouse will suffise, I thinke good to passe them ouer, referring the rest to the practisioner, who shal by his experience (well vnderstanding the pre∣misses) inuent manyfold mo, euen as occasion shall be offred in viewing the ground with situation of places.

The 13. Chapter. A note hovv to suruey an vvhole Region or playne champion Coun∣trey by the ayde of a playne pullished glasse.

IT behoueth you to resorte vnto the firste booke where you were taught by the ayde of a playne pullished glasse of stéele) vpon an high hill or leueled platfourme) to set foorth the syme∣trie or proportion of an whole countrey with al his partes. Ye shall therfore by the arte there shewed, get the proportion or plat of that fielde, wood, marshe, or other inclosure, whiche you desire to suruey, and therewith worke euen in like maner as you were taught in the last••

chapter, to do with the platte drawen with your instrument Topogra∣phicall, one ground, one reason, and like operation serueth them both, the former well vnderstande maketh this so manyfest as it néedeth no o∣ther example, onely this I muste giue you warning of, that you attempt not this kinde of measuring, but onely in champion and playne leuell Countreys, for in vnleuell and hilly places, without moste exquisite ob∣seruation and learned handeling great errour may ensue.

Thus hauing declared seuerall rules for euery kinde of rightlyned Superficies, it seemeth méete somewhat to say of suche playne Super∣ficies as are enuironed with curue lynes or mixte of bothe, and firste of the circle and his partes.

The .14. Chapter. Of Circles.

BEfore I entreate of the mensuration of circles, it shall be requisite to declare Archimedes rules, concerning the proportion of the circumfe∣rence to his diameter, and of the Superficies to his dimetientes square. The rules ensue.

The fyrst Theoreme.

〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉.

EVery circle is equall to that rightangled triangle, of vvhose conteyning sides the one is equall to the semidiameters, the other to the perimetrie or circumference.

The seconde Theoreme.

〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉.

THe proportion of euerye circle to the square of his dimetiente is as 11 to 14.

The thirde Theoreme.

〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉.

THe circumference of a circle is more than triple his diameter by suche a parte as is lesse than 1/7, and more than 12/7•• thereof.

Hovv the Area of a circle is founde.

BY the former Theoremes ye may collecte these rules. Multiplye the circumference of any circle by 7, and diuide by 22, your quotient is the circles diameter, whose medietie multiplied in halfe the circumference yeldeth the Area, or multiplying the square of the circles dimetient by 11, and diuiding the ofcome by 14, your quotient will declare the same.

Example.

Admit the circumference of the circle 44, which en∣creased

[illustration]

by 7 yeldeth 308, this diuided by 22 bringeth in the quotiēt 14, the diameter, whose medietie being 7 aug∣mented with 22 the halfe of the circumference, produ∣ceth 154. Likewise the square of 14 multiplyed in 11, maketh 2156, whiche diuided by 14, bringeth 154, the superficiall capacitie of that circle.

Of the halfe circle.

EUen as I haue declared that in multipleing the halfe diameter of any circle in his halfe circumference, the producte to vtter his contente, so by augmenting the halfe diameter in the fourth parte of the circumfe∣rence, that is in halfe the arke of the semicircle, ye haue the whol summe of that halfe circle.

Example.

ADmit the semicircle DEF were to be measured, whose diameter DF dra∣wen

by the centre C is 14, as afore, the Arcke

[illustration]

DEF 22. Nowe multiplie the semidiameter 7 in 11 riseth 77, the playne of that semicircle, euen so of all portions or partes of a circle. Althoughe to the wittie this may suffise, yet to satisfie also the meane witted, I shall not thinke it tediouse to pro∣pounde an other example.

Of the fragments or partes of a circle.

SVppose GHIA were a portion of a circle to be moten, the whole circumfe∣rence

[illustration]

of the circle wherof this is a por∣tion is as afore 44. Now you shall multi∣plye 7 the semidiameter HA in the halfe arke 15, so haue ye 105, whiche is the su∣perficial content of that portion GHIA, Yf you desire to know the Area of GIB, conteyned of the corde GI, and arcke GIB, ye muste adde to the number before found, the Area of the triangle GIA which I suppose (found by the rules tofore giuen) 22, that maketh 127 the Area of the segment GHI to the corde GI, which subtracted from 154, the whole circle leueth 27 the area of the segment GIB.

I thinke none will doubt how these figures folowing are measured, bicause they are made of parts or portions of a circle, whose plaine is ga∣thered as I haue alredy declared. In the left figure there is at eche end a semicircle, euery of them conteyning (found by the art afore mentioned) 77, which added make 154, the quadrangle hath 84, as the rules of qua∣drangles declareth, that ioyned with 154, bringeth 238, the whole summe of that figure.

[illustration]

The other are but two suche segmentes as GIB, which measured as is before shewed and ioyned together bringeth 54, the Area of that figure. Likewise y^e last figure called a Lunula, ABCD is mesured by deducting the segment ADC (found by the former rules) 27 from ABC known in lyke maner 56 4/•• there remayneth 29 4/7 the Area or Superficies of the Lu∣nula ABCD. Forasmuche as it is necessarye in measuryng of these por∣tions or segmentes of circles to knowe the diameter of the whole circle, wherof they are the fragmentes. I thynke it not amisse herevnto to ad∣ioyne a rule for the same purpose.

Admit ABCD a portion or fragment of a circle, I desire to know the lon∣gitude of his circles diameter, first I measure AC whiche I fynde 12, lykewyse the length of the lyne DB from D the middle of the

[illustration]

streyght lyne to B the middle or highest poincte of the arke whiche I suppose 4, Now diuide the square of AD that is 36 by 4▪ so haue ye 9, which added to 4 bringeth 13, and that is the length of the circles dimetient, whose parte or fragment this figure ABCD is.

These preceptes well noted there can no playne Superficies offer it self whether it be regularor irregular, framed of right lines enuironned with circular or mixte of both, but ye shal readily by diuiding it into triangles & cular fragmentes, finde out the Area and content superficiall thereof. I will nowe therfore only, adioyne certayne questions for the partition and diuision of grounde. And so passe on to the thirde kynde of Geometrie, where you shall haue rules to measure, not onely the solide, but also the superficiall contentes of all maner bodies, but to returne to these parti∣tions my first question shall be of triangles, and so on to figures of more difficultie.

The .15. Chapter. There is a trianguler field hauing on the one side a vvell or foun∣tayne, this fielde must be equally diuided betvvene tvvo partie••, and that in such sort that either of them may haue cōmoditie of that fountayn, not cōming on the others lande. I demaunde hovv that partition shall be made.

LEtte ABC represente the triangular fielde, hauing on the syde BC a well or fountayne at D, from whence I woulde directe a hedge or ditthe in suche sorte that it parteth the Triangle in two euen partes, first therefore I searche the myddle of the syde BC, whiche I suppose E, then my instrument Topographicall at the fountayne situate as he oughte to bee, I turne the Ashidada till I can espye thorough the sightes the op∣posite

[illustration]

angle here represented by A, then re∣mouinge myne instrument to the foresayde middle at E, the Ashidada remayning im∣mouable on the degrees tofore cutte, the in∣strument agayne duely situate, I view tho∣roughe the sightes what parte of the syde AC I can espye, Admitte it F. Now say I that a hedge dyke or other partition runnyng from D the fountayne to F the marke espyed in the syde AC, wyll diuide that triangular figure exactely into two equall portions, and thus maye you withoute supputation onely by the ayde of youre instrument Topographical diuide any triangle in two equal parts, and that from any parte or poincte in any of hir sydes assigned.

The .16. Chapter. To cut off from any triangular fielde as many acres, rodde or other measures, as shall be required, and that by a lyne dravvne from any angle assigned.

YOu shall fyrste measure the syde subtendyng the Angle assigded whiche here I wyll call the base. Secondely, you shall measure the Area of the whole triangle: Then multiplye the number of Acres or Roddes whiche you woulde cutte off from the triangle, by the lengthe of the base, and the producte diuide by the Area of that Trian∣gle, the Quotiente sheweth howe many pearches you shall measure in the base from one of the Angles to cutte of youre desyred number of roddes▪

Example.

Suppose ABC a triangular peece of ground, hauyng his three sides AB 30▪ AC 40, CB 50, my desire is with a right lyne from the angle A to cut of one ••cre of grounde, to performe this by the

[illustration]

••rte before taught, I searche the Area of that figure which is 600 roddes. Now bycause I would cut off from that figure one acre, and an acre cōteineth 160 rods: I multiply 160 in 50, the whole side subtē∣ding the assigned angle, the product being 8000, I diuide by 600 the triangles Area, my quotient is 13 1/•• so many perches I recken in the base BC from B to D, finally extending a lyne from A to D, I conclude the Superficies ABD one exacte acre, and this is there from that tri∣angular field one acre cut of with the lyne AD.

The .17. Chapter. To cut off from any triangular piece of grounde vvhat quantitie of perches ye lyste vvith a lyne equidi∣stant to one of the sides.

TO perfoorme this conclusion it shall be requisite to serch out what parts of the other two sides this equidistant line shal cut, wherof I will giue you two rules, forasmuche as this partition may two seuerall ways be made: for the portion to be cut of eyther is adioyning to the parallele syde, and then is it a quadrangular figure, or else to his subtēding an∣gle, and then is it a triangle. For the triangle ye shal thus doe, square the sides that the parallele line shal cut, and the products seuerally multiplie in the number of perches to bée taken away, the surmountyng summes di∣uide by the Area of the whole triangle, the rootes quadrate of the quotien∣tes is the number of pearches to be accompted in the correspondent sides,

from the angle that subtendeth the Parallele side. But if ye would haue the portion cut off, lye adioyning to the paralele syde, then shall you first deduct that portion from the whole triangular Area, and with the remain∣der woorke in like maner as ye were taught in the former rule, so will your quadrate rootes (accompted from the paralele syde his subtendente angle) shewe to what part of eyther syde the lyne shall be drawne, to cut of the forenamed portion.

Example.

Suppose ABC the triangle from which I would cut off one acre, that is to say 160 rodde by a paralele lyne to the side AB, first therefore I measure the Area of that whole triangle as was before taught in this boke, fynding AB 50 perch, AC 120, BC 130, and so consequentely the Area of that triangle 3000 rodde, the square of AC is 14400, the square of BC is 16900, these augmented by 160, bring 2304000, and 2704000, and these diuided by 3000, produce in the quotientes 768 and 901 ⅓, the roote of 768 perches) being 27 perches 12 foote) I mesure out in the side AC,

[illustration]

beginning from C, admit it ende at D, likewise the roote of 901 ⅓ perches is 30 pear∣ches, and betwene 4 and 9 in∣ches, measuring therefore 30 perches from C in the other side CB I sette vp a marke at E. Nowe if you drawe a streyght lyne from D to E it shall be a paralele to AB, and the peece of grounde, represented by DEC, an exacte acre. But if you desire to laye oute this acre at one of the sides as ye may see in the figure signified by the quadrilater Superficies ABGF, Then must ye deducte the aforesayde 160 roddes fro the Area of that triangle, the re∣mayne is 2840, whiche I multiplye as before in the square of AC, so haue I 40896000, Lykewise the same 2840 augmented by the square of BC produ∣ceth 48016000, these products seuerally deuided by 3000, the Area of the whole triangle will yelde in the quotientes 13632 and 16005 ⅓ theyr quadrate rootes are 116 perches 12 ½ foote, the length of the lyne CF, and 126 perches 8 ½ foote, the lyne GC, Or if yee deducte those rootes from the whole sides AC, and BC there wil remayn 3 rodde 4 foote from A to F, and 3 rodde 8 foote from B to G And thus may you in all Triangular peeces of ground, exactly lay forth an acre,

••r any other quantitie of grounde ye will require, and that either against the side 〈◊〉〈◊〉 Angle, euen as you will desire.

The .18. Chapter. For partition of Paralelogrammes vvhat kinde so euer they be of, note these Rules ensuing.

BIcause partition may sundry wayes be made according to the seuerall situation of the line wherewith it is deuided, I wil first entreat of that diuision that is made by a Pa∣rallel vnto two of the Paralellogrammes sides.

Admit therfore ABCD the peece of ground which by men∣suration I finde to containe 65. Acres 16 Perches, and that I would cut of with a Parallel to AC 5 Acres, as ye may see done by the line EF, whiche were easily brought to passe if I knewe the quantitie of the lines AE or CF, for knowledge of

[illustration]

them therfore I worke thus, first with my square Geometricall or otherwise as hath ben declared in Longimetra, I finde the lēgth of AB 308 rodde, I multiplie ther∣fore 800 for so many rodde are there in 5 Acres, by 308, there amounteth 246400 which diuided by 10416 the number of Perches contained in the whole quadran∣gle, yeldeth in the Quotient 23 Perches 11 fote, the length of the two lines AE & CF. Thus by making the partition EF, the portion AECF, shall conteyne ex∣actly 5 acres, this rule is generall for all quadrangular peeces of grounde, whose sides be parallele, whether it be square rectangle, Rhombus or Rhomboides.

But if you desire to make like partition by a righte line issuyng out of one angle you shall thus woork: first consider whether the portion ye wold cut of be greater or lesse than halfe the parallelogramme: if lesse multiply as before the numbre of perches, that ye wold separate from the quadran∣gle in one of the opposite sides to that angle whence your diuiding lyne is∣sueth, and the product diuide by the medietie of the Area, the quotient will shewe on what parte of that opposite side your diuiding lyne will fal. But if the aforesayd portion to be separate be greater than the medietie of the Parallelogramme ye shal deduct it from the whole Area therof, and with the residue procéede in like maner as was before taught.

Example.

Admit ABCD a Quadrangular peece of ground, which being measured, is founde to containe 50 Acres, and that my desire is to cut of 10 acres from that figure, with a right line passing foor••he from the Angle C, first therefore I mea∣sure the length of the side AB, finding it 200 rodde, 200 therefore I augment by 1600 the number of Perches in 10 acres, there ariseth 320000, whiche diuided by 4000 the number of rodde contained in halfe that figure, your Quotient wil be 80, the number of Perches from A to E, so that

[illustration]

by the Partition CE, ye shall seperate the Triangle ACE containing exactly 10 a∣cres, but if the summe of acres had ben grea∣ter than halfe, ye should haue deducted it from the whole Area, and with the re∣sidue woorke in all respectes as ye haue done with this, the only difference is, that at the ende of your operation, whereas heere the Triangle ACE is the portion seperate, there it shoulde be the residue or Quadrangle ECDB, for that there is no difference in the working, it were superfluous to vse mo Examples.

The .19. Chapter. To cut of from any Trapezium or Quadrangular peece of ground, vvhat part therof ye list.

Example.

Admit ABCD a Trapezium, or Irregular Quadrangle, and that I woulde cut of a quarter of his Area, with a line issuing from the Angle A, first

[illustration]

by the Rules tofore geuen, I measure his content superficiall, which I sup∣pose 72 acres, secondlye I meete the Area of the Triangle ADC, fin∣ding it 50 acres. Nowe bicause I would cut of a quarter of that Tra∣pezium, I diuide his content that is 72 by 4, my Quotient is 18, and

that multiplied in DC, (which heere I suppose founde by mensuration 146 Per∣ches,) produceth 2520, and this diuided by 50 the number of acres in the Triangle ADC, yeldeth in the Quotient 50 Perches 6 ½ foote, which measured out from D in the line DC, sheweth where the line of partition issuing forthe of the angle A shall fall: and thus may you conclude that ADE, is the foresaide quarter of the whole Trapezium, and containeth 18 Acres.

The .20. Chapter. To diuide the superficies of any irregular Pollygonium, vvith a straight line proceeding from any one of the Angles assigned in suche sorte, that the partes shal retaine any proportion appointed.

WHen anye proportion is geuen, there are two Numbers wherewithall it is expressed, and they are called Termini, those two you shall adde together, reseruing the Producte for a Diuisor, then measure the Area of that whole Irre¦gulare Polligonium, which you shall multiply in the lesser of those Termini, the Producte shall you diuide by your re∣serued Diuisor, the Quotient is the content of the lesser Portion, whiche Deducted from the whole, leaueth the Superficies of the greater parte. But to drawe the line or Partition that shall diuide this Pollygonium, it were necessary first to learne on what side and part therof this line should fall, which you shall thus doe: Drawe or imagine right lines extended from the assigned Angle to euery Angle on either side, so shall ye make se∣uerall Triangles, whereof by the Rules tofore geuen, you must search the content Superficiall. This done, ye shall compare the Area of the first tri∣angle, with the Superficial content of the lesser portion found as is before declared. If the Triangle be greater, then may you by the preceptes before geuen, cut of from that Triangle so many Acres or Perches as that les∣ser Portion shoulde containe, but if the Triangle be not greater, ye shall deducte it from the foresayde lesser portion, and the Remaine compare wyth the nexte Triangle, whiche if it be greater than that Tri∣angle

also, subtract from it the Superficies of the triangle, & so procéede on til ye finde a triangle greater than your remayn, then may you say, that your partition or diuiding right line proceding from the assigned angle shall fall on the base or side subtendent of the assigned angle in that last triangle, but to know on what part therof, you must worke with your last remayn and the Area of your last triangle, as you were before taught in the diuision of triangles. And in like manner draw your diuiding line which shal exactly, seperate that Irregular Polligonum, in two Superficies retaining the pre∣scribed proportion: for more plainnesse peruse the example ensuing.

Admit ABCDEF, an irregular Hexagonum which I would diuide with a right lyne issuing foorth of the angle A in suche sort that the greater part might be triple to the lesser; this proportion triple may be expressed with these two num∣bers 3 and 1, and they are called Termini or boundes of that proportion, these two added make 4, wherewith diuide the whole Superficies of that Pollygonum, which I suppose by mensuration founde 64 Acres, my Quotient will be 16. Ad∣mit

[illustration]

also I finde the Area of my first triangle ABC 12 acres, & the Area of the second ACD, 18 acres. Nowe comparing 16 with the Triangle ABC, I finde it greater, deducting ther∣fore the one from the other, ther resteth 4 Acres, whiche foras∣much as it is lesse than the Area of the Triangle ACD, I con∣clude the partition or diuiding line shal fall on the side CD, but to learne on what part thereof, I worke as I was taught in the partition of Triangles thus, first with my square Geometticall or otherwise, I measure the side CD. Admit it 36 Perches, whiche multiplied in 4 the laste Remaine bryngeth 144, and that diuided by 18 the acres contained in the triangle ACD, produceth in the quotient 8, so many rod length shall you meete from C to G, in the side CD, finally extending a right line from the assigned angle A, to the poynt G, you may conclude AFEDG Triple to AGCB.

The 21 Chapter. To diuide any irregular Pollygonium into as many equall partes as ye vvill desire, vvith right lines dravven from any poynt vvithin the superficies therof assigned.

FIrste get the Area of the whole irregulare Pollygonium, which you shall diuide by the number of partes, whervnto ye would disseuer it, and the quotient reserue, then from your poynt assigned deducte or imagine lines to euery an∣gle of that figure, so shall ye parte it into sundry triangles, then compare your reserued quotient with some one trian∣gle beginning where ye li••te, and yf ye finde the triangle greater, then cut of from it a portion equall to your quotient, with a streight line pro∣céeding from the angle adiacent to the assigned poynt, but if your quoti∣ent be greater than the firste triangle, deducte the one from the other, and compare the remayne with the seconde triangle, which triangle yf ye finde greater, cut of from it a portion equall to your remayne, with a line issuing out of the angle ioyning to your assigned poynte, and then compare ageyne your reserued quotient with the remayning triangle, yf your quotient be greater, seperate from the triangle next ensuing a portion equall to the excesse or ouerplus, and that alwayes with a lyn•• issuing oute of the angle adiacent to the poynte firste giuen. Thus pro∣céeding till ye haue circulate the figure, alwayes drawing streyght lines from the poynt assigned, ye shall in the ende departe the whole figure in∣to as many equall portions as ye determined. This one thing is to be noted, that so ofte as your remayne or reserued quotient falleth out e∣quall to any of the triangles, then shal you not draw any line from the asigned poynt, for the side of that triangle serueth for the partition or diuiding lyne.

Example.

LEt ABCDEFG represent an irregular Pollygonium, or seuen sided peece of ground, which I would diuide with streight lines issuing out of the poynt H into three equall partes, first therefore I imagine streight lines from H to euery

angle, so is it diuided into 7 seueral triangles: now by the rules tofore giuen I mete the area of euery triangle, finding HAB 5 acres, HBC 5 acres two roodes, HCD 4 acres, HDE 6 acres 1 rode, HEF 5 acres, HFG 5 acres 12 perches, HAG 4 acres 2 roodes. Thus by adding all these triangles togither you shal find the area of that whole figure 35 acres, 1 roode, 12 perches, which resolued into perches yel∣deth this number 5652, and that diuided by 3 bringeth 1884 perches, and that is the contente of euery portion that this figure should be parted into. First there∣fore

[illustration]

comparing it with the triangle ABH, I finde the triangle the lesser, the one deducted frō the other, there remayneth 1084 perches, and this re∣maine I fynde likewise greater than the triangle CHB, detracting there∣fore the one from the other, there re∣mayneth 204, which compared with the third triangle CHD, forasmuch as it is lesse than the area thereof, I searche by the rules tofore giuen in diuision of triangles, on what parte of the base, CD the diuiding line shal fall that proceedeth from the angle H: admit it cut CD in the poynt I, from I to H extende a right line, and that shalbe the first partition. Now seeing HID ioy∣ned to DEH the fourth triāgle, is lesse than 1884 perches my reserued quotient, I deduct the one frō the other, so there remayneth 448 perches, I serch therfore by the rules tofore giuen on what parte of EF the line shall fall, proceeding from H to cut of 448 perches from HEF: but heere you must consider that this portion of 448 roddes or perches must be adiacent to the side HE, bicause the quotient doth excede the triangles by somuch. This considred admit the partition HK. now bicause my desire was to diuide it onely into three partes, I nede proceede no fur∣ther, for those two lines do exactly departe that whole irregular Pollygonium into three equall partes, wherof the first is that rightlined figure IHABC, the seconde is IHKED, the third is KHAGF, and this partition is made with right lines issuing from the poynt H assigned, whiche was required. In this maner you may diuide any Superficies into as many equal or vnequall partes as you liste, or into what proportion ye will desire, as by these few examples the ingeniouse will redily conceaue of any other: manyfolde rules might I giue you in these and like partitions, but knowing the premisses to the wittie are sufficient; I wil onely

adioyne certayne questions of some more difficultie, with the resolution of them, and so finish the seconde kinde of mensuration.

The .22. Chapter. To diuide any circle vvhose semidiameter is knovven, vvith an other circumference concentricall, in tvvo suche partes that the one por∣tion to the other shal retayne any proportion assigned.

WHen any proportion is giuen, it consisteth of two num∣bers, as I haue before saide, that are called Termini ra∣tionis, those numbers ye shall adde togither, reseruing the producte for a diuisor, then multiplie the square of these midiameter knowē by the lesser of those Termini, and the ofcome diuide by the reserued diuisor, from the quotient thereof, resulting, extracte the quadrate roote, for that is the se∣midiameter of the concentricall circle, whose circumference shall diuide the former giuen Circle in two partes, retayning the proportion assigned.

Example.

ADmit 120 the semidiameter of the circle, BCDE, whose area I would diuide with a concentrirall circumference in

[illustration]

suche sorte that one parte might be triple to the other, this proportion consisteth of 3 and 1, which added together make 4, now the square of the semidiameter beeing 14400, augmented by 1 the lesser of those Termini, produceth the same summe agayn whiche diuided by 4 tofore reserued for that purpose yeldeth 3600, whose quadrate roote is 60, so muche is AF the semi∣diameter of that inwarde Circle, whose circumference hath diuided the Circle

BCDE in two partes, the lesser is the circle FGH, and the greater is the anular Superficies conteined betweene those circumferences, the one beeing triple to the other. In like maner may you diuide that anular Superficies into three other, eue∣ry one of them equall to the same inwarde circle, whereof it were superfluous to adioyne any farther example, forasmuche as the proportion of the partes once knowen, the operation is in all poyntes agreable with the former.

The .23. Chapter. The three sydes of a triangle knovven, by supputation to get the greatest circles semidimetiente that may be described vvithin that circle.

YE shall firste adde the two shorter sides together, reseruing the producte, then substract those sides the one from the other, the re∣mayne multiply in the former product, and the amounting summe diuide by the thirde or longest side, the quotient detracte from that lon∣gest side that was your last diuisor, the medietie of this remayne you shall multiplye in it selfe, and deducte the ofcome from the square of the shortest side, the roote quadrate of the remaynder is the perpendiculare falling from the greatest angle to the greatest side, whiche retayneth the same proportion to the semidiameter of the inscribed circle, that the pe∣rimetrie of the triangle doth to the base or greatest side, working there∣fore by the rule of proportion by thrée knowen, ye may redily finde the fourth.

Example.

LEt ABC be the triangle, AB 36,

[illustration]

BC 48, AC, 60, nowe to get the greatest circles semidiameter that may be described within that triangle, I woorke thus, fyrst 36 added to 48, bringeth 84, then 36 detracted from 48 leueth 12, whiche multiplyed in 84 bringeth 1008, whiche diuided

by 60, the longest side yeldeth in the Quotient 16 ⅘. This deducted from 60 the Diuisor, the remayne will be 43 ⅕, the medietie thereof is 21 ⅖: whose square de∣••ucted frō the square of 36 leueth 829 11/25, Therfore do I conclude √{powerof2}829 11/25, ••he perpendicular BG, for the semidiameter EF, I vse the rule of proportion ••aying 144 the permetrie, giueth 60 the base, what shall √{powerof2}829 11/25 by multiplica∣••ion of the latter and diuision with the first, ye shall finde the fourth proportional number 12, and that is the line EF or semidimetiēt of the greatest circle that may ••e described within that triangle.

This is also to be noted, that the medietie (whose square ye subtracte from the square of the left side to get the perpendiculare) is the line AG: which deducted from the whole base, leueth the other line GC. and those two partes of the bases are called Casus.

The .24. Chapter. To finde the greatest squares side that may be described vithin any triangle vvhose sides are knovven.

SUndrie rules might be giuen to resolue this question, but to auoyde confusion, I will only declare the easiest: ye shal therfore (as is before in this booke taught) searche the per∣pendicular line falling from the greatest angle to the lon∣gest side: this perpēdicular it behoueth you to diuide in such sorte, that the partes retaine such proportion as doth the base or longest side to the sayd perpendiculare, so doing the greater portion is y^e squares side. But Arithmetically to attayne the quantitie of this longer portion, ye shall thus worke: Multiplie the perpendicular in it selfe▪ and diuide the producte by the Base or longest side of the triangle and the perpendi∣cular ioyned togither, the quotient detracted from the whole perpendicu∣lare leueth your desire. Or multiplie the base in the perpendicular, and di∣uide by them bothe ioyned togither as before, so shall your Quotient pro∣duce the squares side also.

Example.

ABC is the triangle whose sides are knowen, AB 24, BC 40, AC 32, the perpendicular founde, a•• is tofore declared, is 19 ⅕, which diuided at I, in suche

sorte that DI to IA retayne the

[illustration]

proportion of BC to DA, DI shall bee the side of the greatest square that may be made within that Triangle: but too finde the length of DI, you muste multi∣plie 40 the base BC in 19 ⅕ the perpendiculare, so haue you 3840/5 which diuided by 206/5 the perpē∣diculare and Base ioyned togi∣ther, yeldeth 12 36/37, the line ID. Likewise, if ye square the perpendicular, the mounting summe will be 9216/25 which diuided by 196/5 the former diuisor bringeth in the quotient 6 ••2/185, which deducted fro 19 ⅕ the perpendicular, there remayneth 12 180/185, the side of the greatest square agreeing with the former operation. In like manner if ye diuide BD, whose longitude you were taught by the laste Chapter to finde, and also AB, the left side of the Triangle in such sorte, that the greater sections to the lesser retayne the same proportion, that the base doth to the perpen∣dicular, the squares of the two greater sections one deducted from the other le∣ueth the Area of the greatest square that maye be described within that Tri∣angle. Also, if ye diuide any of the two lesser sides in two partes retayning the fo∣resayd proportion of the perpendiculare to the Base, the lesser of those Portions augmented by the base and the product diuided by the side bringeth in the quoti∣ent that greatest squares side. Thus also an other way, you maye attayne the same: Diuide both the Casus, that is ••o say, BD, and DC the distance of ey∣ther Angle from the perpendicular, in like manner as hath been said of the Per∣pendicular, then adde both the smaller sections togither, the resulting summe is the squares side: Euery of these wayes working, ye may resolue this question, great pleasure shall the Arithmetrician (especially if he be seene in surde and radi∣call numbers) receyue, when he shall perceyue so diuerse intricate and different o∣perations alwaye in fine to produce the selfe same certayntie.

The ende of the second Booke.

The thirde kynde of Geome∣trie named Stereometria. (Book 3)

IN THIS thirde booke ye shall receyue sundrie rules to mea∣sure the Superficies and Crassitude of solide bodies, vvhereof, although an infinite sorte of differente kyndes might be ima∣gined, yet shall I only entreate of such as are both vsually re∣quisite to be moten, and also may sufficiently induce the inge∣nious to the mensuration of all other solides vvhat forme or figure soeuer they beare. And forasmuch as in setting foorth their seuerall kyndes, I haue chosen to vse the accustomable and auncient names vvell knovven to any trauelled in Geometrie, rather than to forge nevve English vvordes vvhich can neither so breefly nor so aptly expresse the like effecte, least to the com∣mon sorte any obscuritie might grovve, I thinke good to adioyne euery of their diffinitions,

DIFFINITIONS.

• 1 A Solide body is that hath lengthe, breadth and thicknesse boun∣ded or limited with Superficies.
• 2 Lyke solides are such as are encompassed with superficies that are lyke and of equall number.
• 3 A Prisma is a solide Figure comprehended of playne Superficies, whereof two are equall, like, and Parallele, the reste Parallelogrāmes.
• 4 A Pyramis is a solide Figure enclosed with many playne Superfi∣cies rysing from one, and concurring or méeting in a pointe.
• 5 A Sphere is a grosse or solide body comprehended of one conuex Superficies. In the middes whereof there is a pointe from whence all right lines to the same superficies extended, are equall.
• 6 That poincte is called his Center, and a streight line by that Cen∣tre passing thoroughe this solide bounded on eyther side with the conuex superficies is called the Diameter of that Sphere.
• ...

• 7 Also intellectually ye may thus conceyue a Sphere to be made. Suppose a semicircle (his diameter remayning immouably fixed) to be turned round about til it returne to the place whence it firste beganne to moue, the figure so described, is a Sphere.
• 8 Likewise, if a right angled triangle (one of the contayning sides re∣mayning fixed) be turned circularly about the Figure so described, it is called a Cone.
• 9 The right line that remayneth fixed is the Axis.
• 10 The circle described by the other contayning side is the Base.
• 11 The third line or Hypothenusa, is the side of the Cone.
• 12 If a right angled parallelogramme (the one of the sides conteyning a right angle remayning immouable) be circularly turned, the Figure so described, is a Cylinder, and the immouable side is his Axis.
• 13 The Bases are the Circles by reuolution of the two opposite Pa∣rallele sides described.
• 14 The altitude or heigth of any solide body, is a line perpendicular∣ly falling from the toppe or highest parte of the solide vppon the playne whereon the body lyeth.
• 15 This perpendicular or line of altitude in directe solides falleth within the body, and vppon the base, but in declyning solides, it falleth without the bodies and bases.
• 16 As the concourse of lines maketh a playne angle, so the concurring or méeting of many superficies in a pointe, maketh a solide angle.
• 17 In euery solide body a right line drawen from one solide angle to one other is called a line Diagonall. But if it passe betwéene opposite an∣gles, it is named the Diameter.

The fyrst Chapter. To measure the contentes Superficiall and solide of any Prisma.

FOr the Superficies, ye shall adde all the Parallelogrammes & Bases their Area togyther, the resulting summe is your desire, but for the crassitude, ye shall augmente the altitude of the So∣lide in the Area of the Base, the producte is the grosse capa∣citie.

Example.

Considering of Solides called Prismata, there are two kindes, the one directe •• vpright, whose Paralelogrammes are Perpendiculare to their Base, the other ••••lique or declining, whose Paralelogrammes are obliquely situate on their bases. ••f either I minde to propound an example, although one rule suffise them bothe. ••dmit therfore ABCDEF a direct or vpright Triangular Prisma, hauing 〈◊〉〈◊〉 three sides of his base ED 3, DC 4, EC ••, his altitude AD 10, so that by 〈◊〉〈◊〉 rules geuen in Planimetra I finde the Area of either Triangle or base 6, the Paralelogrammes AFED 30, ABCD 40 FBCE 50, all these ioyned toge∣ther,

[illustration]

make 132, the Superficies of that Prisma. Likewise for the superficies of the declining solide IKLMNO, by the preceptes geuē in Planimetra I finde the Area of the right angled Paralelogrammes or bases either of them 18, the Area of the other Pa∣ralelogrammes IGNO 39, IGLK 72, KLMP 39, OPNM 72. and these ioyned ••••gether yeldeth 258, that is the content superficiall therof.

