oid the blows of a prize-fighter. As in many cases of this kind,
the reasoning was enveloped in a mass of verbiage which it was very
difficult to strip off so as to see the real framework of the logic.
When this was done, the syllogism would be found to take this very
simple form:--

The ratio of the circumference to the diameter is the same in all
circles. Now, take a diameter of 1 and draw round it a circumference
of 3 1/5. In that circle the ratio is 3 1/5; therefore, by the
major premise, that is the ratio for all circles.

The three famous problems of antiquity, the duplication of the cube,
the quadrature of the circle, and the trisection of the angle, have
all been proved by modern mathematics to be insoluble by the rule
and compass, which are the instruments assumed in the postulates
of Euclid. Yet the problem of the trisection is frequently attacked
by men of some mathematical education. I think it was about 1870
that I received from Professor Henry a communication coming from
some institution of learning in Louisiana or Texas. The writer
was sure he had solved the problem, and asked that it might receive
the prize supposed to be awarded by governments for the solution.
The construction was very complicated, and I went over the whole
demonstration without being able at first to detect any error.
So it was necessary to examine it yet more completely and take it
up point by point. At length I found the fallacy to be that three
lines which, as drawn, intersected in what was to the eye the same
point on the paper, were assumed to intersect mathematically in
one and the same point. Except for the complexity of the work,
the supposed construction would have been worthy of preservation.

Some years later I received, from a teacher, I think, a supposed
construction, with the statement that he had gone over it very
carefully and could find no error. He therefore requested me to
examine it and see whether there was anything wrong. I told him in
reply that his work showed that he was quite capable of appreciating
a geometric demonstration; that there was surely something wrong in
it, because the problem was known to be insoluble, and I would like
him to try again to see if he could not find his error. As I never
again heard from him, I suppose he succeeded.

One of the most curious of these cases was that of a student, I am not
sure but a graduate, of the University of Virginia, who claimed that
geometers w