So let the second of Ptolemy to be taught, which in our
Table aforegoing is the fifth. I say therefore that the reason of io, unto oa; is made of the reason of iy, unto yu, and the reason of ue, unto ea. For now againe, from the beginning of the Antecedent of the Product i, let a line be drawne parallell unto ea, the Consequent of the second of the Makers, which shall meete with eo, drawne out at length, in n: therefore, by the [32. e.] as io, is to ao; so is en, unto ea. Therefore now again the multiplied proportions are thus:
| ue, | ea, | ue, | ea. |
| iy, | yu, | en, | ue; |
by the [32. e]. Therefore, by multiplication of proportions, the reason of en, unto ea, that is, of io, unto oa, is made of the reason of iy, unto yu, by the reason of ue, unto ea.
It shall not be amisse to teach the same in the examples of Theon. Let us take therefore the reason of the Reflex, unto the Segment; And of the segments betweene themselves; to wit, the 4. and 6. examples of our foresaid draught: I say therefore, that the reason of oe, unto ey, is made of the reason oa, unto ai, by the reason of iu, unto uy. For from the end o, to wit, from the beginning of the Antecedent of the product, let the right line no, be drawne parallell to uy. It shall be by the [32. e.] as oe, is to ey: so the parallell no, shall be to uy: but the reason of no, unto uy, is made of the reason of oa, unto ai, and of iu, unto uy: for the multiplied proportions are,
| iu, | uy, | iu, | uy. |
| oa, | ai, | on, | iu. |
by the [32. e.]
Againe, I say, that the reason of ey, unto yo, is compounded of the reason of eu, unto ua, and of ai, unto io.
Theon here draweth a parallell from o, unto ui. By the generall fabricke it may be drawne out of e, unto ui.
It shall be therefore as ey, is unto yo, so en, shall be unto oi. Now the proportions multiplied are,