For the Crassitude I search the altitude of either Solide, which in the vpright Prisma is, the ereared side of any of his Paralelogrammes, as AD, BC, or FE. ••or they are all equall, euery one of them being 10, but the altitude of the declining Quadrangular Prisma is the Perpendicular OR, falling from O the top of the Prisma perpendicularly on MR a line drawne in the plaine wheron the body re∣••••eth, 10 therfore multiplied in 6, produceth 60, the Solide quantitie of that Pris∣••a: Also the altitude OC being founde by mensuration 12. multiplied in 18 the ••ase, bringeth 216, and that is the Solide capacitie of the declining Prisma IKLMNO.

Although these common péeces. K.L. are moten as is tofore taught, yet 〈◊〉〈◊〉 may readely thus measure them, multiplye the length with the bredth, ••nd the Product in the thicknesse, so haue ye the content or Crassitude.

Beholde the Figure.

The content of K 216 Cubicall foote, the Crassitude of L 216 square foote.

[illustration]

The .2. Chapter. Hovv the contents Superficiall and Solide of a Pyramis may be measured vvhether it be direct or declining.

FOr as muche as euery Pyramis is enuironed with triangular plaines ri∣sing fro the base, and concurring or méeting at the toppe, ye may by the rules geuen in Planimetra measure the Area of euery Triangle▪ whose cō∣tents ioyned together, and the Product to the base, yeldeth the Superficies of that whole Pyramis. But for the content Solide whether the Pyramis be direct or inclinate, ye shall first measure the length of the line Perpendicu∣larly falling fro the top to the plaine wheron the base resteth, which multi∣plied in the third part of the bases Area wil produce the Crassitude of that Pyramis.

Example.

Suppose ABCDE an vpright Quadrangular Pyramis, whose base being a square, I measure only the side BC finding it 12, which multiplied in it selfe, brin∣geth 144 the Area of the base, then do I measure the length of the line AG 10, for so am I taught in Planimetra, that multiplied in 6, halfe DE bringeth 60, the Area of the triangle ADE, but seeing the other triangles are equall to the same, considering they haue equall bases and altitudes, I neede make no more adoe, but multiply 60 by 4 so resuiteth 240 the superficies of the foure triangles concurring as A▪ and this added to 144, the base bringeth 384 the whole content superficiall

[illustration]

[illustration]

of that Pyramis. But for the Crassitude I measure the Altitude or line AF falling from A Perpendi∣cularly vpon the base, this Perpendiculars length I suppose 8, which multipli¦ed in 144 produceth 1152, and that is the Solide con¦••••nt of that Quadrangular Pyramis. In like manner to attaine the Crassitude of 〈◊〉〈◊〉 declining Pentagonal Pyramis IKLMNO, I measure the altitude or Per∣••endicular IH 20, likewise the Area of the Pentagonal basis, whose side is 14, which by the precepts geuen in Planimetra I finde 337, these multiplied together ••elde 6740, the Solide content of the declining Pyramis.

The .3. Chapter. Hovv Cylinders and Cones are measured.

THe Cylinders altitude multiplied in the Circumference of his base, or the side of the Cone augmented in halfe the Peripherie of his base, & the Productes added to their bases, bring the contents Superficiall.

The Solide content of a Cylinder is gotten by augmenting the base in ••is altitude. Likewise the altitude of the Cone multiplied in the third part ••f his base yeldeth his Crassitude.

Example.

Admit the Area of the Cylinders

[illustration]

••ase found by the rules in Planime∣••ra 78 4/7 the circūference being 31 3/7, which multiplied in 21 the Cylinders ••ltitude yeldeth 660, which added ••nto 157 1/7 bringeth 817 1/7, the whole superficies of that Cylinder, but if 〈◊〉〈◊〉 augment 21 in 78 4/7 the bases A∣••ea, it produceth 1650, and that is ••he Crassitude or Solide capacitie ••f that Cylinder. Likewise for ••he Superficies of the Cone GHE, I augment 26 the side HG in 31 ••/•• halfe

the Circumference GFE, therof resulteth 817 1/7, wherunto if ye adioyne 314 2/7 that basis Area there amounteth 1131 3/7, the Cones content Superficiall, and by multi∣plying 24 the altitude in 104 16/21, the third part of the Circle GFE. there ariseth 2514 2/7, the Solide content of the Cone GHE.

A note to finde the altitude of a Solide Cone.

Multiplie the side of the Cone in it selfe, and likewise the Semidiame∣ter of the Circular base, these deducted the one from the other the quadrat•• roote of the Remainder is the Axis or Altitude of the Cone.

The .4. Chapter. Hovve excauate or holovve tymber. &c. is measured.

YOu shall by the Rules afore, searche the content or Crassitude, as though it were not holow, then measure the capacitie of that hollowe, the one subtracted from the other, the Remaine vtte∣reth the magnitude of that excauate body.

Example.

Admit this round holowe timber logge ABC were to be measured the heigth being 14 foote, the Diameter of the outwarde Circle 7 foote, the Dimetient of the

[illustration]

inwarde Circle 5 foote, the circumfe∣rēce of the bigger 22 foote, the cōtent su∣perficiall or plaine of it 38 ½ the Circū¦ference of the lesse 15 foote and ••/7, hys plaine 19 ••/14. Now multiply 38 ½, in 14,

•• haue ye the crassitude of the whole round peece of Timber which is 539 foote. ••hen augment 19 ••/14 into 14 ryseth 275 foote, whiche subtracted from 539, re∣••ayneth 264, so many foote is in the crassitude aboute that hollow piece of tym∣••er. Or thus ye may bring it to passe: Pul 19 9/14 from 38 ½, that is the one super∣••••ciall content from the other, and the remayne multiplye in the heigth 14, So ••aue ye as afore 264 foote. An other example of tymber hollowed and somewhat ••ashioned lyke vnto a troughe: Suppose it to be DEFG, whose outward bredth DE is 10 foote, the length EF 14 foote, the heigth FG 8 foot, the inward bredth ••f the hollow HI 5 foote, the length HK 7 foote, the heigth as before. Now mul∣••iplie 10 in 14, so haue ye 140. This augmented in 8 ryseth 1120. Then multi∣••lye 7 in 5, that maketh 35, whiche augmented in 8, surmounteth 280, that ••umme by subtraction taken from 1120, leaueth 840, so many foote is the crassi∣••ude of this hollow troughe DEFG, euen thus of all other. By that whiche is ••oken ye may gather howe to get the capacitie of all maner regular vessels, and ••hat is performed in multiplying the playne or content superficiall of the inward ••ase or bottom in the profunditie or heigth. Example 35 the inward bases playn ••ugmented in 8, the heigth produceth 280. So many cubicall feete is in the hol∣••owe vessell. These 280 augmented in 51, bringeth foorth 14280 pints of water, farther diuided by 8 sheweth 1785 gallons. Thus of al other, and that for liquour ••onteyned in any maner vessell.

The .5. Chapter. Hovve the crassitude and Superficies of a sphere is te bee measured.

FOr the superficies ye shal multiplie the diameter in his circumference or get the plaine of that circle, as before in Planimetra I haue taught, and encrease it by 4, so haue yée the quantitie superficiall, and that multiplied in the sixt part of the diameter produceth the crassitude. Likewise the diameter multiplied in his square, and the offcome in 11, this product diuided ••y 21 produceth in the quotient also the solide content of the sphere. Many ••o rules coulde I gyue you for the lyke effecte. But for better vnderstan¦••ing of these, beholde the example ensuyng.

Example.

ABCD the globe whose diame∣ter

[illustration]

AC is 14, his circumference be∣ing 44, the one multiplyed by the o∣ther, bringeth 616, and that is the Su∣perficies of the globe whiche encreased by 2 ⅓ bringeth 1437 ⅓ and that is the solide content whiche is likewyse thus produced 14, multiplyed in hys square yeldeth 2744, thys agayne in 11, maketh 30184, whiche diuided by 21 giueth in the Quotiente 1437 ⅓, the Crassitude agreeable to the former worke.

The .6. Chapter. Hovve fragmentes or partes of a Globe are measured.

YE shall augment the whole superficies of the globe by the altitude or thicknesse of the fragment, and the offcome de∣uide by the dimetient, your quotient is the conuex spheri∣call superficies of the fragmente, wherevnto addyng his base or circle ye produce the whole superficies.

For the crassitude if it bée lesse than halfe the globe yée shall firste detracte the altitude of the fragment from the semidiameter of the sphere, the remayne yée shall augmente by the circular base. This producte shall ye subtracte from a number amountinge by the multiplica∣tion of the semidiameter in the conuex sphericall superficies of the frag∣ment, the thirde parte of the remayne is the crassitude or contente solide of that fragment. But if the fragment be greater than half the sphere, then shall ye deducte the spheres semidimetiente from the altitude of the frag∣mente and the remainder multiplied in the circular base the product must be added to the number produced by multiplication of the semidiameter, in the conuex sphericall superficies of the fragment, the thirde part of this resulting summe is the desyred crassitude.

Admit ABCD a fragment of the former globe, whose diameter is 14, and ••hat BD the altitude therof be 4, the spheres whole Superficies being 616, mul∣••••plyed by 4, bringeth 2464, and that diuided by 14, yeldeth in the quotiēt 176, 〈◊〉〈◊〉 conuex Superficies, whervnto if ye adioyn 126, the Area of the base or cir∣••le, whose dimetient is AC, there amounteth

[illustration]

••02, the whole Superficies of that portion ABCD, by the lyke woorking yee shall fynde the ••uperficies of EFGH 566. But for their cras∣••••tude I worke thus, In the lesser portion I finde 〈◊〉〈◊〉 altitude 4, which deducted from 7, the sphe∣••es semidimetient, leaueth 3, whiche augmen∣••ed by 126, the bases Area bryngeth 378, and ••his deducted from 1232 (a number produ∣••ed by multiplication of the spheres semidime∣••••ent in the conuex Superficies of the segment) ••••aueth 854, whose thirde parte is 284 ⅔, so ••uche is the solide contente of ABCD. But ••or the crassitude of the other portion, I aug∣ment 440 his conuex superficies in 7, there a∣••yseth 3080. Likewyse I multiplye 3, the difference betweene the semidiameter ••nd the fragmentes altitude in 126, thereof commeth 378, these added together, ••ake 3458, whiche diuided by 3, yeldeth for a quotient 1152 ⅔, the solide cras∣••••tude of the other fragmente EGFH. But for as muche as the spheres dia∣meter is here supposed knowne, I shall giue you a rule readily by supputation to ••earne the quantitie therof.

The .7. Chapter. Any sphericall segment propouned to atteyne the spheres dia∣•••• meter vvherof it is the fragment.

FIrste measure the altitude of the fragmente: Secondely, the semidia∣meter of the circular base, whiche yée maye also attayne by knowledge ••f hys circumference as was taughte in Planimetra, then square the se∣midiameter and diuide the product by the fragments altitude, the quotient ••dded to your diuisor yeldeth the spheres diameter. Or thus, adde the

square of the altitude to the square of the semidimetient, and diuide that producte by the altitude, so wyll youre quotient expresse the diameters quantitie.

Example.

ABCD the sphericall segment his altitude BD 4, the semidiameter of his circular bases AD nigh 6 ⅓ the square therof

[illustration]

is 40, whiche diuided by 4, yeldeth in the quotient 10, whiche added to 4, bryngeth 14 the Diameter of the Sphere. Lykewyse the square of 4 is 16, added to 40 the square of AD bringeth 56, whiche diuided by 4, giueth for the quotient 14, the sphere 5 dimetient agreeing with the former working.

The .8. Chapter. Hovve a Pyramis cutte off, maye bee measured.

IF the Pyramis bée vnperfecte, yea cut off in the toppe, con∣tinue it by rule and line wittyly layde to the two contra∣ry sides▪ and where the ioyning and common méetynge is there that Pyramis is whole and perfect, then measure that whole by the Arte afore, and also the Pyramis goyng from the toppe of the vnperfecte to the common méetyng. Thys doone ye shall subtracte the crassitude of the lesse from the whole continued Pyramis, so the remayne without doubte is the magnitude of the vnperfect Pyramis.

Example.

Lette ABCDE represente an Hexagonall imperfecte Pyramis encom∣passed with twoo equiangle Hexagonal Parallele playnes and 6 Isoscheles Tra∣pezia,

euery of their Bases being the sides of the

[illustration]

••reater Hexagonum 12, euery side of the lesser Hexagonum 6. Now, applying rules or lines ••nto the sides BF, GE, I finde them to con∣••urre at H. Thus haue you two Pyramides, ••he greater hauing his base an equiangle Hexa¦••onum, whose side is 12: the other is the Pyra∣mis HFG, whose base is the Hexagonū FG, ••uery of his sides being 6, the altitude of the les∣ser Pyramis is 15 the altitude of the greater ••0. Also by the rules giuen in Planimetra, yee ••hall finde the area of the lesser Hexagonum ••3 ⅗, the area of the greater 294 ⅖, and so con∣sequently by multiplication of them in the third ••arte of their altitudes as was taughte before in ••he mensuration of whole Pyramides, the solide contente of the lesser will be 468, the crassitude of the greater is 3744, the one deducted for the other, leueth 3276, the magnitude of that vnperfecte Pyramis FBCDE.

A note how by supputation to get the perpendiculares or altitudes of those Pyramides.

FOr that it were tedious & painfull in a solide of great quantitie, by rule or line to searche out these perpendiculares, ye shall by Arithmetike at∣tayne them thus: multiplie the greater Hexagonall side by the side of the vnperfect Pyramis, and the producte diuide by the difference or ouerplus of the sides Hexagonall, the quotient sheweth the side of the whole great Pyramis, from whose square (if ye deduct the square of the circles semidi∣ameter that encloseth the greater Hexagonum whiche ye were taught in Planimetra how to finde) the roote quadrate of the remainder is the alti∣tude of the greater Pyramis, which augmented agayn by the lesser Hexa∣gonall side, and the offcome diuided by the greater, bringeth in the quoti∣ent, the altitude of the lesser Pyramis.

Example.

The vnperfecte Pyramis side FB being nighe 16 ⅛, I multiplie by 12 the grea∣••er Hexagonall side, there ariseth 193 ½, and that diuided by 6, the lesser side

[illustration]

[illustration]

Hexagonall produceth in the Quotiente 32 1/••, the Pyramidall side HB, from the square thereof, if ye deducte the square of BL, the circles semidiameter that enclu∣deth the greater Hexagonum beeing 144, there will remayne 900, whose roote is 30, the altitude of the greater Pyramis, which aug∣mented by 6, the lesser Hexagonall side, and diuided by 12, produceth in the Quotient 15 the altitude of the lesser Pyramis. These lines for that they are some of them incommensu∣rable, can not exactly be expressed, saue only in surde numbers: but so nighe as is requisite for any Mechanicall mensuration this ope∣ration declareth them: such as are expert in Algebra by the former rules with irrational numbers may precisely shew their quantitie. The very like operation is required in per∣fect and vnperfecte Cones, for measuring their Perpendiculars and crassitude. Wherof to the ingeniouse there need no other Example.

The .9. Chapter. To cut off from any Cone or Pyramis vvhat parte or portion thereof ye vvill desire, vvith a playne equedistante to the base, and to finde on vvhat parte of the solides side the section shall fall.

THis diuision of Cones and Pyramides is in two respectes to be made, either in consideration of their contentes superfi∣ciall, or in regarde of their solide quātities, for either of them I shall giue you seuerall rules. First, it behoueth ye to mea∣sure the side of the Cone or Pyramis, that side shal ye diuide by the quadrate roote of the number expressing the parte assigned, youre quotient is the line or distance from the toppe of the Cone or Pyramis to the section or place where the playne parallele to y^e base shall passe, which will diuide or cut off from the assigned solides superficies the desired part. The same line or distance is also founde with diuision of the Cones side squared by the number expressing the part to be cut off, for the roote qua∣drat of the quotient shall be your desire. But if ye would from any Cone

or Pyramis in like manner make separation, in such sorte that their so∣••ide contentes might retayne like proportion, then shall ye multiplie the side of the Cone or Pyramis, firste in himselfe, and then againe in the off∣come, and this last producte shall ye diuide by the number expressing the parte ye would seperate, the roote cubicall of your Quotiente is the side of the lesser Cone or Pyramis to be cut off from the toppe of the greater.

Example.

[illustration]

Admit ABC the Cone, from whose sum∣mitie or toppe, I woulde cut off a portion, for ex∣ample, the Cone AEF whose superficies should bee the 9 parte of the great Cone ABC, his superficies. I take there∣fore the roote quadrate of 9, that is 3, wherwith I diuide 100 the side of the great Cone there ari∣seth in the quotient 33 ⅓ the line AF or side of the lesser Cone, the same nūber I produce also by the second rule thus, I diuide 10000 the square of AC, the Cones side by 9, the quotiēt yeldeth 1111 1/9, whose roote is likewise 33 ⅓, the side AF. But for operation in solides, I shal giue you an other example: Suppose HK the side of the Pyramis HIK 120, frō the which Pyramis I would cut off an other portion, for example, HLM that might be an eight part of the great Pyramis HIK, the nūber of the side 120, I multiplie in it selfe therof ariseth 14400, which augmented agayn by 120 maketh 1728000, and this diuided by 8 (for that is the nūber wherwith the part is expres¦sed) I finde in the quotient 216000, whose roote cubike is 60, and that is the length of HM or side of the Pyramis HLM, whose solide content is the eight part exactly of the great Pyramis HIK. Euen so shall ye do•• in all other regulare Pyramides, how many equal sides soeuer his base haue: but if the sides of his base be vnequal, and so consequently the sides of the Pyramis also vnequall, then shall ye worke with euery side seuerally as I haue by exāple shewed ye in one: or else work by the rule of three for all the sides are proportionally diuided: this to the wittie will suffise.

Here follovveth other fashioned bodies somevvhat straunge in figure, but in effecte those that tofore are measured.

THe first ABCD appeareth to be two round Pyramides ioyned togither, whose Diameter AC is as the round Pyramis afore measured 12 foote, the heigthe or perpendiculare line 20 foote and

[illustration]

5/40, the side 21, wherfore I conclude the cras∣situde of the one parte or halfe of it to be euē as the round Pyramis before measured, that is 759 foote, whiche doubled, make 1518, the whole content of this Figure, the other Fi∣gure ensuing is like to an egge, and not very vnlyke to the Figure aboue. Wherefore to measure him, ye shall worke as ye did in the other, the crassitude of that so hadde, and then doubled, bringeth the whole contente of him. Also this Figure EFGH is like in contente to the other ABCD, accor∣ding to that forme whose crassitude is 1518, this doubled maketh 3036 foote, so much yee maye affirme the crassitude or quantitie of that figure like vnto an egge. Now to get the quantitie superficiall of him, ye shall multiplie the arcke FEH in the halfe circumference of the circle whose Diame∣ter is EG, or the whole circumference in halfe the arcke, so haue ye the superficies.

The .10. Chapter. Hovv vvyne vesselles or barrels are measured.

SVppose ABCD were the barell to be moten: first, ye must take the iust measure of both the heades seuerally, the one head AB, I haue imagined 7 foote, the other CD as many. Nowe, take the true heigth of the middes where the barell swelleth, which is EF, being here 10 foote. These three diameters kepe, thē note how many foote, or other measure is cōtained frō the mids of either head to the middle of the swelling G in a streight line moten within the Barel. Behold frō the hed H to G the middle swelling, is 6 foote, from G to this letter I as many: now ye must set the

three diameters (tofore reserued) vpō some charde, paper, or other playn, the one differing from other according to their measures, as ye may see in the figure, then enclose them with an arke on both sides, cutting the middle line, which line cros∣seth the diameters squirewise in the middle, the one arke is KBECL, the other is KAFDL, the arkes drawen as ye see, lo a figure is made like vnto an egge:

[illustration]

Truely all suche fashioned fygures I haue taught you to measure immediately afore. In him ye may finde 1047 13/21, now if ye pull awaye by any arte from this whole summe the number of feete conteyned in the peeces of the figure superflu∣ouse and more than the barel at either ende, then the magnitude or capacitie of that vessell must needes remayne, as by the example is playnely perceiued, al∣though it would seeme I had entreated sufficiently, yet I thinke it worthy of re∣membrance to tell you how to serche the contente of these peeces at the endes of the figure, I must suppose that ye know the measuring of a round Pyramis which is entreated before, ymagine the diameter BA which is 7 foote to be the base, KH the heigth of the peece heere imagined a Pyramis: now if ye (as is before mentio∣ned) multiplye the playne of the base in the thirde parte of the heigth, the con∣tente of that figure will come according to that fashioned meeting, which is 51 ⅓, with this number by the precepts of proportion ye shall worke as foloweth: fyrst know the length of HL that is 16 foote, ageine GL beeing 10 foote, both these ad∣ded make 26. Now say 16 giueth 26, what shal 51 ⅓ bring, so haue ye 53 ••/2, so ma¦ny foote is the pece KBA, and the other CDL as many, which added make 166 ⅚▪ pul this out of the number conteyned in the whole figure afore founde, whiche is 1047, so haue ye lefte 880 11/14 the capacitie of your Barrell. Thus the learned

and famouse Archimedes hath taught in the 31 proposition of his boke de Sph••∣roibus & Conoidibus.

An other mensuration of vessels more common.

SOme vse when the base plaine of the extremes and mids of any ves∣sell are not one, to subtract the lesse from the bigger Superficies, the medietie that remayneth deducte from the bigge playne: then whatso∣euer is lefte adde to the lesser Superficies, the halfe of the producte she∣weth the true base to be multiplied in the heigth, of the which commeth the contente.

Example.

THe playne of the bigger Superficies conteyneth 40 foote, the lesser 24, the difference is 16, the halfe 8 subtracted from 40, leueth 32, this added to the lesse superficies 24 make 56, the half is 28, which multiplied in 24 bringeth 67••▪ the whole contente. Lo the figure.

[illustration]

Another note of measuring.

YF the Diameter of the vessell be in the middes 10 foote, but at the endes eyght, adde them, and take halfe, so haue ye nine, thus nine foote shall be thy diameter, whose playne or base (as before) multiplyed in the longitude or heigth, bryngeth a true contente. Note well this kinde for pillers and other bodies, when they haue diameters or bases not agreable.

The .11. Chapter. One rule general exactly to measure al kinde of vvine vessels.

FOrasmuche as there are sundrie kindes of wine ves∣sels, as the tunne, the pipe, the punshion, hogsheads, buttes, barrels, &c. euery of them differing from o∣ther aswell in quantitie as in fashion, to teach seue∣rall rules for euery sorte it were ouer tediouse, le∣uing therefore manyfolde precepts and examples that I might in this case prescribe, I shall for breui∣••ie sake set foorth one onely rule generall, whereby ye shall be able with the ayde of a small proportionall vessell, not onely to measure the whole ••apacitie of any maner wine vessell, but also the default and quantitie of ••icour therein conteyned, when it is partly emptie. And firste I will be∣ginne with this small proportionall vessell, whiche it behoueth you to procure by some skilfull Artificer, so made that their lengthes retayne ••he same proportion that their circumferences or circles, aswell at the ••ndes and middes, as also at other like and proportionall distances ta∣ken in the length of either vessel: as by the example folowing shal more playnly appeare.

Example.

Admit S and

[illustration]

•• the two vessels, •• the greater, S the lesser: nowe ••o examine whe∣••her these two 〈◊〉〈◊〉 proportionall, 〈◊〉〈◊〉 searche firste ••heir lengthes, ••••••nding HD 60 ••••ches, and RQ 〈◊〉〈◊〉 inches, likewise 〈◊〉〈◊〉 search the lon∣••••tude of euerye

depth or diameter, that is to say, BF and LO the diameters or depths of the mid∣des, and AG, CE, KP, MN, the circles of the endes, now if BF, AG and CE re∣tayne the same proportion to KP, LO, MN, (euery one particularly compared to his like) that HD the length of the greater doth to RQ the longitude of the lesser, then are those vessels likely proportionall: but for greater certentie (bicause errour may grow vpon small difference) it were requisite also to measure in ei∣ther of them one other circumference or hoope, as for example in the greater, th•• circle passing by TV exactly in the middle betweene AG and BF, and in the les∣ser the circle XY situate in like maner betwene KP and LO. Now if ye find the proportion of these two circumferences also agreable with the former lengthes and circumferences, ye may assuredly gather these vessels to be proportional. But if ye finde any discrepance or variaunce betweene them, ye shall by the ayde of some skilfull Artificer refourme it in the lesser, till ye haue brought it fully agre∣ing with the proportion of the greater, whiche is readily proued by the rule of proportion. Whereof I minde not here to vse mo words, leste in seeking neede∣lesse playnesse, I become ouer tediouse: your smal vessell thus prepared, ye shall in the next chapter be taught how to vse it.

The .12. Chapter. Hovv by this small prepared vessell to measure the quantitie of the greater.

WHensoeuer ye will measure any maner of wyne vessels, it behoueth you to consider what sorte they are of, and ac∣cordingly to prepare your lesser proportionall vessell, then shall ye in either of them measure the profunditie or greatest diameter, and also the diagonall or crosse lynes from the bung holes to the opposite or lowest parte of ei∣ther base, these diagonall lines ye shall square, and the productes seue∣rally multiplye in his correspondent Diameter, of these surmounting summes ye shall diuide the greater by the lesser, your quotient openeth how many times the lesser vessell is conteyned in the greater, or yf ye augment the greater of those laste surmounting summes by the crassi∣tude of the lesser vessell, and diuide by the lesser summe, your quotient will declare the solide content of the greater vessell.

Example.

Suppose ABCD the wyne vessell, whose capacitie I would knowe EFGH my lesser proportionall vessel, first I measure the diameters EG and AC. Ad∣mit the first 4, the other 40, agayne I mete the diagonall lines. Admit I finde AB 50 EF 5, the square of 4 is 16, which augmēted by 5, bringeth 80, the square of 40 is 1600, & that

[illustration]

augmented by 50, yel∣deth 8000, which di∣uided by 80, bringeth in the quotient 1000, so many tymes is the lesser conteyned in the greater. Now to lerne howe many pottles or gallons is conteyned in that great vessell, I trye firste howe mu∣che liquoure my little prepared vessell will holde: Admit it conteyne ••/4 of a pinte, this augmented by 80000 produceth 60000, whiche diuided by 80, bringeth 750 so many pintes ye may conclude in that great wine vessel which reduced to gallons diuiding by 8 yeldeth 93 gallons 3 quartes, the exact quantitie of liquour that such a vessell wyll conteyne, but if it happen that your vessell be not throughly filled, and that ye desire as well to know how muche liquour would suffise to fill it, as also what quantitie is therein conteyned, ye shall in the next Chapiter receiue therin perfect instruction.

The .13. Chapter. Hovve both the liquour and default or emptinesse in vvine ves∣sels partly filled is to be moten.

FOr greater exactnesse ye shall prouide a fine streight rodde of 4 or 5 foote in lengthe, exactly diuided into 1000 equall portions. Then shall yée moue the vessell whose liquour or defaulte yée woulde measure, till it lye leuell the bung hole directely vp∣warde, thys doone take youre rodde diuided as is before declared, and let

it descende perpendicularly downe thorough the bung, till it come to the bottome of the vessell. Thys doone, note what parte▪ the bung or vp∣moste heigth of the wyne vessell wyll touche of the rodde, and lyke∣wyse howe many partes thereof is wette with the liquour, and laste of all howe many partes are conteyned betwéene the liquour and the bung hole. The fyrst I call the diameter or profunditie of the vessell, the seconde the partes of liquour, and the third shall be named (for distinction) the par∣tes of defaulte or emptinesse. This doone yée shall also measure wyth your diuided rodde how many partes the diameter of youre lesser vessell conteyneth, these parts augmented by the partes of liquoure tofore founde, and the product diuided by the diameter of the wyne vessel, yeldeth in the quotient the partes of liquour, for your little prepared vessell, which it be∣houeth ye so to situate that it may also lye leuell as dyd the greater vessel, then poure in so muche water that by prouyng with your diuided redde ye may finde the partes of liquour exactly agréeing with youre former quoti∣ent: This doon augment the diameter of the wine vessell cubically, that is to say, by hys owne square, and the producte in the quantitie of liquoure now béeing in your little prepared vessell, the producte diuided by the cube of your lesser vessels dimetient yeldeth in the quotient the true quantitie of liquor contayned in the wine vessell, and that againe deducted from the whole contente of the vessell found, as was in the former Chapters decla∣red, leaueth the default or emptinesse, that is, how many gallons or other measures is requisite to fill vp the sayd vessell.

Example.

Admitte ABCD the wyne vessell partly filled, ADCE the liquour, ABCE the defaulte or emptynesse, FGHI my smaller proportionall vessell pre∣pared as I haue tofore in the last Chapiter declared, LM the straight rodde di∣uided into 1000 parts, which being lette perpendicularly fall from B till it touch the opposite side at D, I fynde BD 800 partes, ED that is so much as is w••tte of the rodde 600 partes, lykewyse placing the rodde in my lesser vessel at G, ad∣mit I finde the diameter or profunditie GI 200 partes, this number multiplyed by 600, the partes of fulnesse yeldeth 120000, whiche diuided by 800, bringeth in the quotient 150, the partes of fulnesse for my smaller vessell: fillyng therefore the proportionall small vessell with liquour till it ryse vnto K, that is to the 150 parte in my diuided rodde LM, finally by measure I make triall what quanti∣tie of liquour is in my little vessell. Admit it 3 pottles: Nowe to lerne the quan∣titie of liquour conteyned in the great, first I multiply BD the greater diame∣ter

beyng 800 in his square, thereof aryseth 512000000, and that agayne by 3, so haue I 1536000000, which diuided 8000000 (the cube of GI the lesse dimetiēt) produceth 192, so many pottles is there in that wyne vessell. Nowe if ye desire to knowe how many pottles or gallons more will fill it full, ye may by the last Chap∣ter serche how many gallons the whole vessell will conteyne, and from that with∣draw 96 gallons, the quantitie of liquour already therin conteyned, the remayn is youre desire. I thynke it

[illustration]

not necessarie in this case to adioyne any farther exam∣ple, for that this was in the former Chapter plainly both by rule and example alredy declared. This kind of men∣suration serueth not only for wine vesselles, whereof there are sundry fourmes, but also for all manner bodies, what kinde of fashion soeuer they be of, and likewyse for their fragments or parts, and it is grounded vppon this Theo∣reme. All lyke solides retain among themselues triple proportion of theyr lyke or correspondente sides, that is to say, looke what proportion the cube of one side in the lesser solide, retayneth to the cube of his correspondent side in the greater solide, the same proportion doth the lesser body retayne to the greater: so that with the ayde of the golden rule the firste three beeing knowne, the fourthe is redyly founde: And as this rule serueth for the whole bodies, so is it also to be applyed to all fragments or partes: so that the Superficies wherwith the solides are diuided be lyke, and make equall angles with the sides and Superficies of eyther solides, for thereupon it must con∣sequētly ensue that those fragmēts correspondētly cōpared are also proportionall.

Thus muche I thought to adioyne touching the reason and demonstra∣tion of this rule, wherby the ingeniouse practizer might the better retaine in memorie the operation, and also bée able to applye the same to sun∣dry other vses not héere mentioned yf occasyon bee offered. And for as muche as thys kynde of measurynge dependeth wholly vppon the smalle proportionall Uesselle, whose quantitie is supposed knowen,

I thinke it not amisse to giue one rule generall for all small vessels how irregular soeuer they bée (their fashion not regarded) exactly to finde their solide quantities.

The .14. Chapter. To measure exactly the solide content of any small body, hovv disordred or irregular so euer it be, the forme or fashion not regarded.

YE shall prepare a hollowe vessell of cubicall forme so large that it may conteyne the small irregular body, whiche being placed therin, ye shall youre in so muche water that it couer altogither the bodye, then make a marke where the Superficies or vpmoste parte of the water toucheth. This doone take out the same irregular body, and marke again directly vn∣der the former where the brimme of the water now toucheth, for the di∣stance of these two marks multiplied by the square of the Cubes side, pro∣duceth the crassitude of that irregular body.

Example.

