its half; and from the remainder more than its half; and fo on there will at length remain a magnitude less than c.

For fince AB and C are each finite magnitudes, it is evident that c may be taken fuch a number of times as at length to become greater than AB.

Let, therefore, DE be fuch a multiple of c as is greater than AB, and divide it into the parts DF, FG, GE, each equal to c.

Also from AB take BH greater than its half; and from the remainder AH, take HK greater than its half, and fo on, till there be as many divifions in AB as there are

in DE.

Then because DE is greater than AB; and BH, taken from AB, is greater than its half, but EG, taken from DE, is not greater than its half; the remainder GD will be greater than the remainder HA.

And, again, becaufe GD is greater than HA, and HK, taken from HA, is greater than its half, but GF, taken from GD, is not greater than its half; the remainder FD will be greater than the remainder AK.

. But FD is equal to c by conftruction, whence c is greater than AK; or, which is the fame thing, AK is less than C, as was to be fhewn,

Similar polygons infcribed in circles, are to each other as the fquares of the diameters of thofe circles.

B

M

Let ABCDE, FGHKL be two fimilar polygons, infcribed in the circles ABD, FGK: then will the polygon ABCDE be to the polygon FGHKL as the fquare of the diameter BM is to the square of the diameter GN.

For join the points B, E and A, M, G, L and F, N:

Then, because the polygon ABCDE is fimilar to the polygon FGHKL (by Hyp.), the angle BAE is equal to the angle GFL, and BA is to AE, as GF is to FL. (VI. Def, 1.)

And, fince the angle BAE, of the triangle ABE, is equal to the angle GFL, of the triangle FGL, and the fides about thofe angles are proportional, the angle ABB will alfo be equal to the angle FLG (VI, 5.)

But the angle AEB is equal to the angle AMB, and the angle FLG to the angle FNG (III. 15.), confequently the angle AMB is also equal to the angle ENG.

And fince these angles are equal to each other, and the angles BAM, GFN are each of them right angles (III. 16.), the angle MBA will alfo be equal to the angle NGF (1.28. Cor.), and BM will be to GN as BA is to GF (VI. 5.).

But

But the polygon ABCDE is to the polygon FGHKL as the fquare of BA is to the square of GF (VI. 17.), therefore the polygon ABCDE is alfo to the polygon FGHKL as the fquare of BM is to the fquare of GN.

Q. E.D.

PRO P. III. THEOREM.

A polygon may be infcribed in a circle that shall differ from it by less than any affigned magnitude whatever.

Let ABCD be a circle, and s any given magnitude whatever; then may a polygon be infcribed in the circle ABCD that fhall differ from it by lefs than the magnitude s.

For, let AC, EG be two fquares, the one defcribed in the circle ABCD, and the other about it (IV. 6, 7.); and bifect the arcs AB, BC, CD, DA, in the points m, n, r and's (III. 23.); and join Am, mB, вn, nc, cr, rD, DS and SA:

Then fince the fquare AC is half the fquare EG (I. 32.), and the fquare EG is greater than the circle ABCD, the fquare AC will be greater than half the circle ABCD.

In like manner, if tangents be drawn to the circle through the points m, n, r, s, and parallelograms be de.

fcribed upon the right lines AB, BC, CD, DA, the triangles AmB, вnc, CrD, DSA will each of them be half the parallelogram in which it stands (I. 32.)

But every fegment is lefs than the parallelogram which circumfcribes it; and therefore each of the triangles AmB, BNC, CrD, DSA is greater than half the fegment of the circle which contains it.

And, if each of the arcs Am, mB, &c. be again divided into two equal parts, and right lines be drawn to the points of bifection, the triangles thus formed, may in like manner, be fhewn to be greater than half the fegments which contain them; and fo on continually.

Since, therefore, the circle ABCD is greater than the space s, and from the former there has been taken more than its half, and from the remainder more than its half, &c. there will at length remain fegments which, taken together, shall be less than the excess of the circle ABCD above the fpace s (VIII. 1.), as was to be fhewn.

PRO P. IV. THEOREM.

A polygon may be circumfcribed about a circle that fhall differ from it by lefs than any affigned magnitude whatever.

Let ABCD be the circle, and s any given magnitude whatever; then may a polygon be circumscribed about

the

the circle ABCD, that fhall differ from it by lefs than the magnitude s.

For let the circle ABCD be circumfcribed by the fquare EFGH (IV. 7.), and bife&t the arcs AB, BC, CD, DA with the lines OE, OF, OG and OH; and to the points of bifection draw the tangents kl, mn, pr, st (III. 10.)

Then fince kl is a tangent to the circle, and OE is drawn from the centre through the point of contact, the angle Exk is a right angle (III. 12.), and Ek will be greater than kx (I. 17.) or its equal ka.

But triangles of the fame altitude are to each other as their bafes (VI. 1.); whence the base Ek being greater than the base ka, the triangle Exk will also be greater than the triangle kxA.

And because the triangle Exk is greater than the trianangle kxA, it will also be greater than half the curvelineal space EXA and the fame may be fhewn of any other triangle and the curvelineal space to which it belongs.

In like manner, if the arcs Ax, xB, &c. be again bifected, and tangents be drawn to the points of bisection, the triangles thus formed will be greater than half the curvelineal spaces to which they belong.

Since, therefore, the excefs of the fquare above the circle is greater than the magnitude s, and from the former there has been taken more than its half, and from the remainder more than its half, and so on, there will at length remain spaces, which, taken together, fhall be less than the magnitude s (VIII. 1.), as was to be fhewn.