similar to the inscribed one whose side is ab. Let M be the middle of the intercepted arc, and draw Ma, Mb, and the tangent BMC; then aM, Mb will be two consecutive sides of an inscribed polygon, having double the number of sides that the polygon has. whose side is ab; and,

B

D

consequently, BC being a tangent at M, meeting the tan-" gents at a and b, must, by the proposition, be the side of a polygon having double the number of sides that the polygon has whose side is ab.

Schol. 2. If polygons be thus successively circumscribed about the circle, their perimeters will decrease as the number of sides increase. For BC is less than AB+AC, and consequently aB+BC+Cb<aA+Ab. Now aB+Cb=BC, and aA+Ab is equal to a side of the first circumscribed polygon: hence two sides of the second circumscribed polygon are together less than one side of the first; and, therefore, the whole perimeter of the second is less than that of the first. It is obvious, that with respect to the inscribed polygons, the perimeters increase in the same circumstance; thus, the two sides aM, Mb being together longer than ab, it follows that the perimeter of the second inscribed polygon exceeds that of the first.

The successive circumscribed polygons that we have been considering, continually approach nearer and nearer

towards coincidence with the circle. For OB is nearer an equality to the radius Oa of the circle, than OA, because in the two right-angled triangles OaB, OaA, each having the common side Oa, we have aA longer than aB, and therefore OB is less than OA; and in every succeeding polygon the difference between the radius of the circle and the distance of the centre from the remotest points in the perimeter will, in like manner, perpetually diminish; so that the perimeters continually approach towards coincidence with the circumference, and we have already seen that the perimeters continually diminish.

Now it is plain that if a series of magnitudes continually approach nearer and nearer towards coincidence with any proposed magnitude, and at the same time continually diminish, the magnitude to which they approach must be smaller than either of the approaching terms; we are, therefore, warranted in asserting that the circumference of a circle is a shorter line than the perimeter of any circumscribed polygon.

In a similar manner, by considering the successive inscribed polygons, it appears that they also continually approach towards coincidence with the circle. For Od is nearer an equality to the radius than OD, since the chord aM is shorter than ab, (B. III, Prop. Iv;) so that in each succeeding polygon the perimeter approaches nearer to coincidence with the circumference, and it has been shown that these perimeters successively increase. Hence we may infer that the circumference of a circle is a longer line than the perimeter of any inscribed polygon.

PROPOSITION X.

THEOREM. Two polygons may be formed, the one within, and the other about a circle, that shall differ from each other by less than any assigned magnitude, however small.

Let M represent any assigned surface. It is to be shown that two polygons may be described, the one within, and the other about the circle whose centre is O, which will differ from each other by a magnitude less than M.

Let N be the side of a square, whose surface is less than the surface M, and inscribe in the circle a chord an equal to the line N. Then, by the methods already explained, inscribe in the circle a square, a hexagon, or indeed any regular polygon: let the arcs which its sides subtend be bisected; the chords of the half arcs will be the sides of a regular polygon, having double the num

ber of sides. Let, now, the arcs subtended by the sides of this second polygon be in like manner bisected; the chords will form a third polygon, having double the number of sides that the second has. Continue these successive bisections till the arcs become so small as to be each less than the arc an, their chords forming the inscribed polygon abcd, &c. Circumscribe the circle with a similar polygon ABCD, &c., (B. V, Prop. 1x ;) then this last will exceed the former by a magnitude less than the proposed magnitude M.

From the centre O draw the lines Oa, OA, Oh, OH, and produce ho to d; then the polygon abcd, &c., is composed of as many triangles equal to Oah as the polygon has sides; and in like manner the polygon ABCD, &c., is composed of as many triangles equal to OAH as this polygon has sides; and as the polygons have each the same number of sides, the inscribed is the same multiple of the triangle Oah that the circumscribed is of the triangle OAH. In the two right-angled triangles OAh, OAa, the side Oh, of the first, is equal to Oa, a side of the second; and OA is a common hypothenuse to both, hence those triangles are equal, (B. II, Prop. VIII, Cor. 2;) and the angle AOh is equal to AOɑ, so that the vertical angle hOa of the triangle hOa is bisected by the line Om, consequently the triangle Oml is half the triangle Oah, (B. I, Prop. v, Cor. 1.) In a similar manner it may be shown that the triangle

OhA is half the triangle OAH. Hence the inscribed polygon is the same multiple of Omh, that the circumscribed polygon is of OhA; and, consequently,

Whence, by division, we have

OhA: OhA-Omh :: circ. pol. : circ. pol.—ins. pol.; that is, OhA: Amh:: circ. pol. : circ. pol.—ins. pol.

Now OhA is a right-angle; and since AO bisects the angle A of the isosceles triangle aAh, it is perpendicular to ah: therefore the triangles OhA, Amh are similar. Consequently, (B. IV, Prop. XII,)

OhA: Amh :: Oh2 : hìn2 :: hd: ha; whence hd ha circ. pol.: circ. pol.-ins. pol.

Now a circumscribed square, that is to say, hd, is greater than the polygon ABCD, &c., since the surfaces of circumscribed polygons diminish as their sides increase in number; so that in the last proportion, the first antecedent is greater than the second: consequently the first consequent is greater than the second; that is, the excess of the circumscribed polygon above the inscribed is less than ha3, and therefore less than N2 or than M.

Cor. As the circle is obviously greater than any inscribed polygon and less than any circumscribed one, it follows that a polygon may be inscribed or circumscribed, which will differ from the circle by less than any assignable magnitude.