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CORRESPONDENCE.

EMINENT MATHEMATICIAN, to the AUTHOR of “The

Question:-Are there any commensurable relations between a Circle and other Geometrical Figures?"

London, July 14th, 1860. SIR,

You have done me the honour to enclose to me a copy of your pamphlet on the Squaring and Rectification of the Circle.

If you will allow me to address you by name, I will in the space of one page of writing, point out that your highly ingenious reasoning rests on a fallacy.

I am, Sir,
Obediently yours,

E. M.

JAMES SMITH, Esq., to EMINENT MATHEMATICIAN.

.

Barkeley House, Seaforth,

Liverpool, 17th July, 1860. SIR,

Your letter of the 14th instant only came into my hands this morning.

My only object is to arrive at the truth on the interesting question of “ The Quadrature of the Circle ;” and if there really be a fallacy in my reasoning, I shall be glad to be convinced of it.

I beg to enclose a copy of the paper read by me, in the Mathematical Section of “The British Association for the Advancement of Science,” at its meeting at Aberdeen last year.

I am, Sir,
Yours obediently,

JAMES SMITH.

EMINENT MATHEMATICIAN, to JAMES SMITH, Esq.

London, July 18th, 1860. SIR,

The fallacy which pervades your argument in the paper respecting the Squaring of the Circle, is, that you treat linear and square units as if they were identical. It is very true, abstractedly, that one, and the square of one, are expressed by the same symbol, but they are very different things. As for instance, if you have an estate bounded by a fence one mile long on each side of a square, it is true to say that the side of the square is 1, it is also true to say that the area is 1, but you speak of i linear unit as to the fence, and I square unit as to the estate, and it would be absurd to say that the fence of one side of the estate is equal to the estate itself ; equally absurd is it to say that the side of any square is equal to its area, or any

area.

To apply this remark to your paper, take the words,

“It is admitted that the circumference of the 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.” (See Appendix B.)

But no such thing can be admitted. All that can be admitted is, that the number of square units in one case corresponds with the number of linear units in the other. You might as well say that 50 pounds is equal to 50 thousand pounds, because the figure 50 is used in each case. The unit is a pound in one case, and a thousand pounds in the other.

But there is another fallacy further down in the same paper, where it is said, “ I may here remark, that if any other number than 7854 be assumed as 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."

The area of a Circle whose diameter is linear unity, is fixed, and cannot have more than one value, measured in If that value be *7854, no other number"

• can be assumed as representing it.

And when you assume the ratio of 1 to 78125, you must abandon the hypothesis that i to •7854 represents the true ratio.

In point of fact, if you take the ratio i to 78125 the curve is not a circle, but an inscribed polygon, and one which is quadrable.

Any doubt which you may have as to the real ratio of diameter to circumference in a circle, is readily removed by any treatise on Trigonometry, where the method of finding the length of the circular measure of any part, or the

square units.

a

a

whole, of a circle, is demonstrated beyond the possibility of cavil. I need hardly add that a fixed ratio cannot be both i to "78125, and also i to "7854. If you confine your

•• reasoning to the simple case, “What is the ratio of the diameter to circumference,” omitting all reference to the subject of areas, I think you will not fail to arrive at the full understanding of the matter.

I am, Sir,
Yours obediently,

E. M.

JAMES SMITH, Esq., to EMINENT MATHEMATICIAN.

Barkeley House, Seaforth,

Liverpool, 24th July, 1860. SIR,

The general tenor of your letter of the 18th instant would seem to imply, that you understand me to hold the opinion, that the area of a circle of which the diameter is linear unity, is not a fixed quantity ; it is therefore necessary, before proceeding to deal with your arguments, that I should lay down, in as distinct terms as possible, certain propositions which shall explicitly embody my views on the interesting question of “The Quadrature of the Circle.”

I affirm, that there is a relation between the diameter and circumference of a circle, that this relation is definite and commensurable, and can be expressed in figures with perfect exactness.

I affirm, that the circumference and area of a circle of which the diameter is linear unity, are definite and

:

a

commensurable quantities, and can be expressed in figures without the least error.

I affirm, that the relation between the diameter and circumference of any circle may be expressed in the following terms :-For every linear unit contained in the diameter, there are three and one-eighth linear units contained in the circumference.

And lastly, I affirm, that the figures 3-125, represent the circumference, and 78125 the area of a circle, of which the diameter is linear unity, and that both are fixed and unchangeable quantities.

In your letter you make the following statement: “The area of a circle whose diameter is linear unity is fixed, and cannot have more than one value, measured in square units. If that value be •7854, no other number' can be assumed as representing it.” You are no doubt aware that on the commonly accepted data, the figures 3:1415926, &c., would represent a nearer approximation to the circumference, and 78539815, &c., a nearer approximation, measured in square units, to the area of a circle, of which the diameter is linear unity, than 3'1416, and -7854, the figures which for practical purposes are usually adopted as sufficiently accurate.

If the area of a circle of which the diameter is linear unity be fixed, and cannot have more than one value, (and I admit the fact), and if you maintain •7854 to be that value, you must abandon the orthodox theory. If, on the other hand, you profess to accept the orthodox theory, you must abandon the figures •7854 as the fixed and unchangeable value of the area of a circle of which the diameter is linear unity.

If

you venture to say the circumference and area of a circle of which the diameter is linear unity, have a fixed

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