Four Books on Conics (Commandino 1566)

4. & 22. Book 6 Referring to Euclid's Elements, Book VI, Propositions 4 and 22.as the square of f e, so is the rectangle a g f to the square of g f. But as the square of f e is to the square of c d, so is the square of f g to the square of g c. Therefore, by equality of ratios, as the rectangle of a b & f e is to the square of c d, so is the rectangle a g f to the square of g c.

L E M M A I V.

LET a b be to b c, as a d is to d c: and let a c be cut into two equal parts at point e. I say that the rectangle b e d is equal to the square of e c: and likewise the rectangle a d c is equal to the rectangle b d e, and the rectangle a b c is equal to the rectangle e b d.

AFOR since a b is to b c, as a d is to d c; by composition, and taking the halves of the antecedents, and by conversion of ratio, it will be as b e is to e c, so is c e to e d. Therefore, the rectangle b e d is equal to the square of c e. Let the common part be taken away, namely the square of e d. Therefore, what remains, the rectangle a d c, is equal to the rectangle b d e. Again, since the rectangle b e d is equal to the square of c e, let both be taken away from the square of b e. Therefore, the remaining rectangle a b c will be equal to the rectangle e b d. All of which it was required to demonstrate.

A geometric line diagram labeled with points a, e, d, c, b. Points are marked along a horizontal line with vertical ticks. To the right are marginal labels A, B, and C.
B

17. Book 6

C

C O M M E N T A R Y.

AAND by composition, taking the halves of the antecedents, & by conversion of ratio.] Since a b is to b c, as a d is to d c; by composition it will be as a b, b c to c b, so is a c to c d; and the halves of the antecedents, as e b is to b c, so is e c to c d. For a e is half of a c. Therefore, by conversion of ratio, as b e is to e c, so is c e to e d.

B

5. Book 2.
3Let the common part be taken away, namely the square of e d.] For the square of c e is equal to the rectangle a d c together with the square of e d: and the rectangle b e d is equal to the rectangle b d e together with the square of e d. Therefore, the common part being removed, there remains the rectangle a d c equal to the rectangle b d e.

C

6. Book 2
2Again, since the rectangle b e d is equal to the square of e c, let both be taken away from the square of b e.] For since the line a c is cut into two equal parts at e, and the line c b is added to it; the rectangle a b c and the square of c e are equal to the square of b e. Again, both rectangles e b d and b e d are equal to the square of b e. If, therefore, equal quantities are taken away from the square of b e itself, namely the rectangle b e d and the square of c e; it remains that the rectangle a b c is equal to the rectangle e b d.

L E M M A V.

LET a have to b a ratio compounded of the ratio of c to d, and of the ratio of e to f. I say that c has to d a ratio compounded of the ratio of a to b, and the ratio of f to e.

A geometric diagram showing several horizontal line segments of varying lengths labeled a, b, c, d, g. To the right, vertical segments are labeled e and f.

FOR let the ratio of d to g be the same as that which is of e to f. And since the ratio of a to b is compounded of the ratio of c to d, and the ratio of e to f, that is d to g: but the ratio compounded of the ratio of c to d, and d to g is the same as that of c to g: it will be as a is to b, so is c to g. Again, since c has to d a ratio compounded of the ratio of c to g, and the ratio of g to d: but the ratio of c to g has been demonstrated to be the same as that of a to b: and by converting, the ratio of g to d is the same as that of f to e: c will have to d a ratio compounded of the ratio of a to b, and the ratio of f to e.