5. The greater segment continued to the halfe of the whole, is of power quintuple unto the said halfe, that is, five times so great as it is: and if the power of a right line be quintuple to his segment, the remainder made the double of the former is cut proportionally, and the greater segment, is the same remainder. 1. and 2. p xiij.
This is the fabricke or manner of making a proportionall section. A threefold rate followeth: The first is of the greater segment.
Let therefore the right line ae, be cut proportionally in i: And let the greater segment be ia: and let the line cut be continued unto io, so that oa, be the halfe of the line cut. I say, the quadrate of io, is in power five times so great, as ys, the power of the quadrate of ao. Let therefore of ao, be made the quadrate iosr: We doe see the quadrate ua, to be once contained in the quadrate si. Let us now
teach that it is moreover foure times comprehended in lmn, the gnomon remaining: Let therefore the quadrate aeiu, be made of the line given: And let ri, be continued unto f. Here the quadrate ae, is ([14. e xij].) foure times so much as is that au, made of the halfe: and it is also equall to the gnomon lmn: For the part iu, is equall to ry; first by the grant, seeing that ai, is the greater segment, from whence ry, is made the quadrate, because the other Diagonall is also a quadrate: Secondarily the complements sy, and yi, by the [19. e x], are equall: And to them is equall af. For by the [23. e x]. and by the grant, it is the double of the complement yi. Therefore it is equall to them both. Wherefore the gnomon lmn, is equall to the quadruple quadrate of the said little quadrate: And the greater segment continued to the halfe of the right line given is of power five fold to the power of ao.
The converse is apparent in the same example: For seeing that io, is of power five times so much as is ao; the gnomon lmn, shall be foure times so much as is ua: Whose quadruple also, by the [14. e xij], is av. Therefore it is equall to the gnomon. Now aj, is equall to ae: Therefore it is the double also of ao, that is of ay: And therefore by the [24. e x]. it is the double of at: And therefore it is equall to the complements iy, and ys: Therefore the other diagonall yr, is equall to the other rectangle iv. Wherefore, by the [8 e xij]. as ev, that is, ae, is to yt, that is ai: so is ai, unto ie; Wherefore by the [1 e], ae, is proportionall cut: And the greater segment is ai, the same remaine.
The other propriety of the quintuple doth follow.
6 The lesser segment continued to the halfe of the greater, is of power quintuple to the same halfe è 3 p xiij.
As here, the right line ae, let it be cut proportionally in i: And the lesser ie, let it be continued even unto o, the halfe of the greater ai. I say, that the power of oe, shall be five times as much as is the power of io. Let a quadrate