THE QUESTION : ARE THERE ANY COMMENSURABLE RELA TIONS BETWEEN A CIRCLE AND OTHER GEOMETRICAL FIGURES ? ANSWERED BY A MEMBER OF THE “BRITISH ASSOCIATION FOR THE ADVANCEMENT OF SCIENCE.” А n H B “Strike, but Hear.” APPENDIX A. THE QUESTION: Are there any commensurable Relations between a Circle and other Geometrical Figures ? To you, Reader, I address this question. You may be a Mathematician, and so, competent to correct or to answer, as one having authority, the propositions I am about to lay before you. Or, you may belong to that larger class which accepts as matter of faith, the conclusions endorsed by the professors of this science. In either case I claim your attention ;-if you are a master, to what has been wrongly taught-if you are a disciple, to what has been wrongly learned. It has long been taught that it is impossible to demonstrate exactly the relation, in magnitude, between the Diameter and the Circumference of a Circle. I affirm that it is not impossible ; and further, that I can do it. I see the smile of derision with which my affirmation is received. I have seen it very often. But it does not assure me that I am wrong. I am told that the highest authorities are against me. I know it. But this is a subject on which I cannot admit that authority is of any value. The “highest authorities” were opposed to the proposition of Galileo. Yet he was right. Like him, I appeal to the evidence. Like him, I am told that it is all against me. And, like him, I ask my opponents how many of them have fairly examined that evidence for themselves. Very few, I believe, can answer in the affirmative. Those who have, will not stop to shelter themselves behind great names. They will know that this is a m er on which there is no room for doubt—that if I am wrong, it is not difficult to prove me so—and that whatever may be the process by which the truth is in this instance to be demonstrated, he who, even though himself in error, induces its production, aids, not retards, the progress of science. It may be as the authorities” say it is. But if it be, let us see the proof. Meanwhile I ask the reader's attention to the following demonstrations : First, as to my reasons for rejecting the data commonly accepted. It is affirmed by authority, that the diameter of a circle is to its circumference, as I to 3'14159 &c., an indefinite and incommensurable relation. The figures assumed for practical purposes, to represent the nearest approximative values of the circumference and area of a circle, of which the diameter is unity, are 3'1416 for the former, and •7854 for the latter. Let the diameter of a circle be 4. Then, 4 3.1416 = 125664, will be the circumference of the circle. Half the circumference, multiplied by half the diameter, 1 (12-5664) x 1 (4), = 6*2832 X 2, = 12-5664; or, the square of the diameter, multiplied by the assumed area of a circle of which the diameter is unity, = a a |