I have said that Whewell was gentle with his pupils; it was the same with all who wanted teaching: it was only on an armed enemy that he drew his weapon. The letter which he wrote to Mr. J. Smith is an instance: and as it applies with perfect fidelity to the efforts of unreasoning above described, I give it here. Mr. James Smith is skilfully exposed, and felt it; as is proved by "putting the writer in the stocks."
"The Lodge, Cambridge, September 14th, 1862.
"Sir,—I have received your explanation of your proposition that the circumference of the circle is to its diameter as 25 to 8. I am afraid I shall disappoint you by saying that I see no force in your proof: and I should hope that you will see that there is no force in it if you consider this: In the whole course of the proof, though the word cycle occurs, there is no property of the circle employed. You may do this: you may put the word hexagon or dodecagon, or any other word describing a polygon in the place of Circle in your proof, and the proof would be just as good as before. Does not this satisfy you that you cannot have proved a property of that special figure—a circle?
"Or you may do this: calculate the side of a polygon of 24 sides inscribed in a circle. I think you are a Mathematician enough to do this. You will find that if the radius of the circle be one, the side of this polygon is .264 etc. Now, the arc which this side subtends is according to your proposition 3.125/12 = .2604, and therefore the chord is greater than its arc, which you will allow is impossible.
"I shall be glad if these arguments satisfy you, and
"I am, Sir, your obedient Servant,
"W. Whewell."
AN M.P.'S ARITHMETIC.
In the debate of May, 1866, on Electoral Qualifications, a question arose about arithmetical capability. Mr. Gladstone asked how many members of the House could divide 1330l. 7s. 6d. by 2l. 13s. 8d. Six hundred and fifty-eight, answered one member; the thing cannot be done, answered another. There is an old paradox to which this relates: it arises out of the ignorance of the distinction between abstract and concrete arithmetic. Magnitude may be divided by magnitude; and the answer is number: how often does 12d. contain 4d.; answer three times. Magnitude may be divided by number, and the answer is magnitude: 12d. is divided in four equal parts, what is each part? Answer three pence. The honorable objector, whose name I suppress, trusting that he has mended his ways, gave the following utterance: