tions. Not only is the differential calculus to be excluded, but even that germ of it which, as given by FERMAT in his treatment of this very problem, made some think that he was entitled to claim the invention. The values of x and of (x + h) are not to be compared; and no process is to be allowed which immediately points out the relation of x to the derived function ox. A mathematician to whom I stated the conditioned problem made it, very naturally, his first remark, that he could not see how on earth I was to find out when it would be biggest, if I would not let it grow. The mathematician will at last see that the question resolves itself into the following:- Required a constant, r, such that ox―r shall have a pair of equal roots, without assuming the development of (x + h) or any of its consequences. "RAMCHUNDRA, the author of this work, has transmitted to me some notes of his own life, from which I collect, as follows. He was born in 1821, at Paneeput, about fifty miles from Delhi. His father, SOONDUR LALL, was a Hindoo Kaeth, and a native of Delhi, and was there employed under the collector of the revenue. He died at Delhi, in 1831-2, leaving a widow (who still survives) and six sons. After some education in private schools, RAMCHUNDRA entered the English Government school at Delhi, to every pupil of which two rupees a month were given, and a scholarship of five rupees a month to all in the first and second classes. 'In this school he remained six years. It does not appear that any particular attention was paid to mathematics in this school; but, shortly before leaving it, a taste for that science developed itself in RAMCHUNDRA, who studied at home with such books as he could procure. After leaving school, he obtained employment as a writer for two or three years. In 1841, changes took place in the educational department of the Bengal presidency; the school was formed into a college; and RAMCHUNDRA obtained, by competition, a senior scholarship, with thirty rupees a month. In 1844, he was appointed teacher of European science in the Oriental department of the college, through the medium of the vernacular, with fifty rupees a month additional. A vernacular translation society was instituted, and RAMCHUNDRA, in aid of its object, translated or compiled works in Oordoo, and also on algebra, trigonometry, &c., up to the differential and integral calculus." A Treatise on Differential Equations. By GEORGE BOOLE, F. R. S., Professor of Mathematics in the Queen's University, Ireland, Honorary Member of the Cambridge Philosophical Society. 485 pp. Crown, 8vo. Cambridge: MACMILLAN & Co., and 23 Henrietta Street, Covent Garden, London. 1859. The works devoted exclusively to a systematic treatment of Differential Equations, are neither so many, nor so exhaustive, that teachers and students will not welcome another, provided it will be well adapted to the wants of elementary instruction. To give the author's idea of these wants, and how they are to be supplied, we give the following extract from the Preface. "It was my object first of all, to meet the wants of those who had no previous acquaintance with the subject, but I also desired not quite to disappoint others who might seek for more advanced information. These distinct, but not inconsistent aims determined the plan of composition. The earlier sections of each chapter contain that kind of matter which has usually been thought suitable for the beginner, while the latter ones are devoted either to an account of recent discovery, or to the discussion of such deeper questions of principle as are likely to present themselves to the reflective student in connection with the methods and processes of his previous course. The principles which I have kept in view in carrying out the above design are the following: "1st. In the exposition of methods I have adhered as closely as possible to the historical order of their development. I presume that few who have paid any attention to the history of the Mathematical Analysis, will doubt that it has been developed in a certain order, or that that order has been, to a great extent, necessary-being determined, either by steps of logical de duction, or by the successive introduction of new ideas and conceptions, when the time for their evolution had arrived. And these are