By Thomas Smith, D.D., LL.D. Charles Scribner's Sons. 1902. 12mo, pp.
CSP, identification: Haskell, Index to The Nation. See also: Burks, Bibliography; List of
Dr. Smith's style is light and easy, plena litteratce senectutis oratio; for the author is one who can speak of the quantification of the predicate as a novelty (it was only introduced in 1827), and who, in the words, "In our time, Gauss has shown," etc., refers to a publication of 1801. He has not forgotten his Greek, for he reads Proclus, and he constantly reminds us most winningly of what the education of a gentleman used to be. Quid enim est jucundius senectute, stipata studiis juventutis?
The little volume is not intended for scholars, but for those who know no more of Euclid and his science than they learned in the high school; and where the reader's attention may threaten to tire, he is refreshed by something of a facetious turn until he is ready to resume the more serious discourse. One will naturally not expect the author to have the least inkling of the way of thinking of modern mathematicians about the 'Elements.' He treats the "theory of parallels" in the good old way, taking his stand with those who were valiantly resolved to demonstrate that the theorem that the three angles of a plane triangle are equal in sum to two right angles, follows as a necessary consequence from certain premises concerning a plane, although it stared them in the face that these premises are equally true of the surface of a sphere, while the sum of the angles of a spherical triangle exceeds two right angles. Stated in this way, their undertaking was manifestly predestined to eternal failure. One-half of this state of things was clear to the mind of Euclid. That is to say, his confusion of thought about one-half of it arose from two subconscious assumptions, the recognition of which would have made him wholly right. One of these was that space is immeasurably great. That he assumed this appears (among other places) in his supposed proof (I. 16) that the angles of a triangle are not greater than two right angles; and that he assumed it irreflectively is shown by the language of his second postulate compared with the use he put it to. It reads that a terminated straight line can be produced continuously (); but he applies it as if it read "can be prolonged beyond any assigned length." His other unconscious assumption is that all the figures with which he deals are finite. This is shown by his axiom (called the eighth) that the whole is greater than its part. For, of course, Euclid knew well enough that a straight line terminated at one end only, and endless in the further direction, is not made any shorter by cutting off a finite part of it, since what remains can be shoved along to cover the extension occupied by the whole, and, being endless, leaves no part uncovered. These two assumptions not being explicitly made, his proof of the sixteenth proposition which we have just (substantially) quoted, re