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."
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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.
