or, as 78125:1:: 20'48: 26'2144; and 252144, will be the area of the square PRSB, circumscribed about the circle e. And, √26.2144 =512, will be the value of the side of square P R S B, and diameter of circle e. Produce B P, a side of the square PRSB to A, making A B equal to parts of P B, = 38 (5'12), = x 25 512, 15625 x 5'12, = 8; and 8 will be the value of A B, the side of the square A B C D. Then, 82 = 64, will be the area of square A B C D. = 35 (8), = 3 × 8, 78125 x 8, 50, will be the area of 32 16 25 circle X; (50), = 18 × 50, = '64 × 50, = 32, will be the area of the inscribed square H I K L, and is equal to half the area of square A B C D. The area of circle X is 50. The area of circle e is 20:48. And √50 × 20°48, = √1024, = 32, is the area of the square H I K L, inscribed in the circle X ; and the area of square HIK L is the mean proportional between the area of circle X, and the area of circle e. Hence, the area of every square, is a mean proportional, between the area of a circle circumscribed about it, and the area of a circle of which the diameter is equal to parts of the diameter of the circumscribed circle.* Again, the area of every circle, is a mean proportional, between the area of a square circumscribed about it, or, described on its diameter, and the area of a square of which the perimeter is equal to the circumference of the circle; and this may be demonstrated, either on a true or false value of the circumference and area of a circle. * At this point I should have directed attention to the fact, that the area of square A BCD, is equal to three and one-eighth times the area of circle e. Hence, the area of every square is equal to three and one-eighth times the area of a circle, of which the diameter is equal to § parts of the side of the square. For example: Let the perimeter of the square EBFG, be equal to the circumference of circle X, and let the circumference of a circle of which the diameter is unity be 31416, and its area 7854. The value of the side of square ABCD, and diameter of circle X, is 8. Then, 82 = 64, will be the area of square A B C D. 8 x 31416251328, will be the circumference of circle X. As I 7854 64 the area of square ABCD, to 7854, also 502656, will be the area 50 2656; or, 64 x = = of circle X. As I 7854 50 2656, the area of circle X, to 39'47860224; or, 50'2656 x 64, also 39'47860224, will = be the area of the square E F G H; √39'47860224 6.2832, must be the value of the side of the square EBFG; and 6.2832 x 4 = 25'1328, must be the value of the perimeter of square E BFG; and is equal to the circumference of circle X. = Then, on this data, the area of square A B C D, is 64. The area of square E B F G, is 3947860224. And √64 × 3947860224,= √252663054386, 502656; and is equal to the area of circle X. And the area of circle X is the mean proportional between the area of the circumscribed square A B C D, and the area of the square EB FG, of which the perimeter is equal to the circumference of circle X. Any other hypothetical data may be taken to represent the circumference and area of a circle, of which the diameter is unity, and if the calculations be worked out, they will be found to produce a similar result. Again, let the side of square A B C D be 99. Then, 9′92 = 9801, will be the area of square A BCD; 31⁄2 (98·01) = × 9801, 78125 × 9801, 76'5703125, will be 35 the area of circle X; 18 (76·5703125), = 13 × 76·5703125, = = = ='64 × 765703125, 49'005, will be the area of the inscribed square H I K L; and is equal to half the area of square A B C D. Let the side of the square M B N O, be equal to the radius of circle X; it will be equal to half the side of square A B CD, = (9′9) = 4'95; and 4'952 = 24.5025, will be the area of the square M N O B. = = Then, the area of square A B C D is 98.01. The area of square M B NO is 24:5025. And √9801 x 24.5025, √2401490025, 49′005; and is equal to the area of the square HI KL. And the area of square H I K L, is the mean proportional between the area of square ABCD, circumscribed about the circle X, and the area of square M B N O, described on a radius of circle X. Hence, if a square be inscribed in any circle, the area of such square is a mean proportional between the area circumscribed about the circle, and the area of a square described on a radius of the circle. of a square In the last example, the comparison has 'reference to three squares, and it is impracticable to adopt false data. In the first example, the comparison has reference to two circles and a square, and it is impossible to arrive at corret results, except on true data. In the intermediate example, the comparison has reference to two squares and a circle, and the facts may be demonstrated on either true or false data.* * I have already directed attention, in a note, to one fact which had escaped my notice when writing the letter, and I may now observe that I have omitted to notice some very important facts, of the particular character to which this diagram has a special reference, and to which I would now request attention, as being of the utmost importance in the consideration of this question. I have stated that the circles a, b, c, d, e, represent a succession of cir The demonstrations by means of diagram No. 15, (see Fig. XXIV.) are of the utmost importance, and merit cles, each containing twice the area of the one that precedes it. Take any three of these adjacent circles, say the circles c, d, and e, and let the area of the intermediate circle d be 32. Then, the area of circle e will be 64, and the area of circle c will be 16; and 64 × 16, √1024, 32, is the area of the intermediate circle d, and is the mean proportional between the area of circle e and the area of circle c. = = = = = × 64, 1.28 × 64, The area of circle e is 64. Therefore, 1? (64), 81 92, will be the area of a square circumscribed about the circle e; and 81.92 will be the value of the diameter of circle e. Therefore, 31(81.92), = 25 (81.92), = √2 × 81.92, = √9.765625 × 81.92, = 625 800, will be the circumference of circle e. The area of square d is 32. Therefore, § (32), = §} × 32, = 1·28 × 32, 40.96, will be the area of a square circumscribed about the circle d; 40.96 6.4, will be the diameter of circle d; and 64 × 3.125 = 20, will be the circumference of circle d. = = The area of circle c is 16. Therefore, 3 (16), = }} × 16, = 1.28 × 16, = = 20·48, will be the area of a square circumscribed about the circle c; and 20.48, will be the diameter of circle c. Therefore, 3 (√20·48), 25(√20·48), = √2 × 20·48, = √9.765625 × 20·48, 200; will be the circumference of circle c. Then, the circumference of circle e is/800. The circumference of circle c is 200. And √(√800 × √200), = √(160000), is the circumference of the intermediate circle d, and is the mean proportional between the circumference of the circle e, and the circumference of the circle c. Again, the diameter of circle e is 81.92. The diameter of circle d is 6'4. The diameter of circle c is √20-48. And √(√81·92 × √20-48) √(√1677-7216) 6-4, is the diameter of the intermediate circle d; and is the mean proportional between the diameter of circle e, and the diameter of circle c. Again, let A, B, C, represent three squares, circumscribed about the circles c, d, e. The area of the intermediate square B, will be a mean proportional between the area of square A, and the area of square C. The perimeter of the intermediate square B, will be a mean proportional between the perimeter of square A, and the perimeter of square C. The side of the intermediate square B, will be a mean proportional between the side of square A, and the side of square C. |