Admitte A the cubicall hollowe vessell whose inwarde syde I suppose 20 ynches, B is the irregular bodye, whose crassitude I desyre, firste there∣fore I putte the solide into my

[illustration]

hollowe cube, and pourynge in water till it be thorough∣ly couered. Admit the brim of the water reach•• vnto C, then takinge oute that ir∣r••gular bodye agayne: Ad∣mytte the Superficies of the water fall to D, I measure the distaunce betweene C and D, suppose it 7 ynches, whyche multiplyed in 400, the square of the cubes side produceth 2800, so many cu∣bicall inches are conteined in

that Irregular body B. Neither is it of great importance whether your vessell A be an exact Cube. For what kinde of Prisma soeuer it be, alwayes rising vniform∣ly, and his Paralelogrammes being Perpendicular vpon their base, ye may (chan∣ging the forme of your Calculatiō) bring the same to passe, only wheras in the cube ye did multiply his sides square in the distance betweene the water markes, ye shall now multiply the same distance or difference of water markes in the base of the Prisma, the resulting summe is the Irregulare Solides Crassitude. And thus may you alwayes frame your containing vessel, according to the forme and quantitie of the Irregulare body that ye desire to measure. Meruellous is the appliance of this kinds of mensurations, and straunge conclusions may be perfourmed therby, wher∣in although I meane not in this treatise to reueale any secretes, reseruing them for an other place, yet to geue some light to the ingenious to proceede in applying them farder, I shall not thinke it tedious to shewe howe it maye be vsed to discouer the waight of such things, as no way possibly by ballance may be found.

The 15. Chapter. Hovve the vvaight of any part or portion of a Solide body may be knovvne vvithout seperation therof from the body, vvherby it mought be paised or vvaighed in Ballance.

FOr as muche as neither by common Ballance, neither by that kinde of ballance which the Italians vse, called Statera, nor any other hitherto inuented, the waight of any fragmēt or parte of a Solide body may be knowne without sepera∣ting or cutting that Fragment of from the whole bodye, wherby it may by it selfe alone be paised in the ballance, I thought it not amisse aswell for the rarenesse of the matter, as for the ne∣cessary vses thereof, to set forth this meane of searching waight by water, with y^e aid of Arithmetick: your vessel therfore being prepared as is tofore declared, whether it be Cube or Rectangular Prisma it forceth not, ye shall first fill it full with water, and throw the Solide body therinto, then softly lift that body out of the water till suche time as there remaine no more in the water than that portion whose waight ye desire to knowe, at that in∣stant make a marke on one side of the vessell where the Superficies of the

water then toucheth, then take out the bodye altogether. This done, mea∣sure the distance from your marke to the Superficies of the water as it is nowe after the body is taken quite out. Likewise measure the distance of the waters Superficies from the toppe of the vessell. This done augment the waight of the whole body by the lesser distance, and diuide by the grea∣ter, your Quotient will shewe the true waight of the fragment or portion.

Example.

Admit BCD a piller of an hundred pounde in waight, being of Brasse, Iron, Siluer, or any other Metall, my desire is to knowe the waight of the fote or

[illustration]

portion from B to C, first ther¦fore putting the hole piller into the vessel A, I fil it ful of water, then lifting it softly vp til al the piller be out of the water, saue only the fote or fragment BC, I finde the Superficies of the water fallen to E, then doe I lift out the whole piller, lea∣uing no parte thereof within the vessell, admit now the wa∣ter falne vnto F, and that by measuring I finde EF 8 in∣ches, and GF 20 inches, 8 multiplied in 100 the whole Pillers waight yeldeth 800, which diuided by 20 (the greter distance) from the top of the vessell to the lowest water marke, bringeth in the Quotient 40, so manye pound waight, I c••nclude the Pillers foote or portion BC.

And thus may you by the rising and falling of the water with the aid of Arithmeticke knowe howe to cutte of from this Piller or any other bo∣dye what portion, quantitie or waight ye will prescribe, alwaye it is sup∣posed that it is Corpus Homogeneum, that is to say of matter and kinde of substance equally dispersed through out: otherwise if it be Corpus Hetero∣geneum, that is to say of partes vnlike in substance, it requireth more cu∣rious

calculation, which I reserue for an other place, minding héere only to make an Introduction for the wittie to procéede farther, and not at large to disclose such secretes as may be shewed héerein. By this meanes did Archimedes finde the exacte quantitie of euery seue∣rall metall that was in the Kings Crowne at Syracusa, without opening or breaking any part therof: and many more and no lesse straunge con∣clusions may be done thereby.

The end of the third Booke called Stereometria.

The Preface.

ALthough the Rules and Preceptes already geuen abun∣dantly may suffice for the exact mensuration of any mā∣ner Solides or vessels that are vsually occupied, or may be proponed to be moten, yet for the satisfaction also of such as deliting in matters only new, rare and difficile, séeke to reache aboue the common sorte, I haue thought good to adioyne this Treatise of the 5 Platonicall bodies, meaning not to discourse of their secrete or mysticall appliances to the Elementall regions and frame of Celestiall Spheres, as things remote and farre distant from the Methode, nature and certaintie of Geometrical demonstration, only héere I intend Mathematically to cōferre the Super∣ficiall and Solide capacities of these Regulare bodies with their Circum∣scribing or inscribed spheres or Solides, & Geometrically by Algebraycall Calculations to search out the sides, Diameters, Axes, Altitudes, and lines Diagonal, together with the Semidimetients of their Equiangle Bases, containing or contained Circles, and as in the Problemes ye shal perceiue by Preceptes and Examples, the quantities and proportions of all these lines, Superficies and Crassitudes, with numbers Rationall and Radicall expressed, so haue I by Theoremes sowne the séede of Rules innumerable, wherin the studious may delite him selfe with infinite varietie: Finally I shall in the last Chapiter by Diffinitions and Theoremes partly set forthe the forme, nature and proportion of other fiue vniforme Geometricall So∣lides, created by the transformation of the fiue bodyes Regulare or Plato∣nicall: but not in so ample manner, as the noueltie or difficultie of the mat∣ter requireth, meaning as I sée these the first frutes of my studies liked and accepted, to bestowe time and trauaile vppon an other peculiare Uolume,

which shall conteyne, not onely the demonstrations of these Theorems of spherall solides, but also of Conoydall, Parabollical, Hyperbollical and Ellepseycal circumscribed & inscribed bodies, & many mo lynes & solides produced by the sections of these, and reuolution of their sectiōs: but to re∣turne to this present Treatise, let no man muse y^e writing in the English toung, I haue retayned the Latin or Gréeke names of sundry lines and figures, as cordes Pentagonall, lines Diagonall, Icosaedron, Dodecae∣dron, or such like, for as the Romanes and other Latin writers, notwith∣standing the copiouse and abundante eloquence of their toung, haue not shamed to borrow of the Gretians these and many other termes of arte: so surely do I thinke it no reproche, either to the English toung, or any English writer, where fitte words fayle to borrow of them both, but rather should we séeme therby to do them great iniurie, these béeing in déede certayne testimonies and memorials where such sciences firste tooke their originall, and in what languages and countreys they chieflye florished, which names or words how straunge soeuer they séeme at the firste acquayntaunce, by vse will grow as familiar as these, a triangle, a circle, or suche like, which by custome and continuance séeme méere En∣glish, yet to auoyde all obscuritie that may grow by the noueltie of them, I haue adioyned euery of their diffinitions, and so proceded to Problems and Theoremes with suche methode, as howe obscure or harde soeuer they appeare at the firste, through the rarenesse of the matter: I doubt not but by orderly reading, the ingeniouse Student, hauing any meane taste of cossicall numbers, shall finde them playne and easie.

Diffynitions.

PRoportion is a mutuall or enterchangeable relation of two magnitudes, being of one kind, compared togither in respecte of their quantities.

The second dffiinition.

When the proportion of two magnitudes is such as may be expressed with numbers, then is it certaine & apparant and here is called rational: But when the proportion is such as cannot be expressed with numbers, but with their rootes onely, then is that proportion certayne also, but not apparante, and therfore here I name it surde or irrationall.

The thirde diffinition.

When there be thrée suche magnitudes or quantities that the first to the second retayne the same proportion that the second doth to the third, those quantities are saide to be proportionall, and the first to the thirde retayneth double the proportion of the first to the second, and the seconde is named meane proportionall betwéene the first and the last.

The fourth Diffinition

When foure magnitudes are likewise in continual proportion, the first & the fourth are the extremes, and the second and thirde the meanes, and the extreames are sayd to haue triple the proportion of the meanes.

The fifth diffinition.

Any lyne or number is sayde to be diuided by extreame & meane pro∣portion, when the diuision or section is suche or so placed, that the whole line or number retayne the same proportion to the greater parte, that the greater doth to the lesser.

The sixth diffinition.

A lyne is sayde to be equall in power with two or moe lynes, when his square is equall to all their squares.

The seuenth diffinition.

A lyne is sayd to matche a superficies in power, when the square of that line is equall to the superficies.

The eyght diffinition.

When any equiangle triangle, square, or Pentagonum is in suche sorte described within a circle, that euery of their angles touche the cir∣cumference, their sides are called the trigonal, tetragonall and pentago∣nall Cordes of that circle.

The ninth diffinition.

About euery equilater triangle, square, or Pentagonum, a circle may be described, precisely touching euery of those figures angles, and that circle shall be called the circumscribing or contayning circle.

The tenth diffinition.

Also within euery of these equiangle figures a circle may be drawen, not cutting but only touching euery of their sides, this is called the in∣scribed circle.

The eleuenth diffinition.

Any right line drawen from angle to angle in those equiangle figures passing through the superficies, I name the line diagonall.

The twelfth diffinition.

A right line falling from any angle of these superficies perpendicu∣larly to the opposite side shalbe named that playnes penpendiculare.

The .13. Diffinition.

TETRAEDRON is a body Geometricall

[illustration]

encompassed with fowre equall equiangle tri∣angles.

The fourtenth Diffinition.

HEXAEDRON or CVBVS is a solide fi∣gure,

[illustration]

enclosed with sixe equall squares.

The fiftenth Diffinition.

OCTAEDRON is a body comprehended

[illustration]

of eight equall equiangle triangles.

The .16. Diffinition.

ICOSAEDRON is a solide Figure, vnder

[illustration]

twentye equall equiangle triangles conteyne••▪

The .17. Diffinition.

DODECAEDRON is a solide com∣prehended

[illustration]

of twelue equall equiangle pentago∣nall Superficies.

The .18. Diffinition.

These fiue bodies are called regular, and about euery of them a sphere may be described, that shall with his concaue peripherie exactly touche e∣uery of their solide angles, and it shall be called their comprehending or circumscribing sphere or globe, and these solides shalbe called the inscri∣bed or conteyned bodies of that sphere.

The .19. Diffinition.

Also within euery of these regulare bodies a sphere may be described that shall with his conuex superficies precisely touche all the centres of those equiangle figures wherewith these bodies are enuironed, and such a one I terme their inscribed or conteyned sphere, and those bodies shall be termed the circumscribing solides of that sphere.

The .20. Diffinition.

The semidiameter of this inscribed sphere, forasmuche as it is the very Axis or Kathetus of euery Pyramis, hauing his base one of the equi∣angle

playnes, and concurring in the centre, of which Pyramides (intel∣lectually conceyued) these bodies séeme to be compounded, it shal be na∣med the Axis or Kathetus of that body.

The 21 diffinition.

Euery of these bodies side, I call any one of those equall righte lines wherewith these equiangle Figures are enuironed that comprehende and include these bodyes.

The 22 diffinition.

Any one of the Figures wherewith these solides be enuironed is cal∣led the base of that solide.

The 23 diffinition.

A line falling from any solide angle of these bodyes perpendicularlye on the opposite playne or base, shall be named that solides Perpendi∣culare.

The 24 diffinition.

The power of any line gyuen is sayde to be diuided into lines re∣tayning extreame and meane proportion, when two suche lines are found as both make their squares ioyned togither, equall to the square of the line giuen, and also holde such proportion one to another as the two partes of a line diuided by extreame and meane proportion.

The 25 diffinition.

One of these regulare solides is saide to be described within an other, when all the angles of the internall or inscribed body at once touche the superficies of the comprehending or circumscribing regulare solide.

The fyrst Probleme. To diuide any line or number by extreame and meane Proportion. (Book 1)

ADmit the line diuisible AB extende him foorth sufficiētly, then erect vpon B a perpēdicular to AB as ye were taught in the first booke. Now, place the one foote of your cōpasse in B, and extending the other to A, describe the semicircle ADC, then diuide AB in halfe at F, and placing the one foote of your compasse immoueable in F (making FD a semidiameter) de∣scribe the arcke DG. This done, set the one foote of your compasse in B &

[illustration]

[illustration]

opening the other too G, drawe the semicircle GEI, so is your line AB parted by extreame and mean propor∣tion in I, the greater porti∣on BI, the lesser IA. But Arithmetically to finde the same portions ye must mul∣tiplie the square of your gy∣uen number in 5, and diuide by 4, from the square roote of the producte, deducte the half of your giuē number, the re∣mainder is the greater por∣tion, which subtracted from the whole leueth the lesser parte.

Example.

THe number giuen 12, his square 144, multiplied by 5 and diuided by 4 yeeldeth 180, the halfe of the giuen number 6 deducted from the square roote of 180, leueth √{powerof2}180—6, vvhiche is the greater portion, and that agayne deducted rom 12, leueth this Apotome 18—√{powerof2}180 the lesser por∣tion.

Yée maye also by any of the partes knowen finde the quantitie of the whole thus: Augmente the parte gyuen by 12, and diuide by his corres∣pondente Portion, the quotient is your desire.

For example, Suppose the lesser parte of a line diuided by extreame and meane Proportion, (vvhose quantitie I desire) 10, vvhiche augmen∣ted by 12, bringeth 120: this diuided by 18—√{powerof2}180 bringeth 15+√{powerof2}125, so muche is the vvhole line vvhereof 10 is the lesser parte. Agayne, dedu∣cting 10 from 15+√{powerof2}25, there remayneth 5+√{powerof2}125, and that is the grea∣ter portion. But forasmuch as bothe this and the former conclusion maye sundrie other vvayes be resolued, heere shall ensue Theoremes vvhich the Geometer vvill applie asvvell to this purpose as diuersly to other ne∣cessarie conclusions, vvhereof some shall hereafter ensue.

The fyrst Theoreme.

ANy number parted by extreame and meane proportion, the square of the greater parte is equall to the rectangle or number produced by multiplica¦tion of the leasse portion in the whole.

The 2 Theoreme.

If any line or number be parted by extreame and meane Proportion, and to the whole ye adioyne the greater portion, that surmounting number or line is al∣so diuided by extreame and meane proportion, his greater portion beeing the firste giuen line, and the lesser the line or number adioyned.

The 3 Theoreme

Any line or number parted by extreame and meane proportion, the square of the whole added to the square of the lesser parte, is triple to the square of the greater portion.

The 4 Theoreme.

A line or number diuided by extreame and meane proportion, if ye abate the lesser portion out of the greater, the square of the remainder adioyned to the square of the greater portion is triple to the square of the lesser.

The 5 theoreme.

The proportion of a line parted by extreame and meane proportion to his partes can not be expressed in rationall numbers, for if the whole line bee ratio∣nall, the partes are either of them an irrationall called Apotome, but in surde numbers the proportion may be thus declared, as √{powerof2}5— to 3—√{powerof2}5, so is the whole to the greater, and the greater to the lesse, for they are Proportionall.

The .2. Probleme. The dimetiente of any circle giuen to searche out the sides or Cordes Trigonal, Tetragonal, Pentagonal & Decagonal. (Book 2)

SUppose y^e Dimetiente AB of the circle ABCD giuē, di∣uide this Diameter equally in the poincte E, wherevpon erecte the Perpendicular EC and from C to B drawe a right line, then fixe the one foote of youre Compasse in A and opening the other to E, drawe the Arcke EF, making

intersection with the circle

[illustration]

in F, from thence extende right lines to AB, This done, diuide the Se∣midiameter BE, as yée were taught by the firste Probleme, by extreame and meane Proportion in the poincte G: Final∣ly, from G to C drawe a streight Line, so haue ye FB the Corde Tri∣gonall, CB the Tetragnall, CG Pentagonall, and EG Decagonall. Nowe, Arithmetically by the Diameter knowen to attayne these Cordes, ye must thus worke: Triple the square of the Semidiameter the roote quadrate of the producte is the side Trigonall, double the Se∣midiameters square and the roote quadrate thereof is the Tetragonall Corde: Diuide the Semidimetient by extreame and meane Proporti∣on the greater Portion is the side Decagonall, whose square ioyned to the Semidimetientes square produceth the Square of the Corde Pentagonall.

Example.

Admitte the Diameter AB 12, the halfe is 6, vvhose square tripled, yel¦deth 108, this surde number therefore √{powerof2}108 is the Trigonall Corde. The square of 6 doubled is 72, vvhose roote is the line Tetragonall. The Semidiameter EB 6 diuided by extreame and meane proportion as vvas taught in the first Probleme, yeldeth EG, the greater Portion √{powerof2}45—3, and that is the line Decagonall: vvhose square ioyned vvith the square of 6, produceth 90—√{powerof2}1620, vvhose rote vniuersal is the line CG or Pentago∣nall Corde.

The Diameter of the circle 12

• The corde Trigonall √{powerof2}108
• The corde Tetragonall √{powerof2}72
• The corde Pentagonall √{powerof2} vni. 90—√{powerof2}1620
• The corde Decagonall √{powerof2}45—3

Novve bicause long vvorking vvith irrationall numbers, may breede con∣fusion in such as are not perfect in the rules of Algebra, ye may by the rule of proportion (supposing the Spheres dimetient 1000) reduce surd numbers to integers, although not so exactly as the subtilitie of geometricall demonstra∣tion requireth (considering these cordes cannot precisely be expressed in ra∣tionall numbers) yet for any Mechanicall operation or manual mensuration, the difference shall not be sensible.

Example.

〈 math 〉〈 math 〉

As in these operations by the Diameter ye are brought in knovvledge of the Cordes, so by the knovvledge of any corde, ye may also finde bothe the Diameter and all other cordes, vvherof to the vvittie there neede no farther Example.

The .3. Probleme. The side of any equiangle Triangle geuen, to finde out the Semi∣diameters of his containing and contained circles, vvith the true quantitie of the Area. (Book 3)

FOr the Semidiameter of the containing Circle, it behoueth you to square the side geuen of the Triangle, and diuide the Product by 3 the roote Quadrate of the Quotient is your de∣sire. The halfe of that roote is the Semidiameter of the cer∣tained circle, which multiplied in the Circuite or Perimetry of the Triangle yeldeth a number, whose medietie is the content Superficiall or Area of that Figure.

Example.

[illustration]

The side geuen AB 6, the square therof diuided by 3 produceth 12, the rote therof being √{powerof2}12, is the Semi∣diameter DC, vvhose half √{powerof2}3 is the Semidimetient DE, vvhich multipli∣ed in 18 the triangles Perimetry pro∣duceth √{powerof2}972, vvhose medietie be∣ing √{powerof2}243 is the quantitie Superfi∣ciall or Area of the Triangle ABC. The Semidiameter of the contained circle is also thus found, fro•• the Se∣miperimetry of the triangle deducte euery side, noting euerye difference seuerally, then multiplye these diffe∣rences together, the Zenzicubike roote of the product is your desire.

The side geuen AB 6.

• ...The containing circle••— √{powerof2}12
• ...Semidiameter BD.— √{powerof2}12
• ...The contained Circles— √{powerof2}3
• ...Semidimetient DE.— √{powerof2}3
• ...The Superficies or Area of the equilater Triangle. √{powerof2}243

Any Equilater triangles Area is also thus gotten, multiply the square of his side by it selfe, and the product by 3, the offcome diuide by 16, the quoti∣ents roote Quadrate is the Area.

Or multiply the square of the side geuen by 3, and diuide the offcome by 4, that Quotient augment againe by the square of halfe the side, the roote Quadrate of the Product is the Triangles content.

Or diuide the Triangles side in three equall partes the square of one third parte multiplye by 3, and diuide the offcome by 4, the roote Quadrate of your Quotient encreased by the triangles Semiperimetrie, produceth his Superficies also.

The .4. Probleme. The side of any square knovvne by supputation to attaine the Semidiamiters of his exteriour and interiour Circles, vvith the content of his plaine or Area. (Book 4)

THe halfe of your geuen side is the Semidiameter of the in∣ternall Circle, and if ye double the square of this Semidia∣miter, and from the Product extract a Quadrate roote, that roote is the Semidiameter of the externall Circle. Now for the Area either multiply the side first geuen in it self, or else your lesser Semidiameter in halfe the Perimetry of your square.

Example.

[illustration]

Admit AB the squares side 10, the half being 5 is the Semidimetient of the in∣vvarde Circle, the square thereof doubled is 50, vvhose Quadrate roote is EG, the containing Circles Semidimetient. Novv for the Superficies I augment 10 by it selfe there ariseth 100. Likevvise 5 the in∣ternall Circles Semidiameter multiplied by 20 the squares Semiperimetry yel∣deth also 100, so muche is the Area of the square ABCD.

AB the side of the square 10.

• EG the containing circles Semidiameter √{powerof2}50
• EF the inscribed circles Semidimetient 5
• The Area of the square ABCD 100

The .5. Probleme. The side of any equiangle Pentagonum geuen by Arithmeticke to learne his circumscribing and inscribed circles Se∣midimetients vvith the exact quantitie of his plaine Superficies. (Book 5)

IT behoueth ye to knowe that all equiangle Pentagonall figures of what capacitie so euer they be, their sides to the Semidiameters of their cōtai∣ning

circles alway retain one proportion, so that hauing alredy by the secōd Probleme (the Semidiameter geuen) found the side, ye may contrariwise by the rule of proportion in any other, by the side geuen finde the Semidia∣meter, from whose square if ye deduct halfe the sides square, the roote Qua∣drate of the Remainder is the Semidimetient of your circle interior, the medietie wherof multiplied by the Perimetrie or circuit of your figure yel∣deth the content Superficiall or Area therof.

Example.

Suppose the side of your Pentagonum ABCDE 14, for the Semidiame∣ter AF, I vvorke by the rule of proportion thus, √{powerof2} vniuers. 90—√{powerof2}1620 geueth 6, vvhat yeldeth 14, the fourthe proportionall is √{powerof2} vniuers.

[illustration]

〈 math 〉〈 math 〉 vvhiche by Reduction is broughte to this number √{powerof2} vniuers 98+√{powerof2}1920 ⅘, and that is the containing circles Semidimetient be∣ing very nighe 11 19/21 for ex∣actly nether by integer nor fraction it can be expressed. Novv for FG, I muste de∣ducte 49 the square of CG fro 98+√{powerof2}1920 ⅘ the square of the Semidiameter tofore founde, so there re∣maineth 49+√{powerof2}1920 ⅘, vvhose roote vniuersall is the line FG or Semidime∣tient of the intrinsical circle, vvhich augmented by 35 the Semiperimetrie, produceth √{powerof2} vniuers 60025+√{powerof2}2882400500, vvhich surde number resteth betvvene 337 and 338 being very ••igh 337 ¼, so muche is the Area of the Pen∣tagonum.

The side of the Pentagonū. 14

• The containing circles Semidiameter—√{powerof2} v. 98+√{powerof2}1920 ⅘
• The internal circles Semidimetient —√{powerof2} v. 49+√{powerof2}1920 ⅘
• The line Diagonal — √{powerof2}245+7
• The Pentagonal Area —√{powerof2} vni. 60025+√{powerof2}2882400500

Other rules to perfourme the same.

DIuide the side giuen by extreme and mean proportion as ye wer taught in the 2 probleme, and to the whole syde, adde his greater portion, the summe that thereof resulteth multiplye in it self, and from the product sub∣tracte the square of halfe the side giuen, with the roote quadrate of this re∣mainder diuide the square of half the side, the quotiente adde to youre di∣uisor, the medietie of this resulting summe is your greater semidiameter, whiche subducted from youre former diuisor, leaueth the semidimetient of the intrinsicall circle, and this augmented by the pentagonall semiperime∣trie produceth his Area.

Or thus, your giuen syde diuided as before by extreame and meane pro∣portion, to the whole adde his greater portion, the product square, and ther∣to adde the square of the syde pentagonall, the offcoome diuide by 5▪ youre quotientes quadrate roote is the greater semidiameter, which augmented by the pentagonal syde, and the summe therof amounting diuided by dou∣ble his greater portion parted as is before sayde by extreame and meane proportion, your quotiente wil expresse the quantitie of the lesser semidi∣metient: for the Area ye may augment the diameter of the contayned circle by 5/4 of the pentagonal syde: Or half that lesser semidimetient in the whole circuite of the fygure, the surmountyng summes declare his superficiall quantitie.

Manye moe rules mighte in this matter bée gyuen, but I thynke it better to adioyne certayne Theoremes, whereby the ingenious shall bée able not onely to conceyue the grounde and reason of these already taughte, but also of hymselfe inuente sundry other no lesse certaynely and perchance more spéedyly performyng the premisses.

Theoremes

The first Theoreme.

THe square of any equiangle triangles side, is triple to the square of his con∣teynyng circles semidiameter.

The second Theoreme.

The square of an equilater triangles syde to the inscribed circles semidime∣tientes square is as 12 to 1.

The third Theoreme.

The semidiameter of a squares conteyning circle, is meane proportionall be∣tweene his syde and the semidiameter of his conteyned circle.

The fourth Theoreme.

In any equiangle pentagonum the square of his side is equall to the square of his containyng circles semidiameter, and the square of the corde decagonal ioy∣ned togither.

The fifth Theoreme.

The Diameter of the circle described within a Pentagonum is equall to the sides hexagonall and decagonall of the comprehending circle.

The sixt Theoreme.

The square of the side pentagonall togither with the square of the lyne sub∣tendyng the pentagonal angle, hath proportion to the conteyning circles semi∣diameters square, as 5 to 1.

The seuenth Theoreme.

The square of halfe the side pentagonall beeing deducted fro the square of the lyne diagonall subtending one of the pentagonall angles, the roote quadrate of the remaynder is equall to bothe semidiameters of the contayning and con∣teyned circles ioyned togither.

The eyght Theoreme.

In euery equiangle pentagonum the conteyning circles semidimetient retay∣neth the same proportion to the conteyned circles diameter that the pentagonal side dothe to his lyne diagonall.

The ninthe Theoreme.

The Diagonall lyne in euery equiangle pentagonum doth diuide the perpen∣dicular (fallyng from his subtendent angle to the opposite side) by extreme and meane proportion.

The tenthe Theoreme.

In euery equiangle pentagonum the lynes diagonall do cut themselues by ex∣treme and meane proportion.

The eleuenth Theoreme.

If within one circle an equilater triangle and a square bee described, that square retaineth the same proportiō to the square of the triāgles inscribed circles diameter, that the semidiameter of the triangles conteynyng circle doth to the

semidiameter of the conteyned.

The tvvelfth Theoreme.

Where as with one circle an equiangle Pentagonum and a equilater triangle are described, the pentagonall inscribed circles semidiameter parted by extreme and meane proportion, the greater portion is equall to the triangles conteyned circles semidimetient.

The thirtenth Theoreme.

An equilater triangle and an equiangle pentagonum beeing within one cir∣cle described, if two squares be made, the first equall to the square of the pen∣tagonall side, and the square of the pentagonall lyne diagonall added togither, the second equall to the square of the same diagonall and the square of the lyne, wherby the diagonal exceedeth the side pentagonal, the side of this first square to the side of the second, reteyneth the same proportion that the pentagonal dia∣gonall dothe to the side of the equilater triangle.

The fourteenth Theoreme.

The Tetragonall Cordes square exceedeth the squares of the cordes penta∣gonall and decagonall, by the square of a lyne meane proportionall betwene the cord decagonal and the excesse or differēce of the cords hexagonal & decagonal.

The fifteenth Theoreme.

An equiangle triangles side beeing rationall, his perpendicular and semi∣diameters as well of the conteyning as conteyned circles, are all irrationall, but theyr squares may be expressed in number.

The sixteenth Theoreme.

The Area of an equilater triangle beareth to the square of his side propor∣tion as his perpendicular to his base doubled.

The seuententh Theoreme.

The Area of an equiangle triangle beareth proportion to the square of his syde, as √{powerof2}3 to 4.

The eyghtenth Theoreme.

Double the Area of an equilater triangle is meane proportionall betweene the square of his side, and the square of the perpendicular fallyng from one of his angles to the opposite side.

The nyntenth Theoreme.

The Area of an equilater triangle to the square that is described within his conteyning circle reteyneth the proportion of √{powerof2}3 to ••/3.

The 20 Theoreme.

The side of an equiangle triangle beeing rationall, his Area is irrational, but

the square therof may with number be declared.

The 21 Theoreme.

The Area of an equiangle Pentagonum, to the Triangles area that is descri∣bed with in his containing circle, retaineth sutch proportiō as fiue times the line Diagonal of the Pentagonum to the perimetrie of the Triangle.

The 22 Theoreme.

If an equiangle Pentagonum and an equilater Triangle be both described in one circle, the rectangle contayned of the line Diagonal and the Triangles per∣pendicular, retaineth proportion to the pentagonall Area, as 6 to 5.

The 23 Theoreme.

The Pentagonal side being rational, the Area is irrational, proportioned to the square of the side, as √{powerof2} vniu. 25/16+√{powerof2}125/64 to an vnitie.

The 24 Theoreme.

That square whose side is meane proportional betwene the internal circles se∣midiameter, and the pentagonal semiperimetrie, is equal to the pentagonal Su∣perficies.

The 25 Theoreme.

When an equiangle Pentagonum and an equilater Triangle be both within one circle described, a meane proportionall betwene ⅚ of the triangles perpendi∣cular, and the lyne diagonall is equall in power to the pentagonall Super∣ficies.

The .6. Probleme. The side of any Tetraedron giuen, to searche out the Se∣midiameters of the circumscribyng and inscribed spheres. (Book 6)

FIrste as is taught in the thirde probleme ye must get the circles semidiameter that conteineth the equiangle triangle whose side ye haue: Multiplie the square of this Semidiameter by 9▪ and diuide by 2, the root quadrate of the quotient is the circumscri∣bing spheres dimetient, the medietie wherof if ye augment by it self, and from the product subtract the square of the semidiameter tofore founde, the roote quadrate of the remayne is the Semidiameter of the conteyned Sphere.

Or thus more easily & spéedily: Multiply the side giuē in it self, the ofcome

diuide by 24 the roote quadrate of the quotient is the internall Spheres semidiameter, which tripled, yeldeth the semidiameter of the cōprehen∣ding Sphere.

Example.

The trigonall side supposed 10, the contayning circles semidiameter is √{powerof2}33 ⅓, vvhose square augmēted by 9, and diuided by 2, yeldeth in the quo∣tient 150, vvhose quadrate roote is the dimetient of the sphere, the medietie hereof squared, is 37 ½, from vvhich if ye vvithdravve 33 ⅓, there remayneth 4 ⅙, vvhose quadrate roote is the Axis of this Tetraedron.

Likevvise, if by the second rule ye diuide 100 vvith 24, the quotient is 4 ⅙, vvhose square roote being the Axis of this tetraedron if ye triple it, the re∣sulting number vvil be √{powerof2}37 ½, and that is the semidiameter of the circum∣scribing sphere, ye may also (by deducting 25 halfe the sides square from 100 the square of the vvhole side) finde the perpendiculare, for the roote of the remainder being √{powerof2}75 is the perpendiculare of the base, for the so∣lides perpendiculare I deducte 33 ⅓, the contayning circles semidimetientes square, from 100 the square of the side giuen, there remayneth 66 ⅔, vvhose roote is the solides perpendiculare.

The side of Te∣traedron 10.

• The contayning circles semidiameter—√{powerof2}33 ⅓.
• The contayned circles semidimetien••—√{powerof2}8 ½.
• The perpendicular of the Base—√{powerof2}75.
• The comprehending spheres semidiameter √{powerof2}37 ½.
• The inscribed spheres semidimetient √{powerof2}4 ⅙.
• The Solides perpendiculare—√{powerof2}66 ⅔.

Thus also an other way ye shal most spéedily finde these spheral semi∣diameters, square the side giuen, that squares medietie ye shal triple, the roote quadrate of the producte is the comprehending spheres dimetiente, which diuided by 3, bringeth the inscribed spheres diameter, y^e medieties of these are the semidiameters whereof the lesser augmented by 4, pro∣duceth the solides perpendiculare.

Theoremes of Tetraedron 1

THe square of Tetraedrons side, is equall to the square of his perpendiculare and the square of his contayning circles semidiameter added togither.

The 2 theoreme.

The square of Tetraedrons inscribed circles Semidiameter withdrawen

from the square of his bases perpendiculare, leueth the square of the solides per∣pendiculare.

The 3 Theoreme.

This solides perpendiculare is equall to his axis, and contayning Spheres Se∣midimetiente.

The 4 Theoreme.

The perpendiculare of Tetraedrons base is equall to his contayning and con∣tayned circles semidiameter.

The 5 Theoreme.

Tetraedrons contayning Spheres dimetiente retayneth the same proportion to his Axis, that the square of his side doth to double the square of his contay∣ned circles semidimetiente.

The 6 Theoreme.

Tetraedrons perpendiculare is ⅔ of his comprehending Spheres dimetiente.

The 7 Theoreme.

Tetraedrons Axis is a sixte parte of his comprehending Spheres Dime∣tiente.

The 8 Theoreme.

Tetraedrons contayning spheres Diameters square retayneth the same pro∣portion to the square of his side, that the perpendiculare of his base dothe to the semidiameter of his contayning circle.

