Archimedis Opera cum Apollonii Conicorum Libri IIII (1675) - Page 24

Prop. IV.

Fig. 8. 9.

Given two unequal magnitudes (A, B) and a circle (CDEF), to inscribe a polygon in the circle, and to circumscribe another, such that the side of the circumscribed polygon to the side of the inscribed polygon may have a smaller ratio than the greater magnitude (A) to the smaller (B).

a 3 of this. b 1. 10. el. c const. 4. 6. & 8. 5. 15. 5.

Let OP : OQ

A : B. And having described a semicircle on OP, let OQ be adapted, and PQ be joined. Then the circumference CDEF is bisected, and its half DCF, and the half of this CD, and so continuously, until the angle DGK, half the angle DGH, is equal to the angle POR, which is the angle POQ. And through K, let a tangent LM be drawn, meeting the radii GD, GH produced at L, M; then DH is joined. From the bisection, it is clear that the straight line LM is a side of a polygon circumscribable about the circle, and DH is a side of an inscribable polygon. Now because angles DGN, ROQ are equal, and angles GND, OQR are right angles, the triangles DGN, ROQ will be similar. Wherefore GD (GK) : GN

OR : OQ, which is OP : OQ. But GK : GN

LK : DN

LM : DH. Therefore LM : DH is less than (OP : OQ, which is) A : B. Q. E. F.

Prop. V.

Again, if there are two unequal magnitudes, and a sector is given, a polygon can be described about the sector, and another inscribed, so that the side of the circumscribed to the side of the inscribed has a smaller ratio than the greater magnitude to the smaller.

It is completed in the same way as the preceding.

Prop. VI.

Fig. 10. 11.

Given a circle (G) and two unequal magnitudes (A, B), to circumscribe a polygon about the circle, and to inscribe another, such that the circumscribed to the inscribed has a smaller ratio than the greater magnitude (A) to the smaller (B).

a 3 of this. b 13. 6. c 4 of this. d cor. 20. 6.

Let the line X : Z :: A : B; and between X and Z let a mean proportional Y be found; then let a polygon be inscribed in the given circle, and another be circumscribed, such that the side LM of the latter to the side DH of the former has a smaller ratio than X : Y. I say it is done. For the ratio of LM to DH is duplicated (that is, the ratio of the circumscribed figure to the inscribed).