causes which operate in perfect harmony. Each new scientific conception gives occasion to new applications of deductive reasoning; but those applications may be only possible through the methods and the processes which belong to an earlier stage. Thus, to take an illustration from the subject of the following work: The solution of ordinary simultaneous differential equations properly precedes that of linear partial differential equations of the first order; and this, again, properly precedes that of partial differential equations of the first order which are not linear. And in this natural order were these subjects developed. Again, there exists large and very important classes of differential equations, the solution of which depends on some process of successive reduction. Now such seems to have been effected, at first, by a repeated change of variables; afterwards, and with greater generality, by a combination of such transformations with others involving differentiation; last of all, and with greatest generality, by symbolical methods. I think it necessary to direct attention to instances like these, because the indications which they afford, appear to me to have been, in some work, of great ability, overlooked; and because I wish to explain my motives for departing from the precedent thus set. "Now there is this reason for grounding the order of exposition upon the historical sequence of discovery, that by so doing we are most likely to present each new form of truth to the minds precisely at that stage at which the mind is most fitted to receive it, or even like that of the discoverer, to go forth to meet it." The plan of composition indicated in the above extract is completely carried out in nearly every chapter, of which there are eighteen. The elementary articles of each chapter, which the beginner should first read, are indicated by the author. A valuable feature of the work is the collection of interesting examples appended to each chapter; in all, over 300. At the end of the volume are found the answers of nearly one half the problems, which will be of great convenience to the student while acquiring confidence in his knowledge of the theory. Besides a general survey of the whole volume, we have read several of the earlier chapters with care, to be able to judge of its adaptation to the wants of students in this country; and we can commend it as being the very best work we have ever seen for a text-book upon the subject of Differential Equations; nor need the mathematician fear that he will fail to find it a quite full compendium of our present knowledge of this important department of analysis. THE following gentlemen have sent us solutions of the Prize Problems in the August number of the MONTHLY :· :- DAVID TROWBRIDGE, Perry City, Schuyler Co., N. Y., answered all the questions. His solutions of the Prize Problems in the July number did not reach us in time; as was also the case with the solutions sent by JAMES M. INGALS, of Delton, Sauk Co., Wis. ASHER B. EVANS, Madison University, Hamilton, N. Y., answered all the questions. III. JAMES F. ROBERSON, Senior Class, Indiana University, Bloomington, answered question We have also received a set of excellent solutions from ASAPH HALL, Esq., Assistant at Harvard College Observatory, who does not wish to compete for the prizes. Prof. WERDEN REYNOLDS, late Principal of the Worcester Academy, has accepted the Presidency of the Worcester Female College. . . . . Prof. B. S. HEDRICK has been appointed to the Professorship of Mathematics in Cooper Institute, N. Y. . . . . Rev. THOMAS HILL will deliver a course of lectures before the Lowell Institute, in Boston, in a few weeks, and return to Yellow Springs, Ohio, about January 1, 1860..... Prof. CHAUVENET, of the U. S. Naval Academy at Annapolis, Md., has accepted a Professorship in Washington University, at St. Louis, Mo..... WILLIAM WATSON, Esq., a graduate of and Tutor in the Lawrence Scientific School, already well and favorably known to our readers, will sail for Paris on the 12th of November, where he proposes to spend a year or more in the study of the Mathematics. Mr. WATSON has kindly consented to be an occasional correspondent of the MATHEMATICAL MONTHLY during his absence, and send us such information concerning the schools and methods of