The 9 Theoreme.

The square of Tetraedrons comprehending Spheres dimetiente contayneth the square of his inscribed circles semidiameter 18 tymes.

The 10 Theoreme.

The square of Tetraedrons contayned circles semidiameter is double to the semidiameters square of his contayned Sphere.

The 11 Theoreme.

Tetraedrons perpendiculare retayneth the same proportion to his basis con∣tayning circles semidiameter, that the contayned circles semidimetient doth beare vnto his Axis.

The 12 Theoreme.

The square of the comprehending Spheres dimetiente to the contayning Cir∣cles semidiameters square, hath proportion as 9 vnto 2.

The 13 Theoreme.

Tetraedrons side being rationall, his contayning Spheres diameter is irratio∣nall, and their proportion is as √{powerof2}2 vnto √{powerof2}3.

The 14 Theoreme.

Tetraedrons side being rationall, the Axis is surde, and it beareth propor∣tion to the side as 1. to √{powerof2}24.

The 15 Theoreme.

The Tetraedrons comprehending Spheres dimetiente, is equall in power to these fiue, the contayning circles semidiameter, the contayning Spheres semidime∣tiente, the inscribed circles semidiameter, the Axis, and the solides Perpendicu∣lare.

The 7. Probleme. The side of any Hexaedron giuen, to finde the semidimetientes of the contayning and contayned Spheres. (Book 7)

INcrease the side by it selfe, the offcome triple, frō the pro∣ducte extracte the roote quadrate the halfe therof is the se∣midimetiēt of the circumscribing sphere, frō whose square if ye detract the square of that circles semidimetiente that comprehendeth one of the Cubes quadrate playnes the roote square of the remainder is y^e semidimetiente of the inscribed sphere: the circles Semidiameter is found by the fourth Pro∣bleme, admitting the line giuen the side of a square.

Example.

The Cubes side 6, the square thereof tripled 108, halfe the roote thereof is √{powerof2}27, and that is the semidiameter of the cōtayning sphere, his cōtayning Circles semidiameter is √{powerof2}18, vvhose square deducted frō 27, leueth 9, his roote quadrate being 3 is the Axis or contayned spheres semidimetient.

The Cubes side 6

• ...The line Diagonal••— √{powerof2}72
• ...The contayning Circles Semid. √{powerof2}18
• The contayned circles Semid. 3
• ...The contayning Spheres Semid. √{powerof2}27
• ...The Axis or inscribed 3
• ...Spheres Semidimetient 3

Theoremes of Hexaedron. 1

THE square of the Cubes comprehending Spheres dimetiente is equall to the square of his side, and the square of his basis line Diagonall.

The 2 Theoreme.

The square of his line Diagonall contayneth the square of his Axis 8 times.

The 3 Theoreme.

The square of the Cubes comprehending Spheres semidiameter is equall to the square of halfe the line Diagonall of his base, and the square of his Axis added togither.

The 4 Theoreme.

The semidimetiente of the Cubes contayned circle is equall to his Axis.

The 5 Theoreme.

The Hexaedrons side is meane proportionall betweene his contayning Cir∣cles semidiameter and his basis lyne Diagonall.

The 6 Theoreme.

The Cubes line diagonall to his Axis retayneth triple the proportion of his side to his contayning circles semidiameter.

The 7 Theoreme.

The comprehending spheres dimetiente to the Cubes side, obserueth the Pro∣portion of √{powerof2}3 to 1.

The 8 Theoreme.

The Cubes side being a line rationall, his Axis is also rationall.

The 9 Theoreme.

The Hexaedrons comprehending Spheres Dimetiente beeing rationall, his Axis is a surde and bereth proportion to the Dimetiente, as 1 to √{powerof2}12.

The 10 Theoreme.

The cōprehending spheres diameter hath the same proportion to the line Dia∣gonall of his base that the Cubes side hath to the semidimetiente of his contay∣ning Circle.

The 8. Probleme. The side of Octaedron giuen, to searche out Arithmetically the con∣tayning spheres Diameter and the Axis therof. (Book 8)

DOuble the square of the side, and from the offcome extracte the roote Quadrate, so haue yée the comprehending Spheres dimetiente the halfe (being the semidiameter) if from y^e square therof, you abate y^e square of that circles semidiameter whiche

containeth one of the equilater triangles wherof the body is framed, the roote quadrate of the remaynder is the Axis.

Or thus more briefly, square the side giuen, the producte diuide first by 2 then by 3, the roote square of the first quotient is the contayning sphe∣res semidimetient, the seconde quotiente deducted fro the first, leueth a remaynder, whose roote quadrate is the Axis.

Or thus more redily, from the sixte parte of the giuen sides square, ex∣tracte the quadrate roote, so haue ye the axis, whose square augmented by 12, produceth the square of the comprehending spheres diameter.

Example of the firste.

THe side giuen 10, his square 100, √{powerof2}50 beeing the quadrate roote of halfe that square, is the comprehending spheres semidimetient, from the square therof abating 33 ⅓ the square of the contayning circles semidia∣meter, the remaynder is 16 ⅔, whose quadrate roote is the Octaedrons axis.

Example of the seconde rule.

100 the square of the side diuided first by 2 yeldeth 50, then by 3 bringeth in the quotient 33 ⅓, √{powerof2}50 is the containing spheres semidiameter, the one quotient deducted fro the other leueth 16 ⅔, his roote being √{powerof2}16 ⅔ is the Axis.

Example of the thirde precepte.

The sides square giuen 100 diuided by 6 produceth in the quotient 16 ⅔, the roote quadrate thereof is the Axis, the same number augmented by 12 maketh 200, wherefore I conclude √{powerof2}200 the comprehending spheres diameter.

The side of Octaedron 10.

• ...The contayning Circles semidiameter. √{powerof2}33 ⅓.
• The contayned Circles semidimetient. √{powerof2}8 ⅓.
• ...The contayning Spheres semidiameter. √{powerof2}50.
• ...The Axis. √{powerof2}16 ⅔.

Theoremes of Octaedron. 1.

The Axis of Octaedron is meane proportionall betweene the conteyning and conteyned circles semidiameters.

The second Theoreme.

The Octaedrons side is meane proportionall betweene the diameter and semi∣diameter of the circumscribing sphere.

The thirde Theoreme.

The Axis square increased by 12 yeldeth the compredending spheres dia∣meters square,

The fourth Theoreme.

The conteyning spheres diameter hath the same proportion to his Axis, that the Octaedrons side hath to the conteyned circles semidiameter.

The fifth Theoreme.

The side of Octaedron beeing rationall, the comprehending spheres Dime∣tiente cannot be expressed with number, but his square to the sides square is double.

The sixth Theoreme.

The side of Octaedron beeing rational, his Axis is not in number to be ex∣pressed, but the square thereof to the square of the side is as 1 to 6.

The seuenth Theoreme.

The Octaedrons conteyning spheres semidiameter is equall in power with his Axis and contayning circles semidiameter.

The eight theoreme.

The side of Octaedron is equall in power with the axis, the comprehending spheres semidiametient, and the comprehending circles semidiameter.

The ninth theoreme.

The diameter of Octaedrons comprehending sphere beeing rationall, his side is a surde, and retayneth suche proportion to the diameter as √{powerof2}½ vnto 1.

The tenth theoreme.

The diameter of Octaedron beeing rationall, his contayning circles semidia∣meter is irrationall, and hath proportion to his diameter, as √{powerof2}⅙ vnto 1.

The eleuenth theoreme.

Octaedrons side beeing rationall his axis is surde, and beareth proportion to the side, as √{powerof2}⅙ vnto 1.

The tvvelfth theoreme.

Octaedrons side being rational his contayned circles semidiameter is irratio∣nal, hauing proportion to the side as √{powerof2}••/1•• to 1.

The thirteenth Theoreme.

The axis of Octaedron beeing rationall, his comprehending spheres semidi∣metient is irrationall and their proportion is as an vnitie vnto √{powerof2}3.

The fourtenth Theoreme.

The side of Octaedron beeing rationall, that line which is equall in power to the semidiameters of his contayned sphere, and circle, is also rationall, and the comprehending spheres diameter to that line, hath triple the proportion of the con∣tayning circles semidiameter, to the axis.

The fiftenth Theoreme.

Octaedrons contayned circles semidimetient his axis, the contayning circles semidiameter, and the inscribed spheres diameter, are 4 lines in continuall geo∣metricall proportion.

The .9. Probleme. The side of an Icosaedron measured, by supputation to finde his axis and contayning spheres dimetiente. (Book 9)

ADmitting the side measured a Corde pentagonall, you shall by the fourth probleme searche out the contayning circles semidiameter, and multiplye the square thereof by 5, the quadrate roote of the producte is the dimetiente of the comprehending sphere, and so consequentlye the halfe therof shalbe the semidiameter, from whose square if ye subtracte his contayning circles semidimetientes square, the roote quadrate of the remaynder is the axis.

Example.

Admit the side of Icosaedron 12, which supposed a Cord pentagonall, the side hexagonall or semidiameter found (as was taught in the fifth pro∣bleme) is √{powerof2} v. 〈 math 〉〈 math 〉 the square of this semidiameter increased by 5 maketh 〈 math 〉〈 math 〉 whose quadrate roote is the contayning spheres dia∣meter, his medietie beeing √{powerof2} v. 〈 math 〉〈 math 〉 is the semidiameter. Nowe to attayne the axis I searche fyrst (as was taught in the thirde probleme) the circles semidiameter that contayneth the Icosaedrons triangulare base, fyn∣ding it √{powerof2}48, whose square deducted from the square of the contayning spheres semidiameter leueth 〈 math 〉〈 math 〉, whose quadrate roote is the axis of this Icosaedron.

The side of Icosaedron 12

• The circles semidiameter, wher∣on Icosaedron is framed. √{powerof2}.〈 math 〉〈 math 〉
• ...The semidiameter of the contay∣ning Circle. √{powerof2}48.
• ...The semidimetient of the con∣tayned circle. √{powerof2}12:
• The comprehending Spheres semid, √{powerof2} 〈 math 〉〈 math 〉
• The Axis or Cathetus √{powerof2} v. 〈 math 〉〈 math 〉

Theoremes of Icosaedron and his partes. 1.

The diameter of Icosaedrons comprehending sphere retayneth such propor∣tion to the diameter of the circle wherō the Icosaedron is framed, as √{powerof2}5 to .2.

The second Theoreme.

The semidiameter of this circle whereon the Icosaedron is framed with two cordes decagonal of the same circle do make the comprehēding spheres dimetient.

The thirde Theoreme.

The comprehending spheres semidiameter, doth match in power the Axis and contayning circles semidimetient.

The fourth Theoreme.

If the semidiameter of that circle whervpon the Icosaedron is framed, be di∣uided by extreame and meane proportion, the square of Icosaedrons side is equal in power to the foresaide semidiameter, and his greater portion.

The fifth Theoreme.

The diameter of the comprehending sphere being a line rationall, the Icosae∣drons side is a line irrationall, called of Euclide Minor.

The sixth Theoreme.

If the side of Icosaedron be a line rationall, the dimetient of the comprehen∣ding sphere shalbe an irrationall line called Maior.

The seuenth Theoreme.

The diameter of the comprehending sphere being a rationall, the semidiame∣ter of the circle whervpon the Icosaedron is made, shal be a line irrationall, but his square is rationall, and the comprehending spheres diameters square contay∣neth it 5 times.

The eight theoreme.

The Icosaedrons side being a line rationall, the semidimetiente of that circle wheron the body is framed will be an irrationall, called of Euclide Maior, and it

retayneth vnto this solids side the proportion of √{powerof2} v. ½+√{powerof2}1/20 vnto an vnitie.

The nynth Theoreme.

The Icosaedrons side being rationall, his conteyning circles semidiameter is irrationall, and beareth porportion to that side, as 1 to √{powerof2}3.

The tenth Theoreme.

The comprehending spheres diameter being rational, his conteyning circles semidiameter is an irrational of that kinde which Euclide calleth Minor, and it beareth proportion to the diameter as √{powerof2} vniuers ⅙—√{powerof2}1/180 vnto an v∣nitie.

The eleuenth Theoreme.

The Icosaedrons side admitted rational, the conteyned circles semidiameter is a surde, and reteyneth suche proportion to the side, as √{powerof2}1/12 vnto 1.

The tvvelfth Theoreme.

Icosaedrons conteyning spheres dimetient being rationall, his conteyned cir∣cles semidimetient is an irrationall called Minor, and the proportion betweene them is as 1 vnto √{powerof2} vniuersalss 1/24—√{powerof2}1/2880.

The 13 Theoreme.

Icosaedrons side supposed rationall, his axis is an irrational called Binomium and beareth proportion to the side, as √{powerof2} vni. 7/24+√{powerof2}5/64 vnto 1.

The 14 theoreme.

The diameter of Icosaedrons conteyning sphere admitted rational, his axis is irrationall, and reteyneth sutche proportion to his dimetient as √{powerof2} vni. 1/120+√{powerof2}1/180 vnto 1.

The 15 theoreme.

The semidiameter of the conteyning and conteined circles, although the pro∣portion of eyther to the spheres dimetient be surde, yet the proportion betwene themselues is rationall, and as 2 to 1.

The 16 theoreme.

Icosaedrons side being parted by extreme and meane proportiō, if ye adioyn to the side his greater portion, the square of that whole lyne added to the square of the side, produceth the square of the comprehending spheres diameter.

The 17 theoreme.

The diameter of that circle whereon Icosaedron is framed, reteineth suche proportion to the conteyning circles semidimetient of the bodye, that the Icosae∣drons side doth to the corde pentagonall of the conteyned circle.

The 18 theoreme.

If from the square of the comprehending spheres dimetient ye abate the Ico∣saedrons sides square, that lyne whiche matcheth in power the remayne, beeyng

diuided by extreme and meane proportion, wyll make his greater segment the Icosaedrons side.

The 19 Theoreme.

The pentagonall lyne diagonall of that circle, wheron Icosaedron is framed, exceedeth in power the contayned spheres diameter, by a lyne matching in po∣wer the excesse of Icosaedrons basis conteyning circles diameters square, about the square of his side.

The 20 Theoreme.

Icosaedrons side is meane proportionall betwene his diameter and the deca∣cagonall corde of that circle wheron Icosaedron is framed.

The .10. Probleme. The side of Dodecaedron giuen, by calculation to finde his axis and conteyning spheres diameter. (Book 10)

VSing the ayde of the fifte probleme if you serche out the tri∣angular or trigonal cord of that circle that containeth this Do∣decaedrons pentagonall base, admitting the same an Icosae∣drons syde, ye may by the last chapiter serche out the axis and containing spheres dimetient, whiche are in all poynctes equall and agrée∣able in proportion with the axis and sphericall Diameter of thys Dode∣caedron.

Or thus more spéedily and with lesse confusion in woorking, by the first probleme diuide this dodecaedrons syde giuen by extreme and mean pro∣portion, adding thereto his greater portion, then triple the square of thys whole line, the roote quadrate of the product is the conteining spheres Di∣metiente. Againe it behoouethe you by the fifth probleme to learne the se∣midiameter of the circle that contayneth one of the dodecaedrons pentago∣nal superficies, and deduct the square therof from the square of the com∣prehendyng spheres semidimetient, the roote quadrate of the remainder is the axis of the body.

Example.

The side of Dodecaedron 10, diuided by extreme and meane proportion by the firste probleme maketh his greater portion √{powerof2}125—5, vvhiche ad∣ded to 10, produceth √{powerof2}125+5, the square hereof tripled is 450+√{powerof2}112500 the roote square hereof is the comprehending spheres dimetiēt, vvhose me∣dietie

being √{powerof2} v. 112 ½+√{powerof2}112500/16 is that spheres semidiameter. Agayn, the semidiameter of this Icosaedrons contayning circle I finde by the fifth probleme √{powerof2} v. 50+√{powerof2}500, vvhose square taken from the square of the conteyning ••pheres semidiameter leaueth 62 ½+√{powerof2}112500/16—√{powerof2}500, vvhose roote quadrate is the axis or inscribed spheres semidimetiente.

The side of Dodecaedrō 10

• ...The conteyning circles semid. √{powerof2} vni. 50+√{powerof2}500.
• The conteyned circles semidimetient √{powerof2} vni 25+ √{powerof2}500.
• The diameter of the conteining sphere √{powerof2} v. 450+√{powerof2}112500
• The axis √{powerof2} vni. 62 ½+√{powerof2}1••2500/16—√{powerof2}500·
• The pentagonall diagonall lyne √{powerof2}125+5.

The axis of this body is also thus founde, I adde the lyne diagonal being √{powerof2}125+5 to 10 the dodecaedrons side, therof aryseth √{powerof2}125+15, this me∣dietie squared maketh 350/4+√{powerof2}28125/4 from vvhence I deduct 25+√{powerof2}500 the square of the conteyned circles semidimetiente, there remayneth 250/••+√{powerof2}28125/4—√{powerof2}500, vvhose roote quadrate is the dodecaedrons axis ex∣actly agreeyng vvith the former operation.

Theoremes of Dodecaedron.

THe square of the diameter of Dodecaedrons comprehending sphere is triple to the square of his basis line diagonall.

The seconde theoreme.

The Dodecaedrons side being added to the diagonall lyne of his basis maketh a lyne, whose square ioyned to the square of the side▪ produceth the square of the conteyning globes dimetient.

The third Theoreme.

The diameter of Dodecaedrons comprehending sphere being rationall, the side of that body shall be an irrationall called Apotome, bearing proportion to the diameter as √{powerof2} vni. ½—√{powerof2}5/36 vnto 1.

The fourth Theoreme.

The side of dodecaedron being rationall, his comprehending spheres dimeti∣ent is an irrational Binomium, hauing the squares of his names or compoun∣ding lynes in proportion as 5 to 1, and this irrational diameter to the Dodecae∣drons side, reteyneth proportion, as √{powerof2} vniuers. 9/2+√{powerof2}45/4 vnto an vnitie.

The fifth theoreme.

If dodecaedrōs side be adioyned to his diagonal line, the medietie therof squared

is equall to the square of the axis and the square of the conteined circles semidi∣ameter.

The 6 theoreme.

Dodecaedrons side beyng rational, the conteyning circles semidimetient is an irrationall called Maior, and beareth suche proportion to the side, as √{powerof2} vni. ½+√{powerof2}1/20 vnto 1.

The 7 theoreme.

The square of Dodecaedrons side added to the square of his basis diagonall produceth a quantitie, whose fifth part is the conteyning circles semidiameters square.

The 8 theoreme.

Dodecaedrons side beyng rationall, the diagonall of his basis shall be an ir∣rationall called Binomium, reteining proportion to the side, as √{powerof2}5/4+½ vnto 1.

The 9 theoreme.

Dodecaedrons axis being deuided in extreme and meane proportion, ma∣keth his greater parte the lesse semidimetient of his basis.

The 10 Theoreme.

The side of Dodecaedron beeyng rationall, his basis conteyned circles dia∣meter is an irrationall Maior, proportioned to the syde, as √{powerof2} v. 1+√{powerof2}frac45; to an vnitie.

The .11. Probleme. The side of Tetraedron knovvne to finde his superfi∣ciall and solide content. (Book 11)

FOr the superficies ye shall augmente the squared square of the side by 3, the productes quadrate roote is youre desire, or multi∣plye the contayning circles semidimetient in the perimetrie of the triangle, or augmente the contayned circles semidiameter in the perimetrie of the triangle and double the product, so haue you by e∣uerye of these operations hys content superficiall. For hys crassitude thus, Multiply the semidimetient of the contayning circle in the axis of the bo∣dy, and the product in the side, Or the semidimetient of the contained cir••le in the diameter of the conteyned sphere, and the offcome in the syde: Or the axis in the perimetrie of the traingle, and the product in the semidia∣meter of the contayning circle, the offcome diuided by 3, yeldeth in the quotient the solide quantitie.

Examples of the Superficiall quantitie.

The side of Tetraedron 10, his squared square 10000 multiplied in 3, brin∣geth 30000, I conclude √{powerof2}30000 the Superficiall content. Likevvise the side being 10, I finde by the thirde Probleme his containing circles Semidia∣meter √{powerof2}33 ⅓, vvhich increased by 30 the triangular Perimetrye yealdeth √{powerof2}90000/3, Also the semidiameter of the intrinsecal circle being √{powerof2}8 ⅓ multi∣plied in 30 bringeth √{powerof2}90000/12 vvhich doubled amounteth to √{powerof2}30000 the vvhole Superficies of that body agreeing vvith bothe the former operations.

Examples of the Solide capacitie.

The side admitted as before 10, by the thirde Probleme the Semidiameter of the containing circle is √{powerof2}33 ⅓, the Axis ye shall by the sixthe Probleme finde √{powerof2}4 ⅙, these multiplied together make √{powerof2}2500/18, and this againe in 10, bringeth √{powerof2}250000/18. Likevvise the Semidiameter of the contained Circle found by the thirde Probleme √{powerof2}8 ⅓, augmented by √{powerof2}50/3, the contayned Spheres Dimetient, there ariseth √{powerof2}1250/9, and that againe in 10 the side tri∣angular bringeth √{powerof2}125000/9. Also the Axis √{powerof2}4 ⅙ multiplied by 30 the tri¦angles Perimetrie, produceth √{powerof2}22500/6, and this againe in √{powerof2}33 ⅓ the containing Circles Semidiameter, maketh √{powerof2}2250000/18, vvhose third part is √{powerof2}250000/18 the Solide capacitie of that bodye exactlye agreeing vvith the former vvorkings.

The side of Tetraedron 10

• The containing circles Semidiameter √{powerof2}33 ⅓
• The contained circles Semidimetient √{powerof2}8 ⅓
• The containing Spheres Semidimetient √{powerof2}37 ½
• The Axis or Kathetus √{powerof2}4 ⅙
• The content Superficiall √{powerof2}30000
• The content Solide √{powerof2}13888 8/9

Theoremes of Tetraedrons super∣ficiall and Solide contentes.

THe side of Tetraedron being Rationall, the line that matcheth in power his superficies is an irrational, retaining proportion to the side as √{powerof2}{powerof2}3 vnto 1.

The second theoreme.

The Tetraedrons superficies to the square of his side beareth proportion as √{powerof2}3 vnto 1.

The third theoreme.

The triangle whose basis is equall to the Perimetrie of this solides basis, and his altitude or Perpendicular to the Diameter of the containing circle is equall to the superficial quantitie of this body.

The fourth theoreme.

The right angled Paralelogramme, whose altitude is the contained circles dia∣meter, and his base the triangular Perimetrie, is equal to the Tetraedrons super∣ficies.

The fifth theoreme.

That equilater triangle is equal to the Tetraedrons superficies, whose side beareth proportion to the solides side as 2 to 1.

The sixth theoreme.

A Quadrangular Prisma hauing his base a square, whose side is meane proportional betwene the Axis and the containing circles semidimetient, and his altitude the Tetraedrons side, is equal to the crassitude of Tetraedron.

The seuenth theoreme.

The solide of Tetraedron may by imagination be parted into 4 equal Trigo∣nal Pyramides, hauing for their bases the triangles of that body, and meeting with their toppes or Vertices in the center of the Sphere.

The eight theoreme.

A triangular Prisma, whose altitude is equal to the third part of the Axis, and his base one of the equilater triangles wherewith the bodye is encompassed, is equal to one of those Pyramides that meete with their toppes or Vertices at the center of the body.

The ninthe theoreme.

A right angled Quadrangular direct Prisma, hauing for his longitude, lati∣tude and profunditie these three lines, Tetraedrons side, his Axis, and containing circles Semidimetient, is equal to the Tetraedron.

The tenth theoreme.

If foure right lines be found in continual proportion, so as the first being equal to the Tetraedrons side, beare proportion to the last, as √{powerof2}72 vnto 1, the Cube of the second is equal to the Tetraedron in crassitude.

The .12. Probleme. The side of a Cube measured, to finde his Su∣perficiall and Solide quantitie. (Book 12)

MUltiplie the side first by 2, then by 3, these Productes multiplied together, declare the Superficiall content. Or double the comprehending Spheres Diameters square. Or triple the Diagonall lines square. Or multi∣ply the Cubes base by 6, euery of these wayes ye haue the superficies of this body.

For the Crassitude thus, Augment the side in the base, or the Axis in the square of the line Diagonall, or the superficies in the sixth part of his altitude, any of the Productes is your desire.

Example.

Admit Hexaedrons side 10, by the seuenth Probleme ye shall finde the Diameter of his containing Sphere √{powerof2}300, his Axis 5, his line Diagonall √{powerof2}200. The side first increased by 2 maketh 20, then by 3 bringeth 30, these together multiplied make 600. Likevvise the square of the comprehending spheres Diameter 300 doubled maketh 600, and the square of the line Dia∣gonall being 200 tripled maketh also 600, I conclude therefore 600 the con∣tent Superficiall.

Example of the Solide content.

10 the side augmented in 100, the base maketh 1000. Also 5 the Axis in 200 the square of the line Diagonall produceth 1000. Likevvise 600 the superfi∣cies in 5/3 the sixth part of the side, yeldeth also 1000, thus I finde by all these rules 1000 the Cubes solide quantitie.

The side of the Cube 10

• The containing circles Semidiameter —√{powerof2}50
• The contained circles Semidiameter —5
• The containing Spheres Dimetient —√{powerof2}300
• The Axis or Kathetus—5
• The line Diagonal —√{powerof2}200
• The cubes Superficiall quantitie —600
• The Solide capacitie—1000

Theoremes of Hexaedrons super∣ficies and Crassitude. Theoreme first.

THe comprehending spheres Dimetients square doubled is equall to the Cubes superficiall content.

The second theoreme.

A Paralelogramme whose altitude is double to the Cubes side, and his base triple, is equall to the superficies of that solide.

The third theoreme.

That square is equall to the Cubes superficies, whose side is meane proportio∣nall betwene the Cubes side and a line 6 times his length.

The fourth theoreme.

Any Prisma whose base is equall to the square of the Cubes basis line Diago∣nall, and his altitude the Cubes Axis, is equall to his solide quantitie.

The fifth theoreme.

The Cube may intellectually be deuided into 6 Quadrate Pyramides, euery of them hauing to his base one of the Cubes square basis, and concurring or mee∣ting at the centre of the sphere, which is a common Vertex to them al.

The sixth theoreme.

Any Pyramis hauing his Base equall to the square of the comprehending spheres Diameter, and his altitude the side of the Cube, is equall to his crassitude.

The seuenth theoreme.

A Tetraedron whose side is proportioned to the Cubes side, as √{powerof2}{powerof2}12 vnto an vnitie, haue equal superficial contentes.

The eight theoreme.

An Octaedron whose side is equal to a line meane proportional betweene the Cubes side and his Diameter, hath his superficies equal to the Cubes.

The ninthe theoreme.

If 7 lines be continually proportional, the last retayning proportion to the first as 72 vnto 1, the Tetraedron of the seconde is equall to the Cube of the first.

The tenth theoreme.

If 4 lines be continually proportionall the last bearing proportion to the first, as 1 to √{powerof2}2/9 the cube of the first is equall to the Octaedron of the second.

The .13. Probleme. Octaedrons side giuen to searche his superficiall and solide contente. (Book 13)

MUltiplye the contayning circles diameter in the circuite or perimetrie of the Triangle, or increase the squared square of the side giuen by 12, the quadrate roote of the producte is the superficies: for the solide contente worke thus, multiplye the diameter of the interior circle in the diameter of the internall sphere, and the producte in the side. Or the dimetiente of the contayning circle in the Axis of the body, and the product in the side. Or the Axis in the perimetrie of the triangle and the resulting summe in the contayning circles dimetient, the product diuided by 3 yeldeth the solide content.

Example.

THe side of Octaedron supposed 10, the contained circles diameter by the thirde probleme I finde √{powerof2}400/3, which augmented by 30, the triangles perimetrie produceth √{powerof2}360000/3

The squared square of the triangulare side is 10000 which augmented by 12 bringeth 120000, whose quadrate roote is this bodyes superficiall quantitie.

The example of the solide content.

THe side being 10 by the thirde probleme I fynd the contayning circles semidimetient √{powerof2}33 ⅓, the contayned spheres diameter by the eight probleme is √{powerof2}66 ⅔, these multiplyed togither make 20000/9, whiche in∣creased by 10 the triangulare side produceth √{powerof2}2000000/9

The diameter of the contayning circle √{powerof2}400/3 augmedted by √{powerof2}50/3 the axis, bringeth √{powerof2}20000/9, and this again in 10 maketh as before √{powerof2}2000000/9

The axis √{powerof2}50/3 multiplied by 30 the triangulare perimetry, bringeth √{powerof2}45000/3 and this agayne in √{powerof2}400/3 the contayning circles dimetiente, produceth √{powerof2}18000000/9 whose thirde parte is √{powerof2}222222 2/9 the solide ca∣pacitie of that Octaedron.

The side of Octaedron 10

• ...The contayning Spheres dimetient √{powerof2}200.
• ...The contayning circles diameter √{powerof2}400/3.
• ...The Axis or Cathetus √{powerof2}16 ⅔.
• ...The internall circles diameter √{powerof2}33 ⅓.
• ...The content superficiall √{powerof2}120000.
• ...The solid capacitie √{powerof2}222222 2/9.

Theoremes of Octaedrons contente superficiall and solide. 1.

THe side of Octaedron admitted rationall, his content superficiall is irratio∣nall, and the lyne that matcheth it in power is a surde called of Euclide a line mediall, retayning suche proportion to the side, as √{powerof2}{powerof2}12 to 1.

The 2 theoreme.

The superficiall quantitie of Octaedron beareth proportion to the square of his side as √{powerof2}12 to an vnitie.

The 3 theoreme.

A rectangulare parallelogramme hauing the one of his contayning sides the perimetrie of Octaedrons triangulare basis, and the other equall to the dimetient of his contayning circle, is equall to the whole superficies of this body.

The 4 theoreme.

An equilater triangle, whose side beareth proportion to the side of Octaedron, as √{powerof2}8 to 1 is equall to the capacitie superficiall of that body.

The 5 theoreme.

The square whose side is meane proportionall betweene the contayning circles diameter, and the perimetrie of this bodies basis, is equall to his superficies.

The 6 theoreme.

A quadrangulare Prisma hauing to his base a square (whose side is meane proportionall betweene the contayned circles diamiter and the contayned spheres diameter, and his altitude the solides side is equall to this Octaedrons crassitude·

The 7 theoreme.

This body may be diuided into 8 equall trigonall Pyramides, whose bases are the equilater triangles wherwith this solide is enuironed, euery of these Pyramides meete at the center of the contayning and contayned sphere, whiche is the com∣mon vertex to them all.

The 8 theoreme.

To euery of these Pyramides that Prisma is equall, whose altitude is the con∣tayned spheres semidiameter, and his base ⅓ of one of those equilater triangles, area, wherewith this body is enuironed.

The 9 Theoreme.

A quadrangulare directe Pyramis hauing his base equall to the square of a line in meane proportion betweene the axis and the perimetrie of the triangle, and his altitude the contayning circles dimetiente, is equall to this regulare solide.

The 10 Theoreme.

Yf foure lines in continuall proportion haue the firste and greatest equall to Octaedrons side, retayning proportion to the laste, as 1 to √{powerof2}2/9 the Cube of the seconde is equall to this solide.

The .14. Probleme. The side of Icosaedron knowen, by supputation to learne the con∣tentes superficiall and solide of that bodie. (Book 14)

COnsidering this body is enuironed with 20 equilater trian∣gles, ye shall by the thirde probleme get one of those triangles Area, which increased by 20 yeldeth your desire. Or augment the squared square of the giuen side by 75, and from the pro∣ducte extracte the roote quadrate. Likewise if you adde the semidiameter of the inscribed circle to the diameter of the circumscribed circle, the of∣coome multiplyed by double the triangles perimetrie, produceth the Ico∣saedrons content superficiall. For the solide capacitie adde the contay∣ning and contayned circles semidimetientes together, the producte in∣crease by double the side knowen, the resulting summe augmented by the Icosaedrons axis (whiche ye were in the ninth probleme taughte to finde) yeldeth the grosse capacitie. Also if ye augment the squared square of the side by 8 ⅓ the quadrate roote of the producte increased by the axis of this body bringeth the desired crassitude. Or if ye augment the sides squared square by 2 1/12 and the resulting summe agayne in the square of the inscribed spheres dimetiente, the roote quadrate of the producte is likewise the Icosaedrons solide capacitie.

Examples of the former rules.

THe Icosaedrons side supposed 12, by the thirde probleme ye shall finde the area of the equilater triangle √{powerof2}3888, which increased by 20 pro∣duceth √{powerof2}1555200. Likewise the squared square of 12 beeing 20736 aug∣mented by 75, yeldeth, 1555200, whose quadrate roote is your desired super∣ficies. In like maner if ye adde √{powerof2}12, the semidimetiente of the inscribed circle, to √{powerof2}192 the dimetiente of the circumscribed circle (for so shall ye finde them by the thirde probleme the triangles side beeing 12) the product will be √{powerof2}300, which multiplyed by 72 double the triangles perimetrie, yeldeth agayne √{powerof2}1555200 the superficies of that whole Icosaedron.

