The Way to Geometry, Book 23
The twenty third Booke of Geometry, of a Prisma,
1 A Pyramidate is a plaine solid comprehended of pyramides.
2. A pyramidate is a Prisma, or a mingled polyedrum.
3. A prisma is a pyramidate whose opposite plaines are equall, alike, and parallell, the rest parallelogramme. 13 dxj.
As here thou seest. The base of a pyramis was but one: Of a Prisma, they are two, and they opposite one against another, First equall; Then like: Next parallell. The other are parallelogramme.
Therefore
4. The flattes of a prisma are two more than are the angles in the base.
And indeed as the augmentation of a Pyramis from a quaternary is infinite: so is it of a Prisma from a quinary: As if it be from a triangular, quadrangular, or quinquangular base; you shal have a Pentaedrum, Hexaedrum, Heptaedrum, and so in infinite.
5. The plaine of the base and heighth is the solidity of a right prisma.
6. A prisma is the triple of a pyramis of equall base and heighth. è 7 p. x i j.
As in the example a prisma pentaedrum is cut into three equall pryramides. For the first consisting of the plaines a e i, a e o, a o i, e i o; is equall to the second consisting of the plaines a o i, a o u, a i u, a u y, by the 10 e vij. Because it is equall to it both in common base and heighth. Therefore the first and second are equall. And the same second is equall to it selfe, seeing the base is i o u, and the toppe a. Then also it is equall to the third consisting of the plaines a i u, a i y, u i y, a u y. Therefore three are equall.
If the base be triangular, the Prisma may be resolved into prisma's of triangular bases, and the theoreme shall be concluded as afore.
Therefore
7. The plaine made of the base and the third part of the heighth is the solidity of a pyramis of equall base and heighth.
The heighth of a pyramis shall be found, if you shall take the square of the ray of the base out of the quadrate of the side: for the side of the remainder, by the 9 e xij, shall be the altitude or heighth, as in the example following.
Here the content of the triangle by the 18 e xij, is found to be 62 44/125 for the base of the pyramis. The altitude is 9 15/19: Because by the 6 e xviij, the side is of treble power to the ray. But if from 144, the quadrate of 12 the side, you take the subtriple i. 48, the remainder 96, by the 9 e x ij, shall be the square of the heighth. And the side of the quadrate shall be 9 15/19. Now the third part of 9 15/19 is 3 5/19. And the plaine of 62 44/125 and 3 5/19, shall be 203 1103/2375 for the solidity of the pyramis.
So in the example following, Let 36, the quadrate of 6 the ray, be taken out of 292 9/1156 the quadrate of the side 17 3/34 the side 16 3/34 of 256 9/1156 the remainder shall be the height, whose third part is 5 37/102; the plaine of which by the base 72 ¼ shall be 387 11/24 for the solidity of the pyramis given.
If the pyramis be unperfit, first measure the whole, and then that part which is wanting: Lastly from the whole subtract that which was wanting, and the remaine shall be the solidity of the unperfect pyramis given: As here, let a o, the side of the whole be 16 5/12: e o, the side of the particular be 8 1/16. Therefore the perpendicular of the whole o u, shall be 15 5/32: Whose third part is 5 5/96: Of which, and the base 93 3/11 the plaine shall be 471 134/1056 for the whole pyramis. But in the lesser pyramis, 9 the square of the ray 3, taken out of 65 1/256 the quadrate of the side 8 1/16 the remaine shall be 56 1/256; whose side is almost 7 ½ for the heighth. The third part of which is 2 ½. The base likewise is almost 22. The plaine of which two is 55, for the solidity of the lesser pyramis: And 471—55 is 416, for the imperfect pyramis.
After this manner you may measure an imperfect Prisma:
8. Homogeneall Prisma's of equall heighth are one to another as their bases are one to another, 29, 30,31, 32 p xj.
The reason is, because they consist equally of like number of pyramides. Now it is required that they be homogeneall or of like kindes; Because a Pentaedrum with an Hexaedrum will not so agree.
This element is a consectary out of the 16 e iiij.
And
9. If they be reciprocall in base and heighth, they are equall.
This is a Consectary out the 18 e iiij.
