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Although so many books on elementary geometry are continually appearing, no apology need be offered for the publication of the present work. It has nothing in common with the ordinary text-book, except that it deals with the same subject. Prof. Halsted yields to none in his reverence for the marvellous work achieved by Euclid; nevertheless, he belongs to that school of mathematicians which maintains that Euclid's system is not infallible; that his theory is, in fact, built up from an imperfect and incomplete set of fundamental axioms to which he himself tacitly and, perhaps even unconsciously, added. In the opinion of Prof. Halsted and kindred thinkers it has become necessary, for the advancement of truth, that the system which has held sole sway for so many centuries should give place to another and a better one. Unlike many of the writers who undertake the task of reforming Euclid, Prof. Halsted shows no tendency to be content with less rigid proof: on the contrary, he urges the necessity for the utmost rigour; and this, we venture to think, is one of the strongest of his many strong claims to consideration. He asserts that the principles which form the groundwork of his book secure both greater simplicity and increased rigour for his demonstrations. Hilbert's Foundations of Geometry furnish the basis for the present treatise. Accustomed as we are to the small number of simply worded axioms which are met with in Euclid, it is somewhat difficult to acquire readily a comprehensive grasp of the five groups of assumptions considered essential by Hilbert, and, seeing that an authority as notable as Poincare failed to detect the redundancy of one of Hilbert's betweenness assumptions, no humbler mathematician need hesitate to reserve for a time any definite expression of opinion as to the exact extent to which Hilbert's assumptions are deserving of being regarded as unimpeachable. None however, will dispute the care and the effort to attain perfection which mark the drawing up, the classification, and the enunciation of the assumptions ; none can fail to recognize how in Prof. Halsted's hands they yield simple and delightful proofs of many of the propositions with which every student of mathematics is familiar. Four only of the five groups of assumptions are used in the present work. viz., those in which the ideas of association, of betweenness, of congruence, and of parallelism claim attention. The Archimedean principle of continuity is avoided in demonstrating the theory of proportion, and in its place stands a sect calculus which furnishes for geometry an analogue to the operations of algebra as applied to real numbers. The associative, commutative, and distributive laws which govern algebra are shown to apply equally to the sect calculus for geometry. The charm of many of the author's methods of proof has been referred to: it exists in a marked degree in the sixth chapter, where the originality displayed in the solution of problems is specially attractive. When Hilbert's Foundations of Geometry appeared there at once arose in the mind a doubt as to the possibility-at any rate, at the present time-of adapting the system to the needs of the immature student; but the production of Prof. Halsted's work shows that no cause for the doubt really existed. -The Education Outlook, Volume 57

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