fact, that this commensurable right-angled triangle, is a link in the chain of commensurable geometrical figures, connecting one with another in a most remarkable manner, so much so, that their connection may be reduced to a system, at once beautiful, harmonious, and thoroughly self-consistent. In the following diagram, (see figure A,) let ABC be a commensurable right-angled triangle, of which the side B C is to the side A B, in the ratio of 3 to 4; and D, a circle, of which the diameter is equal to the perpendicular of the triangle. In the first place, let the diameter of the circle, and perpendicular of the triangle, be 8. On the orthodox hypothesis, 8 X 3*1416 = 25*1328, will be the approximative value of the circumference of the circle. On my hypothesis, which, for the sake of distinction, I shall call the heterodox hypothesis, 8 X 3'125 = 25, will be the exact circumference of the circle. The perpendicular of the triangle is 8. Therefore, À (8) 6, will be the base of the triangle. (8), or (6 + 8?), = 10, will be the hypotheneuse of the triangle. And 6 + 8 + 10 = 24, will be the value of the perimeter of the triangle. Then, on the former hypothesis, the ratio between the circumference of the circle, and perimeter of the triangle, will be, as 25'1328 to 24. And on the latter hypothesis, the ratio between the two, will be, as 25 to 24. And these ratios hold good, no matter how peculiar may be the decimal selected as the diameter of the circle and perpendicular of the triangle. For example: Let the diameter of the circle, and perpendicular of the triangle, be 7.7. On the orthodox hypothesis, 7*7 x 3.1416 = 24'19032, will be the approximative value of the circumference of the circle. On the heterodox hypothesis, 7*7 X 3'125 = 24'0625, will be the exact value of the circumference of the circle. The perpendicular of the triangle is 77. Therefore, * (7.7) = 5775, will be the base of the triangle; (707), or ✓(57752 +7.7), = 9:625, will be the hypotheneuse of the triangle ; and 5-775 + 7*7 + 9625 = 23'1, will , be the value of the perimeter of the triangle; and, as 25'1328 :24: :24:19032:23'1 ; or, as 25 : 24 : : 24'0625 :23:1; and so far one hypothesis would appear to be just as good as the other. But, reverse the operation, and let the circumference of the circle be the given quantity, say 60; and let it be required to find the values of the diameter of the circle, and the perimeter of the triangle. On the heterodox hypothesis, 60 = 3125 = 19-2, will be the exact value of the diameter of the circle, and perpendicular of the triangle ; (192) = 14:4, will be the base of the triangle; and (19:2), or, J(14:42 + 19:22), = 24, will be the hypotheneuse of the triangle ; and, 19:2 + 144 + 24 = 57.6, will be the value of the perimeter of the triangle; and, as 25:24::60:57.6 exactly; and on this hypothesis the ratio holds good, whether the circumference of the circle be the given quantity, to find the diameter of the circle and the perimeter of the triangle; or, the perpendicular of the triangle be the given quantity, to find the circumference of the circle. On the orthodox hypothesis, 60+ 3'1416= 19'0985&c. will be the approximative value of the diameter of the circle, and perpendicular of the triangle ; À (19'0985) 14:323875, will be the base of the triangle; and (19'0985) = 23.873125, will be the hypotheneuse of the triangle ; and, 19:0985 + 14'323875 + 23.873125 = 57.2955, will be the value of the perimeter of the triangle. But, these figures are less than those required by the orthodox ratio. For, as 25'1328 : 24::60: 57-2956, and it appears to me to be about as absurd to attempt to maintain the orthodox hypothesis, as it would be to maintain that 6 x 8 48, but that 8 X 6, is only equal to 47 and a fraction. I have endeavoured to bring this subject under the notice of several gentlemen, who pride themselves on their high mathematical attainments, and judging from the reception I have met with at their hands, I am disposed to think that the philosophical world is not much wiser now, than it was in the days of Galileo, but, “ Magna est veritas, et prevalebit,” and it is not in the power of ten thousand men of the highest genius, by their united efforts to annul this glorious fact. Political, religious, and social freedom, have, however, made rapid strides even in our own day, in this privileged and happy nation, and it is now the right of a Briton, to be able to freely express an opinion, without incurring the risk of torture, or the danger of a prison ; and in virtue of this privilege, I venture to address you thus freely, as a “great authority” on this subject, conceiving that the importance of it fully justifies me in doing so. In conclusion, I may remark that my position in life is happily one of the most perfect independence. I have gone into this enquiry from a pure love of science, and a disinterested desire to promote it, and this course I shall continue to pursue. To myself personally, therefore, it is a matter of little consequence what course you may be pleased to adopt. I shall be glad however to find, that you do not consider the subject unworthy your attention, and in this respect form an honourable exception, among those of your professional brethren, with whom I have come into contact. I remain, SIR WILLIAM, JAMES SMITH. Sir Wm. R. Hamilton, L.L.D., &c., Observatory, near Dublin. P.S.—Having referred to an incident in connection with Mr. Airy, the Astronomer Royal of England, it might be said, if he were not made aware of it, that I was making a false charge, without having given him the opportunity of refuting it. To prevent this, I shall write him and enclose a copy of this letter, and if he should have changed his opinion, it will also give him the opportunity of admitting it. I received from Sir W. R. Hamilton, by return of post, the following very courteous reply: Observatory, near Dublin, February 15th, 1860. Sir, I have received your letter, with its printed enclosure ; another copy of this latter, (namely of the printed Paper,) had indeed reached me some months ago; but I did not understand that you required me to acknowledge it: nor would it have been a pleasant task to inform an ingenious gentleman, without necessity, of my entire disagreement from his views. But since, while reminding me of my having had the honour to preside in the absence of Lord Rosse, when you read your paper in Section A of the British Association, at Aberdeen, last year, you are pleased “ again to direct my attention to the subject, and to give me an opportunity of communicating with you, if I should think |