Page:A budget of paradoxes (IA cu31924103990507).pdf/349

distinction between the two birds. Nauticus lays down—quite correctly—that the sine of an angle is less than its circular measure. He then takes 3.1416 for 180°, and finds that 36′ is .010472. But this is exactly what he finds for the sine of 36′ in tables: he concludes that either 3.1416 or the tables must be wrong. He does not know that sines, as well as $\pi$, are interminable decimals, of which the tables, to save printing, only take in a finite number. He is a six-figure man: let us go thrice again to make up nine, and we have as follows:—

Circular measure of 36′	.010471975…
Sine of 36′	.010474784…
Excess of measure over sine	.000000191…

Mr. Smith invites me to say which is wrong, the quadrature, or the tables: I leave him to guess. He says his assertions 'arise naturally and necessarily out of the arguments of a circle-squarer:' he might just as well lay down that all the pigs went to market because it is recorded that 'This pig went to market.' I must say for circle-squarers that very few bring their pigs to so poor a market. I answer the above argument because it is, of all which Mr. James Smith has produced, the only one which rises to the level of a schoolboy: to meet him halfway I descend to that level.

Mr. Smith asks me to solve a problem in the Athenæum: and I will do it, because the question will illustrate what is below schoolboy level.

Let $x$ represent the circular measure of an angle of 15°, and $y$ half the sine of an angle of 30° = area of the square on the radius of a circle of diameter unity = .25. If $x-y=xy$, firstly, what is the arithmetical value of $xy$? secondly, what is the angle of which $xy$ represents the circular measure?

If $x$ represent 15° and $y$ be 1/4, $xy$ represents 3° 45′, whether $x-y$ be $xy$ or no. But, $y$ being 1/4, $x-y$ is not $xy$ unless $x$ be 1/3, that is, unless $12x$ or $\pi$ be 4, which Mr. Smith would not admit. How could a person who had just received such a lesson as I had given immediately pray for further exposure, furnishing the stuff so liberally as this? Is it possible that Mr. Smith, because he signs himself Nauticus, means to deny his own very regular, legible, and peculiar hand? It is enough to make the other members of the Liverpool Dock Board cry, Mersey on the man!

Mr. Smith says that for the future he will give up what he calls sarcasm, and confine himself, 'as far as possible,' to what he