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.

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.

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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.

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

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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]

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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.

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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.

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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.

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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.

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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.

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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.

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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.