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 to the extended converse must be
identical with some one or other of the cases under the universal
affirmative proposition with which we set forth, which is absurd."
{296}
On this I cannot help transferring to my reader the words of the Pacha when
he orders the bastinado,--May it do you good! A rational study of logic is
much wanted to show many mathematicians, of all degrees of proficiency,
that there is nothing in the _reasoning_ of mathematics which differs from
other reasoning. Dr. Day repeated his argument in _A Treatise on
Proportion_, London, 1840, 8vo. Dr. Ritchie was a very clear-headed man. He
published, in 1818, a work on arithmetic, with rational explanations. This
was too early for such an improvement, and nearly the whole of his
excellent work was sold as waste paper. His elementary introduction to the
Differential Calculus was drawn up while he was learning the subject late
in life. Books of this sort are often very effective on points of
difficulty.