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 ae be the right line whose power is sesquialter unto ai 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 ao, be perpendicular to the equilater triangle uys, unto the center r, by the [7 e xxj]. And let lr be knit with the angles, by lu, ls, ly. I say that the triangles uys, usl, uyl, 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 ur, by the [12 e xviij]. But the higher one also is of treble power to the same ray, as is manifest in the first figure of the ray oi, which is for the ray of the second figure ur. For as ao, is to oi, so by the [9 e viij], is oi, unto oe: And by the [25 e iiij], as the first rect line ao, is unto the third oe: so is the quadrate ao, unto the quadrate oi. And by compounding ao with oe; As ae is to oe; so are the quadrates ao; and oi, that is, by the [9 e xij], the quadrate ai, unto the quadrate oi, But ae is the triple of oe. Therefore the quadrate ai, is the triple of the quadrate oi. Wherefore the higher side equall to ai, is of treble power to the ray: And therefore also all the sides are equall: And therefore againe the triangles themselves are equall.