a paraphrased abridgment of the larger work of the senior author, his Applied Mechanics for the higher technical schools. The work is planned by him, and mainly carried out by his old pupil and assistant. The result is one of the best elementary treatises on the subject that has yet appeared. It is full and comparatively symmetrical in its pure mathematics, and, as a work of application, is unsurpassed. The introductory matter is that usually found in such books; but includes, I note, Simpson's rule for measurement of areas, which has come to be important in view of its frequent use in the engineering problems which now find place in these later works. Part I, "The Principle of Work," has a truly scientific form, beginning with kinematics, the science of pure motion, then taking up problems in pure stress, and finally considering problems of energy-the result of the combination of the two preceding elements. The simple machines are studied in Part II, with and without friction; and the direct-acting steam-engine is taken as the type of the whole system of more intricate machinery. This last is studied in all its essential features, including the effects of friction, and of the inertia of its reciprocating parts, and the energy-problems involved in the action of fly-wheels and in its regulation. Dynameters, brakes, and governors are taken up and examined in all their mechanical principles; and well-chosen examples, as in the preceding part of the work, illustrate their action and their computations.

The second part includes also the study of the strength of materials and the applications of the principles deduced, as illustrated in the work of the engineer; while a carefully made collection of experimentally derived data supplies the needed. contents for the formulas.

Part III is devoted to hydraulics, and presents the principles of static and dynamic action of fluids, and their illustration in hydraulic machinery, as in overshot and undershot waterwheels and pumps; but omitting turbines.

The book contains many of those modern features of the later works in engineering which have hitherto not appeared in the works of writers unfamiliar with applied mechanics, as actually in application in construction and practical work. The treatment of the steam-engine, for example, includes a discussion of great value, exhibiting the effects of the inertia and kinetic energy of moving parts in modifying the action of the machine, in turning the crank, and thus affecting for good

or ill the regularity of its motion as a whole, and its regulation. This book is more nearly up to date in its subject than any work of its class that we have yet seen. It is admirably adapted to its purpose, the instruction of students in schools, especially in such as are preparatory to the great engineering colleges. The book is well made, though a larger type and page would be preferable.

SIBLEY COLLEGE, Cornell UNIVERSITY.

R. H. THURSTON.

A Treatise on Trigonometry.-By Professor JAMES E. Oliver, Associate Professor LUCIEN A. WAIT, and Assistant Professor GEORGE W. JONES, of Cornell University. Fourth edition. Ithaca, N. Y. : Published for the Authors, pp. viii,

160.

Logarithmic Tables.-The same—pp. 96.

Trigonometry has passed far beyond its original field and wholly outgrown its name. In all developments, however, the trigonometric functions retain their original character, and so make proper the use of the limited name over the entire subject. The new discussions have not only modified definitions but have made a marked line between elementary and advanced treatment. The Trigonometry of the Cornell Professors proposes to give both elementary and advanced discussion. There is grave doubt whether discussions of value only to advanced readers should have place at all in an elementary book; they certainly should not at the cost of full consideration of the elements. The student beginning trigonometry enters a new field; there are new elements, new symbols, new combinations. Definitions must be clearly stated and each relation made distinctly secure. The work before us is in some respects a grave disappointment. The use of linear and polar elements of position as a basis for the definition of trigonometric functions is excellent. But in the text the discussions of rectangular and polar co-ordinates are so placed as to appear ends rather than means, in this field, to a further end. The first editionalso before us-was in this particular much better.

The elementary discussion of the angle is particularly obscure where it should be especially clear. Angle is definedsomewhat awkwardly-by description, as an opening between two intersecting straight lines, one fixed, the other rotating on the point of intersection. The discussion implies that continuous rotations, giving periodically the same positions, make the

same angle, which is, in fact, one of an indefinite number of angles. To one now first meeting the motion of an angle beyond 360°, there comes a puzzle. Are 30°, 390°, 750°, etc., different angles or the same angle? If they are the same angle, why talk about "congruent angles"? If they are different angles, how do they make, as they clearly do, one opening and therefore, by the definition, one angle?