Nowe for the solide contente yf you adde √{powerof2}192 (for so is the dime∣tiente of the contayning circle) to √{powerof2}12, the semidimetiente of the con∣tayned circle, there will amounte √{powerof2}300, which augmented by 24 the tri∣angles side doubled bringeth √{powerof2}17280, and this by √{powerof2} v. 720/10—R. 20—48, (for so is the axis as ye may by the ninth probleme perceiue) multiplied, produceth √{powerof2} vni. 7257600+√{powerof2}48372940800000, and that is the solide capacitie of this body. The same summe is also produced by multiplication of 20736, the squared square of 12, in 8 1/••, and the producte thereof in √{powerof2} v. 42+√{powerof2}1620, the axis. Also if ye multiplye 20736 the sides squared square by 2 1/12 the ofcoome will be 43200, whiche augmented agayne by 168+√{powerof2}25920 the square of the inscribed spheres dimetiente produ∣ceth 7257600+√{powerof2}48372940800000, whose vniuersall quadrate roote is the solide capacitie of this Icosaedron, and by reduction to numbers ratio∣nall it falleth out betweene 3769 and 3770.

The side of Ico∣saedron. 12.

• ...The contayning circles diameter. √{powerof2}192.
• ...The contayned circles semid. √{powerof2}12.
• The comprehending Spheres semid. √{powerof2} v 〈 math 〉〈 math 〉
• The Axis. √{powerof2} v. 〈 math 〉〈 math 〉
• ...The superficiall contente √{powerof2}1555200.
• The solid capa. √{powerof2} v. 7257••00+√{powerof2}48372940800000.

Theoremes of Icosaedrons contents superficiall and solide. 1.

THe superficies of Icosaedron to the square of his side, beareth the proportion of √{powerof2}75 to 1.

The second Theoreme.

Any parallelogramme whose base is equall to double the perimetrie of the Icosaedrons triangulare base, and his altitude to the circumscribing Circles

Diameter and the inscribed circles semidiameter added togither, is equall to the Icosaedrons 20 triangles.

The 3 Theoreme.

When Tetraedrons side retayneth suche proportion to Icosaedrons side, as Icosaedrons diameter doth to the semidiameter of the circle wheron it is framed, that Tetraedrons superficies is equall to Icosaedrons.

The 4 Theoreme.

When the square of Octaedrons side contayneth the square of Icosaedrons sides medietie 10 times, then is that Octaedrons superfiiciall quantitie equall to Icosaedrons.

The 5 Theoreme.

A directe quadrangulare rightangled Prisma that hath for his three di∣mensions, these three right lines, double the side, the Axis, and a line componed of the Diameter of the contayning circle, and the semidiameter of the contayned circle, is equall to the solide contents of Icosaedron.

The .15. Probleme. The side of Dodecaedron giuen, to searche out by Arithmeticall calculation the superficiall and solide contente. (Book 15)

THis body as was before declared among the Diffinitions is encompassed with 12 equal equiangle Pentagonal Su∣perficies, so that, if ye searche the Area of any one (as was taught in the fift Probleme) and increase the same by 12, the Dodecaedrons superficies resulteth, Or by the forenamed fift Probleme, ye shal learne the quantitie of the contayning circles semidiameter, and also of the line Diagonall, the semidiameter ye shall augmente by 3, the line Diagonall by 5, these productes multi∣plied togither, bring the desired Superficies also.

In like manner for the crassitude if ye multiplie the semidiameter of the contayning circle in the line Diagonall and the offcome in the Axis of this body (which ye were in the tenth Probleme taught to finde) the producte augmented by 5, yeeldeth youre desire. Or the Semidia∣meter of the contayned circle in the Dimetient of the contayned sphere, and the product in the pentagonall perimetrie bringeth the solide con∣tente also.

Example.

Admit the Dodecaedrons side 1, by the 5 and 10 Probleme, I finde hys contayning spheres semidimetient, √{powerof2} vni. 9/8+√{powerof2}180/256, the Axis √{powerof2} v. ⅝+√{powerof2}180/256—√{powerof2}1/20, the cōtaining circles semidimetient √{powerof2} v. ½+√{powerof2}1/20, the line Diagonall of the base √{powerof2} vniuers. 3/2+√{powerof2}5/4. Novve, for the Dodecaedrons superficies, I searche firste the area of one Pentago∣num, vvhich I finde by the fifth probleme √{powerof2} vni. 25/16+√{powerof2}625/320, this aug∣mented by 12, produceth √{powerof2} v. 900/4+√{powerof2}810000/20. Likevvise the contayning circles semidimetiente tripled, maketh √{powerof2} vni. 9/2+√{powerof2}81/20, and the line dia∣gonall of the base increased by 5, yeldeth √{powerof2} vni. 150/4+√{powerof2}12500/16, these multiplied togither, produce √{powerof2} vni. √{powerof2}253125/16+√{powerof2}253125/80+√{powerof2}455625/16+√{powerof2}455625/80, vvhich contracted, maketh √{powerof2} v. 225+√{powerof2}40500, so much is the Dodecaedrons superficiall quantitie, and beeing reduced to rationall numbers, it falleth out betvvene 20 and 21 very nighe 20 13/20. Novve, to attayne the solide quantitie I augment √{powerof2} vni. ½+√{powerof2}1/20 the contayning circles semidimetiente, in √{powerof2} v. 3/2+√{powerof2}5/4 the line Diagonall, there ariseth √{powerof2} vni. 1+√{powerof2}5/16+√9/80, and this agayne multiplied in √{powerof2} vni. ⅝+√{powerof2}45/64—√{powerof2}1/20 the Dodecaedrons Axis, the producte aug∣mented by 5, maketh √{powerof2} vni. 10 ⅝+√28125/1024+√78125/1024+√28125/64+√{powerof2}50625/1024+√140625/1024—√625/20, so much is the Dodecaedrons crassitude. Agayne, by the other rule I augmente √{powerof2} v. ¼+√1/20 the semidimetiente of the cōtayned circle, in √{powerof2} v. 5/2+√45/4—√⅘ the Diameter of the inscri∣bed sphere, therof resulteth √{powerof2} v. 1 7/40+√95/64+√5/16—√1/20, vvhich mul∣tiplied agayne in 5, the Pentagonall perimetrie produceth √{powerof2} vni. 29 ⅜+√28125/64+√3125/16—√625/20 this is also the Dodecaedrons solide capacitie ex∣actly agreeing vvith that former, operation, and by reduction is founde to reste betvvene √{powerof2}58 and √{powerof2}59, being very nighe 7 9/14.

The side of Dode∣caedron 1

• The contayning circles √{powerof2} v. ½+√{powerof2}1/20
• Semidimetiente √{powerof2} v. ½+√{powerof2}1/20
• The contayned circles semidiameter, √{powerof2} v. ¼+√{powerof2}1/••
• The line Diagonal, √{powerof2} vni. 3/2+√{powerof2}5/4
• The contayning spheres semidimetiente, √{powerof2} vni. 9/8+√{powerof2}180/256
• The Axis or Kathetus, √{powerof2} v. ⅝+√{powerof2}180/256—√{powerof2}1/20
• The contente superficiall, √{powerof2} vni. 225+√{powerof2}40500
• The crassitude or solide capacitie, √{powerof2} v. 29 ⅜+√{powerof2}55125/64

Theoremes of Dodecaedrons Su∣perficiall and Solide quantities. 1.

A Pentagonall equiangle superficies (whose side to the side of Dodecaedrons retayneth the proportion of the comprehending Spheres Diameter to the medietie of the line Diagonall) is equall to the Dodecadrons contente super∣ficiall.

The 2 theoreme.

A rightangled Parallelogramme contayned of the Pentagonall perimetrie, and triple the Diameter of the contayned Circle, is equall to Dodecaedrons su∣perficies.

The 3 theoreme.

That Cube whose side is meane proportionall betweene the semidiameter of the internall Circle, and the perimetrie of the Pentagonall base, hath a superfi∣ciall contente equall to Dodecaedrons.

The 4 theoreme.

When an Icosaedrons side is meane proportionall betweene the Pentagonall bases Diagonall and his circūscribing circles Trigonall corde, the Superficies of that Icosaedron is equall to the superficies of Dodecaedron.

The 5 theoreme.

If two meane proportional lines be founde betwene the Pentagonall Diago∣nall and his circumscribing circles corde Trigonall, that Icosaedron whose side is the meane proportionall nighest to the corde Trigonall, is equall to the Dode∣caedron.

The .16. Probleme. The diameter of any sphere knovven, to searche out the sides, Axes and contayning or contayned circles semidiameters of all suche bodyes regulare as are therein to be described, both Arithmeti∣cally and Geometrically. (Book 16)

COnsidering that in any Sphere all the regulare bodyes maye be described as it is by Euclide sufficiently demon∣strated, and that both the proportion of their sides Axes and perpendiculars to the spheres dimetient, and also the maner of working in euery of them is differēt, I thynk it best to adioyn seueral rules for euery of their operations, & first for y^e in∣uention

of their sides peruse the preceptes following: square the spheres diameter, and from the producte deducte ⅓, the roote quad•••••••• of the re∣mainder is the Tetraedrons side: Double the square of the Spheres se∣midiameter, so haue ye the square of Octaedrons side, diuide the Diame∣ters square by 3, the roote quadrate of your quotient is the Cubes side, For the Icosaedrons side ye shall diuide the Diameters square by 5, the rote of the quotient note, and supposing the same a circles semidiameter, by the second Probleme searche out the circles pentagonall Corde, for that is the Icosaedrons side. Finally, for the side of Dodecaedron, ye shal diuide the square of the spheres Diameter by 3, and from the producte extracte the quadrate roote, this roote diuided by extreame and meane proportion, (as ye were taught in the firste Probleme) yéeldeth for hys greater parte the Dodecaedrons side. Nowe the sides of euery regulare body thus founde, for their perpendiculares, Axes, and Semidiameters of their contayning and contayned circles, ye shall resorte for Tetrae∣dron to the sixte Probleme, for Octaedron to the seauenth, for Hexaedrō to the eight, and so in order for the reste, wher ye are taughte by the side to searche out all those forenamed lines, and so shall ye finde the exacte quantitie of all perpendiculares Axes and semidiameters of any regu∣lare body in that sphere contayned.

Example.

It vvould be ouer tedious to shevve the calculation for finding of euery particular line, vvherfore I shall only giue examples of the sides, for that the practize of their rules hath not bene yet set forth in any former Probleme. Admit the spheres Diameter that shall comprehende these regulare bodies 10, the square thereof diuided by 3, yeldeth 33 ⅓ vvhich deducted from 100, leueth 66 ⅔, the quadrate rote thereof is the inscribed Tetraedrons side. The square of the spheres semid. is 25, vvhich doubled maketh 50, the Zēzike rote thereof, is the Octaedrons side. 100 diuided agayne by 3, yeeldeth as before 33 ⅓, the roote square thereof is the Cubes side, vvhich diuided by extreame and meane proportion (as vvas taught in the firste Probleme) maketh the greater segment √{powerof2}41 ⅔—√{powerof2}8 ⅓, so much I affirme the contayned Do∣decaedrons side. Novv, for the side of Icosaedron I diuide 100 by 5, thereof ariseth 20, vvhose roote quadrate admitted a side Hexagonall, his corres∣pondente Corde Pentagonall by the seconde Probleme ye shall finde

√{powerof2} v 50—√{powerof2}500, the sides thus knovvne for the Axis, Semidiameters, and other lines, 〈…〉〈…〉 vse the same supputation that you did in those 5 Pro∣blemes past, vvhere ye vvere taught by the sides knovvne to attaine all the other lines, and for more plainnesse I shal at the end of this chapiter adioyne a Table containing the true quantitie of all the rest, vvhich ye may vse in stead of an Example to direct you, if happely you erre in your supputations, and for the farther satisfaction of such as seeke to reach beyond the commō sort, and vvill not content them selues vvith bare rules and preceptes, vnlesse they may also conceiue some grounde and reason of their vvorkings, I haue thought good to euery of these Problemes ensuing, to adioyne his peculiare figure, vvith meanes Geometricall (no regarde had to Irrationall numbers vvithout aide of Arithmeticall supputation) to searche out the sides, Diame∣ters, and Axis, of al the regular bodies inscribed or circumscribed of spheres, by knovvledge of their Diameters, or mutually cōferred together by knovv∣ledge of some side, according to the tenure of the Chapiter vvherin they are placed. And although breuitie (vvhich in this trifeling treatise I haue chefely affected) compell me not to stay in making demonstration of euery rule and Theoreme, yet the very construction of the figures vvell vvayed and confer∣red vvith Euclides 5 last bookes of Solides, vvill geue sufficient light to the in∣genious both to vnderstand the cause of these, and to inuent many mo vvher∣of there is no ende.

Geometrically vvithout aide of Arithmeticall calculation, to attaine the quantitie of all these forenamed lines ye shall thus vvorke.

Admit the Diameter of the comprehending sphere geuen AB, vvhich ye shall as vvas taught in the first booke diuide in tvvo equal partes at C, and in three at E, AE being a third part, vppon either of those sections errear Per∣pendiculars, and (describing a Semicircle vpon the Diameter) note their in∣tersections vvith FD, dravving lines from either of them to AB, so is AF the Cubes side, AD Octaedrons side, FB the side of Tatraedron, AF di∣uided by extreame and meane proportion (as ye vver taught in the first Pro∣bleme at G) maketh AG the Dodecaedrons side, vvhich extended out to H, ye shall make FH equall to FG, dravving the right line HB, and from F extend a Paralele to HB, till it crosse the Diameter in I, erecting there∣vppon the Perpendicular IK, so is the Cord KB the inscribed Icosaedrons side, IL is a third part of IB, ML a Paralele to IK, MB is the Diame∣ter of Icosaedrons basis containing circle, vvhose medietie MN is the Dia∣meter of the contained circle, the halfe therof MS parted by extreame and

meane proportion in V, so as SV be the greater segment, VN vvil be the Semidiameter of Dodecaedrons contained circle, and NB the Semidimeti∣ent of his containing circle, NC the Axis both of Icosaedron and Dodeca∣edron, VB is the Perpendicular of Dodecaedrons basis, and MA his So∣lides altitude, BS the perpendicular of Icosaedrons basis, and MA like∣vvise his Solides altitude, AF (the inscribed cubes side) is also Dodecae∣drons basis line Diagonal, EF is the greater Semidiameter of Tetraedrons base, and EP his medietie the lesser Semidiameter, EC Tetraedrons Axis, EB his Perpendicular or altitude, FB the cubes line Diagonall, OB his medietie the greater Semidiameter of the cubes base, OC the lesse Semidia∣meter, and Hexaedrons axis Octaedrons containing circles Semidimetient, OF the Sediameter of his contained circle RO, his Axis CO, and AF his altitude. Thus haue ye Geometrically in one figure the exacte quantities and proportions of all the regular bodyes sides, Diameters, Axes, Perpendi∣culars, and lines Diagonall, vvhereby ye may also be able bothe to conceiue some reason of such rules as are past, or Theoremes that shall ensue. And also inuent diuers meanes to abreuiate suche painful calculation as by the former rules ye shalbe forced to enter into, vvhile ye laboure vvith irrationall num∣ber to searche out the hidden proportions of these vnknovvne lines, as by proofe the industrious vvill soone perceiue.

[illustration]

The comprehen∣ding Spheres Di+metient geuen AB 10

• ...Tetraedrons
  • Syde, BF √{powerof2}66 ⅔
  • Basis greater semid, FE, √{powerof2}22 2/••
  • Basis lesse semid. PE. √{powerof2}5 5/••
  • Axis, EC, 1 ⅔
  • Altitude, EB, 6 ⅔
• ...Hexaedrons
  • Syde, FA, √{powerof2}33 ⅓
  • Basis greater Semid. OF, √{powerof2}16 2/••
  • Basis lesse semid. OC, √{powerof2}8 ⅓
  • Axis CO, √{powerof2}8 ⅓
  • Altitude AF, √{powerof2}33 ⅓
• ...Octaedrons
  • Syde AD, √{powerof2}50
  • Basis greater semid. OF, √{powerof2}16 ⅔
  • Basis lesse semid. OR √{powerof2}4 ⅓
  • Axis QF, √{powerof2}8 ⅓
  • Altitude AF √{powerof2}33 ⅓
• ...Icosaedrons
  • Syde KB, √{powerof2} v. 50—√{powerof2}500
  • Basis greater sem. MN, √{powerof2} v. 16 ⅔—√{powerof2}55 5/••
  • Basis lesse semid. N.S √{powerof2} v. 4 ⅙—√{powerof2}3 17/36
  • Axis, CN, √{powerof2} v. 8 ⅓+√{powerof2}55 5/9
  • Altitude MA, √{powerof2} v. 33 ⅓+√{powerof2}888 8/••
• ...dodecaedrōs
  • Syde AG, √{powerof2}41 ⅔+√{powerof2}8 ⅓
  • Basis greater semi. NB, √{powerof2} v. 16 ⅔—√{powerof2}55 5/9
  • Basis lesse semid. NV, √{powerof2} v. 4 ⅙+√{powerof2}3 17/36
  • Axis CN, √{powerof2} v. ⅓+√{powerof2}55 5/9
  • Altitude MA, √{powerof2} v. 33 ⅓+√{powerof2}888 8/9

Theoremes of the Regular bodyes in one containing sphere described. Theoreme first.

THe cōtaining spheres diameters square to the square of the inscribed Tetrae∣drons side is as 3 to 2.

The second theoreme.

The spheres Dimetient is in power double to Octaedrons side.

The third theoreme.

The spheres Diameter is in power triple to the Cubes side.

The fourth theoreme.

The spheres Dimetient being rationall, Icosaedrons side is suche an irra∣rationall

as Euclide calleth Minor, and beareth proportion to the Diameter, 〈◊〉〈◊〉 √{powerof2} vni. ½—√{powerof2}1/20 to 1.

The fifth theoreme.

The spheres Dimetient rationall, Dodecaedrons side is an irrationall Apo∣tome, retaining proportion to the dimetient, as √{powerof2}5/12—√{powerof2}1/12 vnto 1.

The sixth theoreme.

Tetraedrons Axis, is a sixth part of his spheres dimetient.

The seuenth theoreme.

Octaedron and the Cubes Axis are equall, and triple in power to the Axis of Tetraedron.

The eight theoreme.

The Axis of Icosaedron and Dodecaedron are equall, either of them being such an irrationall as Euclide calleth Maior, the spheres dimetient admitted ra∣tionall.

The nynth Theoreme.

Tetraedrons side, is the meane proportionall betweene his Perpendicular and his diameter.

The tenth Theoreme.

Octaedrons containing circles semidimetient is meane in proportion betweene his inscribed circles semidiameter and Tetraedrons side.

The eleuenth Theoreme.

The Hexaedrons side is a meane proportional betwene Tetraedrons side, and Octaedrons basis containing circles semidiameter.

The tvvelfth theoreme.

The cubes side being rational, the dodecaedrons side is an irrational Apotome, bearing proportion to the side of the Cube as √{powerof2}1 ¼—½ vnto 1.

The thirtenth theoreme.

Octaedrons side being rationall, the side of Icosaedron is such an irrationall as Euclide nameth Minor, and retaineth proportion to Octaedrons side as √{powerof2} vn. 1—√{powerof2}⅕ to an vnitie.

The fourtenth theoreme.

Octaedrons side being rationall, the Dodecaedrons side is an irrationall Apo∣tome, retaining proportion therunto as √{powerof2} v. 1—√{powerof2}5/9 vnto an vnitie.

The fiftenth theoreme.

The semidiameter of Dodecaedrons bases containing circle, hathe the same proportion to the Cubes side, that Icosaedrons side hath to the Diameter of his containing sphere.

The 16 theoreme.

Hexaedrons side beyng rationall, the conteyned Dodecaedrons circles semi∣diameter is an irrationall named by Euclide Maior, hauyng proportion to the Cubes side, as √{powerof2} v. ⅛+√{powerof2}••/320 vnto 1.

The 17 theoreme.

Tetraedrons side admitted rationall, the conteyned circles semidiameter of Icosaedron is an irrationall Minor, bearing proportion to the Tetraedrons side as √{powerof2} vniuers. ⅛—√{powerof2}1/320 vnto √{powerof2}2.

The 18 Theoreme.

Octaedrons conteyned circles diameter to the dimetient of dodecaedrons con∣teyned circle, hath proportion as 1 to √{powerof2} v. 1+√{powerof2}⅕.

The 19 theoreme.

Dodecaedrons conteyned circles semidiameter beeing an irrationall Maior, Icosaedrons conteyned circles semidimetient shall be an irrationall Minor, bea∣ring proportion as √{powerof2} v. 1+√{powerof2}⅕ to √{powerof2} vni. 1—√{powerof2}⅕.

The 20 Theoreme.

Dodecaedrons side, to octaedrons axis, retaynethe sutche proportion, as the greater part of a line diuided by extreame and meane proportion, to the medie∣tye of the whole.

The 21 Theoreme.

If from the square of dodecaedrons dimetient, ye subtracte the square of his side, the roote of the remaynder (diuided by extreme and meane proportion) maketh his greater part the Cubes side, and his lesse the dodecaedrons side.

The 22 theoreme.

A right lyne equall in power to the diameter and semidiameter of dodeca∣edrons basis conteyning circle, reteineth the same proportion to their conteyning spheres diameter, that dodecaedrons side doth to the side of Icosaedron.

The 23 theoreme.

Dodecaedrons side reteineth the same proportion to the side of the cube, that the semidiameter of dodecaedrons conteyning circle dothe to the diameter of his conteyned circle.

The 24 theoreme.

Dodecaedrons basis internall circles semidiameter diuided by extreme and meane proportion, maketh his greater part the Icosaedrons inscribed circles se∣midimetient.

The 25 theoreme.

If two lines equall in power to the spheres diameter retaine the proportion of

a lyne (diuided by extreme and meane proportion) to his greater part, the lesser of those lines is Icosaedrons side: but if their proportion bee as the whole to the lesser part, then is that lesse part the side of dodecaedron.

The .17. Probleme. Arithmetically and geometrically to search out all the sides, diameters, perpendiculars, and lines Diagonall, vvith the bases semidiame∣ters, of all suche regular bodies as shall circumscribe or cōprehend any sphere vvhose dimetient is knovvn. (Book 17)

SEing these bodies as it is demonstrated by Euclide, ar•• of sutche vniforme composition that they will bothe re∣ceiue an inward sphere touching with his conuex super∣ficies euery of their bases centers, and also an outward sphere inclosing and with his concaue peripherye tou∣ching euery of their angles, for resolution of this Pro∣blem it shall be requisite to shewe how the diameter of the containing sphere may be found by knowledge of the cōtayned spheres dimetient: whiche doone, by the proposition laste past, the sides, and semidi∣ameters, may consequently be knowne. But bycause these 5 bodies being described without one sphere, are not also inuironed of one, but haue thrée seuerall differente comprehendyng spheres, the largest enuironyng Te∣traedron, the nexte encompassing the cube and Octaedron, and the laste in∣closing Icosaedron and dodecaedron (for this as it is by demonstration ap∣proued of Euclide, so is it also apparant by these former problemes) it sée∣meth therefore requisite to prescribe thrée seueral rules for y^e inquiring out of their Dimetientes. And first for Tetraedrons Diameter, ye shall only increase the diameter geuen by 3, the producte is youre desire. And for the spheres dimetiente that contayneth the cube and Octaedron, ye shall di∣uide the diameter geuen by √{powerof2}••/3 the quotiente or resultyng summe is likewise your demaunde. Or diuide the same diameter by √{powerof2} vniuers. ⅓—√{powerof2}8/90, So haue yée the spheres dimetiente that comprehendeth Dodecaedron and Icosaedron.

Example.

Admit the diameter of the sphere vvheron these bodies shall be described 10, this augmented by 3, maketh 30, the diameter of tetraedron: Againe the square of 10 augmented by 3, bringeth 300, so is √{powerof2}300, the diameter of octaedron and the cube: I diuide 100 by ⅓—√{powerof2}4/45 therof ariseth 1500, —√{powerof2}1800000, I cōclude therfore √{powerof2} v. 1500—√{powerof2}1800000, the diameter of the sphere that shal include the dodecaedron & Icosaedron, vvhose inscri∣bed spheres diameter is 10, and proceedyng by the laste probleme yee shall fynde the dodecaedrons syde √{powerof2} v. 1250.—√{powerof2}1512500, Icosaedrons side √{powerof2} vniuers. 1050—√{powerof2}1012500, and so foorth of al the other solides sides, and semidiameters: for consideryng theyr operation is nothynge differente from that vvas shevved in the 16 probleme, it vvere in vaine here agayne to make therof a superfluous recitall.

Geometrically to perfourme the same.

Vpon AB the diameter giuen of the sphere (vvhich al these regular bodies shal circūscribe) describe the semicircle, AKB, and vpon the same center C, describe the semicircle DNE, hauing his diameter DE triple to AB the dia¦meter giuen, and frō A, errere the perpendicular AF, vvhich diuided in tvvo equall parts at ω, maketh A ω the lesse semidiameter of Tetraedrons base, and AF the greater: from F extende cordes to ED, EF is the tetraedrons side, and EA his perpendiculare. Novve yf ye fixe one foote of the com∣passe in C (and openyng the other to ω) describe the semicircle HLI, it vvill touche the medietie of FE at G, vvhereby ye haue HI the diameter of Hexaedron and octaedron, HL, the side of Octaedron, AB the side of the cube, A ω the semidimetient of Octaedrons basis conteyning circle, AK, the semidiameter of the Cubes conteynyng circle, AC the se∣midimetient of the cubes inscribed circle, and AM the medietie of AK is the octaedrons basis lesser semidimetiente. Thus haue you founde the diameters, sides, and circular semidiameters, of these firste 3 bodyes: for the other tvvo ye shall thus proceede, from N as ye vvere taughte in the firste booke, dravve NO paralele to CE equall to the medietie thereof, and (couplyng CO togyther vvith a strayght lyne) from P vvhere it cut∣teth the greatest semicircles circūference let fal the perpendicular PQ, and vpon S (leauing SQ equal to 23 part of QE) erect the perpēdicular SR, thē

from R and P, extende straight lines to C & E, and to the middle of RE ex∣tēd the right line CT, cutting the left semicircle in φ, from thēce dravv a pa∣ralele to RE, cutting RC, CE in π, x, and (making Cx a semidiameter) des∣cribe the semicircle β π x crossing CP in ρ, frō thēce to x dravv the right line x ρ, for that is the circumscribing Icosaedrons side, and φ x is the greter semi∣dimetient of his base, vvx the lesser, β x the diameter of Icosaedrōs compre∣hēding sphere: This done from R dravv a paralele to the semicircles diame∣ter, c••ossing the greatest circumferēce in Y, fro thence to the end of the gre∣test dimetient, dravv the line DY, from vvhose medietie θ, dravv a straight line to the center C, then as vvas taught in the first booke, cut D θ in half at μ, and diuide D μ in extreme & mean proportion at λ by the first probleme, from these points λ, FY, to the cēter C extend right lines, cutting the cir∣cumference of the last described semicircle in the points α z, from thē to β, dravv lines again, and diuide the cord α β in extreme and mean proportion at γ: Likevvise β z is diuided in halfe at δ, being both the contact of the least circūference, and also the intersection made vvith θ C, but vvhere λ C con∣reth vvith β z place this letter ε, so is β γ the circūscribing dodecaedrōs side β x his dimetient, β δ the semidiameter of his conteyning circle, ε δ the les∣ser semidiameter of the dodecaedrons pentagonall basis, and α β his line dia∣gonall. Thus haue ye in one figure all the sides and diameters both circular and spherall of all suche regulare solides as comprehende or circumscribe the assigned sphere.

[illustration]

The diame∣ter of the in∣ternall giuen Sphere 10

• ...Tetrae∣drons
  • Diameter DE, 30
  • Syde EF, √{powerof2}600
  • Contayning circles sedimiameter, AF, √{powerof2}200
  • Inscribed circles semidimetient A ω √{powerof2}50
  • Perpendiculare or Altitude, AE 20
• ...Hexae∣drons
  • Diameter HI √{powerof2}300
  • Syde AB, 10
  • Basis greater semidiameter, AK, √{powerof2}50
  • Basis lesser Semidimetient, AC 5
  • Basis line Diagonall, √{powerof2}200
  • Altitude AB 10
• ...Octae∣drons
  • Diameter HI √{powerof2}300
  • Syde HL, √{powerof2}150
  • Basis greater semid. AK √{powerof2}50
  • Basis lesse semidiameter, AM, √{powerof2}25/2
  • Perpendicular AB, 10
• ...Icosae∣drons
  • Diameter β x √{powerof2} v. 1500—√{powerof2}1800000
  • Syde ρ x. √{powerof2} viii. 1050—√{powerof2}1012500
  • Basis greter semidiamet. ω φ √{powerof2} v. 350—√{powerof2}112500
  • Basis lesse semidiameter φ w, √{powerof2} v. 87 ½, —√{powerof2}7031 ¼
  • Altitude 10
• ...Dodeca∣edrons
  • Diameter β x √{powerof2} v. 1500—√{powerof2}1800000
  • Syde γ β √{powerof2} vni. 1250—√{powerof2}1512500
  • Basis greater semid. δ β—√{powerof2} v. 350—√{powerof2}112500
  • Basis lesse semidiameter ε δ √{powerof2} v. 37 ½—√{powerof2}781 ¼
  • Altitude AB, 10

Theoremes of the internall Spheres circumscribing Solides. 1

TEtraedrons diameter is rationall and triple to the contayned Spheres dia∣meter, the internall Spheres diameter supposed rationall.

The 2 Theoreme.

Tetraedrons sides square retayneth the same proportion to the square of the internall spheres Diameter, that his contayning spheres Diameter doth to his Axis.

The 3 Theoreme.

Tetraedrons Basis contayning circles semidiameter, is double in power to the contayned spheres Diameter.

The 4 Theoreme.

The square of Tetraedrons inscribed circles semidiameter to his Axis, hol∣deth the same proportion in power, that his perpendiculare dothe to his inscribed Spheres diameter.

The 5 Theoreme.

Tetraedrons side, is double to Octaedrons side, one Sphere beeing contayned of them both.

The 6 Theoreme.

The square of Tetraedrons side, to the square of the Cubes side, retayneth the same proportion, that Tetraedrons comprehending spheres Diameter dothe to his Axis.

The 7 Theoreme.

The diameter of Tetraedrons inscribed circle, is equall to the Diameter of Octaedrons contayning circle.

The 8 Theoreme.

Tetraedrons diameter to the diameter of the sphere that comprehendeth Octaedron and the Cube, hath the proportion of 3 to his quadrate roote.

The 9 Theoreme.

Octaedron and the Cubes comprehending spheres diameter beeing rationall, the diameter of Icosaedron and Dodecaedrons contayning sphere is an irratio∣nal Minor, bearing proportion to the forenamed diameter, as √{powerof2} v. 5—√{powerof2}20 vnto 1.

The 10 Theoreme.

Icosaedron and Dodecaedrons axis beeing rationall, their comprehending spheres dimetiente, shall bee an irrationall Minor, bearing proportion thervnto, as √{powerof2} vni. 60—√{powerof2}2880 vnto an vnitie.

The 11 theoreme.

Icosaedrons axis beeing rationall, his side is an irrationall Apotome propor∣tioned therevnto, as √{powerof2} v. 42—√{powerof2}1620 vnto an vnitie.

The 12 theoreme.

Dodecaedrons Axis rationall, his side is a surde Minor proportioned to the Axis, as √{powerof2} v. 50—√{powerof2}2420 vnto an vnitie.

The 13 Theoreme.

Icosaedrons side being an irrationall Apotome of the sixte order, Dodecae∣drons side is an irrationall Minor, retayning suche Proportion thereto, as √{powerof2} v. 50—√{powerof2}2420 vnto √{powerof2}27—√{powerof2}15.

The 14 Theoreme.

Icosaedrons axis being rational, his basis cōtayning circles semidiameter is an Apotome of the first order bearing proportiō to the Axis, as 3—√{powerof2}5 vnto an vnitie.

The 15 Theoreme.

Dodecaedrons axis being rationall, his Pētagonall basis greater semidiameter is an Apotome of the first order proportionall to his side, as √{powerof2} v 7—√{powerof2}45 to √{powerof2} v. 25—√{powerof2}605.

