THE present edition of this work is of necessity little more than a reprint of its predecessor. Mistakes have been corrected and omissions supplied, but the nature of the subject does not admit of any very important changes either in substance or arrangement.

There have been many criticisms upon the First Edition, and these for the most part most favourable and encouraging. Among the rest, special acknowledgments are due to the writer of the notice in the Philosophical Magazine for October of last year, for pointing out a grave blunder in the section on the length of curved lines, whereby a proposition has been described as a definition. In drawing attention to this mistake, the writer of the notice adds some observations upon the general subject, which appear to call for a little detailed consideration here. Не appears to make light of the difficulty involved in the accurate definition of the length of a curved line, and suggests the introduction of the conception of a flexible line pulled straight. Practically we are, without doubt, familiar enough with the conception of the length of a curved line, but familiarity with a conception does not diminish the difficulty of accurate definition ; on the contrary, it frequently enhances that difficulty.

In the case before us, suppose the reviewer's suggestion adopted, and the length of a curved line defined as being that of the straight line with which it coincides when pulled out. In the process of pulling out the identity of form is destroyed, and nothing but identity of length remains, that is to say we must provide that in the process of rectification the length of the curve is to remain unaltered, and thus the thing to be defined (viz. the length of the curve) is introduced into the definition of itself. The suggested definition does, in fact, tacitly assume the conception of a curve as composed of infinitesimal rectilinear elements, and conceives that the rectification takes place by turning each element in succession round one extremity until it comes into the same straight line with the last preceding element. If such a conception of a curve be admitted, any definition of its length is, of course, superfluous, being included in the introductory definitions of the addition of straight lines, but such a conception cannot well find place in an elementary work like the present.

Another and less friendly criticism, the only one in fact which has appeared of a markedly unfavourable character, pronounces the demonstrations to be cumbersome and tedious, and especially adduces Props. il and 17 of Bk. I. in illustration of this assertion.

The general criticism does not, of course, admit of refutation; it must be left to the judgment of the public. But concerning Prop. II, Bk. I., it may be well to state, what should indeed have been mentioned in the Preface to the First Edition, that the present longer demonstration was intentionally preferred to the shorter one based on the principle of continuity, and that because this principle, though most obvious and natural to advanced geometricians, is by no means so to boys and learners generally. It was thought therefore desirable to make the demonstrations in Bk. I. depend as much as possible upon the principle of superposition alone, and not to introduce the idea of continuity and continuous motion until later on in the Second Book.


October 1872.





The study of Elementary Geometry (at least in England) has been for a long time identified with one particular treatise, accepted as a standard. At the present moment there is a wide-spread dissatisfaction with that treatise; but there is very little agreement as to the manner in which it may be best improved. The most suitable Preface, therefore, to a new work on Geometry would appear to consist in an enumeration of the main features of its agreement or disagreement with Euclid, and an attempt as far as possible to justify such agreement or disagreement in each particular instance.

I. The Work agrees with Euclid in retaining the syllogistic form throughout. Many objections, strong and ably urged, have been alleged against this method of treatment by modern writers. It is said that the study of Geometry for its own sake is thereby made subordinate to its study as a logical discipline; and that the detailed syllogistic form into which all the demonstrations are thrown is a source of obscurity to beginners, and damaging to true geometrical freedom and power.

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