ure is the representation which
comes before the mind: this is not true; we might as well say the same of
the object itself. In July 1831, reading an article on squaring the circle,
and finding that there was a difficulty, he set to work, got a light denied
to all mathematicians in--some would say through--a crack, and advertised
in the _Times_ that he had done the trick. He then prepared this work, in
which, those who read it will see how, he showed that 3.14159... should be
3.0625. He might have found out his error by _stepping_ a draughtsman's
circle with the compasses.

Perspective has not had many paradoxes. The only other one I remember is
that of a writer on perspective, whose name I forget, and whose four pages
I do not possess. He circulated remarks on my notes on the subject,
published in the _Athenaeum_, in which he denies that the stereographic
projection is a case of perspective, the reason being that the whole
hemisphere makes too large a picture for the eye conveniently to grasp at
once. That is to say, it is no perspective because there is too much
perspective. {295}

ON A COUPLE OF GEOMETRIES.

Principles of Geometry familiarly illustrated. By the Rev. W.
Ritchie,[640] LL.D. London, 1833, 12mo.

A new Exposition of the system of Euclid's Elements, being an attempt
to establish his work on a different basis. By Alfred Day,[641] LL.D.
London, 1839, 12mo.

These works belong to a small class which have the peculiarity of insisting
that in the general propositions of geometry a proposition gives its
converse: that "Every B is A" follows from "Every A is B." Dr. Ritchie
says, "If it be proved that the equality of two of the angles of a triangle
depends _essentially_ upon the equality of the opposite sides, it follows
that the equality of opposite sides depends _essentially_ on the equality
of the angles." Dr. Day puts it as follows:

"That the converses of Euclid, so called, where no particular limitation is
specified or implied in the leading proposition, more than in the converse,
must be necessarily true; for as by the nature of the reasoning the leading
proposition must be universally true, should the converse be not so, it
cannot be so universally, but has at least all the exceptions conveyed in
the leading proposition, and the case is therefore unadapted to geometric
reasoning; or, what is the same thing, by the very nature of geometric
reasoning, the particular exceptions