The .18. Probleme. The side of any Tetraedron giuen, to finde the sides, Diameters, and Axes, of all such regulare bodyes as maye therein bee described. (Book 18)

HAuing heretofore at large set foorthe by Problemes sundrie wayes (the sides of these bodyes gyuen) to finde the semidiameters of their contayning and con∣tayned circles, the diameters of their comprehending and comprehended spheres, with their contentes su∣perficiall and solide: hauing also by Theoremes she∣wed manifolde diuersitie of proportions rational and surde of these bodyes, their Superficies and lines compared with theirs comprehending and contayned spheres, there remayneth only nowe to conferre these bodyes mutually inscribed or circumscribed one with an other, and to search out by the side of any one knowen, the sides and dia∣meters both circulare and spherall, with the capacities superficiall and solide of all such bodyes as may within or without the same bodye be de∣scribed, I shall therefore first beginne with Tetraedron, and so procéede with the reste. Tetraedron receyueth only Octaedron and Icosaedron, for the Cube and Dodecaedron cannot possibly therein be so placed, that all their angles at one instante might exactly touche his superficies, the Tetraedrons side therfore giuen parted in two equall portions, either medietie is the inscribed Octaedrons side: Likewise the medietie of Tetraedrōs sides square, is y^e square of Octaedrons diameter which diui∣ded by 12, produceth a nūber, whose quadrate roote is y^e Octaedrons Axis.

For the inscribed Icosaedrō ye shall diuide the medietie of Tetraedrons side by extreame and meane proportion, and double the square of the lesse portion, the quadrate roote of the producte is the side. Or deducte one of these portions from the other, & adde the square of the remaynes medietie to the square of Octaedrons sides medietie, the rote quadrate of the producte doubled, is the Icosaedrons Diameter: Or if from the laste producte, ye abate the thirde parte of Icosaedrons sides square, the roote quadrate of the remayne is the Icosaedrons Axis.

Example.

Tetraedrons side supposed, Octaedrons side is ½, √{powerof2}½ his Diameter, vvhose square diuided by 12 bringeth 1/24, the roote being √{powerof2}1/24, is the Oc∣taedrons Axis. Likevvise for Icosaedron the medietie of Tetraedrons side diuided in extreame and meane proportion by the first Probleme, maketh the lesser portion ••/4—√{powerof2}5/16, the square hereof doubled, hath for his roote √{powerof2} vni. 14/8—√{powerof2}180/64, so much is the inscribed Icosaedrons side. Agayne, the difference of Tetraedrons sides medieties Portions diuided by ex∣treame and meane proportion is √{powerof2}20/16—1, the square of halfe this diffe∣rence is 9/16—√{powerof2}80/256, vvhich added to the square of Octaedrons sides me∣dietie, produceth ⅝—√{powerof2}5/16, the roote thereof doubled, is √{powerof2} v. 5/2—√{powerof2}5 the true quātitie of Icosaedrōs dimetiēt. Novv by subtracting 7/12—√{powerof2}45/144 the third parte of Icosaedrons sides square, from ⅝—√{powerof2}5/16, the laste pro∣ducte vvhose roote ye doubled to make the Diameter youre remainder vvil be this number 1/24+√{powerof2}45/144—√{powerof2}5/16, vvhose roote vniuersall is the inscri∣bed Icosaedrons Axis.

The con∣tayning Tetrae∣drons side 1

• ...Tetraedrons
  • Diameter. √{powerof2}3/2
  • Axis, √{powerof2}1/24
• ...The inscribed Octaedrons
  • Syde ½
  • Diameter √{powerof2}½
  • Axis, √{powerof2}1/24
• ...The cōtayned Icosaedrons
  • Syde √{powerof2} vni. 7/4—√{powerof2}45/16
  • Diameter √{powerof2} vni. 5/2—√{powerof2}5
  • Axis, √{powerof2}1/24.

Or thus geometrically without respect of number.

Admit AB the Tetraedrons side giuen, thereon I describe the semicir∣cle ACB, erecting the perpendiculare DC, and drawing the line AC, that Corde AC is the inscribed Octaedrons dimetiente, and AD his side. Now diuide DC by extreame and meane proportion as was taught in the firste probleme at E, and from E erect the perpendiculare EF, cutting AC in F, conclude FC the inscribed Icosaedrons side: againe parte AC in halfe at H, and from H to E extende a straight line, for that shal be the Icosaedrons semidiameter. For the Axis ye shall cut of from DB his twelfth parte, as ye were taught in the first booke: suppose it BI, vpon I erecte a perpendicu∣lare, cutting the circumference in K, so is the Corde KB the inscribed Ico∣saedron and Octaedrons Axis. As for their bases circulare semidiameters they are founde as was taught in the fifth and sixtenth problemes, foras∣much as both their sides and dimetientes are knowen.

[illustration]

Theoremes of Tetraedrons inscribed solides. 1.

Tetraedrons side is double to the side of his contayned Octaedron.

The second Theoreme.

Octaedrons diameter is a meane proportionall betweene his side and the side of his contayning Tetraedron.

The thirde Theoreme.

If any equilater triangle be so described and situate within the Tetraedrons equiangle triangulare base, that the angles of the inscribed triangle diuide the sides of the base by extreame and meane proportion, the medietie of this inscri∣bed triangles side, is the side of this Icosaedron.

The fourth Theoreme.

Tetraedrons side diuided by extreame and meane proportion, his lesser parte

is double in power to this Icosaedrons side.

The fifth Theoreme.

The dimetient of Octaedron diuided by extreame and meane proportion ma∣keth his lesser portion the Icosaedrons side.

The sixth Theoreme.

Icosaedrons diameter is equall in power with Octaedrons side, and the excesse or difference of his two partes, the one deducted fro the other.

The seuenth Theoreme.

The side of Tetraedron beeing rationall, his inscribed Icosaedrons side is an Apotome, bearing proportion to the Tetraedrons sides, as √{powerof2} v. 1 ¾—√{powerof2}••••/16 vnto 1.

The eight theoreme.

The Axes of these inscribed bodies Octaedron and Icosaedron are equall.

The ninth theoreme.

The side of Tetraedron beeing rationall, the contayned bodies axes are ratio∣nall, in power commensurable, and beare proportion to the side as √{powerof2}24 to 1.

The tenth theoreme.

Octaedrons diameter beeing rationall, Icosaedrons dimetient is an irrationall Minor, proportioned to the dimetiente of Octaedron as √{powerof2} v. 5—√{powerof2}20 vnto an vnitie.

The .19. Probleme. The side of a Cube giuen, to finde the sides, diameters and axes of all suche regulare bodies as may therin be described. (Book 19)

THe Cube is capable of thrée bodies, Tetraedron, Octaedron, and Icosaedron, for the gretest Dodecaedron that may with∣in a Cube be imagined, will only with 12 of his angles touche the Cubes superficies, his other eighte angles remayning within the body vnder euery of the Cubes angles, diuiding his semidi∣metients by extreame and meane proportion: this body therfore omitted bicause his inscription is vnperfect, I shall giue rules for the other thrée, and first of Tetraedron, double the square of the Cubes giuen side, the product roote quadrate is the Tetraedrons side, the same giuen sides square tripled bringeth his comprehending spheres diameters square, whiche diuided by 36 yeldeth in the quotient a number, whose quadrate roote is the Tetraedrons axis. The Cubes sides squares medieties roote quadrate, is the inscribed Octaedrons side.

The Octaedrons diameter is equall to the Cubes side,

The Cubes sides square diuided by 12, bringeth the contayned Octae∣drons Axis square. For the contayned Icosaedrons side, ye shall parte the contayning Cubes side giuen by extreame and meane proportion, as ye were taught in the firste probleme, the greater parte is your desire. And if you adde the square thereof to the square of Hexaedrons side, the roote quadrate of the resulting summe is the Icosaedrons dimetient. And yf ye square the medietie of this dimetient, and from it deduct a third part of the square of this inscribed Icosaedrons side, the roote quadrate of the remayne is the Icosaedrons Axis.

Example.

HExaedrons side giuen 1, his square doubled is 2, √{powerof2}2 is the contayned Tetraedrons side, the giuen sides square tripled is 3, his quadrate roote is tetraedrōs diameter, that square diuided by 36 yeldeth 1/12, √{powerof2}1/12 is the axis. Likewise √{powerof2}½ beeing the quadrate roote of halfe the Cubes side is the inscribed Octaedrons side, the Octaedrons diameter beeing equall to the Cubes side is 1 also, and the Cubes sides square diuided by 12 yeldeth 1/12, I conclude √{powerof2}1/12 the axis. Againe for Icosaedron I diuide 1 by extreame and meane proportion, the greater parte is √{powerof2}5/4—½, so muche is the Ico∣saedrons side, the square of this side added to the square of the Cubes side, maketh 2 ½—√{powerof2}5/4, whose roote vniuersal is the diameter: from the square of this rootes medietie being ⅝—√{powerof2}1/64, if ye deducte ½—√{powerof2}5/36 the third parte of the sides square, there will remayne ⅛+√{powerof2}5/36—√{powerof2}5/64 the roote vniuersall therof is Icosaedrons axis.

The Cubes side 1. His inscribed.

• ...Tetraedrons
  • ...Syde √{powerof2}2.
  • ...Diameter √{powerof2}3
  • Axis √{powerof2}1/12.
• ...Octaedrons
  • Side √{powerof2}½.
  • ...Diameter. 1.
  • Axis √{powerof2}1/12.
• ...Icosaedrons
  • Side √{powerof2}5/4—½.
  • Diameter √{powerof2} v. 2 ½—√{powerof2}••/••.
  • Axis √{powerof2} v. ⅛+√{powerof2}5/36—√{powerof2}••/64.

Ye may also with the compasse finde out all these sides, diameters and axis, no regarde hadde to any number, so the contayning Cubes side be knowen. Admitte the Cubes side giuen AB, extende the same line out to D, and vppon A as a centre, making AB the semidiameter, describe the semicircle BDC, and vppon A erecte the perpendiculare AC, as ye were taughte in the firste book: concurring with the circumference in C, agayne as ye were taughte in the firste booke cutte of from AB 1/24 parte. Admitte it FB, vppon F erecte an other perpendiculare crossing the circumference in G, diuide also CB in two equall partes in I, then open your compasse to the length of AB, and fixing one foote in B, with the other crosse the circumference in E. Laste of all by the fyrste probleme diuide AB by ex∣treame and meane proportion in H, so as AH may be his greater portion, and extende right lines from H to C, from B to C, from E to BD, and from G to B. Thus haue ye the diameters sides and axes of those inscribed bodies, for CB is the inscribed Tetraedrons side, DE the Tetraedrons dimetiente, GB the Tetraedrons axis, BI is the inscribed Octaedrons side, AC the Octaedrons dimetiente, GB also Octaedrons axis, AH is the side of the inscribed Icosaedron, and CH his diameter, as for the axis ye may geometrically finde it by the sixtenth probleme, the comprehending spheres dimetiente beeing knowen, whereof I minde not heere to make any newe recitall, considering it is sufficient playnly declared before.

[illustration]

Theoremes of Hexaedrons inscir∣bed regulare Solides.

THe inscribed Tetraedrons side is double in power to the conteynyng cu∣bes side.

The 2 theoreme.

The conteyning Cube and his conteined tetraedron haue one comprehending sphere, but their axes are different, and their proportion in power triple.

The 3 theoreme.

The conteyning cubes axis is equall to his inscribed Octaedrons semidiame∣ter, and their axes in power retaine triple proportion.

The 4 Theoreme.

Hexaedrons side diuided by extreame and meane proportion, maketh his greater parte the Icosaedrons side.

The 5 Theoreme.

The inscribed Icosaedrons dimetient is in power equall to his conteyning cu∣bes side, and the greater portion therof, it being diuided by extreme and meane proportion.

The 6 Theoreme.

The Cubes side being rationall, his inscribed Icosaedrons side is an apotome bearing proportion to the cubes side, as √{powerof2}1 ¼—½ vnto an vnitie.

The 7 Theoreme.

Hexaedrons side is meane proportionall betweene his inscribed Tetraedron and Octaedrons side.

The 8 theoreme.

The Cubes side is equall in power to his axis, and the conteyned tetraedrons semidiameter.

The 9 theoreme.

Hexaedrons diameter to his inscribed Tetraedrons perpendicular, retayneth the proportion of 3 to 2.

The 10 Theoreme.

The comprehending cubes side beeyng rationall, his conteyned Icosaedrons side is an irrationall Binomye, bearyng proportion vnto the cubes side as √{powerof2} vniuersalis ⅛+√{powerof2}5/576 to 1.

The .20. Probleme. Octaedrons side giuen, to searche out all his conteyned bo∣dies, sides, diameters and axes. (Book 20)

DOdecaedron within this body may not aptly be described, for as much as only eight of his solide angles touche the superficies of this body béeing situate in the centers of octaedrons bases, the other 12 falling quite within this body, not touching his super∣ficies on any parte. But Tetraedron may be inscribed and all his solide angles placed in the centers of octaedrons triangles. Likewise the cubes solide angles take their place in the centers of his bases, only Icosaedron hath his situation somwhat more straunge and intricate, euery of his 12 angles resting in a side of octaedron, and diuide his 12 sydes by extreame and meane proportion. Now to attain the diameters axes &c. Peruse these rules ensuing.

For Tetraedron.

Augment octaedrons side by 2, and diuide by 3, the quotient is tetrae∣drons side, the square therof augmented by 3, and parted by 2, yeldeth his diameter, which diuided by 6▪ declareth the axis.

The square of octaedrons side multiplied by 2 and diuided by 3 produ∣ceth this Cubes diameters square, which againe diuided by 3, sheweth his sides square, half the side is the axis.

For Icosaedron.

Diuide the side giuen by extreme and meane proportion, the square of the lesser parte double, and from the product extract the roote quadrate, so haue yée the inscribed Icosaedrons side, deduct one of those former founde parts of the giuen syde from the other, and square the difference, for that added to the square of Icosaedrons side bringeth the square of his dime∣tient. Nowe for the axis ye may deduct the third part of Icosaedrons sides square from the square of his semidimetiente, the roote quadrate of the remainder is his ••athetus.

Examples of Tetraedron.

Octaedrons side giuen 1▪ increased by 2, and diuided by 3, maketh ⅔, the Te∣traedrons side, the square therof being ⅘ augmēted by 3 and parted by 2, yeldeth ⅔√{powerof2}⅔ is his diameter, which diuided by 6, bringeth √{powerof2}1/54 the axis.

Of the Cube.

The square of Octaedrons side 1, augmented by 2, and diuided by 3, produ∣ceth ⅔ the diameters square, that again diuided by 3, yeldeth 2/9, √{powerof2}2/9 is the cu∣bes side, whose halfe being √{powerof2}1/18 is the axis.

Of Icosaedron.

Octaedrons side 1 diuided by extreme and meane proportion, the greater part •• √{powerof2}5/4—½, the lesser 3/2—√{powerof2}5/••▪ this latter partes square doubled is 7, —√45, the roote quadrate vniuersall therof is Icosaedrons side. Againe by subtrac∣tion of the partes ye shall find the difference √{powerof2}5—2, the square therof added to the square of Octaedrons side giuen, maketh 10, —√{powerof2}80. I cōclude √{powerof2} vni. 10, —√{powerof2}80 the Icosaedrons diameter. Finally for the axis I deduct 7/3—√{powerof2}5 the thirde parte of Icosaedrons sides square, from 5/2—√{powerof2}5, the semidi∣metient of Icosaedrons containing sphere, there remayneth ⅙√{powerof2}⅙ is the axis.

The comprehen∣ding Octaedrons side 1

• ...Tedraedrons
  • Syde ⅔
  • Diameter √{powerof2}⅔
  • Axis, √{powerof2}1/54
• ...Hexaedrons
  • Syde √{powerof2}2/9.
  • Diameter √{powerof2}⅔.
  • Axis √{powerof2}1/18.
• ...Icosaedrons
  • Syde √{powerof2} vni. 7—√{powerof2}45.
  • Diameter √{powerof2} vni. 10—√{powerof2}80.
  • Axis √{powerof2}⅙.

Or thus vvithout ayde of numbers, admit AB the Octaedrons side giuen, thereon (makyng C the medietie a center) I describe the semicircle ADB, and from C, I erect the perpendicular CD, dravving lines from D to AB, the semidiameter CB, I diuide in 3 equall partes at E and I, vppon I, I errere the perpendicular IK, concurring vvith the circumference in K, from E, I dravv EM paralele to DB, cutting AD in M, as vvas taughte in the firste booke of this treatise: againe by the first probleme I diuide AB by extreme and mean proportion in F, and thervpon I raise the perpendicular FG, cros∣sing AD in G, and from F to D, I extend a streyght line▪ finally I dravve the ryght lyne KB, cuttyng of a thirde parte at H, and thus is the fygure fully fynyshed, contaynyng all these bodies, sydes, Diameters, and Axes: For AE is Tetraedrons syde, KB hys Semidiameter, HB his axis, AM

the cubes side, KB his semidiameter, MD his axes, AG Icosaedrons side, F D his semidiameter, and KB his axes.

[illustration]

Theoremes of Octaedrons inscribed regular bodyes. 1.

THe conteynyng Octaedrons side beareth proportion to the side of his con∣teyned tetraedron, as 3 to 2.

The seconde theoreme.

Octaedrons side, to his inscribed cubes side, beareth the proportion of √{powerof2}4 ½ vnto 1.

The third Theoreme.

The side of tetraedron, to the side of the cube, is as 2 vnto √{powerof2}2.

The fourth Theoreme.

Octaedrons conteyning dimetient, is triple in power to his inscribed Icosae∣drons axis.

The fifth theoreme.

The conteyaing octaedrons diameter is triple to the conteyned cubes side.

The sixte Theoreme.

The axes of these 3 inscribed bodies are proportionall, and the cubes axes it the meane proportion betwene the other two.

The seuenth Theoreme.

The side of Octaedron diuided by extreme and mean proportion, that quan∣titie which is double in power to the lesser part, is equall to the inscribed Icosae∣drons side.

The eight theoreme.

Icosaedrons diameter is equall in power to the side of the conteyning Octae∣dron, and the d••fference or excesse of Octaedrons sides partes being deuided in extreme and meane proportion.

The ninth Theoreme.

Octaedrons dimetient diuided by extreame and meane proportion maketh his lesser segment the inscribed Icosaedrons side.

The tenth theoreme.

Octaedrons side being rationall, Icosaedrons side is an Apotome proportioned to the Octaedrons side, as √{powerof2} vniuers. 7—√{powerof2}45 vnto an vnitie.

The .21. Probleme. Icosaedrons side geuen, in line or number to set forth all the sides, Diameters and Axes of his contained regulare bodyes. (Book 21)

ICosaedron is a bodye of suche vniforme capacitie, that he recei∣ueth all the other foure regulare bodyes, whereof Tetraedron, the Cube, and Dodecaedron, haue their solide angles al resting in the centers of his bases, and Octaedrons angles are situate in the medieties or middle sections of Icosaedrons opposite sides, and his thrée diameters crosse them selues at right angles vpō the center of his spheres. Now for the inuention of their sides, Diameters, and Axes, note the rules ensuing.

Of Tetraedron.

DIuide the geuen side by extreame and meane proportion, adioyning therunto his greater part the producte square, and from it subtracte the third part of the square of the geuen side, the roote Quadrate of the remain∣der is the Tetraedrons containing spheres Diameter, and the third part of his square doubled is the square of the Tetraedrons side, whose third parte deducted fro the fourth part of the Diameters square, leaueth the square of the Axis, or diuide the Dimetient by sixe, so haue ye the Axis also.

Of Octaedron.

THe side of Icosaedron geuen, being parted by extreame and meane pro∣portion, hauing his greater parte to him adioyned, yeldeth the Octae∣drons dimetient, whose squares medieties roote is the side, and the thirde parte of his sides square abated from the square of the Semidimetient, lea∣ueth the square of his Axis.

Of Hexaedron.

THe Cubes Diameter is equall to the Diameter of Tetraedron, which found as before, ye shal diuide the square therof by 3 the quotientes roote Quadrate is the Cubes side, whose medietie is his Axis.

The inscri∣bed regular Solides lines.

• ...AB The comprehending Icosaedrons side geuen 1
• EB The comprehending Icos. Diameter √{powerof2} v 5/2+√{powerof2}5/4
• SK The comprehending Icosaedrons Axis, √{powerof2} v. 7/24+√{powerof2}5/64
• CP Tetraedrons side, √{powerof2} v. 7/9+√{powerof2}5/9
• CK Tetraedrons Diameter, √{powerof2} v. 1 ⅙+√{powerof2}¼
• SN Tetraedrons Axis, √{powerof2} v. 7/216+√{powerof2}5/5184
• CR Hexaedrons side, √{powerof2} v. 7/18+√{powerof2}5/36
• CK Hexaedrons Dimetient, √{powerof2} v. 1 ⅙+√{powerof2}1 ¼
• CT Hexaedrons Axis √{powerof2} v. 7/72+√{powerof2}5/576
• CF Octaedrons side, √{powerof2} v. ¾+√{powerof2}5/16
• CB Octaedrons Dimetient, √{powerof2}5/4+½
• CT Octaedrons Axis √{powerof2} v. ⅛—√{powerof2}25/2880
• CM Dodecaedrons side √{powerof2}5/36+⅙
• CK Dodecaedrons Diameter √{powerof2} v. ⅙+√{powerof2}5/4
• VZ dodecaedrōs axis, √{powerof2} v. ⅛+√{powerof2}1/320+√{powerof2} V.D. 1/18√{powerof2}1/102••

Theoremes of Icosaedrons inscribed bodyes. Theoreme first.

THe containing Icosaedrons dimetient is in power equall with his side and the diameter of his inscribed Octaedron.

The second theoreme.

The inscribed Octaedrons dimetient, is in power equal to the dimetient of the contained Cube and the diameter of Icosaedrons contained circle.

The third theoreme.

Octaedrons dimetient, (diuided by extreme and meane proportion) his grea∣ter segment is the comprehending Icosaedrons side.

The fourth theoreme.

Icosaedrons dimetient in power is equall with the diameter of his containing circle, and the dimetient of his contained cube.

The fifth theoreme.

Tetraedrons side, is double in power to the side of the contained cube.

The sixth theoreme.

Tetraedrons diameter, is triple in power to the inscribed Cubes side.

The seuenth theoreme.

Dodecaedrons inscribed side is the third part of such a line as diuided by ex∣treame

and meane proportion maketh his greater parte the contayning Icosae∣drons side.

The eyght Theoreme.

Octaedrons diameter retayneth the same proportion to the side of Dodecaedrō that the diameter of Tetraedron doth in power to the side of the Cube.

The ninthe Theoreme.

Icosaedrons diameter retayneth the same proportion to the dimetient of Octa∣dron, that the side of Icosaedron bereth to the semidiameter of that circle, wher∣on Icosaedron is framed.

The tenth Theoreme.

Icosaedrons axis is triple in power to the axis of the contayned Cube.

The .22. Probleme. The side of any Dodecaedron giuen, both Arithmeticallye and Geo∣metrically to serche out the sides diameters and axis of all the re∣gulare bodies therin described. (Book 22)

SUch is the resemblance and mutual conformitie betwene Icosaedron and this body, that the 12 solide angles of Ico∣saedron will precisely rest in the centers of Dodecaedrons pentagonall bases, the angles of the inscribed Octaedron haue their place in the medietie of the sixe opposite sides of Dodecaedrō, which coopled togither with right lines make 12 sides contayning his 8 triangles, and 3 diameters, crossing themselues at right angles on the center of his spheres. But the solide angles of the internall Cube and Tetraedron are situate in the angles of the contay∣ning Dodecaedron, and one sphere comprehendeth them all thrée. As for the longitude of their axes, sides and diameters, peruse these preceptes folowing.

Of Tetraedron.

DIuide the side giuen by the firste probleme into extreame and meane proportion, and to the whole side adioyne his greater part, the square of the resulting quantitie tripled maketh the Tetraedrons dimetientes square, whiche augmented by 2 and diuided by 3 bringeth the square of Tetraedrons side, which diuided by 24 yeldeth in the quotient the square of Tetraedrons Axis: extracte the rootes quadrate of these squares, so haue ye the desired lines.

Of Hexaedron.

DIuide Tetraedrons sides square by 2, so haue ye the square of the Cu∣bes side, which augmented by 3 sheweth the diameters square, and that diuided by 12 produceth the axis square, extracte the Zenzike rootes of these numbers, so haue ye the lines.

Of Octaedron.

DOdecaedrons side giuen added to the side of Hexaedron late foūd, ma∣keth the diameter of Octaedron, the roote quadrate of his squares me∣dietie is the side, and the roote quadrate of the dimetientes twelfth parte is the axis.

Of Icosaedron.

THe side of Dodecaedron béeing giuen, first search the diameter of his containing circle by the fifth & tenth problemes, and deduct the square therof from the square of the Cubes dimetiente founde as is before decla∣red, the roote quadrate of the remaine is the Icosaedrons dimetient: Now for his side ye shall adde the squares of the Cubes side and Dodecaedrons sides together, the roote vniuersall of the producte ye shall reserue for a diuisor, then multiplye Icosaedrons dimetiente by Dodecaedrons side, and the product diuide by your reserued diuisor, the quotient is the Ico∣saedrons side, the square of this side deducte from triple the square of Ico∣saedrons semidimetiente, and fro the thirde parte of the remaynder ex∣tracte the Zenzike roote, for that is the Icosaedrons Axis.

Bicause these rules of them selues are apparaunt ynough, I shal only adioyne an example for the last, with a table contayning the numbers redy calculate of all the rest, which shall supplye the place of examples for the other.

An example of Icosaedron.

THe Dodecaedrons side giuen 20 diuided by extreame and meane pro∣portion maketh his greater parte √{powerof2}500—10, whiche added vnto 20, is √{powerof2}500+10, the square thereof tripled is 1800+√{powerof2}1800000, so is the square of the Cubes dimetient, and from it deducting 800+√{powerof2}128000 Dodecaedrons basis contayning circles diameters square, there wil remayne 1000+√{powerof2}968000, the roote quadrate vniuersall therof is the diameter of the contayned Icosaedron, which multiplied by the Dodecaedrons side cre∣ateth √{powerof2} v. 400000+√{powerof2}154880000000, and that diuided by √{powerof2} v. 1000+√{powerof2}200000 (for so much ariseth for a diuisor by additiō of the Cubes sides square to the square of Dodecaedrons side) there will resulte of that diui∣sion

√{powerof2} v. 280+√{powerof2}72000, so muche is the contayned Icosaedrons side, whose square deducted fro triple the square of Icosaedrons semidiameter, 17600/80+√{powerof2}70560000/320, whose thirde partes roote is √{powerof2} v. 1880/12+√{powerof2}102000/16 so muche is the contayned Icosaedrons axis, and that doubled sheweth the quantitie of his contayned spheres dimetiente.

The containing Dodecaedrons side 20.

• ...Dodecaedrons
  • Diameter √{powerof2} v. 1800+√{powerof2}1800000
  • Axis √{powerof2} v. 250+√{powerof2}60500.
• ...The contayned Tetraedrons
  • Syde √{powerof2} v. 1200+√{powerof2}800000
  • Diameter √{powerof2} v. 1800+√{powerof2}1800000
  • Axis √{powerof2} v. 50+√{powerof2}1388 8/9
• ...The inscribed Hexaedrons
  • Syde √{powerof2} v. 600+√{powerof2}200000
  • Diameter √{powerof2} v. 1800+√{powerof2}1800000
  • Axis √{powerof2} v. ••50+√{powerof2}12500
• ...The included Octaedrons
  • Syde √{powerof2} v. 700+√{powerof2}450000
  • Diameter √{powerof2} v. 1400+√{powerof2}1800000
  • Axis √{powerof2} v. 116 ⅔+√{powerof2}12500
• ...The internall Icosaedrons
  • Syde √{powerof2} v. 280+√{powerof2}72000
  • Diameter √{powerof2} v. 1000+√{powerof2}968000
  • Axis √{powerof2} v. 156 ⅔+√{powerof2}24500

For the geometricall searching out of all these lines, beholde the figure folowing, where AB is the Dodecaedrons side giuen, BC such a line as be∣ing diuided into extreame and meane proportion maketh his greater parte equall to BA, and therfore the inscribed Cubes side AC is the inscribed Octaedrons diameter, and AFC a semicircle described thereon; the centre beeing E, DC is equall to AB, and perpendiculare to AC, AD is the dia∣meter of Dodecaedron, the contayned Cube, and Tetraedron, AGD a se∣micircle described vpon his medietie the centre N, BG is an arke described vpon the centre D, GH is a perpendiculare let fall on AD, HK is the fifth part of HD, KI an other perpendiculare ereared on that poynt cut∣ting the semicircle in I, AI is the diameter of the inscribed Icosaedron, and AVI a semicircle theron described, BL equall to AI, LM a parallele to DC,

falling perpendicularly vpon AC extended to M, AV is equall to LM, it is the contayned Icosaedrons side, VX a perpendiculare from V to AI, XY is a thirde parte of AX, YZ a perpendiculare to AI, cutting the semicircle at Z, ZI is the contayned Icosaedrons inscribed spheres diameter, WI his medietie is the Icosaedrons axis, AO is a thirde parte of AD, OP a per∣pendiculare vpon AD, PD the inscribed Tetraedrons side, PA the Cubes side, OR a parallele to DC, RC the diameter of the inscribed Octaedrons contayning circle, S his middle poynt, ES Octaedrons Axis, EF a perpen∣diculare ereared on the diameter, AC cutting his semicircles circumference in F, AF Octaedrons side, TW is the semidiameter of Icosaedrons con∣tayning circle, TN the semidiameter of Dodecaedrons contayning circle, OP the semidimetient of Tetraedrons base, and DQ the circles semidia∣meter that contayneth the Cubes square. Thus haue ye in one figure all the sides, diameters, and axes, bothe of the contayning Dodecaedron, and his inscribed Tetraedron, Octaedron, Hexaedron, and Icosaedron, with the se∣midiameters also of the circles inuironing their bases, but lest the intricate crossing and concurring of lines and arkes might breade confusion to suche as haue not ben trayned in geometricall demonstration. I shall adioyne a ta∣ble contayning the sides, axes, and diameters, both circulare and spherall of euery body perticularly by it selfe, expressed also exactly with numbers and Algebraicall charecters, the contayning Dodecaedrons side admitted an v∣nitie wherein the skilfull Arithmetrician shall finde matter abundante to exercise or delight himselfe.

[illustration]

The compre∣hending Do∣decaedrons

• Syde AB, 1
• Basis line diagonall, √{powerof2}1 ½+√{powerof2}5/2 BC.
• Contayning spheres diameter, √{powerof2} v. 4 ½+√{powerof2}11 ••/•• AD.
• Basis contayning circles Semid. √{powerof2} v. ½+√{powerof2}1/20 TN.
• Axis, √{powerof2} vni. ⅝+√{powerof2}121/320, AT.

Dodecae∣drons in∣scribed re∣gulare

• ...Tetrae∣drons
  • Syde √{powerof2} vni. 3+√{powerof2}5 DP.
  • Diameter, √{powerof2} vni. 4 ••/2+√{powerof2}11 ••/4, AD.
  • Basis greater semid. √{powerof2} v. 1+√{powerof2}5/9 OP.
  • Axis, √{powerof2} vni. ⅛+√{powerof2}5/••76, ON.
  • Perpendiculare √{powerof2} vni. 2+√{powerof2}2 2/9 OD.
• ...Octae∣drons
  • Syde, √{powerof2} v. 1 ¾+√{powerof2}••5/16, AF.
  • Diameter √{powerof2} vni. 7/2+√{powerof2}45/4, AC.
  • Basis greater semid. √{powerof2} v. 7/12+√{powerof2}45/144, RS.
  • Axis, √{powerof2} v. 7/24+√{powerof2}5/64, ES.
• ...Hexae∣drons
  • Syde √{powerof2} vni. 3/2+√{powerof2}5/4, AP.
  • Diameter, √{powerof2} vni. 4 ½+√{powerof2}11 ¼, AD.
  • Basis diagonall, √{powerof2} vni. 3+√{powerof2}5. PD.
  • Contayning circles semid √{powerof2} v. ¾+√{powerof2}5/16, DQ.
  • Axis, √{powerof2} vni. ⅜+√{powerof2}5/64 NQ.
• ...Icosae∣drons
  • Syde, √{powerof2} vni. 7/10+√{powerof2}9/20, LM.
  • Dimetient, √{powerof2} v. 2 ½+√{powerof2}121/20, AI.
  • Basis cont. circles semid. √{powerof2} v. 7/30+√{powerof2}1/20, Tw.
  • Axis, √{powerof2} v. 47/120+√{powerof2}49/320, wI.
  • The Circles Semidiameter whereon Icos. is framed▪ √{powerof2} v. ½+√121/500 VX.

Theoremes of Dodecaedrons inscribed regulare Solides.

TEtraedrons inscribed sides square ioyned to the Cubes sides square maketh the diameters square of Hexaedron, all beeing described within one Dode∣caedron.

The 2 Theoreme.

Tetraedrons side is double in power to the cubes side, and equal to his line dia∣gonall.

The 3 Theoreme.

Dodecaedrons Diameter is equall in power to the dimetiente of Octaedron and Dodecaedrons side.

The 4 Theoreme.

Icosaedrons dimetiente is equall in power to his owne side, and suche an other line also (as beeing by extreame and meane proportion diuided) maketh his greater parte equall to Icosaedrons side.

