### Association of Teachers of Mathematics in the Middle States and Maryland. Bulletin No. I

1904 Mathematical Gazette
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more » ... EMATICAL GAZETTE. if the following method of detection is mastered-Take the right answer from the book and the wrong answer; form a ratio or fraction of the two answers, placing the incorrect one in the numerator and the correct one in the denominator; then the fraction thus formed will be (1) a square quantity if one ratio is misplaced; (2) the product of two square quantities if two ratios are misplaced." No explanation of this fact is vouchsafed. We might now understand why many teachers prefer their students not to have access to the answers. But the book is full of good points. This content downloaded from 129.16.69.49 on Wed, 30 Dec 2015 04:18:53 UTC All use subject to JSTOR Terms and Conditions REVIEWS. 69 Aufgaben aus der Niederen Geometrie. By I. ALEXANDROFF. Pp. v, 123. 1904. (Teubner.) This is the German edition of a useful introduction to methods of attacking the solution of geometrical riders. It was translated from the Russian into French some years ago, and was at the time reviewed in these columns. We are glad to call attention to the German edition, as to the student who knows French and wishes to pick up German mathematical terminology, the two volumes may prove of considerable value, quite apart from the intrinsic merits of the book. Lehrbuch der Differenzenrechnung. By D. SELIWANOFF. Pp. vi, 92. (Teubner.) This is a handy little introduction to the Calculus of Finite Differences, simply and clearly written. The first part deals with introductory matter, interpolation, and the approximate values of certain integrals. The second treats of summation, the J. Bernouilli Function, and Euler's summation formula with its applications. The third is devoted to linear difference equations of the first order, and linear difference equations with constant coefficients-the whole being subject matter of Boole's first eleven chapters, but treated lightly and on a smaller scale. Lehrbuch der Analytischen Geometrie. By 0. FORT and 0. SCHLOMILCH. Vol. I. Analytische Geometrie der Ebene. 7th edition. Revised by R. HEGER. 1904. (Teubner.) This, the seventh edition of a book, the special characteristic of which is its clearness and simplicity of exposition, is revised and improved in various details by Professor Heger of the Polytechnic at Dresden. The second, third, and fourth editions were edited by O. Fort, and it is one-and-twenty years since the fifth edition was revised by Professor Heger. The ninth and tenth chapters deal with lines of a higher order than the second, with transcendental lines in general, and spirals and the usual roulettes in particular. M6thodes de Rdsolution et de Discussion des Problmes de Gdom6trie. By G. LEMAIRE. Pp. 224. 1904. (Vuibert et Nony.) This is a good representative of a type of compilation of which we have but few instances in this country. It covers much the same ground as Petersen and Alexandroff, both of which are available in French. There is something to be said for the view that a boy taught, as until recently he has here been taught, will not necessarily take "in his stride" and without preparation in some detail, such a grasp of the different methods by which a problem may be attacked as he will after the study of such a volume as this. The names of the sections will sufficiently indicate the lines on which the book is constructed: Loci; method of the intersection of loci; determination of a line (envelopes, lines through a fixed point, lines parallel to a given direction); translation, rotation, symmetry, method of similar figures; figures similarly and similarly situated, inversion; transformation and division of figures. Many of the solutions of problems are accompanied by discussion in detail, most unusual with us, but a process especially commendable in the case of intelligent boys, who often find this method a source of much inspiration. Advanced Course in Algebra. By W. WELLS. Pp. viii, 581. 6s. 6d. (Heath.) The course that is covered in this volume is hardly what we should call "advanced" algebra in this country, and yet we are told it is intended to meet the needs of "Colleges and Scientific Schools of the highest rank." For the book contains no more than is to be found in Smith's Treatise, or Hall & Knight's Higher Algebra-not so much, indeed, for the subject is not treated with the same detail. It takes nearly four hundred pages to bring the student to the end of the chapter on the binomial theorem, and only one hundred and fifty pages are devoted to permutations and combinations, probability, continued fractions, summation of series, theory of numbers, determinants, and theory of equations. Simple, simultaneous, and quadratic equations take up one-fifth of the book. Graphs are well to the fore in dealing with equations generally, the theory of the irrational number, derivatives, multiple roots, Sturm's theorem, and discontinuous functions. The chapter on involution contains the binomial theorem for positive integral REVIEWS. 69 Aufgaben aus der Niederen Geometrie. By I. ALEXANDROFF. Pp. v, 123. 1904. (Teubner.) This is the German edition of a useful introduction to methods of attacking the solution of geometrical riders. It was translated from the Russian into French some years ago, and was at the time reviewed in these columns. We are glad to call attention to the German edition, as to the student who knows French and wishes to pick up German mathematical terminology, the two volumes may prove of considerable value, quite apart from the intrinsic merits of the book. Lehrbuch der Differenzenrechnung. By D. SELIWANOFF. Pp. vi, 92. (Teubner.) This is a handy little introduction to the Calculus of Finite Differences, simply and clearly written. The first part deals with introductory matter, interpolation, and the approximate values of certain integrals. The second treats of summation, the J. Bernouilli Function, and Euler's summation formula with its applications. The third is devoted to linear difference equations of the first order, and linear difference equations with constant coefficients-the whole being subject matter of Boole's first eleven chapters, but treated lightly and on a smaller scale. Lehrbuch der Analytischen Geometrie. By 0. FORT and 0. SCHLOMILCH. Vol. I. Analytische Geometrie der Ebene. 7th edition. Revised by R. HEGER. 1904. (Teubner.) This, the seventh edition of a book, the special characteristic of which is its clearness and simplicity of exposition, is revised and improved in various details by Professor Heger of the Polytechnic at Dresden. The second, third, and fourth editions were edited by O. Fort, and it is one-and-twenty years since the fifth edition was revised by Professor Heger. The ninth and tenth chapters deal with lines of a higher order than the second, with transcendental lines in general, and spirals and the usual roulettes in particular.