instruction, especially in the Mathematics, as it may be in his power to give. Instead of the hasty observations of the mere tourist, we shall be informed of every day school life abroad, by one whose education and experience as a teacher of the mathematics will enable him to judge of those things which will most benefit his fellow teachers at home. Our best wishes go with him. . . . . It gives us pleasure to add the following names to our list of coöperators and contributors: E. A. STRONG, Esq., Grand Rapids, Mich.; HENRY WARD POOLE, Esq., Boston; LUCIUS BROWN, Esq., Fall River, Mass.; SAXE GOTHA LAWS, Esq., Dover, Delaware; CHAUNCEY SMITH, Esq., Boston. The portrait of Dr. BOWDITCH, which we have the gratification of presenting to our readers, needs no commendation at our hands, as a work of art; and we will also add that it is recognized by his family as a faithful likeness. It is eminently proper that the first portrait given in the MONTHLY should be of the "FATHER OF AMERICAN GEOMETRY." We commend the list of his writings, with the interesting and valuable series of notes by his son, N. I. BOWDITCH, Esq., to the attention of all. We trust it may not be long before we shall have the pleasure of presenting the portraits of other eminent mathematicians, both of our own and foreign countries. .... BOOKS RECEIVED. Tables of Victoria, Computed with regard to the Perturbations of Jupiter and Saturn. By F. BRÜNNOW, Ph. Dr., Professor of Astronomy in the University of Michigan, and Director of the Observatory at Ann Arbor. Printed by order of the Board of Regents. New York: B. WESTERMANN & Co. London: TRÜBNER & Co. 4to. pp. 75. 1859. Linear Perspective Explained. By WILLIAM N. BARTHOLOMEW, Author of BARTHOLOMEW's Sketch Book, and Series of Drawing Books, in six numbers. Boston: SHEPARD, CLARK & BROWN. 1859..... Résumé de Leçons de Géométrie Analytique et de Calcul Infinitésimal. By J. B. BELANGER, Ingénieur en chef des Ponts et Chaussées, Professeur de Mécanique à l'Ecole Impériale Polytechnique et à l'Ecole Centrale des Arts et Manufactures. Seconde Edition. Paris: MALLET-BACHELIER. 1859..... Journal de Mathématiques Pures et Appliquées; Publié by JOSEPH LIOUVILLE. Juillet. 1859..... Nouvelles Annales de Mathématique Rédigé Par MM. TERQUEM et GERONO. Septembre, 1859. THE MATHEMATICAL MONTHLY. Vol. II.... DECEMBER, 1859.... No. III. PRIZE PROBLEMS FOR STUDENTS. I. IF two circles touch each other, any straight line passing through the point of contact cuts off similar parts of their circumferences. II. Find the four roots of the recurring equation 1 III. If 2 cos 0 = u+prove that 2 cos 20 = u2 + + 1 из 2 cos n 0 = u* + 1; and then find the sum of the series, cos + cos 20+ cos 30..... + cos n o. IV. Having given the Right Ascensions and Declinations of two stars, to find the formula for the distance between them. Also, find what the distance becomes, when for one star A. R. is 8h 12m 38o.17, and Dec. 17° 23′ 49′′.8 north, and for the other A. R. is 13h 28m 19.92, and Dec. 21° 12′ 37′′.2 south. V. In a frustum of any pyramid or cone, the area of a section, parallel to the two bases and equidistant from them, is the arithmetical mean of the arithmetical and geometrical means of the areas of the two bases. The solutions of these problems must be received by February 1, 1860. REPORT OF THE JUDGES UPON THE SOLUTIONS OF THE PRIZE PROBLEMS IN No. XI. Vol. I. THE first Prize is awarded to GUSTAVUS FRANKENSTEIN, Springfield, Ohio. The second Prize is awarded to ASHER B. EVANS, Madison University, Hamilton, New York. Solve the equations PRIZE SOLUTION OF PROBLEM I. By GUSTAVUS FRANKENSTEIN, Springfield, Ohio. x+y=a (213+y3) (x2+y3) = b, and give a discussion of the values of the roots. Squaring and cubing x + y = a, we get x2+ 2xy + y2 = a2, x2 + 3 x2 y + 3 xy2+y3 = a3. · x2 + y2 = a2 — 2 x y, x3 + y3 — a3 — 3 x y (x + y) = a3 — 3 a xy. · · (x2 + y3) (x2 + y2) = (a3 — 3 a x y) (a2 — 2 x y) = b. Hence, knowing the sum, a, of x and y, and their product, q, their values will be given by the quadratic 22-ax + q = 0. DISCUSSION. Case I. When a and b have the same signs. Since - a2 is negative, whether a be positive or negative, it is evident that two of these values of x will always be imaginary. If a and b are of the same sign, q is real, since a (24b+a) is positive; and x |