The Way to Geometry, Book 22
The twenty second Booke, of P. Ramus Geometry, Of a Pyramis.
1. The axis of a solid is the diameter about which it is turned, e 15,19,22 d x j.
THe Axis or Axeltree is commonly thought to be proper to the sphere or globe, as here a e: But it is attributed to other kindes of solids, as well as to that.
2. A right solid is that whose axis is perpendicular to the center of the base.
Thus Serenus and Apollonius doe define a Cone and a Cylinder: And these onelyonly Euclide considered: Yea and indeed stereometry entertaineth no other kinde of solid but that which is right or perpendicular.
3. If solids be comprehended of homogeneall surfaces, equall in multitude and magnitude, they are equall. 10 d x j.
Equality of lines and surfaces was not informed by any peculiar rule; farther than out of reason and common sense, and in most places congruency and application was enough and did satisfie to the full: But here the congruency of Bodies is judged by their surfaces. Two cubes are equall, whose sixe sides or plaine surfaces, are equall, &c.
4. If solids be comprehended of surfaces in multitude equall and like, they are equall, 9 d x j.
This is a consectary drawne out of the generall difinition of like figures, at the 19 e. iiij. For there like figures were defined to be equiangled and proportionall in the shankes of the equall angles: But in like plaine solids the angles are esteemed to be equall out of the similitude of their like plaines: And the equall shankes are the same plaine surfaces, and therefore they are proportionall, equall and alike.
5 Like solids have a treble reason of their homologall sides, and two meane proportionalls. 33. p xj. 8 p xij.
It is a consectary drawne out of the 24 e. iiij. as the example from thence repeated shall make manifest.
6 A solid is plaine or embosed.
7 A plaine solid is that which is comprehended of plaine surfaces.
8 The plaine angles comprehending a solid angle, are lesse than foure right angles. 21. p x j .
For if they should be equall to foure right angles, they would fill up a place, by the 22 e, vj. neither would they at all make an angle, much lesse therefore would they doe it if they were greater.
9 If three plaine angles lesse than foure right angles, do comprehend a solid angle, any two of them are greater than the other… And if any two of them be greater than the other, then may comprehend a solid angle, 21. and 23. p xj.
It is an analogy unto the 10 e vj. and the cause is in a readinesse. For if two plaine angles be equall to the remainder, they shall with that third include no space betweene them: But if thou shalt conceit to fit the plaine to the shankes, with their congruity they should of two make one: but much lesse if they be lesser.
The converse from hence also is manifest.
Euclide doth thus demonstrate it: First if three angles are equall, then by and by two are conceived to be greater than the remainder. But if they be unequall, let the angle a e i, be greater than the angle a e o: And let a e u, equall to a e o, be cut off from the greater a e i: And let e u, be equall to e o. Now by the 2 e, vij. two triangles a e u, and a e o, are equall in their bases a u, and a o. Item a o, and e i, are greater than a i, and a o: And a o, is equall to a u. Therefore o i, is greater than i u. Here two triangles, u e i, and i e o, equall in two shankes; and the base o i, greater than the base i u. Therefore, by the 5 e vij. the angle o e i, is greater than the angle i e u. Therefore two angles a e o, and o e i, are greater than a e i.
10 A plaine solid is a Pyramis or a Pyramidate.
11 A Pyramis is a plaine solid from a rectilineall base equally decreasing.
As here thou conceivest from the triangular base a e i, unto the toppe o, the triangles a o e, a o i, and e o i, to be reared up.
In the pyramis a e i o u, thou seest from the quadrangular base a e i o, unto the toppe u, foure triangles in like manner to be raised.
Therefore
12 The sides of a pyramis are one more than are the base.
The sides are here named Hedrae.
And
13 A pyramis is the first figure of solids.
For a pyramis in solids, is as a triangle is in plaines. For a pyramis may be resolved into other solid figures, but it cannot be resolved into any one more simple than it selfe, and which consists of fewer sides than it doth.
Therefore
14 Pyramides of equall heighth, are as their bases are 5 e, and 6. p xij.
And
15 Those which are reciprocall in base and heighth are equall 9 p xij.
These consectaries are drawne out of the 16, 18 e. iiij.
16 A tetraedrum is an ordinate pyramis comprehended of foure triangles 26. d xj.
As here thou seest. In rectilineall plaines we have in the former signified, in every kinde there is but one ordinate figure: Amongst the triangles the equilater: Amongst the quadrangles, the Quadrate: so now of all kinde of Pyramides, there is one kinde ordinate onelyonly, and that is the Tetraedrum And yet not every Tetraedrum is such, but that only which is comprehended of triangles, not onelyonly severally ordinate, but equall one to another altogether alike.
Therefore
17 The edges of a tetraedrum are sixe, the plaine angles twelve, the solide angles foure.
For a Tetraedrum is comprehended of foure triangles, each of them having three sides, and three corners a peece: And every side is twise taken: Therefore the number of edges is but halfe so many.
And
18 Twelve tetraedra's doe fill up a solid place.
Because 8. solid right angles filling a place, and 12. angles of the tetraedrum are equall betweene themselves, seeing that both of them are comprehended of 24. plaine right-angles. For a solid right angle is comprehended of three plaine right angles: And therefore 8. are comprehended of 24. In like manner the angle of a Tetraedrum is comprehended of three plaine equilaters, that is of sixe third of one right angle: and therefore of two right angles: Therefore 12 are comprehended of 24.
And
19 If foure ordinate and equall triangles be joyned together in solid angles, they shall comprehend a tetraedrum.
This fabricke or construction is very easie, as you may see in these examples: For if thou shalt joyne or fold together these triangles here thus expressed, thou shalt make a tetraedrum.
20. If a right line whose power is sesquialter unto the side of an equilater triangle, be cut after a double reason, the double segment perpendicular to the center of the triangle, knit together with the angles thereof shall comprehend a tetraedrum. 13 p xiij.
For a solid to be comprehended of right lines understand plaines comprehended of right lines, as in other places following.
As here, Let first a e be the right line whose power is sesquialter unto a i the side of the equilater triangle, as in the forme was manifest at the 13 e xij. And let it be by the 29 e v. be cut in a double reason in o: And let the double segment a o, be perpendicular to the equilater triangle u y s, unto the center r, by the 7 e xxj. And let l r be knit with the angles, by l u, l s, l y. I say that the triangles u y s, u s l, u y l, are equilater and equall, because all the sides are equall. First the three lower ones are equall by the grant: And the three higher ones are equall by the 9 e xij. And every one of the higher ones are equall to the under one. For if a Circle bee supposed to bee circumscribed about the triangle, the side shall be of treble power to the ray u r, by the 12 e x viij. But the higher one also is of treble power to the same ray, as is manifest in the first figure of the ray o i, which is for the ray of the second figure u r. For as a o, is to o i, so by the 9 e viij, is o i, unto o e: And by the 25 e iiij, as the first rect line a o, is unto the third o e: so is the quadrate a o, unto the quadrate o i. And by compounding a o with o e; As a e is to o e … so are the quadrates a o: and o i, that is, by 9 e xij, the quadrate a i, unto the quadrate o i, But a e is the triple of o e. Therefore the quadrate a i, is the triple of the quadrate o i. Wherefore the higher side equall to a i, is of treble power to the ray: And therefore also all the sides are equall: And therefore againe the triangles themselves are equall.