value, but that this value cannot be expressed exactly in figures; this I deny, and maintain, on the contrary, that 3.125 exactly represents the one, and 78125 exactly represents the other; and I cannot discover any thing in your letter, at all calculated to throw a doubt into my mind as to the truth of my figures, much less to convince me that I am in error. You will of course now understand, that I maintain, as the only true ratio of diameter to area in a circle, of which the diameter is linear unity, that of I to 78125, and abandon all others.

You are of opinion that the fallacy which pervades my argument, is that of treating linear and square units as if they were identical. I really cannot plead guilty to this charge. You illustrate your statement by supposing an estate in the form of a square, and containing an area of one square mile. The fence on each side of the estate will of course be one linear mile. You say, "It is very true, abstractedly, that I, and the square of 1, are expressed by the same symbol, but they are very different things."

Now, symbols may be either arithmetical or algebraical, the notation of the former is figures, the notation of the latter is letters, and by an application of the two, to the consideration of the question at issue, I think the fallacy of your reasoning may be made apparent.

Let the fence on one side of the estate be represented by the algebraical symbol A, and let the area of the estate be represented by the algebraical symbol B, and let the value of the algebraical symbol A, be represented by the arithmetical symbol 1. It may be one yard, one furlong, one mile, or one any thing. It would be very absurd if I were to say that A and B were the same thing, but it would be equally absurd, if I were to attempt to express the value of B, by any other arithmetical symbol

than I, until I am informed whether the algebraical symbol A, stands for one yard, one furlong, one mile, or one something else, and then it ceases to be a geometrical, and is resolved into an arithmetrical problem.

Let D represent the diameter, C the circumference, and A the area of a circle, and let the value of D be represented by the arithmetical symbol 4.

=

=

2

Then, on the orthodox data, 4x 3'1416=125664, will be the arithmetical value of C ; (C) × 1 (D),=1(12·5664) X (4), 6·2832 x 2, 12.5664, will be the arithmetical value of A. Or, on the writer's hypothesis, 4 × 3′125 = 12.5, will be the arithmetical value of C; (C) × 1 (D), 1 1 = (125) × (4), 625 × 2, 125, will be the arithmetical value of A. It would be very absurd if I were to say, that C and A represented the same thing, but it would be equally absurd, if I were to attempt to express the values of C and A, by any other than the same arithmetical symbols, as ascertained either on one hypothesis or the other.

In the enclosed diagram (see Fig. I.) let A B C D be a square, E a circle circumscribed about it, and F a circle inscribed in it, the area of which is admitted to be equal in arithmetical value to half the area of circle E. Let X represent the circumference of circle E, Y the area of circle E, and Z the area of circle F, and let the diameter of circle E, be represented by the arithmetical symbol 8.

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Then, on the orthodox data, 8 x 3'1416 = 25'1328, will be the arithmetical value of X; (X) x 1 (8), = 1 (25.1328) × (8), = 12·5664 × 4, = 50′2656, will be the 1 arithmetical value of Y; (Y), (50'2656), 251328, will be the arithmetical value of Z. Or, on the writer's hypothesis, 8 x 3'125 = 25, will be the arithmetical value of X; (X) × (8), = 1 (25) × (8), = 12′5 × 4, 50, 1 1

(Y), = (50), 25, Again, it would no

say that X, which

will be the arithmetical value of Z. doubt be very absurd, if I were to represents the circumference of circle E, and Z, which represents the area of circle F, were the same thing; but it would be equally absurd, if I were to attempt to express the values of X and Z, by any other than the same arithmetical symbols, as ascertained either on one hypothesis or the other.

In the first part of the paper read by me at “The British Association for the Advancement of Science," I have simply directed attention to the fact, that whatever hypothetical data be taken to represent the area of a circle, of which the diameter is linear unity, whether 7854 or any other number-I might say 78539815, or 78125— the ratio that exists between the area of any square and the area of a circle inscribed in it, exists likewise between the numerical value of the four sides of such square, and the circumference of the inscribed circle; between the area of the circle and the area of a square, of which the four sides are together equal in numerical value to the circumference of the circle; and between the side of the square containing the inscribed circle, and the side of a square of which the numerical value is equal to one-fourth of the circumference of the circle; and for this purpose the paragraph to which you refer is superfluous, and might have been omitted altogether. But suppose the paragraph to have run thus :— "It cannot be denied that the arithmetical symbols which represent the circumference of circle B, are equal in numerical value to four times the arithmetical symbols which represent its area,"-who could have disputed the fact, and where is the fallacy in the reasoning?