The Probability of Hit when the Probable Error in Aim is Known with a Comparison of the Probabilities of Hit by the Method of Independent and Parallel By Mansfield Merriman, Professor of Civil EnginPamphlet, 12 pages. Reprinted from the Journal

Fires from Mortar Batteries. eering in Lehigh University.

of the U. S. Artillery, Vol. VIII, No. 2.

The problem considered in this paper is, To find the probability of hit on the target or deck of a ship whose area is 4a1, where 2a is the width of the target in azimuth and 2.4 its length in range, a shot being fired with the intention of hitting the center. B. F. F.

Contributions to the Geometry of the Triangle. By Robert J. Aley, A. M., Professor of Mathematics in the University of Indiana. Pamphlet, 32 pages.

This thesis was accepted by the Department of Mathematics of the University of Pennsylvania in partial fulfillment of the requirements for the degree of Doctor of Philosophy, which is a sufficient testimonial of its importance and value. B. F. F.

Periodico di Mathematica Per L'Insegnamento Secondario. Dott. G. Lazzeri. November-December number.

The Mathematical Gazette. Edited by F. S. Macauley, M. A., D. Sc. October number.

Bollettino della Associazione "Mathesis” Fra Gl'Insignanti di Mathematica delle Scuole Medie.

Revue Semestrielle des Publications Mathomatiques Rédigée sous les auspices de la Sociéte Mathematique d'Amsterdam. Par P. H. Schoute, D. J. Korteweg,

J. C. Kluyver, W. Kapteyn, P. Zeeman.

The American Monthly Review of Reviews. An International Illustrated Monthly Magazine. Edited by Dr. Albert Shaw. Price, $2.00 per year in adSingle numbers, 25 cents. The American Monthly Review of Reviews Co., 13 Astor Place, New York.

vance.

The December number of the American Monthly Review of Reviews has several interesting features. Mr. Ernest Knaufft, editor of the Art Student, contributes an elaborate study of "John Gilbert and Illustration in the Victorian Era"; Dr. Clifton H. Levy tells "How the Bible Came Down to Us," with a number of reproductions from anciens Biblical manuscripts and printed texts; Lady Henry Somerset pays a tribute to the late Duchess of Teck; an English officer in the Indian service writes about the Ameer of Afghanistan; Mr. E. V. Smalley discusses Canadian reciprocity, and Mr. Alex. D. Anderson summarizes the progress of the American Republics. There is also a 23-page illustrated department devoted to the season's new books, with an introductory chapter, by Albert Shaw, on "Some American Novels and Novelists." Altogether, the Review is not lacking in novelty or variety. B. F. F.

THE

AMERICAN

MATHEMATICAL MONTHLY.

VOL. IV.

Entered at the Post-office at Springfield, Missouri, as Second-class Mail Matter.

EONHARD EULER (oi'ler), one of the greatest and most prolific mathematicians that the world has produced, was born at Basel, Switzerland, on the 15th day of April, 1707, and died at St. Petersburg, Russia, November the 18th (N. S.), 1783. Euler received his preliminary instruction in mathematics from his father who had considerable attainments as a mathematician, and who was a Calvinistic* pastor of the village of Riechen, which is not far from Basel. He was then sent to the University of Basel where he studied mathematics under the direction of John Bernoulli, with whose two sons, Daniel and Nicholas, he formed a life-long friendship. Geometry soon became his favorite study. His genius for analytical science soon gained for him a high place in the esteem of his instructor, John Bernoulli, who was at the time one of the first mathematicians of Europe. Having taken his degree as Master of Arts

*The Encyclopedia Brittanica says Euler's father was a Calvinistic minister, while W. W. R. Ball, in his History of Mathematics, says he was a Lutheran minister. Euler himself was a Calvinist in doctrine, as the following, which is his apology for prayer, indicates: "I remark, first, that when God established the course of the universe, and arranged all the events which must come to pass in it, he paid attention to all the circumstances which should accompany each event; and particularly to the dispositions, to the desires, and prayers of every intelligent being; and that the arrangement of all events was disposed in perfect harmony with all these circumstances. When, therefore, a man addresses God a prayer worthy of being heard it must not be imagined that such a prayer came not to the knowledge of God till the moment it was formed. That prayer was already heard from all all eternity; and if the Father of Mercies deemed it worthy of being answered, he arranged the world expressly in favor of that prayer, so that the accomplishment should be a consequence of the natural course of events. It is thus that God answers the prayers of men without working a miracle.''

in 1723, Euler afterwards applied himself, at his father's desire, to the study of theology and the Oriental languages, with the view of entering the ministry, but, with his father's consent, he returned to his favorite pursuit, the study of mathematics. At the same time, by the advice of the younger Bernouillis, who had removed to St. Petersburg in 1725, he applied himself to the study of physiology, to which he made useful applications of his mathematical knowledge; he also attended the lectures of the most eminent professors of Basel. While he was eagerly engaged in physiological researches, he composed a dissertation on the nature and propagation of sound. In his nineteenth year he also composed a dissertation in answer to a prize-question concerning the masting of ships, for which he received the second prize from the French Academy of Sciences.

When his two close friends, Daniel and Nicholas Bernoulli, went to Russia, they induced Catherine I, in 1727, to invite Euler to St. Petersburg, where Daniel, in 1733, was assigned to the chair of mathematics. Euler took up his residence in St. Petersburg, and was made an associate of the Academy of Sciences. In 1730 he became professor of physics, and in 1733 he succeeded his friend Daniel Bernoulli, who resigned on a plea of ill health.

At the commencement of his astonishing career, he enriched the Academical collection with many memoirs, which excited a noble emulation between him and the Bernouillis, though this did not in any way affect their friendship. It was at this time that he carried the integral calculus to a higher degree of perfection, invented the calculation of sines, reduced analytical operations to greater simplicity, and threw new light on nearly all parts of pure or abstract mathematics. In 1735, an astronomical problem proposed by the Academy, for the solution of which several eminent mathematicians had demanded several months' time, was solved by Euler in three days with the aid of improved methods of his own, but the effort threw him into a fever which endangered his life and deprived him of his right eye, his eyesight having been impaired by the severity of the climate. With still superior methods, this same problem was solved later by the illustrious German mathematician, Gauss.

In 1741, at the request, or rather command, of Frederick the Great, he moved to Berlin, where he was made a member of the Academy of Sciences, and Professor of Mathematics. He enriched the last volume of the Mélanges or Miscellanies of Berlin, with five memoirs, and these were followed, with astonishing rapidity, by a great number of important researches, which were scattered throughout the annual memoirs of the Prussian Academy. At the same time, he continued his philosophical contributions to the Academy of St. Petersburg, which granted him a pension in 1742.

The respect in which he was held by the Russians was strikingly shown in 1760, when a farm he occupied near Charlottenburg happened to be pillaged by the invading Russian army. On its being ascertained that the farm belonged to Euler, the general immediately ordered compensation to be paid, and the Empress Elizabeth sent an additional sum of four thousand crowns. The despotism of Anne I. caused Euler, who was a very timid man, to shrink from public