And
10. If a Prisma be cut by a plaine parallell to his opposite flattes, the segments are as the bases are. 25 p. xj.
The segments are homogeneall because the prismas. Therefore seeing they are of equall heighth (by the heighth I meane of plaine dividing them) they shall be as their bases are: And here the bases are to be taken opposite to the heighth.
11. A Prisma is either a Pentaedrum, or Compounded of pentaedra's.
Here the resolution sheweth the composition.
12 If of two pentaedra's, the one of a triangular base, the other of a parallelogramme base, double unto the triangular, be of equall heighth, they are equall 40. p xj.
The canse is manifest and briefe: Because they be the halfes of the same prisma: As here thou maist perccive in a prisma cut into two halfes by the diagoni's of the opposite sides.
Euclide doth demonstrate it thus: Let the Pentaedra's a e i o u, and y s r l m, be of equall heighth: the first of a triangular base e i o: The second of a parallelogramme base s l, double unto the triangular. Now let both of them be double and made up, so that first be a e i o u n … The second y s r l v f. Now againe, by the grant, the base s l, is the double of the base e i o,: whose double is th… base e o, by the 12 e x. Therefore the bases s l, and e o, are equall: And therefore seeing the prisma's, by the grant, here are of equall heighth, as the bases by, the conclusion are equall, the prisma's are equall; And therefore also their halfes a e i o u, and y s n l r, are equall.
The measuring of a pentaedrall prisma was even now generally taught: The matter in speciall may be conceived in these two examples following.
The plaine of 18. the perimeter of the triangular base, and 12, the heighth is 216. This added to the triangular base, 15 18/3…. or 15, ⅗, almost twise taken, that is, 31 ⅕, doth make 247 ⅕. for the summe of the whole surface. But the plaine of the same base 15 ⅖, and the heighth 12. is 187 ⅕, for the whole solidity.
So in the pentaedrum, the second prisma, which is called Cuneus, (a wedge) of the sharpnesse, and which also more properly of cutting is called a prisma, the whole surface is 150, and the solidity 90.
13 A prisma compounded of penta…dra's, is either an Hexaedrum or Polyedrum: And the Hexaedrum is either a Parallelepipedum or a Trapezium.
14 A parallelepipedum is that whose opposite plaines are parallelogrammes ê 24. p xj.
Therefore a Parallelepipedum in solids, answereth to a Parallelogramme in plaines. For here the opposite Hedrae or flattes are parallell: There the opposite sides are parallell.
Therefore
15 It is cut into two halfes with a plaine by the diagonies of the opposite sides. 28 p xj. It answereth to the 34. pj.
Let the Prisma be of sixe bases a i, y o, y e, u i, r i, a u. The diagonies doe cut into halfes, by the 10. ex. the opposite bases: And the other opposite bases or the two prisma's cut, are equall by the 3 e. Wherfore two prisma's are comprehended of bases, equall both in multitude and magnitude: therfore they are equall.
And
16 If it be halfed by two plaines halfing the opposite sides, the common bisection and diagony doe halfe one another 39. p xj.
Because here the diameters (such as is that bisection) are halfed betweene themselves [or doe halfe one another.] Let the parallelepipedum a e i o u y, be cut in to y the halfs by two plains, frō s r l m, u i v f, halfing the opposite sides: Here the common section t s, and the diagony a o, doe cut one another.
17 If three lines be proportionall, the parallelepipedum of meane shall be equall to the equiangled p…rallelepipedum of all them. è 36. p x j.
It is a consectary out of the 8 e.
18 Eight rectangled parallelepiped's doe fill a solid place.
19 The Figurate of a rectangled parallelepipedum is called a solid, made of three numbers 17. d vij.
As if thou shalt multiply 1,2,3. continually, thou shalt make the solid 6. Item if thou shalt in like manner multiply 2,3,4. thou shalt make the solid 24. And the sides of that solid 6 solid shall be 1,2,3. Of 24, they shall be 2,3,4.
Therefore
20 If two solids be alike, they have their sides proportionalls, and two meane proportionalls 21 d vij, 19. 21. p viij.
It is a consectary out of the 5 e xxij. But the meane proportionalls are made of the sides of the like solids, to wit, of the second, third, and fourth: Item of the third, fourth, and fifth, as here tho… seest.