who will give the subject a candid and careful consideration, that the circumference of a circle is, in fact, neither more nor less than three and one-eighth times its diameter. It is only necessary for the writer's present purpose to give one more illustration. Let the diameter of a circle be 768. On the Orthodox data, 768 × 3'1416=2412'7488, will be the circumference of the circle. On the writer's hypothesis, 768 x 3'125 2400, will be the circumference of the circle. By hypothesis, let the circumference of the circle be equal to the perimeter of a polygon of 40000 sides. Then, on the former data, 24127488 40000 = *06031872. On the latter data, 2400 ÷ 40000 = '06. If 40000 equidistant radii be supposed to be drawn in the circle, the value of that part of the circumference of the circle contained by any two radii, will be represented by one or other of the above sets of figures, as one or other of the assumed data may be adopted. It is quite certain that no subdivision of the circumference of a circle, however minute, can resolve that part of it contained by any two radii into a straight line; but for the purpose of this demonstration, it is sufficient that it be admitted, that the part of the circle contained by two radii, if the subdivision be very minute, will describe a figure very nearly equal to an isosceles triangle of which the two sides are equal to the radius of the circle, and the base equal to that part of the circumference of the circle contained by any two radii. The radius of the circle, in this instance, is 384; a definite and admitted quantity. And if the part of the circle contained by two radii be supposed to represent a figure very nearly equal to an isosceles triangle, of which the value of the sides is 384, the data, though only hypothetical, by which we can most nearly approach these figures (an admitted quantity), must be (if not true) the nearest approximation to the truth. It will be admitted that, on either data, the sides of the (supposed) isosceles triangle will be equal to a radius of the circle, and will be represented by the number 384. On the Orthodox data, the base of the triangle will be 06031872. On the writer's hypothesis, the base of the triangle will be ‘06. Then: Then If the angle of the triangle be bisected, and a : 3842—03015936o,=147456—0009095 869956096, = 147455 9990904130043904, the square root of which is 383.999998815, will be the value of the third side. Or: 384-'03, 147456— 0009, = 147455′9991, = the square root of which is 383-999998828, will be the value of the third side. If, on the other hand, we take the bisecting line of the (supposed) isosceles triangle (admitted to be equal to the radius) and the base, to be the two sides of a triangle adjacent to the right angle. Then: : 3842+030159362,=147456+0009095869956096, Or: = 147456.0009095869956096, the square root of which is 384'000001184, will be the value of the third side. 384032, 147456+0009, 1474560009, = = the square root of which is 384000001171, will be the value of the third side. In the one case, on the writer's hypothesis, we are carried more nearly to the radius, and in the other are less in excess of it, than on the Orthodox data; and it is consequently demonstrated that the part of the circumference of the circle contained by two radii is more nearly resolved into a straight line, on the writer's hypothesis, than on the Orthodox data; and must, if not absolutely true, be admitted to present the nearest approximation to the truth. And now, Reader, if you think me wrong, let me ask you to say where, and how. I have bestowed much time and labour on this impeachment of what is supposed to be a truth beyond all doubt. What, after doubting, and examining, I now deny to be true, others also may at least doubt. If it be very truth, let it be proved to be so. The proof cannot be difficult; or, if it be so, should be so handled that it may be difficult no longer. To simplify the subject will be a public benefit. And as to doubt is a duty, where palpable proof is not supplied; so to supply such proof is also a duty-and one especially incumbent on that learned fraternity who claim to be regarded as our guides. In their hands, then, I leave it. I have done my duty. Let them do theirs. 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., AstronomerRoyal of Ireland, in the chair. ON THE RELATIONS OF A CIRCLE INSCRIBED IN A SQUARE. 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 am 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 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, 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 to 4. And, as 17854: 4: 31416, which is said to be the circumference of a circle of which the diameter is unity. 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 : 78547854 61685316, the area of square C. Then, √61685316= = 7854, must be the value of the side of |