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constructed on one plan throughout, that of always giving in the simplest possible form the direct proof from the nature of the case. The axioms necessary to this simplicity have been assumed without hesitation, and no scruple has been felt as to the increase of their number, or the acceptance of as many elementary notions as common experience places past all doubt. For in an experimental science we are bound to have recourse to nature for as many principles as are necessary to the clearest exposition of that science, subject only to the conditions that these principles be well chosen and beyond reasonable dispute.

The book differs most from established teaching in its constructions, and in its early application of Arithmetic to Geometry.

The arbitrary restrictions of Euclid involve him in various inconsistencies, and exclude his constructions from use. When, for instance, in order to mark off a length upon a straight line, he requires us to describe five circles, an equilateral triangle, one straight line of limited, and two of unlimited length, he condemns his system to a divorce from practice at once and from sound reason. The constructions given in this book are theoretically consistent, and are employed by practical men.

In the application of arithmetical methods, all that has been done has been to anticipate a step which sooner or later is necessary to progress. Valuable results are seldom attained without the bold and skilful combination of the best means at our command, and it is to such a combination that the range and power of Modern Geometry are mainly due. Measurement is besides indispensable to every practical enquiry, and it seems unwise to deprive the learner of the power and interest given by its employment.

Moreover some method of dealing with questions of proportion is absolutely necessary to replace the abandoned Fifth book of Euclid, and to prevent the confusion inseparable from the present thoroughly irrational manner of studying the Sixth book.

Continental practice sanctions the innovations which the Author has ventured to adopt.

As to the book itself, though so small it contains all that is usually read of the first six books of Euclid, with considerable additions. It has been designed as a sufficient introduction to Trigonometry, Conic Sections, and applied Mathematics. It is not intended for private reading, but for a class-book, and the master in using it should work into his teaching the theorems and constructions given at the end of each part. These have been selected with care, and are not mere enigmas but propositions of some importance either in principle or application. The time spent upon them will certainly not be wasted.

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We acquire a clear idea of“the magnitude of an angle by

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A

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supposing it to be described by the revolution of one of the sides OB about the end o of the other 0A, which remains. fixed. When OB has revolved so far as to be in the same line as OA but not in the same direction it has passed over two right angles. When OB again coincides with 0 A, it has passed over four right angles. The minute-hand of a watch passes over four right angles in an hour.

All right angles are equal, and the space occupied by: two right angles is always the same; we shall often denote: it by 2R

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