Euclid assumes what he wants to disprove, and shows that his assumption leads to absurdity, and so upsets itself. Mr. Smith assumes what he wants to prove, and shows that his assumption makes other propositions lead to absurdity. This is enough for all who can reason. Mr. James Smith cannot be argued with; he has the whip-hand of all the thinkers in the world. Montucla would have said of Mr. Smith what he said of the gentleman who squared his circle by giving 50 and 49 the same square root, Il a perdu le droit d'être frappé de l'évidence.[[217]]
It is Mr. Smith's habit, when he finds a conclusion agreeing with its own assumption, to regard that agreement as proof of the assumption. The following is the "piece of information" which will settle me, if I be honest. Assuming π to be 3⅛, he finds out by working instance after instance that the mean proportional between one-fifth of the area and one-fifth of eight is the radius. That is,
| if π = | 25 8 | , | πr2 5 | · | 8 5 | = r. |
This "remarkable general principle" may fail to establish Mr. Smith's quadrature, even in an honest mind, if that mind should happen to know that, a and b being any two numbers whatever, we need only assume—
| if π = | a2 b | , to get at | πr2 a | · | b a | = r. |
We naturally ask what sort of glimmer can Mr. Smith have of the subject which he professes to treat? On this point he has given satisfactory information. I had mentioned the old problem of finding two mean proportionals,
as a preliminary to the duplication of the cube. On this mention Mr. Smith writes as follows. I put a few words in capitals; and I write rq[[218]] for the sign of the square root, which embarrasses small type:
"This establishes the following infallible rule, for finding two mean proportionals OF EQUAL VALUE, and is more than a preliminary, to the famous old problem of 'Squaring the circle.' Let any finite number, say 20, and its fourth part = ¼(20) = 5, be given numbers. Then rq(20 × 5) = rq 100 = 10, is their mean proportional. Let this be a given mean proportional TO FIND ANOTHER MEAN PROPORTIONAL OF EQUAL VALUE. Then
| 20 × | π 4 | = 20 × | 3.125 4 | = 20 × .78125 = 15.625 |
will be the first number; as