The 5 theoreme.

A line equall in power to Dodecaedron and his inscribed Cubes side, hathe such proportion to Icosaedrons diameter, as Dodecaedrons side, hath to the side of his contayned Icosaedron.

The 6 theoreme.

Octaedrons diameter diuided by extreame and meane proportion maketh his greater parte the Cubes side, and his lesser the side of Dodecaedron.

The 7 theoreme.

Hexaedrons sides square is equall to the square of Dodecaedrons side, and a rightangled Parallelogramme contayned of the Cubes side and Dodecaedrons side.

The 8 theoreme.

Octaedrons diameters square, is equall to the square of Tetraedrons side, and a right angled Parallelogramme contayned of Dodecaedrons side and his inscribed Cubes side.

The 9 theoreme.

Octaedrons diameter being by extreame and meane proportion diuided, the excesse or difference of these partes squares deducted fro the square of the dia∣meter, leueth the square of the inscribed Tetraedrons side.

The 10 theoreme.

Hexaedrons diameter is equall in power to Dodecaedrons Basis contayning circles diameter of the inscribed Icosaedron.

The 11 theoreme.

The excesse of Dodecaedrons basis contayning circles diameters square a∣boue the square of his line Diagonall, added vnto his contayned Spheres dia∣meter, createth the square of his inscribed Tetraedrons side.

The 12 theoreme.

Dodecaedrons dimetient is equall in power with these 3 lines, the greater dia∣meter of his owne basis, the diameter of Icosaedrons contayned sphere, and the greater diameter of Icosaedrons basis.

The 13 theoreme.

Octaedrons dimetiente retayneth the same proportion to the diameter of hys basis, that Dodecaedrons diameter doth to the side of his inscribed Tetraedron.

The 14 theoreme.

Octaedrons contayning circles diameter retayneth the same proportion to his contayned spheres dimetiente, that Tetraedrons side dothe to the side of the Cube.

The 15 theoreme.

Dodecaedrons side being rationall, the side of his contayned Icosaedron is an irrationall Binomie of the 3 order, bearing proportion to the contayning dode∣caedrons side, as √{powerof2}9/20+√{powerof2}5/20 to an vnitie.

The 16 theoreme.

Octaedrons diameter, to the side of Icosaedron, is proportioned in power, as 5 to 1.

The 17 theoreme.

Octaedrons diameter beareth such proportion to the dimetient of Icosaedron. as a Corde Pentagonall doth to a Corde Hexagonall of the same circle.

The 18 theoreme.

Icosaedrons diameter diuided by extreame and meane proportion maketh his greater parte the diameter of his comprehending dodecaedrons Basis internall Circle.

The 19 theoreme.

Octaedrons dimetient exceedeth in power the dimetiente of Icosaedron by the diameter of Dodecaedrons basis internall circle.

The 20 theoreme.

A line in power equall to the sides of Dodecaedron and the Cube to the dia∣meter of Icosaedron, retayneth such proportion as the lesse segmente of Dodecae∣drons basis perpendiculare (diuided in extreame and meane proportion) doth to the lesse semidiameter of the same basis.

The 21 Theoreme.

Icosaedrons side holdeth the same proportion to Octaedrons dimetiente, that the semidiameter of Dodecaedrons contayned circle doth to the perpendiculare of his base.

The 22 Theoreme.

If any line be diuided in extreame and meane proportiō, and from the whole ye abate the medietie of the greater parte, the remayne beeing by extreame and meane proportion agayne diuided, the lesser segmente of this laste diuided line, to the medietie first abated, retayneth such proportion, as Dodecaedrons side, to the side of his inscribed Icosaedron.

The 23 Theoreme.

If Icosaedrons side doubled be diuided in extreame and meane Proportion, and to the greater parte the Icosaedrons side adioyned, the whole line thereof re∣sulting, is the diameter of the inscribed Octaedron.

The 24 theoreme.

The contayned Icosaedrons semidiameter to the lesse Semidimetiente of the comprehending Dodecaedrons base, retayneth such proportion as Dodecaedrons basis line Diagonall to the side.

The 25 theoreme.

Icosaedrons inscribed diameter to the dimetiente of Dodecaedrons contay∣ning circle, beareth such proportion, as the medietie of Dodecaedrons basis dia∣gonall, to the greater portion of his side by extreame and meane Proportion di∣uided.

The .23. Probleme. The side, Diameter or Axis of any regulare bodye knovven, to searche out all those forenamed lines in any regulare bodye that shall include or circumscribe that proponed solide. (Book 23)

ALthough this question of circumscribing bodyes mighte be diuided into 5 Chapters, and in euery one as many dif∣ferente rules, preceptes, and Theoremes taught, as was in the former of bodyes inscribed, yet for breuitie sake, I thinke beste to remitte the more ample handeling here∣of to the ingeniouse studente, who comparing the rules, and well way∣ing the Theoremes alreadie giuen, shall easely applie them to this purpose, and inuente many m••e perhappes of greater facilitie and no lesse certaintie: and in this Chapter I will only open one way, leauing a large fielde for others to inuent and exercise them selues in at plea∣sure. It shall therefore be requisite when the side, Diameter or Axis of any regulare bodye is proponed, to consider by the fifte Problemes

last past what bodies may conteyne or circumscribe the same, and resor∣ting to theyr peculiar chapters, serche out the numbers appropriate to the conteyning and conteyned solides sides, axes and diameters, which found by thrée quantities knowne, vsing the rule of proportion, ye may readi∣ly finde the fourth, as by the example shall more plainly appere.

Example.

Suppose the side of a tetraedron giuen 10, for the sides, diameters and axes I repaire first to the 19 probleme, vvhere I find the conteining cubes side, be∣ing 1, the conteined tetraedrons side √{powerof2}2, saying therfore by the rule of pro∣portion √{powerof2}2, the conteined tetraedrons side foūd heretofore in the 19 pro∣bleme, giueth 1 for his conteining cubes side, vvhat yeldeth 10 the side giuen your fourth proportional number vvill be √{powerof2}50, the containing cubes side. Likevvise in the 22 probleme, I find the conteining dodecaedrons side being 1, the cōtained tetraedrons side √{powerof2} v. 3+√{powerof2}5. augmenting therfore 10, the side giuen by tetraedrons side in that probleme found, and diuiding by √{powerof2} vni. 3+√5, your quotient vvill be √{powerof2}62 ½—√{powerof2}12 ½, so mutch conclude the side of a dodecaedron that shal conteyn or comprehend this tetraedron, vvhose side is 10. In like maner may ye serche out the other sides, diameters, and axes, of all the comprehending bodies, vvhereof I leaue to giue any far∣ther examples, these tvvo being sufficient to the ingenious to proceede vvith lyke order in the rest.

But for sutch as not contented with one kinde of working wil delighte themselues in the diuersitie of rules and kyndes of calculation, I haue thought good to adioyne these Theoremes ensuing, which wel wayed and compared with sutch as are already past, shal yeld matter abundantly for the inuention of many mo conclusions and strange operations, than hither∣to hath ben vsed or published by any.

Theoremes of these bodies mutually circumscribed and confer∣red vvith their inscribed regular bodies. 1.

TEtraedron may be conteyned or circumscribed of all the other foure regular bodies, and his side being rationall, his containing Octaedrons side is also ra∣tionall, proportioned thervnto, as 3 to 2.

The 2 theoreme.

Tetraedrons comprehending cubes side is equall to the dimetient of his in∣scribed Octaedron.

The 3 theoreme.

Tetraedrons conteyning Octaedrons side to the side of his inscribed Octae∣dron is triple.

The 4 Theoreme.

Tetradrons side being rationall, his encompassing Icosaedrons side is an Apo∣tome, and triple to tetraedrons inscribed Icosaedrons side.

The 5 theoreme.

Tetraedrons side being rationall, his circumscribing dodecaedrons side is an Apotome of the 6 order, proportioned to the side as √{powerof2} vni. ¾—√{powerof2}5/16 vn∣to an vnitie.

The 6 Theoreme.

Hexaedron hath only 3 circumscribing regular bodies, for no tetraedron may be so placed about a cube but that his superficies shal either cut or not touch some of the Hexaedrons angles.

The 7 Theoreme.

Hexaedrons conteyning Octaedrons side is triple to his conteyned Octae∣drons side, and to the Hexaedrons side it beareth sutche proportion, as tetrae∣drons diameter to his containyng circles semidimetient.

The 8 theoreme.

Hexaedrons side being rationall, his comprehending Icosaedrons side, is an irrationall Minor, proportioned to the cubes side, as √{powerof2} v. 31 ½√{powerof2}911 ¼ vnto 1.

The 9 theoreme.

Hexaedrons side being rational, his comprehending Dodecaedrons side is an Apotome of the sixt he order proportioned to the side as √{powerof2} v. 1 ½—√{powerof2}¼ to an vnity.

The 10 Theoreme.

Hexaedrons Dimetient is meane proportionall betwene his side and the Di∣metiente of his contayninge Octaedron.

The 11 theoreme.

Octaedron may be comprehended of all the other regular bodies, and his side is the medietie of his encompassing tetraedrons side.

The 12 Theoreme.

Octaedrons externall cubes side is equall to his diameter, and double in po∣wer to his side.

The 13 Theoreme.

Octaedrons side being rationall, his including Icosaedrons side is an Apoto∣me of the 6 order, hauing proportion to Octaedrons side, as √{powerof2} vniuer. 3—√5 vnto 1.

The 14 theoreme.

Octaedrons side being rationall, his enuironing dodecaedrons side is an Apo∣tome,

proportioned to the Octaedrons side, as √{powerof2} vni. 7—√{powerof2}45 vnto 1.

The 15 theoreme.

Octaedrons comprehending Dodecaedrons side is equall to his conteyned I∣cosaedrons side.

The 16 Theoreme.

Icosaedron may be comprehended perfectely of all the other regulare solides, and his contayning tetraedrons side is triple to the side of his contained tetrae∣dron.

The 17 theoreme.

Icosaedrons side being rationall, his comprehendyng cubes side is a Binomy, and the Icosaedrons Diameter in power equall to them bothe.

The 18 theoreme.

Icosaedrons side rational his encompassing Octaedrons side is a Binomye and equall to the medietie of his externall Tetraedrons side.

The 19 theoreme.

Icosaedrons side rationall, his comprehending dodecaedrons side is an irrati∣onal Apotome, bearing proportion to the side as √{powerof2} v 17 ½—√{powerof2}281 ¼ vn∣to 1.

The 20 theoreme.

Icosaedrons comprehendyng cubes side is double in power to his compre∣hended Octaedrons side.

The 21 theoreme.

Dodecaedron can perfectly be comprehended of no regulare bodie saue onely of Icosaedron, whose side reteyneth sutche proportion to the side of Dodecae∣dron, as √{powerof2} vni. 13 ½—√101 ¼ vnto an vnitie.

The 22 theoreme.

Dodecaedron and Icosaedron hauyng equall sides, the cubes side that con∣teyneth Icosaedron is equall to the cubes side conteyned of dodecaedron.

The 23 theoreme.

Dodecaedron and octaedron hauing equal and rationall sides, the side of octa∣drons comprehending Icosaedron is an Apotome of the same order, that Dode∣caedrons inscribed Tetraedrons side is a Binomye, and their names or compo∣ning quantities equall.

The 24 theoreme.

Dodecaedron and the cube hauyng equall sides, whether they be rationall or surd, the dodecaedrons conteyning Icosaedrons side is triple to the cubes com∣prehending dodecaedrons side.

The 25 theoreme.

Dodecaedron and Tetraedron hauyng equall Sydes, whether they

be rationall or irrationall, the dodecaedrons conteyning Icosaedrons side to the Tetraedrons comprehending Dodecaedrons side, reteyneth sutch proportion, as Tetraedrons diameter doth to the lesse semidiameter of his basis.

The .24. Probleme. The side, diameter, Axis, or altitude, of any regular body, or any se∣midiameter, perpendicular or line diagonall of their base giuen, to search out the content Superficiall and solide, not on∣ly of that body, but also of any other regular so∣lide that shall inscribe or containe that body or any of his spheres. (Book 24)

FOr resolution of this Probleme I might prescribe rules in∣numerable of moste straunge and intricate operation, sutch is the varietie of proportions betwéene these bodies sides, di∣metientes, axes, and other their superficiall and solide lynes, that if I should sette foorthe for euery peculiar question that might herin be proponed, but the tenth part of sutch precepts as offer themselues vpon consideration of these solides formes, proportions & nature, this only chapter shuld grow to an huge treatise. But to conclude it in shortest forme and fewest wordes, the whole varietie may be reduced to these principall poyntes, what manner line is proponed or giuen, and what content or capacitie is demaunded. If the line giuen be none of the bodies sides, but some diameter, axis, perpendicular or other lyne before named, ye haue in the 6 probleme and those other foure that immediatly ensue, the proportion of al sutch lines to the solides sides in rationall and irrationall numbres expressed, so that by conuersion of the proportions there founde, vsing the rule of proportion (as hathe bene partely before declared, and shall in thys Chapiter by example be somwhat more plain∣ly shewed) yée may finde out the correspondent syde of that solide, whose diameter, axis or other lyne is giuen. Then if it bée the content superficial or solide of the same body that is required, yée shall resorte to the 10 pro∣bleme, and the other foure followyng, where yée shall fynde in the correspondente Chapiter by the syde tofore knowne, the meanes howe to searche his forenamed contentes.

But if those contents be not demaunded of the same body, they are either of bodyes inscribed or circumscribed, and that either of the proponed regu∣lare solide, or some of his spheres, for bodyes described within his spheres, it behoueth ye first by the side knowne to learne the Diameter of that Sphere, and then repaire vnto the sixtenth Probleme, where ye shall be taught to finde all the inscribed Solides sides, but if it be an inscribed body of your proponed Solide, then haue ye preceptes for the inuention of his side in the eightenth Probleme, or some other of the foure immediately insuing. Likewise if it be the content of a circumscribing bodye that is re∣quired, that body doth either circumscribe some Sphere, and then haue ye Precepts in the seuententh Probleme for the inuention of his side, or else it containeth the Solide, whose side is knowne, and then may ye by the rules geuen in the last Probleme search out his side, and the side of euery Solide so found (resorting to the sixthe, seuenthe, eigth, ninthe and tenthe Problemes) ye shall there in his answerable Chapiter, receiue rules for the inuention of his capacitie superficiall and Solide, or else worke for e∣uery seuerall body according to the rules ensuing.

Rules for the content Superficiall and Solide of the fiue Regular bodyes.

For Tetraedron.

AVgment the square of the side founde by √{powerof2}3, and the Cube of the side by √{powerof2}1/••2, the first producte is the Superficies, the seconde is the Solide content of Tetraedron.

For Octaedron.

MVltiplie the sides square by √{powerof2}12, and the sides cube by √{powerof2}2/9, so haue ye tvvo productes equall to the tvvo contentes Solide and Superficiall of Octaedron.

For Hexaedron.

THe square of the side augmented by 6, geueth his Superficies, vvhiche augmented by ⅙ of the side, produceth the crassitude.

For Icosaedron.

INcrease the sides squared square by 75, the roote Zenzike of the producte is the Superficies, and the Cube of the side augmented by √{powerof2}7.2 31/••2+√{powerof2}5 245/576 yeldeth the crassitude.

For Dodecaedron.

MVltiplie the sides square by √{powerof2} v. 225+√40500, so haue ye the superfi∣cies, or augment the sides Cube by √{powerof2} v. 29 ⅜ +√{powerof2}861 21/64, the product is the crassitude.

For more plainnesse I shal adioyn v. questions and examples of these prin∣cipal varieties, vvherunto al other peculiar questions may be referred.

The fyrst question. I haue a Dodecaedron, vvhose comprehending spheres diameter I knovv to be fiue, I demaund his ca∣pacitie superficial and solide.

COnsidering the Dimetient of this solide is knowne vnto me, I repaire vnto the tenth Probleme, where I finde the Dodecae∣drons side being 10, the comprehending spheres diameter √{powerof2} v. 450+√{powerof2}112500, conuerting therefore the limites or boundes of this proportion, I say thus √{powerof2} v. 450+√{powerof2}112500 geueth 10, what yeldeth 5, working by the rule of proportion, ye shall finde the fourthe proportionall number √{powerof2}10 5/12—√{powerof2}2 1/12, so much is the side of the proponed Dodecae∣dron, which knowne I resort to the fiftenth Probleme, there am I willed to searche out the Semidiameter and line Diagonall of his basis, and the Axis or Semidimetient of his inscribed sphere, his Axis is √{powerof2} v. 2 1/12+√{powerof2}3 17/36 his cōtaining circles semidimetient √{powerof2} v. 4 ⅙ —√{powerof2}3 17/36, his basis line Diagonall is √{powerof2}8 ⅓, so that by multiplication and diuision of these num∣bers according to the Precepts there geuen, I finde the Dodecaedrons su∣perficies √{powerof2} v. 7812 ½—√1220703•• ¼ and his crassitude √{powerof2} vn 1808 97/216+√{powerof2}654097 28403/46656—√{powerof2}523278 1033/11664. In like manner if ye augment 12 ½—√{powerof2}86 29/3•• the square of the Dodecaedrons founde side, in 225+√{powerof2}40500. Or √{powerof2} v. 3038 7/36+√4709502 1033/1296 √{powerof2}8477105 5/144—√{powerof2}5128125 2125/11664 be∣ing the Cube of the same side, in √{powerof2} v. 29 ⅜+√{powerof2} v 861 1/64, for so are ye taught to doe by the rule of this Chapiter, the first Producte will be the Dodecae∣drons superficies, and the last his crassitude, exactly agréeing with your for∣mer operations.

The second question. A cube is proponed, vvhose Diameter is the Zenzike roote of 108▪ I vvoulde knovve the superficiall and solide con∣tentes of such a Tetraedron as this Cubes con∣tayned sphere should circumscribe.

BIcause the diameter of this Cube is geuē, I resort to the sixtenth Probleme, by the rules ther prescribed, I finde the Axis 3, which doubled maketh 6, the Diameter of this cubes contained sphere, and considering the Tetraedron, whose capacitie is required, must be circumscribed of this sphere, I search out againe by the same cha∣piter this spheres inscribed Tetraedrons side, finding it the zenzike roote of 24 ye may also thereby learne the Axis and containing circles semidiame∣ter, and so consequently the capacities of this Tetraedron, wherof ye haue examples in the eleuenth Probleme: Or by the first rule of this Probleme if ye augment 24 the square of Tetraedrons side by √{powerof2}3 there ariseth √{powerof2}1728 for his superficies, and by multiplication of √{powerof2}13824 the Cube of this inscribed Tetraedrons side by √{powerof2}12/72, there amounteth √{powerof2}192, so muche conclude the Tetraedrons crassitude,

The thirde question. I demaund the superficiall and Solide capacitie of a Dodecaedron, circumscribing such an Octaedrons containing sphere, as hath for his side this irrationall Maior √{powerof2} v. 20+√{powerof2}387 ⅕.

BY the eigth Probleme I finde this Octaedrons cōprehending spheres dimetient √{powerof2} v. 46+√{powerof2}7744/5 the medietie hereof being √{powerof2} vniu. 10+√{powerof2}484/5 is the Axis of the Dodecaedron, whose cōtents are required, which knowne by the rules of the seuententh Probleme, I finde the side √{powerof2} v. 500—√{powerof2}234256, and by extraction of this irrationall numbers roote, it appeareth 4, whose square augmēted by the roote Ʒenzike vniuer∣sal of 2••5+√{powerof2}40500 produceth √{powerof2} v. 57600+√{powerof2}2654208000, so muche is the Dodecaedrons superficies. Likewise by multiplication of √{powerof2} vniu. 275/8 +√{powerof2}••••125/64 in the Cube of 4 there ariseth √{powerof2} v. 962560/8+√{powerof2}92484403200••/64 for the crassitude of this Dodecaedron.

The fourth question. There is a Dodecaedron, vvhose side is this irrationall Apotome √{powerof2} v. 1120—√{powerof2}1152000, my desire is to knovve his inscri∣bed Icosaedrons superficies and crassitude.

FOrasmuch as this Icosaedron whose contentes are required, is described within the Dodecaedren, whose side is knowne, I search out by the rules geuen in the xxij. Chapiter, the side of his contained Icosaedron. Or with more spéede ye may thus work, saying a containing Dodecaedrons side being an vnitie, geueth for the side of his contained Icosaedron √{powerof2} v 7/10+√{powerof2}9/20, what shall √{powerof2} vniuersalis 11200—√{powerof2}1152000, by multiplication of these latter numbers ye shal pro∣duce √{powerof2} v. 784—√{powerof2}518400, which ye shal finde (by extraction of the vni∣uersal ••enzike roote) 8, the side thus found, ye may search out his contents by the fiftenth Probleme, or by the fourth rule tofore prescribed thus, I augment 64 by √{powerof2}75 there amounteth √{powerof2}307200 the inscribed Icosae∣drons superficies. Or increasing 512 the side cubically multiplied by √{powerof2} v. 2 21/72+√{powerof2}5 245/576 there amounteth √{powerof2} v. 637155 5/9 +√{powerof2}372827022222 2/9 the de∣sired cassitude.

The fyfth question. An Icosaedron is offered, vvhose basis contayning circles semidimetient is this surde Binomye √{powerof2} vni. 14 14/15 +√{powerof2}204 ⅘ the superficies and crassitude of his contayning Dodecaedron is required.

COnsidering the semidiameter of this Icosaedrons basis is giuen, I repare to the 9 Probleme, wherein I receaue rules for the inuention of the comprehending Dodecaedrons side, or more particularly to set forth the whole operation, I am refer∣red to the 9 Probleme, where I perceaue by the 9 Theoreme, that the square of this knowen semidiamiter tripled, maketh the square of the Icosaedrons side: I augmente therefore the geuen Binomyes square by 3, there aryseth 44 ⅘ +√{powerof2}36864/20, the roote zenzike vniuersall therof is the Icosaedrons side. And for his contayning Dodecaedrons side I

worke by the precepts of the last Chapter, saying: √{powerof2} v. 2••0+√{powerof2}72000 giueth 20, what yeeldeth √{powerof2} vni. 44 ⅕+√{powerof2}36864/20 youre fourth Pro∣portionall number will be 8. Or thus by the fifteenth Theoreme of the 22 Probleme: Diuide √{powerof2} vni. 44 ⅘+√{powerof2}36••6••/20 by √{powerof2}9/20+√{powerof2}5/••••••, so will the quotiente bee 8 also, and that is the comprehending Dodecae∣drons side. Nowe for his contentes solide and superficiall, I repaire to the fiftenth Probleme, where I am referred to the 10 Probleme, and working by the preceptes there giuen, I finde the Dodecaedrons basis line Diagonall √{powerof2} v. 96+√{powerof2}5120, his basis greater Semidiameter √{powerof2} vni. 32+√{powerof2}4006/20▪ his Axis √{powerof2} vni 40+√{powerof2}495616/320, triple the Se∣midiameter is √{powerof2} vni. 288+√{powerof2}••2••776/20, the line Diagonall increased by 5, maketh √{powerof2} vni. 2400+√{powerof2}3200000, these multiplied togither, create this surd rote √{powerof2} v. 691200+√{powerof2}53084160000+√{powerof2}265420800000+√{powerof2}95552488000, resting betwene the Ʒenzike rootes of 1745904, and 1745905, being very nyghe 1321 ••63/2642 so much is the desired contente Su∣perficiall. Agayne by multiplication of √{powerof2} vni. 96+√{powerof2}5120 the line Diagonall, in √{powerof2} vni. 3••+√{powerof2}4••••6/20, there amounteth √{powerof2} v. 4096+√{powerof2}5242880+√{powerof2}1887436 ⅘, whiche increased by √{powerof2} vni. 40+√{powerof2}495616/320, the Axis produceth √{powerof2} vniuers. 163840+√{powerof2}••0305289240576/320+√{powerof2}66520453480448/3200, and this multiplied by 5, createth √{powerof2} vni 7700480+√{powerof2}59190018048000 the contayned Dodecaedrōs contente Solide. The like is brought to passe by the laste rules, for 64 the Dodecaedrons sides square augmented by √{powerof2} vni ••25+ √{powerof2}40500, maketh √{powerof2} v. 92160••+√{powerof2}679477248000 the superficies, and 512 the Cube of 8 multiplied in √{powerof2} vni. 29 ••/8+√{powerof2}••••125/64, createth √{powerof2} v. 7700480+√59190018048000 the Solide capacitie of a Dodecaedron described within the Icosaedron, whose side was giuen exactly agréeing with the former calculations.

Theoremes of the superficiall and solide quantities of these regulare bodyes conferred by mutuall inscription and circumscription, and firste:

Of Tetraedron.

TEtraedrons superficies is double to the superficies of his inscribed Octae∣dron.

The 2 theoreme.

Tetraedrons superficies to the superficies of his contayning Cube beareth pro∣portion, as 110√{powerof2}3.

The 3 theoreme.

Tetraedrons circumscribing Cubes contente superficiall, to the contente su∣perficiall of his contayned Octaedron, retayneth such proportion as the same Octaedrons superficies doth to the square of his side.

The 4 theoreme.

Tetraedrons incompassing Octaedrons superficies contayneth the superfi∣cies of his inscribed Octaedrons 9 tymes.

The 5 theoreme.

One base of Tetraedrons contayning Icosaedron, is equall vnto 9 of his in∣scribed Icosaedrons triangulare bases.

The 6 theoreme.

Tetraedron to his contayned Octaedron retayneth such proportion as their sides.

The 7 theoreme.

Tetraedrons contayning Octaedron to the Cube described without the same Tetraedron, retayneth such proportion as the perpendiculare of the contay∣ning Octaedrons basis to the lesse semidiameter of the Tetraedrons base.

The 8 theoreme.

Tetraedrons circumscribing Octaedron contayneth his inscribed Octaedron 27 times.

The 9 theoreme.

Tetraedrons contayning Cube holdeth the same proportion to Tetraedrons inscribed Octaedron, that the Cubes diameter doth to the Octaedrons Axis.

The 10 theoreme.

Tetraedrons comprehending Icosaedron, contayneth his inscribed Icosaedron 27 tymes.

Of Hexaedron.

The 11 theoreme.

HExaedrons superficies to the superficies of his inscribed Tetraedron, holdeth such proportion as Tetraedrons dimetiente doth to the Cubes side.

The 12 theoreme▪

Hexaedrons superficial•• contente to the con••ente superficiall of his inscribed Octaedron, retayneth the proportion of the Cubes diameter to Octaedrons Se∣midimetiente.

The 13 theoreme.

Hexaedrons contayning Octaedrons contente superficiall hath suche Pro∣portion to the superficiall quantitie of his inscribed Tetraedron, as the Octae∣drons contayned spheres dimetientes square to the greater semidiameters square of Tetraedrons base.

The 14 theoreme.

Hexaedrons comprehending Octaedrons superficies contayneth his inscri∣bed Octaedrons superficies •• tymes.

The 15 theoreme.

The proportion of Hexaedrons comprehending Dodecaedrons superficies▪ to the superficies of his inscribed Icosaedrō, is cōpounded of double, the proporti∣on betweene a Corde trigonall and a corde Pentagonall of one ••ircle, adioyned to the proportion of the Pentagonal, Diagonall to the same Corde Trigonall.

The 16 theoreme.

Hexaedrons solide capacitie to his inscribed Tetraedron, beareth suche pro∣portion as the Hexaedrons comprehending spheres semidimetiente dothe vnto the Tetraedrons Axis.

The 17 theoreme.

Hexaedron to his contayned Octaedron, retayneth the Proportion of his Diameter to the Octaedrons axis.

The 18 theoreme.

Hexaedrons comprehending Octaedron to his contayned Tetraedron, bea∣reth the proportion of 27 vnto 2.

The 19 theoreme.

Hexaedrons externall Octaedron contayneth his internall Octaedron 27 tymes.

The 20 theoreme.

Hexaedrons comprehending Dodecaedrons solide capacitie to the contente solide of his inscribed Icosaedron, retayneth the same Proportion that the line Diagonall of the Dodecaedrons Basis doth vnto that parte of his side where∣vnto the whole side is proportioned as the square of a Corde Trigonall to the square of a Corde Pentagonall.

Of Octaedron.

The 21 theoreme.

Octaedrons base doubled, is equall vnto 9 of his contayned Tetraedrons tri∣angular bases.

The 22 theoreme.

Octaedrons superficies to the contente superficiall of his inscribed Cube, bea∣reth suche proportion as the Octaedrons axis to the Cubes sides thirde parte.

The 23 theoreme.

Octaedrons circumscribed tetraedrons superficies contayneth the superficies of his inscribed tetraedron 9 tymes.

The 24 theoreme.

Octaedrons comprehending Cubes Diagonal lines square, is triple to the su∣perficies of his comprehended Cube.

The 25 theoreme.

Octaedrons external tetraedrons superficies to the superficial quantitie of his internall Cube, is proportioned, as √{powerof2}27 vnto an vnitie.

The 26 theoreme.

Octaedron is proportioned to his inscribed tetraedron, as 13 ½ vnto 1.

The 27 theoreme.

Octaedron to his inscribed Cube retayneth double the Proportion of theire sides.

The 28 theoreme.

Octaedrons external tetraedron to his internal tetraedron, retayneth triple the proportion of their sides that is 27 vnto 1

The 29 theoreme.

Octaedrons circumscribing Cube to his inscribed Cube, retayneth double the proportion of the comprehending Cubes Diameter to the internal Cubes side.

The 30 theoreme.

Octaedrons comprehending Dodecaedron to his included Icosaedron, bea∣reth suche Proportion as the square of a corde Trigonal doth to the greater part of a Pentagonal cordes square (of one Circle) diuided by extreame and meane Proportion.

Of Icosaedron.

The 31 Theoreme.

ICosaedrons comprehending Tetraedrons superficies contayneth the superficies of his internall Tetraedron 9 times.

The 32 Theoreme.

Icosaedrons contayning Hexaedrons superficies to the superficies of his con∣tayned Octaedron is proportioned as the Cubes diameter to his axis.

The .33. Theoreme.

Icosaedrons circumscribing Octaedrons superficies to the superficies of his in∣ternall Cube, beareth suche proportion as 9 vnto √{powerof2}12.

The .34. Theoreme.

Icosaedrons comprehending Tetraedrons superficies is double to the superfi∣cies of his comprehending Octaedron.

The .35. Theoreme.

Icosaedrons comprehending Octaedrons superficies hath the same proportion to his inscribed Tetraedrons superficies that Octaedrons lesser spheres dimetien∣tes square hath to the square of the greater semidiameter of Tetraedrons base.

The .36. Theoreme.

Icosaedrons comprehending Tetraedron contayneth his inclosed Tetrae∣dron 27 times.

The 37 Theoreme.

Icosaedrons contayned tetraedron is a thirde parte of his contayned Cube.

The .38. Theoreme.

Icosaedrons incompassing tetraedron to hi•• inclosed Cube, ••oldeth the propor∣tion of their diameters squares.

The 39 Theoreme.

Icosaedrons external Hexaedron contayneth his internal Octaedrons 6 times.

The 40 Theoreme.

The proportion of Icosaedrons solide capacitie to his comprehended Dode∣caedron, is equall to the proportion of the Icosaedrons semidiameter to the Dode∣caedrons axis, and the difference (of a pentagonall diagonals proportion to his Corde trigonall of one selfe same circle deducted from the proportion of these so∣lides diameters) ioyned together.

Of Dodecaedron.

The .41. Theoreme.

THe proportion of Dodecaedrons superficies to his inscribed Icosaedrons su∣perficies, is compounded of the proportion of Dodecaedrons basis pentagonall diagonall, to the Corde trigonall of his contayning circle, and the proportion of dodecaedrons greater spheres semidiameter to his inscribed Icosaedroms axis.

The .42. Theoreme.

Yf from the proportion of Dodecaedrons externall Icosaedrons greater spheres semidiameter to Dodecaedrons axis, ye deducte the proportion betweene the Dodecaedrons pentagonall basis diagonall and his contayning circles corde Trigonall, there remayneth the proportion of their contentes superficiall.

The .43. Theoreme.

The superficies of Dodecaedrons contayning Icosaedron to the superficies of his contayned Icosaedron, retayneth quadruple the proportion of Dodecaedrons comprehending spheres semidiameter to his axis.

The .44. Theoreme.

Dodecaedrons contayned Cubes superficies and side, and his contayned Te∣traedrons superficies and diameter, are reciprocally proportionall, that is to saye, the cubes superficies retayneth the same proportion to the Tetraedrons superficies that the Tetraedrons diameter doth to the Cubes side.

The .45. Theoreme.

Dodecaedrons contayned Icosaedron and Octaedron haue equall superficiall quantities.

The 46 Theoreme.

The proportion of Dodecaedron to his inscribed Icosaedron is equall to the proportion of Dodecaedrons greater spheres semidiameter to Icosaedrons axis, and the proportion of Dodecaedrons basis diagonal line to the inscribed Icosae∣drons side.

