The Way to Geometry, Book 24
Of Geometry the twentie fourth Book. Of a Cube.
1 A Rightangled parallelepipedum is either a Cube, or an Oblong.
2 A Cube is a right angled parallelepipedum of equall flattes, 25. d xj.
As here thou seest in these two figures.
Therefore
3 The sides of a cube are 12. the plaine angles 24. the solid 8.
Therefore
4 If sixe equall quadrates be joyned with solid angles, they shall comprehend a cube.
As here in these two examples.
Therefore
5 If from the angles of a quadrate, perpend…culars equall to the sides be tied together aloft, they shall comprehend a Cube. è 15 p xj.
It is a consectary following upon the former consectary: For then shall sixe equall quadrates be knit together:
6 The diagony of a Cube is of treble power unto the side.
For the Diagony of a quadrate is of double power to the side, by the 12 e, xij. And the Diagony of a Cube is of as much power as the side the diagony of the quadrate, by the same e. Therefore it is of treble power to the side.
7 If of foure right lines continually proportionally the first be the halfe of the fourth, the cube of the first shall be the halfe of the Cube of the second è 33 p xj.
It is a consectary out of the 25 e, iiij. From hence Hippocrates first found how to answer Apollo's Probleme.
8 The solid plaine of a cube is called a Cube, to wit, a solid of equall sides. 19, d vij.
Therefore
9 It is made of a number multiplied into his owne quadrate.
So is a Cube made by multiplying a number by it selfe, and the product againe by the first. Such are these nine first cubes made of the nine first Arithmeticall figures.
This is the generall invention of a Cube, both Geometricall and Arithmeticall.
10 If a right line be cut into two segments, the Cube of the whole shall be equall to the Cubes of the segments, and a double solid thrice comprehended of the quadrate of his owne segment and the other segment.
As for example, the side 12, let it be cut into two segments 10 and 2. The cube of 12. the whole, which is 1728, shall be equall to two cubes 1000, and 8 made of the segments 10. and 2. And a double solid; of which the first 600. is thrise comprehended of 100. the quadrate of his segment 10. and of 2. the other segment: The second 120. is thrice comprehended of 4, the quadrate of his owne segment, and of 10. the other segment. Now 1000—600—120. —8, is equall to 1728: And therefore a right &c.
But the genesis of the whole cube will make all this whole matter more apparant, to wit, how the extreme and meane solids are made. Let therefore a cube be made of three equall sides, 12, 12, and 12: And first of all let the second side be multiplied by the first, after this manner: And not adding the severall figures of the same degree, as was taught in multiplication, but multiply againe every one of them by the other side; and lastly, add the figures of the same degrees severally, thus:
Therefore
11. The side of the first severall cube is the other side of the second solide: And the quadrate of the same side is the other side of the first solide, whose other side is the side of the second cube; and the quadrate of the same other side is the other side of the second solid.
In that equation therefore of foure solids with one solid, thou shalt consider a peculiar making and composition: First that the last cube be made of the last segment 2… Then that the second solid of 4, the quadrate of his owne segment, and of 10, the other segment be thrise comprehended: Lastly that the first solid of 100, the square of his owne segment 10 and the other segment 2, be also thrice comprehended: Lastly, that the Cube 1000, be made of the greater segment 10. Out of this making &c.
And thus much of the Cube: Of other sorts of parallelepipedes, as of the Oblong, the Rhombe, the Rhomboides, and of the Trapezium, and many flatted pentaedra's there is no peculiar stereometry. The measuring of a Prisma hath in the former beene generally declared, and is now onelyonly farther be made more plaine by speciall examples; as here:
The plaine of the perimeter of the base 20, and the altitude 5 is 100. This added to 25 and 25, both the bases that is to 50, maketh 150, for the whole surface. Now the plaine of 25 the base, and the heighth 5 is 125, for the whole solidity.
So in the Oblong, the plaine of the base's perimeter 20, and the heighth 11, is 220, which added to the bases 24 and 24, that is 48, maketh 268, for the whole surface. But the plaine of the base 24, and the height 11, is 264, for the solidity.
The same also Geodesie or manner of measuring is used in the measuring of rectangled walls or gates and doores, which have either any window, or any hollow or voyde space cut out of them, if those voyde places be taken out of them; as here thou seest in the next following example. The thickenesse is 3 foote; the breadth 12, the heighth 11. Therefore the whole solidity is 396. Now the Gate way is of thickenesse 3 foote, of breadth 4: of heighth 6. And therefore the whole solidity of the Gate is 72 foote. But 396—72 are 3…4. Therefore the solidity of the rest of the wall remaining is 324.
In the second example, the length is 10. The breadth 8, the heighth 7. Therefore the whole body if it were found, were 560 foote. But there is an hollow in it, whose length is 6, breadth 4, heighth 7. Therefore the cavity or hollow place is 168. Now 560 — 168 is 492, for the solidity of the rest of the sound body.
Thus are such kinde of walls whether of mudde, bricke, or stone, of most large houses to bee measured. The same manner of Geodesy is also to be used in the measuring of a Rhombe, Rhomboides, Trapezium or mensall, and any kinde of multangled body. The base is first to be measured, as in the former: Then out of that and the heighth the solidity shall be manifested: As in the Rhombe the base is 24, the heighth 4. Therefore the solidity is 96.
In the Rhomboides, the base is 64 3…/12…: The heigh 11. Therefore the solidity is 1028 44/1…9.
The same is the geodesy of a trapezium, as in these examples: The surface of the first is 198: The solidity 192 ½.
The surface of the second is 158 3/49; The solidity is 91 29/4….
The same shall be also the geodesy of a many flatted Prisma: As here thou seest in an Octoedrum of a sexangular base: The surface shall bee 762 6/11: The solidity 1492 4/11.
And from hence also may the capacity or content of vessells or measures, made after any manner of plaine solid bee esteemed and judged of as here thou seest. For here the plaine of the sexangular base is 41 1/7; (For the ray, by the 9 e xviij, is the fide:) and the heighth 5, shall be 205 5/7. Therefore if a cubicall foote doe conteine 4 quarters, as we commonly call them, then shall the vessell conteine 822 6/7 quartes, that is almost 823 quartes.