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.

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

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

DIFFINITIONS.

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

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

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

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

Example.

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

[illustration]

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

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

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

Beholde the Figure.

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

[illustration]

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

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

Example.

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

[illustration]

[illustration]

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

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

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

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

Example.

Admit the Area of the Cylinders

[illustration]

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

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

A note to finde the altitude of a Solide Cone.

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

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

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

Example.

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

[illustration]

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

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

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

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

Example.

ABCD the globe whose diame∣ter

[illustration]

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

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

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

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

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

[illustration]

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

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

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

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

Example.

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

[illustration]

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

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

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

Example.

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

euery of their Bases being the sides of the

[illustration]

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

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

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

Example.

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

[illustration]

[illustration]

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

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

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

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

Example.

[illustration]

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

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

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

[illustration]

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

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

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

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

[illustration]

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

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

An other mensuration of vessels more common.

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

Example.

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

[illustration]

Another note of measuring.

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

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

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

Example.

Admit S and

[illustration]

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

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

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

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

Example.

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

[illustration]

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

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

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

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

Example.

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

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

[illustration]

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

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

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

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

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

Example.

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

[illustration]

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

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

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

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

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

Example.

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

[illustration]

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

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

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

The end of the third Booke called Stereometria.