Let therefore the two right lines be ae, and ai: and from the ends of these other two reflected, be iu, and eo, cutting themselves in y; and the two former in u, and o. The reason of the particular right lines made shall be as
the draught following doth manifest. In which the antecedents of the makers are in the upper place: the consequents are set under neathe their owne antecedents.
| The I. is Ptolemeys and Theons I. | |||
| The makers: | The reason made. | ||
| iu, | ye. | ||
| uy, | eo, | ia, | ao. |
| The II. is Theons VI. | |||
| au, | ey. | ||
| ue, | yo, | ai, | io. |
| The III. is Theons III. | |||
| ea, | ui. | ||
| au, | iy, | eo, | oy. |
| The IIII. is Theons II. | |||
| oa, | iu. | ||
| ai, | uy, | oe, | ey. |
| The V. is Ptolemys, II. Theons IIII. | |||
| iy, | ue. | ||
| yu, | ea, | io, | ao. |
| The VI. is Theons V. | |||
| eu, | ai. | ||
| ua, | io, | ey, | yo. |
The businesse is the same in the two other, whether you doe crosse the bounds or invert them.
Here for demonstrations sake we crave no more, but that from the beginning of an antecedent made a parallell be drawne to the second consequent of the makers, unto one of the assigned infinitely continued: then the multiplied proportions shall be,
The Antecedent, the Consequent; the Antecedent, the
Consequent of the second of the makers; every way the reason or rate is of Equallity.
The Antecedent; the Consequent of the first of the makers; the Parallell; the Antecedent of the second of the makers, by the [32. e]. Therefore by multiplication of proportions, the reason of the Parallell, unto the Consequent of the second of the makers, that is, by the fabricke or construction, and the [32. e.] the reason of the Antecedent of the Product, unto the Consequent, is made of the reason, &c. after the manner above written.
For examples sake, let the first speciall example be demonstrated. I say therefore, that the reason of ia, unto ao, is made of the reason of iu, unto uy, multiplied by the reason of ye, unto eo. For from the beginning of the Antecedent of the product, to wit, from the point i, let a line be drawne parallell to the right line ey, which shall meete with ae, continued or drawne out infinitely in n. Therefore, by the [32. e], as ia, is to ao: so is the parallell drawne to eo, the Consequent of the second of the makers. Therefore now the multiplied proportions are thus iu, uy, in, ey, by the 32. e: ye, eo, ey, eo. Therefore as the product of iu, by ye, is unto the product of uy, by eo: So in, is to eo, that is, ia, to ao.