APPENDIX B. Paper read in the Mathematical Section of the British Association for the Advancement of Science, at their Twenty-ninth Meeting, held at Aberdeen, on Wednesday the 21st September, 1859. SIR WM. R. HAMILTON, LL.D., M.R.I.A., Astronomer Royal of Ireland, in the chair. ON THE RELATIONS OF A CIRCLE INSCRIBED IN A SQUARE. а am In drawing the attention of this section of the Association to the subject of “The Relations of a Circle inscribed in a Square,” I shall confine myself simply to the development of a few facts. These facts appear to me of importance, and to be well worthy of consideration. I satisfied there is contained within them the germ of truth, and that if further inquiry be instituted into the subject, it must lead to the discovery of other facts of great value to the advancement of science. Without further preface I shall direct the attention of. the section to the following diagram : Diagram No. 1. A B to 4. In this diagram, let the side of square A be unity, and the side of square C equal in value to one-fourth of the circumference of circle B inscribed in square A. Then, the diameter of circle B, inscribed in square A, A must be unity. The area of circle B is said to be represented by the figures 7854, and it necessarily follows that the ratio between the area of square A, and the area of circle B, will be, as I to 7854. The four sides of square A are together equal in value And, as 1 : 7854 :: 4:3-1416, which is said to be : the circumference of a circle of which the diameter is unity. a I am not aware that the following fact has ever been observed, viz. : That, as I : -7854, so is the area of circle B to the area of square C, of which the side is equal in value to onefourth of the circumference of circle B; or in other words, of which the four sides of the square are together equal in value to the circumference of circle B. Therefore, as I : *7854 : : 7854 : 61685316, the area of square C. Then, 7:61685316 = "7854, must be the value of the side of square C, and is equal in numerical value to the area of circle B. It is admitted that the circumference of circle B is equal to four times its area, and the side of square C being equal in numerical value to the area of circle B, the circumference of circle B is also equal in value to four times the side of square C. Again : as 1 : •7854, so is the side of square A to the side of square C; or, as 1 :"7854::1:7854, the side of square C. This illustration may be made plainer and more striking by varying the figures. Let me assume the side of square A to be 8, then the diameter of circle B inscribed in it must also be 8. Now, the usual mode of ascertaining the circumference of circle B, is to multiply the diameter by the figures 3:1416, said to be the circumference of a circle of which the diameter is unity. Therefore, 8 x 31416 = 25:1328, will be the circumference of circle B; and half the circumference, multiplied by half the diameter, or, the square of the diameter multiplied by •7854, will give the area of circle B. Therefore, 1 (25-1328) ~ } (8) = 12-5664 x 4, or 64 X •7854= 50°2656, will be the area of circle B. But these values may also be obtained by means of the ratios, as follows: The four sides of square A are together equal in value to 32; and, as 1 : 07854:: 32 : 25*1328, the circumference of circle B. The area of square Ais 64, and, as 1:7854::64:50-2656, the area of circle B. But further : As 1:"7854, so is the area of circle B to the area of square C, of which the four sides are together equal in value to the circumference of circle B. Therefore, a as 1 : 7854 :: 50*2656:: 3947860224, the area of square C. Then, 239-47860224 = 6:2832, must be the value of the side of square C; and, as 1 : 7854, so is the side of square A to the side of square C; or, as I : -7854 :: 8 : 6'2832, the side of square C. Then, 4 times the side of square C, = 6'2832 X 4, = 25'1328, is the circumference of circle B. And it is demonstrated that the four sides of square C are together equal in value to the circumference of circle B, and that the ratio of 1 to •7854 holds good in four distinct respects, as regards this diagra I may here remark, that if any other number than 7854 be assumed to represent the area of a circle of which the diameter is unity, the only effect of it would be, to change the values of the circumference of circle B, and the side and area of square C, but would not, in any respect whatever, affect the principle involved in the ratios. For example : If the area of circle B be assumed to be •78125, then the ratio between the area of square A and the area of circle B would be as i to •78125, and the effect of the alteration would be to give the value of the side of square C as •78125, and its area :6103515625, instead of 7854 and 61685316, as in my first illustration. I must now direct the attention of the section to another diagram, in connection with what I have already adduced, and in doing so I shall assume the diameter of circle B, in diagram No. I, to be unity, and its area to be •78125. In the diagram No. 2, let the diameter of circle A be the half of unity, or -5. Then, if circle A be circumscribed by a square, the value of the side of such square must be •5, and the value of its area '25. And, as I : 78125 :: 25: 1953125; or, -25 X 78125, is also equal to 1953125, the area of circle A. Diagram No. 2. А. с It is admitted that a circle described with any radius, is four times the value of a circle described with half that radius; and agreeably with this admitted proposition, the area of circle B, in diagram No. 1, is equal in value to sour times the area of circle A, in diagram No. 2. I must now direct the attention of the section to a fact, which I believe has never before been observed, viz. : that the ratio between the diameter of a circle and the side of a square, of which the area is equal in value to twice the area of the circle, is as 4 to 5. Therefore, as 4:5, so is the diameter of circle A to the side of square B; or, as 4:5:: 5:625, the side of square B. And, '6252 *390625, will be the area of square B, and is equal in value to twice the area of circle A. Then, as 1 : 78125, so is the area of square B, to the area of circle C inscribed in it; or, as I : 78125 :: '390625: : |