The 47 Theoreme.

The proportion betweene dodecaedron and his comprehending Icosaedron, is componed of the proportion betweene Icosaedrons contayning spheres dimetient and dodecaedrons internall spheres diameter, added to the proportion betweene Icosaedrons side, and dodecaedrons basis line diagonall.

The 48 Theoreme.

Dodecaedrons inscribed Icosaedron to his inscribed Octaedron, retayneth suc••e proportion, as Icosaedrons axis doubled to the diameter of Octaedrons

contayned sphere.

The 49 Theoreme.

Dodecaedrons comprehended Tetraedron to his internall Hexaedron, beareth the proportion of tetraedrons contayned spheres dimetient to the Cubes comprehending spheres diameter.

The 50 Theoreme.

Dodecaedrons comprehending Icosaedron to his contayned Icosaedron hol∣deth sixefolde the proportion of his contayning and contayned spheres diameters.

The .25. Probleme. A Metamorphosis or transformation of the fiue regulare bodies. (Book 25)

HItherto haue I onely intreated of the fiue regulare bodies, Theorically and practically opening sundrie meanes to search out the proportion and quantities of their sides, diameters, axes, perpendiculares, altitudes, and contentes both superficiall and solide, and that not onely in these so∣lides considered by them selues, but also conferred with other, aswell by inscription as circumscription, both of themselues and their spheres: so as I suppose hardely any question may be proponed concerning these bo∣dies, whiche by the form•••• problemes or theoremes may not be resol∣ued. Yet before I finishe this Treatise I thought good to adioyne one Chapter of the transformation of these Solides into suche bodies (as though they may not be tearmed regulare, for that Euclide hath demon∣strate onely fiue, and no mo possibly to be founde or imagined) yet haue they such vniforme composition and conuenience with th••se solides, that they are not onely enuironed with equilater and equiangle superficies as they be, but also haue all their sides equall, and one comprehending sphere exactly and at once touching all their solide angles, they are also capable of the regulare bodies, and onely herein different, that whereas the regulare solides be inuironed with one kinde of playnenes and re∣ceiue one internal sphere, and so consequently one a••is. Thei•• Transfor∣med bodies are incompassed with seuerall kind•• of playnesse, and haue se∣uerall axes and contayned spheres, of which their diuersitie aryseth ma∣nyfolde

mo straunge, rare, and differente kinds of proportions, than may in a very large volume (I will not saye be demonstrate, but onely by Theoremes) be declared. Transformed regulare solydes I call them, both to auoyde the forging of newe names, which I should be inforced to vse for distinction sake, and also bicause they séeme to be created by the vnifourme section of the regulare bodies, and may by addition of certaine proportionall equall Pyramides, be reduced to the regulare solides, and are in sundry proportions and proprieties so agreable and resemblante to those regulare solides, whose names they beare, that they séeme onely to lose the fourme, and yet still to retayne the nature of them. I meane not here in so ample maner as the noueltie of the matter requireth to in∣treate of them, but onely by diffinitions and Theoremes open so muche as may be sufficient to explane the composition, fourme, nature, and pro∣portion of these, and also giue light to the ingeniouse infinitely to pro∣céede for inuention of the like, whose vse and appliance may be manyfold to conclusions no lesse straunge than necessarie: but thereof in due place for the nature of these transfourmed solides peruse their seuerall diffini∣tions and Theoremes immediately ensuing.

Of Tetraedron transformed. The firste Diffinition.

TEtraedron transformed is a solide incompassed with foure equall equilater triangles, and foure equall equiangle playnes Hexagonall, hauing equall sides with the triangles.

[illustration]

The firste Theoreme.

This solide hath 18 equall sides, 36 playne angles, and 12 solide angles, and may be circumscribed of a sphere as exactly as any of the forenamed regulare bodies: the right line drawen from this spheres center to one of the solide angles I call the transformed tetraedrons semidiameter.

The 2 theoreme.

This solide also receyueth two intrinsicall spheres, the lesser touching all the centers of his hexagonall playnes, the greter touching the centers of the trigonal playnes, and making circulare sections concentricall with the hexagonall bases, the semidiameters of these spheres I call the solides Axes.

The 3 theoreme.

Within this solide may Octaedron be described, and his sixe angles shall reste in the medieties of those his sides that are peculiare to the hexagonall playnes and not common both to the hexagonall and trigonall bases.

The 4 theoreme.

Within this solide may two Tetraedrons of differente quantitie be described, the greater hauing his solide angles resting in the centers of the trigonal playnes, and the lesser hath his angles situate in the hexagonall bases centers.

The 5 theoreme.

Icosaedron may within this transfigured Tetraedron be described, hauing his solide angles placed in the hexagonall playnes, and foure of his trigonall ba∣ses concentricall, with those hexagonall bases wherein they are situate.

The 6 theoreme.

This solides semidiameter is equall in power to the semidiameter of his tri∣angulare playnes contayning circle, and his greater Axis.

The 7 theoreme.

The square of this solides side deducted from the square of his semidiameter, leueth the square of his lesse axis.

The 8 theoreme.

The squares of the axes deducted one from an other, the Zenzike roote of the remayne is the semidiameter of the hexagonall basis concentrical circle tofore mentioned in the seconde theoreme.

The 9 th••or••me.

The altitude of this transformed solide is equall to his Axes bothe ioyned together, and beareth proportion to his side, as foure vnto the roote Zenzike of sixe.

The 10 theoreme.

This solide semidiameter beareth to his side the proportion of √{powerof2}11 to √{powerof2}8.

The 11 theoreme

The greater axis to the lesser is proportioned as 5 vnto 3.

The 12 theoreme.

The medietie of this solides altitude is equall to the semidiameter of the con∣centricall circle made in the Hexagonall bases by the peripherie of the solid••s greater internall sphere.

The 13 theoreme.

The lesse semidiameters of this solides hexagonoll and trigonall bases are proportioned one to an other, as the squares of their greater semidiameters.

The 14 theoreme.

The greater diameter of this solides hexagonall basis, to the greater dimeti∣ent of his trigonall base, is in power triple.

The 15 theoreme.

The side of this solide is equall to the side of his lesser inscribed tetraedron, and to his greater internall tetraedrons side it retayneth suche proportion as 3 to 5.

The 16 theoreme.

This solides inscribed Octaedrons side is triple to the medietie of his contay∣ning transfigured Tetraedrons side.

The 17 theoreme.

The side of this solides inscribed Octaedron (beeing by extreame and meane proportion diuided) the lesser segmentes square is halfe the square of the inscribed Icosaedrons side.

The 18 theoreme.

The superficies of this transfourmed Tetraedron is equall to bothe the su∣perficies of an Octaedron and Icosaedron, hauing equall sides with this solide ioyned together.

The 19 Theoreme.

This solides crassitude is equall vnto two Pyramides, whereof the one hath for his base a triangle whose side is double to the side of this solide, and his axis equall to the greater axis of this body, the other is an hexagonall Pyramis, whose base is an equiangle hexagonum, hauing his side double to the side of this solide, and his altitude equall to this transformed tetraedrons lesser axis.

The 20 Theoreme.

This transfigured solide may be resolued into 23 equall tetraedrons, euery of them of equall sides vnto this figure, and all ioyned together make his exacte ••rassitude, whiche is proportioned to the Cube of his side, as 23 to the quadrate roote of 72.

Of the transfigured Cube. The seconde Diffinition.

A Transfourmed Cube is a figure geometrical enuironed with 6 equi∣angle Octogonall and 8 equilater triangular playnes or bases, whose sides are all equall.

[illustration]

The 21 Theoreme.

This figure hath 36 equall sides, 72 playne angles, and 24 solide angles, it may be incompassed of a sphere, exactly touching with his concaue peripherie euery of their solide angles, and the semidimetiente of that comprehending sphere is called also this solides semidimetiente.

The 22 theoreme.

This transfygured bodye is also capable of two internall spheres, the lesse touching onely the centers of the Octogonall playnes, the greater both touching

exactly with his conuex circumference the triangulare bases centers, and also cut∣ting the playnes Octagonall describing therein concentricall circulare sections, the semidimetientes of these spheres are the axis of the solide.

The 23 theoreme.

Within this solide may a Tetraedron be described, whose angles shall reste in the centers of the trigonall bases.

The 24 theoreme.

Octaedron may also herein be placed, his angles resting in the centers of the Octogonall playnes.

The 25 theoreme.

Hexaedron may likewise be inscribed by drawing streight lines conioyning all the trigonall bases centers.

The 26 theoreme.

This transformed figure receyueth also Icosaedron, whose 12 solide angles are placed in his 6 Octagonall bases, euery of them receyuing two of his angles.

The 27 theorome.

Dodecaedron may not exactly be within this figure described, for 12 of his angles beeing situate in the 6 Octagonall bases, diuiding their lesse semidiame∣ters by extreame and meane proportion, the other 8 angles will not touche his superficies, but rest precisely in the greater Axes of this solide, directly vnder the centers of the 8 trigonall playnes, vniformly and proportionally cutting euery of these forenamed greater Axes.

The 28 theoreme.

The side of this transfigured body is triple in power to the greater semidiame∣ter of his triangulare base, and to the lesse semidiametiente of that base it is pro∣portioned as 1 vnto √{powerof2}1/12.

The 29 theoreme.

The side of this transfourmed bodye beeyng rationall, the greater semi∣diameter of his Octogonall bases is an irrationall Maior, bearing proportion to the solides side, as √{powerof2} vni. 1+√{powerof2}½ vnto an vnitie.

The 30 theoreme.

This solides side being rationall, his Octogonall bases lesse semidimetiente is an irrational Binomie, retayning such proportion to the side, as √{powerof2} v. ¼+√{powerof2}½

The 31 theoreme.

This solide side being rationall, his greater axis is an irrationall Binomie proportioned to the side, as √{powerof2}⅔+√{powerof2}¾ vnto an vnitie.

The 32 theoreme.

The lesse Axis of this Figure is equall to the losse semidimetiente of his Octo∣gonall basis, and retayneth vnto his side the proportion of √{powerof2}½+••/2 to an vni∣tie.

The 33 theoreme.

This transformed Figures side being rationall, his comprehending spheres di∣metiente is an irrationall Maior, proportioned to the side, as √{powerof2} v. 7+√{powerof2}32 to 1.

The 34 theoreme.

This solides contayned Tetraedrons side is proportioned to the greater Axis, as √{powerof2}8/3 to an vnitie.

The 35 theoreme.

The side of the inscribed tetraedron is double in power to the side of the in∣scribed Cube.

The 36 theoreme.

This transfigured Cubes lesse altitude is double in power to the side of his in∣scribed Octaedron.

The 37 theoreme.

The Octogonall bases lesse Diameter diuided by extreame and meane Pro∣portion, maketh his greater segmente, the inscribed Icosaedrons side.

The 38 theoreme.

The side of this transformed Solide, being by extreame and meane Proporti∣on diuided, if to the lesse portion ye adioyne a line in power double, that whole line thereof resulting, is the side of the tofore mentioned inscribed Dodecae∣dron.

The 39 theoreme.

The Cube of the transfigured Solides Octogonall bases lesser Diameter, hath a superficies exceeding the superficiall contente of all this Solides Octogonall bases, by a superficies proportionate to the superficiall quantitie of his trigonall bases, as the Cubes dimetiente to his side.

The 40 theoreme.

The Cube whose side is equall to the lesse altitude of the transfigured hexae∣dron, excedeth in solide capacitie this transformed figure, by the solide content of an Octaedron described vpon the same side with this transfygured hexaedron.

Of Octaedron transformed. The 3 Diffinition.

A Transfigured Octaedron is a Geometricall Figure▪ incompassed with 14 bases, whereof 8 are equall equiangle Hexagonall playnes, and the other 6 are equall squares.

[illustration]

The 41 theoreme.

A transformed Octaedron hath 36 equall sides, 72 playne angles, and 24 solide angles, and may be inclosed with a contayning sphere exactly touching all his Angles.

The 42 theoreme.

This solide also receyueth two internall spheres, the one onely touching all the Hexagonall bases centers, the other both touching all the centers of the squares▪ and also with his peripherie cutting the Hexagonall playnes, delineating in thē concentricall circles, the semidiameters of these spheres are called the Axes.

The 43 theoreme.

Within this solide may Tetraedron be described, and his angles situate in the centers of the Hexagonall bases.

The 44 theoreme.

This Figure also receyueth the Cube, with his 8 solide angles residente in the •• ce••••ers of his Hexagonall playnes.

The 45 Theoreme.

The inscribed Octaedrons 6 solide angles reste in the 6 centers of the Qua∣drate playnes.

The 46 theoreme.

This solides inscribed Icosaedron hath all his Angles situate in those his 12 sides which are peculiare to the Hexagonall bases, not perticipating with the quadrate playnes.

The 47 theoreme.

This solides side is equall to the greater semidiameter of his Hexagonal ba∣sis, and double in power to the greater semidimetientes of his quadrate playnes.

The 48 theoreme.

The greater dimetient of the Hexagonal basis, is equal in power to his lesse diameter and side.

The 49 theoreme.

The lesse diameter of the Hexagonal basis, is triple in power to his lesse dia∣meter of the quadrate basis.

The 50 theoreme.

The greater Axis of this solide, is double in power to his side.

The 51 theoreme.

This solides semidimetiente, is equal in power to his lesse Axis, and side.

The 52 Theoreme.

The semidimetiente of the concentrical circle, is equal to the greater semidia∣meter of the quadrate playnes, and the square thereof added to the square of the lesse Axis produceth the square of the greater.

The 53 Theoreme.

The inscribed tetraedrons side is double to the side of this transfigured So∣lide.

The 54 theoreme.

The side of the inscribed Cube, is meane proportionall betwene the side of the inscribed tetraedron, and the side of this solide.

The 55 theoreme.

The diameter of this body is fiuefolde in power greater than the side of his inscribed Cube.

The 56 theoreme.

This solid••s inscribed Octaedrons side, is equall to the side of his inscribed te∣traedron, and double in power to the inscribed Cubes side, or the greater Axis, his equall.

The 57 theoreme.

If the greater diameter of this transformed Figures quadrate bases be diuided by ext••eame and meane proportion, triple the lesse parte is the inscribed Icosae∣drons side.

The 58 Theoreme.

The side of an Octaedron transformed being rationall, the righte line that matcheth in power his superficies, is an irrationall called of Euclide, Potens rationale & mediale, proportioned to the square of the side, as √{powerof2} v. √{powerof2}432+8 vnto an vnitie.

The 59 Theoreme.

This transfigured Octaedron may intellectually bee diuided into 41 Pyra∣mides rysing fro the superficies, and concurring with their toppes or vertices in the center of this Solide, whereof 8 are Hexagonall, and 6 Tetragonall, accor∣ding to the bas••s from whence they ascende, and the Hexagonall Pyramides ioyned togither, are triple to the Solide capacitie of the tetragonall Pyramides.

The 60 theoreme.

Octaedron transformed to the Cube of his side retayneth the Proportion, of √{powerof2}28 vnto an vnitie, and may bee resolued into 24 equal Octaedrons, hauing euery of their sides equall to this transformed Figure.

Of the transfigured Icosaedron. The fourth Diffinition.

ICosaedron transfigured, is a solide bodye incompassed with 32 Equi∣angle and equilater bases, whereof 20 are Hexagonall, and the other 12 Pentagonall playnes.

[illustration]

The 61 theoreme.

This transformed figure hath 90 equall sides, 180 plaine angles, and 60 solides, one comprehending sphere inuironeth all his angles, and the semidiameter of that sphere, is the semidimetient of this solide.

The 62 theoreme.

This solide also conceiueth two internall spheres, the one only touching all the Hex••gonal bases centers, the other both touching all the centers of the Pentago∣nall bases, and also cutting the plaines Hexagonal, deliniating in them concentri∣cal circles, the spheres semidimetients are this solides Axes.

The 63 theoreme.

Within this body may Tetraedron be described, and his angles situate in 4 of the Hexagonal bases centers.

The 64 theoreme.

This solides inscribed cube hathe his solide angles also placed in the centers of his Hexagonal plaines.

The 65 Theoreme.

Icosaedron transfigured, receiueth an internal Octaedron, whose solide angles rest in the medieties of 6 suche sides of his Hexagonall bases, as communicate not with his Pentagonal playnes, but are situate Paralel, Perpendicular, and oppo∣site one to an other.

The 66 Theoreme.

Icosaedron may also herein be inscribed, his 12 solide angles resident in the cen∣ters of his 12 Pentagonall bases.

The 67 theoreme.

Dod••caedron may also perfectly be described within this solide, all his solide angles being placed in the centers of all the Hexagonall bases.

The 68 theoreme.

The side of this transformed figure being rationall, his semidiameter is an ir∣rational Maior, proportionate to his side, as √{powerof2} vniuers. 3 ⅝+√{powerof2}6 2••/6•• to an vnitie.

The 69 theoreme.

The greater semidimetient of the Hexagonal bases, is equal to the side of this transfigured body, and to the greater semidiameter of the Pentagonal plaines, it beareth proportion as 1 vnto √{powerof2} v. ½+√{powerof2}1/20.

The 70 theoreme.

The lesse semidimetients of these Pentagonal and Hexagonal bases, are pro∣portionate, as √{powerof2} v. ⅓+√{powerof2}4/45 vnto an vnitie.

The 71 Theoreme.

This transfigured solides side being rational, his greater Axis is a Binomye proportioned to the side, as √{powerof2} v. 3 ⅛+√{powerof2}1681/320 vnto an vnitie.

The 72 theoreme.

The side of this solide being rational, his lesser axis is also a Binomye, & to the side proportionate as √{powerof2} v. 21/8+√{powerof2}405/64 vnto an vnitie.

The 73 theoreme.

The greater axis in power exceedeth the lesser, by the semidimetient of the con∣centrical circle, which retaineth the same proportion to the side of this trāsfor∣med solide, that √{powerof2} v. ½—√{powerof2}1/26 doth vnto an vnitie, and it is the greater part of this solides Pentagonal bases semidiameter, diuided by extreame and meane proportion.

The 74 theoreme.

This transformed figures inscribed Tetraedrons side, is double in power to the side of the inscribed cube, and to the side of this solide, it beareth the proportion of √{powerof2} v. 7+√{powerof2}45 vnto 1.

The 75 theoreme.

The side of Octaedron described within a transformed Icosaedron, whose side is rational, shall be a Binomye proportional to the rational side, as √{powerof2} vniu. 6 ¾+√{powerof2}405/16 to 1.

The 76 theoreme.

This transformed figures side being rational, his inscribed Icosaedrons side is an irrational Binomye, retaining such proportion to this transformed solides side, as triple his greater altitude to the diameter of his comprehending Icosaedron, that is as √{powerof2} v. 4 ⅕+√{powerof2}3 ⅕ to an vnitie.

The 77 theoreme.

The side of a Dodecaedron described within this transfigured Icosaedron, being diuided by extreame and meane proportion, maketh his greater parte this transformed solides side, and is proportioned thereunto, as √{powerof2} v. 3/2+√{powerof2}••/4 vnto an vnitie.

The 78 theoreme.

The superficies of this transformed Icosaedron, is equal to the superficial ca∣pacities of one Dodecaedron, and 6 Icosaedrons, hauing their sides all equall to the side of this transformed figure.

The 79 theoreme.

The superficies of a transfigured Icosaedron, is equal to the superficial ca∣pacitie of a dodecaedron and an Octraedron of equal sides to this solide, and the

superficies of suche a transformed Tetraedron, as hath a side equal to bothe the sides of those two bodyes ioyned together.

The 80 theoreme.

A transfigured Icosaedron may be resolued into 12 Pentagonal and 20 hex∣agonal Pyramides, concurring with their toppes or vertices all in the centers of this transformed body, and their bases the Pentagonal and Hexagonal equian∣gle superficies tofore mentioned in the diffinition of this solide. The 12 Pentagonal Pyramides crassitude, to a Dodecaedron created vpon this transfigured solides side, is proportioned, as his greater axis, to that line, whose greater segment is this Pentagonal basis lesse semidimetient. And the crassitude of the other 20 Hexa∣gonal Pyramides, is ⅔ of this transfigured solides containing Icosaedron.

Of Dodecaedron transformed. The fifthe Diffinition.

A Transformed Dodecaedren is a massie or solide figure, comprehēded of 12 equiangle decagonal, and 20 equilater triangular bases.

[illustration]

The 81 Theoreme.

A transfigured Dodecaedron hathe 90 equal sides, 180 plaine Angles, and ••0 solide Angles, comprehended of one sphere, with his concaue Peripherie

touching all those angles, whose semidimetient is the semidiameter of this transfor∣med solide.

The 82 Theoreme.

This transformed figure hathe two internall sphere••, the greater with his con∣uex superficies touching all the trigonall bases centers, and cutting the bases or playnes Decagonall, describing in them concentricall circles, the lesse sphere only toucheth all the Decagonall bases centers resting wholy within this solide, and the semidimetientes of these spheres are called the Axes of this transformed fi∣gure.

The 83 theoreme.

This transfigurate body receiueth an internall Tetraedron, whose solide an∣gles rest in the centers of his trigonall bases.

The 84 theoreme.

Hexaedron may also within this body be described, his angles likewise remai∣ning in the Trigonall bases centers.

The 85 theoreme.

Within a transfigured Dodecaedron may an Octaedron be framed, and his angles will be resident in the medieties of those his sides that be peculiar to the De∣cagonall, not communicating with the trigonall bases, but opposite, paralel, and Orthogonal, one to an other.

The 86 theoreme.

Icosaedron within this transfigurate body may be described, and all his 12 an∣gles in the centers of his 12 Decagonal bases situate.

The 87 theoreme.

Dodecaedron may also within this transfigurate body be described, all his 20 solide angles placed in the 20 trigonal bases centers.

The 88 theoreme.

This solides semidiameter is equal in power to his lesse Axis, and the greater semidiameter of his Decagonal bases.

The 89 theoreme.

The side of this solide being rationall, his Decagonall bases lesse semidiameter is an irrational Maior, proportioned to the greater semidimetient of the same bases, as √{powerof2} v. 5/4+√{powerof2}5/4 vnto √{powerof2} v. 1 ½+√{powerof2}1 ¼.

The 90 theoreme.

The side of this transfigurate body, is triple in power to the greater semidia∣meter of his trigonall bases, and to the lesse semidimetient of the same bases, it is proportioned, as the Diameter of a Cube to his Axis.

The 91 theoreme.

The dimetient of this solide, is equall in power to these three, his side, his lesse altitude, and the lesser diameter of his decagonall bases.

The 92 theoreme.

The square of this solides greater axis, added to the thirde parte of his sydes square, maketh the square of his semidimetient.

The 93 theoreme.

The square of the lesse altitude of this solide, deducted from the square of the greater, leaueth the square decagonall basis concentricall circles diameter.

The 94 theoreme.

Twentie direct or vpright equ••later triangular Pyramides may vppon these solides 20 triangular bases of s••tch al••••tude be erected, and to this solide in sutch sorte adioyned, that the wh••••e body thereof resulting, shall be a dodecaedron: whose side conteyneth in power fiu••fold the side of this transformed figure.

The 95 theoreme.

Euery of these adioyned Pyramides sides, are equall to the greater portion of this transfigured dodecaedrons side, diuided by extreame and meane pro∣portion.

The 96 theoreme.

The square of this transfourmed figures trigonall bases greater semidimeti∣ent, deducted from the square of his sides greater portion diuided by extreame and meane proportion, leaueth the square of his adioyned Pyramides altitude.

The 97 theoreme.

The greater altitude of this solide, is equall in power to the sydes of his in∣scribed cube, and tetraedron, and if the power therof be diuided in two lines re∣tayning the proportion of a line diuided by extreame and meane proportion to his lesse parte, the lesser of these lines is this transfigured solides inscribed Do∣decaedrons side.

The 98 theoreme.

This transformed Dodecaedrons inscribed Octaedrons dimetient, is equall in power to the inscribed Icosaedrons diameter, and the lesse dimetient of his deca∣gonall bases, and the power of the lesse altitude of this solide, (being diuided in two ryght lynes retaynyng extreame and meane proportion) the lesser of those lynes is the inscribed Icosaedrons side.

The 99 theoreme.

The superficies of a transfigured Dodecaedron dothe exceede the superficies of an Icosaedron made vppon this transformed solides side, by the superficies of

••n equiangle Decagonum, whose side retaineth the same proportion to the side of this solide, that the side doth to his Triangular bases lesser semidiameter, and the superficies of all this solides Decagonal bases, to the superficies of all his Trigonal bases, retaineth the proportion of his decagonal basis lesser diameter, to the semi∣dimetient of his trigonall bases contained circle.

The 100 Theoreme.

A transformed Dodecaedron may be resolued into 32 Pyramides, rising from his equilater bases, and concurring with their toppes or Vertices at his Center, whereof 12 are Decagonal, and 20 Trigonal, according to the bases from whence they rise, and the proportion of all the decagonal Pyramides crassitude, to the so∣lide content of all the trigonal, is compounded of the proportion betweene the De∣cagonal basis lesse diameter, & the trigonal basis semidiameter, added to the pro∣portion of the lesse altitude to the greater.

Theoremes of these transformed bodyes conserred both vvith their circumscribing regular bodyes, and also betvveene them selues.

The 1 theoreme.

THe square of a trāsformed tetraedrons diameter, added to 8 times the square of his side, produceth the square of his comprehending tetraedrons diameter.

The 2 theoreme.

The superficies of a transfigured Tetraedron, is proportioned to the superfi∣cies of his comprehending Tetraedron, as 7 to 9.

The 3 Theoreme.

If the side of a transformed Octaedron, to the side of a transfigured Tetrae∣dron, retaine the proportion of √{powerof2}7 vnto √{powerof2}12, then is the superficial capaci∣tie of the transformed Octaedrons Hexagonal playnes, equal to the whole su∣perficies of Tetraedron transformate.

The 4 Theoreme.

The solide capacitie of a transformed Tetraedron, to his circumscribing Te∣traedron, holdeth the proportion of 23 vnto 27.

The 5 Theoreme.

The solide quantitie of a transfigured Tetraedron, to the content solide of a transfigurate Octaedron framed on the same side, is proportioned as 23 to 96.

The 6 Theoreme.

Hexaedron transformed, may of a cube be circūscribed, and his side deducted

from the cubes side, leaueth a line double in power to the transformed Hexaed••ōs side.

The 7 Theoreme.

If ye adioyne √{powerof2}⅙ of a trāsfigured Hexaedrons side vnto his greater Axis, the product is the semidimetient of his containing cube.

The 8 Theoreme.

The containing cubes superficies, exceedeth the content superficial of his inscri∣bed transformed Hexaedrons Octogonal bases, by the square of a line retaining such proportion to the transfigured solides side, as a Cubes diameter, to the grea∣ter semidimetient of his quadrate bases.

The 9 Theoreme.

A transfigured cube, with foure Tetraedrons made on his side, are equal to the crassitude of his comprehending Hexaedron.

The 10 theoreme.

A transformed Tetraedron, and transfigured Hexaedron of equall sides, haue the differenees betweene their crassitudes and the solide capacities of their comprehending regular bodyes equal.

The 11 theoreme.

A transfigured Octaedron, may within an Octaedron be described, and the dimetient of that containing Octaedron to the side of the transformed body, re∣taineth the same proportion, that the side of a transfigured cube dothe to the de∣fault or difference betweene his greater Axis, and the semidimetient of his com∣prehending Hexaedron.

The 12 theoreme.

A transformed Octaedron, and a transformed Tetraedron hauing equall sides, their comprehending regular solides sides are equal also, and their superficial quantities in double proportion.

The 13 theoreme.

Octaedrons superficies, doth surmeunt the content superficial of his inscribed Octaedron transfigured by the difference of an equiangle Hexagonum, (whose side is double to the transformed solides side) from the square of the same transfi∣gured solides lesse altitude.

The 14 theoreme.

If two meane proportional lines be found betwene the sides of Octaedron trās∣formed, and his circumscribed regular body the Octaedron hauing to side the lesse of those meane proportional lines, deducted from the crassitude of the containing Octaedron, leaueth the solide quantitie of the transfigured bodye.

The 15 theoreme.

If two meane proportional lines be found betweene the side of a comprehēding

Octaedron, and the greater Axis of his inscribed transfigured Octaedron, and likewyse two other meane proportionals betwene the inscri••••d transformed soli∣des side and his quadrate bases greater diameter: A cube h••••yng for ••his side the greater of the former two meane proportionall lynes, is equall to the circum∣scribing Octaedron. And if from that cube yee deduct a cube, hauyng his side the lesser of the latter meane proportionals, the residue or remayning quantitie, is the crassitude of the inscribed transfigured octaedron.

The 16 theoreme.

Icosaedron transformed may be inclosed with an Icosaedron, and all the 180 playne angles of the transfigured body, concurring and resting in the 30 sides of the conteyning Icosaedron, whose side is triple to the syde of the contayned trans∣fourmed body.

The 17 theoreme.

The semidimetient of the comprehending Icosaedron is equall in power to the lesser axis, and a meane proportionall betwene the transfigured body, and his contayning Icosaedrons sides.

The 18 theoreme.

The greater diameter of a transfigured Icosaedrons pentagonall basis, diui∣ded by extreme and mean proportion, and the greater part adioyned to his grea∣ter altitude, produceth his comprehending Icosaedrons dimetient.

The 19 theoreme.

The superficies of an Icosaedron surmounteth the superficiall quantitie of all his inscribed transfigurate Icosaedrons hexagonall playnes, by a superficies re∣tayning sutche proportion to the superficiall capacitie of all his pentagonal play∣nes, as the lesse d••met••ent of the comprehending Icosaedrons basis dothe to the lesse diameter of the transformed solides pentagonall basis.

The 20 theoreme.

Two sutche lynes being founde, that the first be to a transfigured Icosaedrons side double, and the seconde lyne equall to the greater segment of this transfour∣med Icosaedrons pentagonall basis greater semidiameter parted by extreme and meane proportion, a pentilater Prisma, hauing for his altitude this second ly••e, and for the syde of his pentagonall equiangle basis the first lyne, shall be equall to the excesse or difference of the conteining Icosaedron aboue the crassitude of his inscribed transfigured Icosaedron.

The 21 theoreme.

Dodecaedron transfigured, may of a regular Dodecaedron bee circumscri∣bed, whose syde beeing diuided in two lynes reteyning double extreame and

meane proportion, if from the greater ye deducte the lesser, the re∣maine is the side of the inscribed transfigured Dodecaedron.

The 22 theoreme.

The side of Dodecaedron transformed, being subtracted from the side of his comprehending regular Dodecaedron, and from the square of that remaynes medietie againe deducting one thirde part of the square of the transfigured Solides side, there remaineth a su∣perficies, whose quadrate roote or line matching it in power, added to the greater Axis of the transfigured body, maketh the semidi∣metient of his comprehending regular Dodecaedron.

The 23 theoreme.

The superficiall content of all transformed Dodecaedrons De∣cagonall bases, to the superficiall capacitie of all his trigonal plaines, retaineth suche proportion as the lesse Diameter of the Decagonal bases, to the lesse semidimetient of his Triangular bases, and if vnto the Perpendicular of the Triangular bases ye adioyne triple the Decagonall bases lesser dimetient. The square whose side is meane proportionall betweene that resulting line, and the circuit or Peri∣metrie of one Decagonall base, is equall to the vniuersall superficies of the transfigured Dodecaedron.

The 24 theoreme.

The superficies of a comprehending regular Dodecaedron, sur∣mounteth the superficial quantitie of all his contained transfigurate Dodecaedrons decagonal bases, by a superficies, retaining the same proportion to the superficiall content of all his trigonall plaines, that the lesser of two such right lines retaining double extreame and meane proportion (as ioyned together make the lesser part of the comprehē∣ding Dodecaedrons Pentagonal Perpendicular diuided by extreme and meane proportion) dothe vnto the lesse semidiameter of the transformate Dodecaedrons trigonall bases.

The 25 Theoreme.

A transformed Dodecaedrons comprehending Dodecaedrons

diameter, being into two right lines retaining quadruple extream and meane proportion diuided, and the lesser of them againe into two lynes retaining double extreame and meane proportion parted: A trigonall Pyramis, hauing for his equilater triangular bases side, a line double to the transfigured Dodecaedrons side, and his altitude fiuefoulde the lesse line produced by the latter section of the Dodecaedrons diameters lesser part, shalbe equall to the excesse or difference, wherby the containing regular Dodecaedron, surmounteth the solide capacitie of his inscribed transfigurate Dodecaedron.

The end of this Mathematicall discourse of the Platonicall regular Solides, and their Me∣tamorphosis compiled and inuen∣ted by Thomas Digges Gentleman. ANNO AET. 25.

DIFFICILIA QVAE PVLCHRA.

[illustration]

HIS DEDIT FVNDAMINA VIRTVS.

❧Imprinted at London by Henrie Bynneman, dwelling in Knightriders streat, at the signe of the Mermaid.

ANNO DOMINI. 1571.