as concise and intelligible as any other, and it has one important claim to be retained in an elementary geometrical treatise, inasmuch as there is absolutely no place for strict logical treatment in any other part of our modern mathematical course. Analysis, even in its simplest form soon, and of necessity, degenerates into a manipulation of symbols. The elements of Algebra must be acquired as a mechanical exercise, as an art rather than a science. The very rudiments of algebraical analysis involve difficulties far beyond a schoolboy's intellect, and necessitate as much profound thought as the highest mathematics. In Geometry, on the other hand, the conceptions involved are plain and easy, 'every step is planted on firm ground,' and the vigorous logical treatment of which these conceptions are susceptible not only supplies a mental discipline of inestimable educational value, but furnishes a test whereby to try the conclusions arrived at by the more rapid processes of analytical reasoning.

II. While, however, the syllogistic form has been retained in this treatise for the reasons which have been mentioned, innovations have been introduced in the following important respects:

I. An extended application of the principle of superposition.

The application of this principle is the great stumbling-block to the young geometrician, and yet it is absolutely indispensable, and hardly a step

can be made without it. Euclid has recognised its importance, and introduced it early enough in his course; but he uses it grudgingly and timidly, and he imparts this feeling of timidity to the learner. Thus, in Proposition 4 of Book I. there is no suggestion of the possibility of the two triangles being so situated as to make it necessary for the plane of one of them to be reversed before superposition can take place; the proof adopted obviously supposes a simple transference of position, the same face remaining uppermost throughout. It is true that the language employed is sufficiently general to include every case, but it is equally true that the ideas suggested to the reader are limited in the manner that has been mentioned. Now in Proposition 5, which is the very first instance of the application of Proposition 4, the triangles are so situated as to require this reversal of the plane of one of them before they can be superposed; and the whole of the long and tedious reasoning of this proposition might have been avoided if that had been done at first explicitly and avowedly which is afterwards done substantially, though in a sense surreptitiously. To avoid the superposition of a triangle upon itself with plane reversed, Euclid has recourse to the artificial and almost disingenuous device of turning one triangle into two by producing the sides, and then applying Proposition 4 to these two triangles, ignoring the

t that the results of Proposition 4 are really

being applied to a case for which the proposition has never been explicitly proved. Since the application of the test of superposition is so very indispensable, it would seem desirable to give as much prominence to it as possible early in the course; thus, by frequent use, rendering it familiar to the learner, and thereby divesting it of its terrors. This method has accordingly been adopted in the following treatise.

2. The introduction of hypothetical constructions. Euclid's Geometry comprises two classes of propositions, entirely different in their aim and nature, but mixed up together without any indication of this difference. The propositions in the one class are theoretical and general, while those in the other are practical and special. The former treat of the science of Geometry, the latter of the application of that science to the art of geometrical drawing. There is no doubt that great light may be thrown upon, and additional interest imparted to, the theorems by the problems; but this advantage is counterbalanced by the disadvantage attending Euclid's treatment. The strictly logical form adopted by him induces, and was intended to induce, the belief that each individual proposition is essential to all that follow. Hence, an inevitable confusion arises in the mind of the reader between that which is possible theoretically and conceivably, and that which is possible in relation to the instruments to which Euclid chooses to restrict him

self, i. e. the compasses and ruler. Many advanced mathematicians even would be puzzled to give an explanation offhand of the impossibility of the timehonoured problems of squaring a circle, bisecting a cube, and trisecting an angle. Nor is this confusion. of ideas the only evil result arising from the exclusion of hypothetical constructions, for the treatise is thereby rendered inconsistent with itself, and properties of the circle are assumed in Book I. which are demonstrated at full length in Book III. (compare, for example, Book I. Proposition 12 with Book III. Proposition 2); and in Book XI. recourse is necessarily had to hypothetical constructions, where lines are supposed to be drawn in space, concerning which all that is known is the possibility of the existence of such lines.

3. The arithmetical treatment of ratio and proportion.

The unanswerable objection to Euclid's treatment of ratio and proportion is that it is practically disregarded. The reasoning is exquisite and profound, it is too exquisite,' it is artificial and remote from our practical common-sense notions on the subject. No teacher dreams of taking his pupils through Euclid's fifth book; and thus the opportunity for acquiring much valuable instruction is for ever lost to the student. It is at this point that the conception of number is properly brought into contact with the conception of continuous magnitude, and no arithmetical treatise on fractions

can adequately supply the omission. In the following treatise the properties of ratio and proportion are, in the first place, explained and proved in a few simple propositions with reference to commensurable magnitudes, and they are afterwards extended by the simple application of the method of limits to incommensurable magnitudes.

4. The admission of axioms even where derivable from other axioms already stated (as Axiom 2, upon straight lines).

It does not seem to be of any great importance what truths are assumed as axiomatic (i. e. whether demonstrable or not), provided they really have that character and require no demonstration to make them clear to the mind of the learner; but it is important that they should not be too numerous, and also that they should be distinctly enunciated as axioms and not tacitly assumed from time to time as intuitions. Many Continental and some English writers appear to be somewhat lax in this respect. They attach so much importance to the rapid acquisition of mere geometrical knowledge that they lose sight of the equal if not greater importance of obtaining a mastery of the processes by which such knowledge may be attained. Without doubt the clearness of intuition acquired by a practised geometrician will frequently make him impatient of the successive steps of detailed reasoning, and he will be eager to conduct his pupil to the desired end by a shorter and an easier