Book VI. fore the point z bifecting A G will fall between E and G. And because the fide DE of the right angled triangle DEZ is greater than DZ, and the fide DA of the right angled triangle AZD is greater than D E. Therefore the arch HE will cut the circumference of the circle below A, and fall at T above the point z. Again, fince the fector EDT is greater than the triangle EDZ: the ratio of the fector EDT to the fector E DH, will be greater than the ratio of the triangle EDZ to the sector ED H. But the ratio of the triangle EDZ to the triangle EDA is lefs than the ratio of the triangle EDZ to the sector ED H. Therefore much more will the ratio of the fector EDT to the fector EDH be greater than that of the triangle EZD to the triangle EDA. But [by 33. 6.] in the fame circle as fector is to fector, fo is the arch of the one to the arch of the other; and as arch is to arch, so is one angle at the centre to another: alfo [by 1. 6.] as the triangle EDZ is to the triangle EDA, fo is z E to E A. Therefore the ratio of the angle EDV to the angle EDA, will be greater than the ratio of GE to E A. But the angle G D B is to the angle BDA [by 33. 6.] as the arch BG, is to the arch A B; and GE is to E A, as BG is to AB. Therefore the ratio of the arch BG to the arch AB is greater than that of the right line BG joining the extremes of one, to the right line A B joining the extremes of the other.

Therefore, &c. Which was to be demonftrated.

This is one of Ptolemy's theorems, with his moft ingenious demonstration.

PROP. XIX. THEOR.

If two circles touch one another at ▲, and thro' A be drawn two right lines cutting the circles in D, E, B, C. I fay the right lines A B, AC, AD, AE will be proportional.

For join the points D, B; E, C; and conceive a F to be

309 angle FAE: and fince, in the firft figure the angle at A is common to both the triangles A DB, A E C, but in the fecond figure, the vertical angles at A are equal to one another, the remaining angles A DB, A EC [by 32. 1.] will be equal too; and fo the triangles A DB, AEC will be equiangular. Wherefore [by 4. 6.] it will be as A B İŞ to A C, fo is AD to AE.

Therefore, &c. Which was to be demonftrated.

PROP. XX.

THEOR.

If the right line KL, or its continuation, joining the centres K, L of two circles whofe diameters are EH, DA, be fo divided in м, that Kм be to LM, as AK is to E L. And if from or thro' м be drawn any right line M B cutting the circles in the points G, F, C, B, I say first the fegments of the circles CB, GF will be fimilar. Secondly the rectangle under M G and M в will be equal to the rectangle under MH and M A. Thirdly the rectangle under мÉ and M D equal to the rectangle under MF and м c. Draw the femidiameters GL, LF, CK, KB; and join the points G, H; C, D.

Then, firft, because [by fuppofition] it is as K M is to

Book VI. LM, fo is K B to FL; the femidiameters B K, FL will [by 2.6.] be parallel. And therefore [by 29. 1.] the angle K BC will be equal to the angle L F G.

In like manner, because [by fuppofition] as Kм is to LM, fo is KC to LG; the femidiameters KC, LG will be parallel; and fo the angle KCB equal to the angle LGF. And therefore the remaining angles BKC, FLG will be equal to one another; that is, the arch B C [by 33. 6.] is the fame part of the circumference of the circle A B C, as the arch GF is of the circumference of the circle FGH. And fo BC, FG. will be fimilar fegments of thofe circles.

Secondly, From the centres K, L draw the perpendiculars KR, LS to MB.

Then because BC, FG are fimilar fegments, and RC, SG are half of the right lines CE, GF [by 3. 3.1. Therefore the triangles KR C, LSG will be fimilar; and fo KC, LG will be parallel; and the angles CKD, GLH equal, and CD, GH parallel. Wherefore as MD is to MC, fo is MH to MG. But as MD is to MC, fo [by fuppofition] is M B to M A. Therefore [by II. 5.] as MB is to M A, fo will мH be to м G. Wherefore [by 16.6.] the rectangle under MB and RG will be equal to the rectangle under MA and MH.

Thirdly, because the four points E, F, G, H are in a circle, it will be [by 36. 3. and 16. 6.] as MH is to MG, fo is MF-to ME. But MH is to м G, as MD is to MC: Therefore [by II. 5.] as M F is to ME, fo will MD be to MC; and fo [by 16. 6.] the rectangle under M E and MD will be equal to the rectangle under MF and Mc.

Therefore, &c. Which was to be demonstrated.

SCHOLIUM,

This propofition, and the converfe of the last, are lemmas of Vieta's, in his Apollonius Gallus. By means of the former he defcribés a circle through two given points to touch a right line given in pofition. By means of the other, he defcribes a circle through a given point to touch two given circles,

PROP.

PROP. XXI. PROBL.

If two circles interfect one another, and any right line be drawn cutting the circles, this will be proportionally divided by the circumferences of the

circles.

Let the circles A CB, A FB interfect one another in the points A and B, and let A B be drawn. Alfo let any right line DC cut both the circles, viz. A CB in the points c and H, and the circle AF B. in the points D and F, and the right line AB in the point G: I fay the right line DC is divided proportionally in the points H, G, F, that is, as F C is to GF, fo will DH be to HG.

A

For because in the circle A CBH, the lines AG, GC, G н, Gâ, [by 35. 3. and 16.6.] are proportional, and the lines A G, GF, DG, GB, in the circle A F D B, the lines G C, GF, DG, GH will [by equality] be proportional. And therefore [by D 17.5.] as F C is to GF, fo will D H be to и G.

H

C

F

B

G

Therefore, &c. Which was to be demonstrated.

EUCLID's

ELEMENTS,

воок XI.

A

Solid is that which has length, breadth, and thicknefs.

2. The bound of a folid is a fuperficies a.

3. A right line is perpendicular to a plane, when it makes right angles with all the right lines that touch it, and are drawn in the faid plane.

4. A plane is perpendicular to a plane, when the right lines drawn in one plane perpendicular to the common fection of the planes be alfo at right angles to the other plane.

5. The inclination of a right line to a plane, is, when a perpendicular is drawn from the highest point of that line to the plane, and another line drawn from the point in which the perpendicular cuts the plane, to the end of the faid line which is in the fame plane, viz, the acute angle contained under the joined and inclining line.

6. The inclination of a plane to a plane, is an acute angle contained under the right lines, which being drawn in either of the planes to the fame point of their common fection is at right angles to the common section.

7. Planes are faid to have fimilar inclinations, when the aforefaid angles of their inclination are equal to one another.

8. Parallel planes are those which being produced will

never meet⚫

a It might be better to fay, the bound or bounds of a folid is one or more fuperficies, for a fphere has but one bound or fuperficies, and a cube fix.

5

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