There is a manifest lack of proportion in the fullness of discussion. Complementary and supplementary angles-definitions of the simplest-are discussed for a full page, while the basic functional relations are stated and demonstrated in less than a half-page. The theorem (4) which establishes the primary relations between trigonometric functions, contains in one statement eight distinct propositions. It is followed by eight demonstrations, given as of the one theorem. The relations given in the theorem are not even of one class, but manifestly of three classes: reciprocal, complex, quadrate. The demonstrations are also of three classes: the reciprocal relations are founded directly upon the definitions; the complex require a single algebraic step; the quadrate, a geometric and an algebraic step. Relations and proof are very simple-to one who knows; but to one who is yet unfamiliar even with the functional symbols, there is, to speak mildly, great danger of confusion. Strangely enough, two-thirds of the next page is taken up with a consideration of the double sign coming directly from the quadratic relations and needing but a word to illustrate the interpretation to the student. The relations. given in theorem 4 are fundamental; they recur constantly; the student should have clear conception and facile use of them; they should be emphasized by distinct statement. The same tendency to pack into limited space diverse statements is shown elsewhere, though there is no theorem so grossly bad as the one noted. With the disposition to economize space, it is somewhat remarkable that the authors deem it necessary (p. 57) to redemonstrate the proposition of Plane Geometry that "The square of any side of a plane triangle equals the sum of the squares of the other two sides less twice the product of one of these sides into the projection of the other upon it." This is as competent as other propositions used in the text without demonstration.

The treatment of the solution of triangles, both plane and spherical, is much more distinct than the discussion of the

theory; though there are throughout examples of a tendency to crowd closely together in some statements and to make minute subdivisions and repetitions in others.

The objections to the text are not a matter of speculation with the writer. He used for two years the earlier edition of this text and was convinced that, despite the recognized mathematical ability of the authors, the forms of statement seriously obscured the subject.

The Logarithmic Tables issued for use with the Cornell. Trigonometry are admirable. Well arranged, well printed in clear type on good paper, they are worthy of full commendation.

HAMILTON COLLEGE,

CLINTON, N, Y.

OREN ROOT.

University of the State of New York; Regents' Bulletin, No. 5, April, 1891. Albany, N. Y.: University of the State of New York, 1891, pp. 156.

Among all the agencies which are to-day at work elevating the standards of public education, the University of the State of New York holds a prominent place, not only from its unique and far-reaching powers, but from the way in which these powers are exercised. Of all the fruitful projects undertaken by this vigorous institution, it would be difficult to find one more wisely conceived or more admirably executed than the system of examinations conducted by the Regents of the University. The syllabus of these examinations, in its fourth edition, is now issued as Regents' Bulletin, No. 5. The first edition of this work was merely a summary statement of the requirements in the examinations conducted by the University. It was published in 1880 as a pamphlet of twentyeight pages, and included thirty-six branches; the edition of 1891 is a true syllabus, containing one hundred and fifty-six pages and including fifty-eight subjects. Unlike the first, which was prepared by the Secretary alone, this edition is not only the outgrowth of three previous editions, but is also the result of the combined labors of the Assistant Secretary, the editor, and the many principals and teachers of the schools of the State, to some of whom credit is given by name in the body of the syllabus. These facts are all interesting and significant. The unity and progress which it is the chief aim of this work to secure can be reached only through such co-oper

ation. The important changes, both in the quantity and in the quality of the requirements, which have been made at the suggestion of representative teachers throughout the State, cannot fail greatly to extend the usefulness and increase the influence of the University.

The classification of subjects, as it appears in this syllabus, could not well be improved, and a greater degree of freedom is manifest in the general plan of the examinations. The following plank, which appears in the platform relating to examinations, will give an idea of the spirit of progress which characterizes the entire syllabus. "The aim in all the examinations is to test, as far as practicable, the thought-power of the candidate. Definitions, rules, principles, and laws are of little value without a knowledge of their application." The most radical and most significant change that has been made in accordance with these statements is to be found in the treatment of physics and chemistry. Not only has their time been nearly doubled, but the requirements as to amount of information have been reduced fully one-half. It is refreshing to find that the proper names,—an even score of them,—from Frauenhoffer to Rhumkorff, that studded the chapters devoted to physics in the edition of 1888, have disappeared to a man; and in their stead we find "the motion of the lawn-sprinkler," the "method of supplying cities with water," and "of heating houses by steam." Similar changes have been made in the section on chemistry. In literature, also, there is a like improvement.

On the contrary, it must be urged, as a general criticism, that on very many subjects there is still too much required. The "information fallacy" has a tendency to intrude itself. It seems inherent in examinations that they should tend to test quantity of knowledge and not quality of mind. Whatever tests of thought-power the question papers may prove to be, it remains a fact that in too many cases the syllabus calls for an amount of information inconsistent with the development of the power to think, and even, in some cases, inconsistent with the gaining of a love of study. In some of the sciences and in the requirements for the first year's work in Greek and Latin, the preparation for examination may easily develop into cramming, and into a verbal memorizing, rather than an intelligent making of rules. In literature, also, where, as a result of twenty weeks' study, there is a demand for knowledge about the lives and characters of no less than thirty-three authors,