r, the required radius O'A' by R', and the apothem O'D' by ', we have therefore, ” is an arithmetic mean between R and r, and R' is a geometric mean between R and r'. MEASUREMENT OF THE CIRCLE. The principle which we employed in the comparison of incommensurable ratios (II. 49) is fundamentally the same as that which we are about to apply to the measurement of the circle, but we shall now state it in a much more general form, better adapted for subsequent application. 28. Definitions. I. A variable quantity, or simply, a variable, is a quantity which has different successive values. II. When the successive values of a variable, under the conditions imposed upon it, approach more and more nearly to the value of some fixed or constant quantity, so that the difference between the variable and the constant may become less than any assigned quantity, without becoming zero, the variable is said to approach indefinitely to the constant; and the constant is called the limit of the variable. Or, more briefly, the limit of a variable is a constant quantity to which the variable, under the conditions imposed upon it, approaches indefinitely. As an example, illustrating these definitions, let a point be required to move from A to B under the following conditions: it shall first move over one-half of AB, that is to C; then over one-half of CB, to C; then over one- 4 half of CB, to C"; and so on indefinitely; then the distance of the point from A is a variable, and this variable approaches indefinitely to the constant AB, as its limit, without ever reaching it. As a second example, let A denote the angle of any regular polygon, and n the number of sides of the polygon; then, a right angle being taken as the unit, we have (8), The value of A is a variable depending upon n; and since n may be taken so 4 great that shall be less than any assigned quantity however small, the value of A approaches to two right angles as its limit, but evidently never reaches that limit. 29. PRINCIPLE OF LIMITS. Theorem. If two variable quantities are always equal to each other and each approaches to a limit, the two limits are necessarily equal. For, two variables always equal to each other present in fact but one value, and it is evidently impossible that one variable value shall at the same time approach indefinitely to two unequal limits. 30. Theorem. The limit of the product of two variables is the product of their limits. Thus, if x approaches indefinitely to the limit a, and y approaches indefinitely to the limit b, the product xy must approach indefinitely to the product ab; that is, the limit of the product xy is the product ab of the limits of x and y. 31. Theorem. If two variables are in a constant ratio and each approaches to a limit, these limits are in the same constant ratio. Let x and Y be two variables in the constant ratio m, that is, let x = my; and let their limits be a and b respectively. Since y approaches indefinitely to b, my approaches indefinitely to mb; therefore we have x and my, two variables, always equal to each other, whose limits are a and mb, respectively, whence, by (29), a mb; that is, a and b are in the constant ratio m. PROPOSITION XVI.-THEOREM. 42. The area of a circle is equal to half the product of its circumference by its radius. Let the area of any regular polygon circumscribed about the circle be denoted by A, its perimeter by P, and its apothem which is equal to the radius of the circle by R; then (22), A D B Let the number of the sides of the polygon be continually doubled, then A approaches the area S of the circle as its limit, and P approaches the circumference C as its limit; but 4 and Pare in the constant ratio R; therefore their limits are in the same ratio (31), and we have 43. Corallary I. The area of a circle is equal to the square of its radius multiplied by the constant number . For, substituting for C its value 2πR in [1], we have S πR2. 44. Corollary II. The area of a sector is equal to half the product of its arc by the radius. For, denote the arc ab of the sector a O b by c, and the area of the sector by s; then, since c and s are like parts of C and S, we have (III. 9), 45. Scholium. A circle may be regarded as a regular polygon of an infinite number of sides. In proving that the circle is the limit towards which the inscribed regular polygon approaches when the number of its sides is increased indefinitely, it was tacitly assumed that the number of sides is always finite. It was shown that the difference between the polygon and the circle may be made less than any assigned quantity by making the number of sides sufficiently great; but an assigned difference being necessarily a finite quantity, there is also some finite number of sides sufficiently great to satisfy the imposed condition. Conversely, so long as the number of sides is finite, there is some finite difference between the polygon and the circle. But if we make the hypothesis that the number of sides of the inscribed regular polygon is greater than any finite number, that is, infinite, then it must follow that the difference between the polygon and the circle is less than any finite quantity, that is zero; and consequently, the circle is identical with the inscribed polygon of an infinite number of sides. This conclusion, it will be observed, is little else than an abridged statement of the theory of limits as applied to the circle; the abridgment being effected by the hypothetical introduction of the infinite into the statement. PROPOSITION XVII.-PROBLEM. 46. To compute the ratio of the circumference of a circle to its diameter approximately. FIRST METHOD, called the METHOD OF PERIMETERS. In this method, we take the diameter of the circle as given and compute the perimeters of some inscribed and a similar circumscribed regular polygon. We then compute the perimeters of inscribed and circumscribed regular polygons of double the number of sides, by Proposition X. Taking the last-found perimeters as given, we compute the perimeters of polygons of double the number of sides by the same method; and so on. As the number of sides increases, the lengths of the perimeters approach to that of the circumference (36); hence, their successively computed values will be successive nearer and nearer approximations to the value of the circumference. Taking, then, the diameter of the circle as given 1, let us begin by inscrib ing and circumscribing a square. The perimeter of the inscribed square=4X ¿ × √ 2 = 2√2 (13); that of the circumscribed square 4; therefore, putting P = 4. p 2√2 = 2.8284271, we find, by Proposition X., for the perimeters of the circumscribed and inscribed regular octagons, 2p X P Then taking these as given quantities, we put P3.3137085, p 3.3137085, 3.0614675. 3.0614675, and find by the same formulæ for the polygons of 16 sides P' — 3.1825979, p ́ 3.1214452. Continuing this process, the results will be found as in the following From the last two numbers of this table, we learn that the circumference of the circle whose diameter is unity is less than 3.1415928 and greater than 3.1415926; and since, when the diameter that π = 3.1415927 1, we have C =π (40), it follows within a unit of the seventh decimal place. SECOND METHOD, called the METHOD OF ISOPERIMETERS. This method is based upon Proposition XI. Instead of taking the diameter as given and computing its circumference, we take the circumference as given and compute the diameter; or we take the semi-circumference as given and compute the radius. Suppose we assume the semi-circumference C = 1; then since C = 2πR, we have that is, the value of r is the reciprocal of the value of the radius of the circle whose semi-circumference is unity. Let ABCD be a square whose semi-perimeter each of its sides. Denote its radius OA by R, and its apothem OE by r; then we have 1; then *The computations have been carried out with ten decimal places in order to ensure the accuracy of the seventh place as given in the table. Now, by Proposition XI., we compute the apothem r and the radius R' of the regular polygon of 8 sides having the same perimeter as this square; we find R + r 0.3017767, R' = √ R × ~ = 0.3266407. Again, taking these as given, we put = 0.3017767, R = 0.3266407, and find by the same formulæ, for the apothem and radius of the isoperimetric regular polygon of 16 sides, the values Continuing this process, the results are found as in the following Now, a circumference described with the radius r is inscribed in the polygon, and a circumference described with a radius R is circumscribed about the polygon; and the first circumference is less, while the second is greater, than the perimeter of the polygon. Therefore the circumference which is equal to the perimeter of the polygon has a radius greater than r and less than R; and this is true for each of the successive isoperimetric polygons. But the r and R of the polygon of 8192 sides do not differ by so much as .0000001; therefore the radius of the circumference which is equal to the perimeter of the polygons, that is, to 2, is 0.3183099 within less than .0000001; and we have 47 Scholium I. Observing that in this second method the value of r=1, for |