Let us now, in order to avoid confusion, denote the inscribed polygon whose side is ab by p, the corresponding circumscribed polygon by P; the inscribed polygon of double the number of sides by p', and the similar circumscribed polygon by P'. Then it is plain that the space Oad is the same part of p that OaA is of P, that OaM is of p', and that OaBM is of P'; for each of these spaces requires to be repeated the same number of times to complete the several polygons to which they respectively belong. Hence then, and because magnitudes are as their like multiples, it follows that whatever relations are shown to exist among these spaces will be true also of the respective polygons of which they form part. Now the right angled triangles ODa, OAa, BMA are similar: the two first furnish the proportion OD: Oa:: Oa: OA, or which is the same thing, OD: OM :: OM: OA; and consequently, since triangles of the same altitude are as their bases, it follows that ODa: OMȧ:: OMa: OAα, that is, the triangle OMa is a mean between ODa and OAa; consequently the polygon p' is a mean between the polygons p and P. Again, the similar triangles ODa, BMA give the proportion OD: Oa:: BM: BA, or which is the same thing, OD: OM :: aB BA; and consequently, since triangles of the same altitude are as their bases, it follows that ODa: OMa:: OaB: OBA, therefore ODa +OMa: 20Da :: OaB+OBA: 20aB, consequently P+p' 2p :: P : P. Scholium. It was proved in proposition XI. that a circle is equivalent to the rectangle contained by its radius, and a straight line equivalent to half its circumference. In order, therefore, to construct a rectangle equivalent to any given circle, it would only be necessary, from having the radius, to draw a straight line equal to half the circumference. But this is a problem that has never yet been effected, so that the equivalent rectangle remains still undetermined, and therefore the quadrature of the circle, as this problem is called, is not capable of being rigorously ascertained. This, however, is a circumstance little to be regretted, for it has been shown (Prop. X. Cor.) that polygons may be inscribed in, and circumscribed about, a circle that shall approach so near to coincidence with it as to : differ from it by a magnitude less than any that can be possibly assigned a degree of approximation obviously equivalent to perfect accuracy, since no magnitude can be found sufficiently small to denote its difference therefrom. The principal object of inquiry then should be, at least in a practical point of view, how we may most expeditiously carry on the approximation alluded to; and the problem above furnishes us with one of the best elementary methods for this purpose that can be given. P = = 2p. P = Let us represent the radius of the circle by 1, and let the first inscribed and circumscribed polygons be squares: the side of the former will be 2, and that of the latter 2, so that the surface of the former will be 2, and that of the latter 4. Now it has been proved in the proposition that the surface of the inscribed octagon, or, as we have denoted it, p', will be a mean between the two squares p and P, so that p'√8=2·8284271. Also from the proportion p+p: 2p: P: P' we obtain the numerical value of the circumscribed octagon, that is, 3.3137085. Having thus obtained numerical expressions for the inscribed and circumscribed polygons of eight sides, we may from these, by an application of the same two proportions in a similar way, determine the surfaces of those of sixteen sides, and thence the surfaces of polygons of thirty-two sides, and so on till we arrive at an inscribed and circumscribed polygon, differing from each other, and consequently from the circle, so little that either may be considered as equivalent to it. The subjoined table exhibits the area, or numerical expression for the surface, of each succeeding polygon carried to seven places of decimals. Number of sides. Area of the inscribed polygon. Area of the circumscribed H It appears then that the inscribed and circumscribed polygons of 32768 sides differ so little from each other that the numerical value of each, as far as seven places of decimals, is absolutely the same and as the circle is between the two, it cannot, strictly speaking, differ from either so much as they do from each other; so that the number 3.1415926 expresses the area of a circle whose radius is 1, correctly, as far as seven places of decimals. We may, therefore, conclude that were the absolute quadrature of the circle attainable, it would exactly coincide with the above number, as far at least as the seventh decimal place, which is an extent even beyond what the most delicate numerical calculations are ever likely to require. Were it necessary, however, the approximation might be continued to double the number of decimals: it has indeed been carried by some to a much greater length than this. Ludolph van Ceulen had the patience to extend the approximation as far as the thirty-sixth place of decimals, by a method somewhat different indeed from that above described, but requiring an equal degree of labour and attention. Since his time the quadrature of the circle has been approached still nearer by other methods. An infinite series was discovered by Machin, by which he reached the quadrature as far as the 100th place of decimals, and which proved to be 3.1415926535,8979323846,2643383279,5028841971, 8628034825,3421170679; and even this number has been extended by later mathematicians thirty or forty figures further. Having then found the numerical expression for the surface of a circle whose radius is 1, we readily find the area of any circle whatever; for since the surfaces are as the squares of the radii, we have only to multiply the square of the radius of any proposed circle by the number 3.14159, &c, and the product will be the area. Also, since the surface of a circle is equivalent to half the circumference multiplied by the radius (Prop. XI.), it follows that when the radius is 1 the half circumference must be 3.14159, &c.; or since the circumferences of circles are as their radii, when the diameter is 1, the circumference will be 3.14159, &c., so that the circumference of any circle is found by multiplying its diameter by 3.14159, &c., or, as is usual, simply by 3-1416. For the ordinary purposes of mensuration the circumference will be determined with suf 7 22 ficient precision by multiplying the diameter by 22, and dividing the product by 7, which is the approximation discovered by Archimedes. The fraction is equal to 3·1428, and consequently the circumference, as determined by this last method, differs from the truth by rather more than a thousandth part of the diameter, which in most practical cases is too inconsiderable to deserve notice. BOOK VIII. PROPOSITION I. PROBLEM. To divide a given straight line into any proposed number of equal parts. Let it be proposed to divide the straight line AB into a certain number of equal parts. From one extremity A draw an indefinite straight vided into the same number of equal parts that AB is to be divided into; let the point E be at the distance of two of those parts from C, then if CBD be drawn, making BD CB, and the points D,E be joined, the line AB will be divided in F; so that BF will be one of the required parts of AB. For draw GB; then since CG GE and CB=BD, the sides CE, CD of the triangle CED are divided proportionally by the line GB; therefore (Prop. V. B. VI.) GB is parallel to ED or EF: therefore, in the triangle AGB, we have the proportion AG: EG:: AB: FB, but AG is a given multiple of EG; therefore (Prop. XI. B. V.) AB is the same multiple of FB. Otherwise as follows:-From one extremity A draw the indefinite straight line AC, making A. any angle with AB, and from the other extremity draw BD, making an equal angle with BA. Upon BD repeat the distance AC as many times, wanting one, E B |