The conclusion of the review of Dr. Clarke's Travels in Russia, is unavoid

THE }

ECLECTIC REVIEW,

For OCTOBER, 1810.

Art. I. Elements of Algebra. By Leonard Euler, translated from the French; with the Additions of Lagrange, and the Notes of the French Translator: To which is added an Appendix, containing the Demonstration of several curious and important numerical Propositions, alluded to, but not investigated, in the Body of the Work, &c. &c. Second Edit. 2 vols. 8vo. pp. xxxviii. 909. Price 24s. Boards. Johnson and Co. 1810.

WHENEVER we reflect upon the sterling excellence of

Euler's Elements of Algebra, it is not, without feeling considerable surprize, that an interval of thirteen years, since their first translation into the English language, should have elapsed before the appearance of a new edition. This profound mathematician was not, like many of the continental geometers, seduced from the love of simplicity, by a vain desire to puzzle, to startle, or to astonish. In all his investigations he pursued truth in the simplest and most direct path; and never wandered from his road to chase butterflies or gather wild-flowers, losing sight of objects of importance in the pursuit of trifles. If, in the course of his researches, he has ever employed abstruse formulæ or intricate processes, it is because the real difficulty of the subject rendered it absolutely necessary; and even in such cases, he more frequently conducts his readers to the result by striking out some happy expedient, than by guiding them through a labyrinth of deep analytical calculation. Euler was a philosopher of indefatigable industry and profound research; and so excursive was the range of his enquiries, that there are very few regions in the mathematical, physical or metaphysical sciences, upon which he has not thrown additional light, and which he has not enriched with new discoveries: insomuch that the catalogue alone of the works of this extraordinary man, extends to the amount of fifty pages. Of these performances several were printed separately, and others inserted in the memoirs of different Academies: viz, in 38 volumes of the Petersburg Acts (from Vol. VI. 3.T

6 to 10 papers in each volume);-in several volumes of the Paris Acts;-in 26 volumes of the Berlin Acts (about 5 papers in each volume);-in the Acta Eruditorum, in 2 vols ;-in the Miscellanea Taurinensia ;-in vol. 9 of the Society of Ulyssingue;-in the Ephemerides of Berlin;-and in the Memoires de la Societé Economique, for 1766.

By far the greater proportion of Euler's writings are addressed to the "primores populi." He has aimed rather to heighten the superstructure of science than to strengthen its foundations; and hence it happens that but few of his numerous productions can be strictly called elementary. The work now on our table, however, is perfectly such; and it is marked, in a most eminent degree, with the first quality of an elementary performance, perspicuity. We do not hesitate to call it the most perspicuous treatise on algebra extant. proof of the remarkable fitness of this book for novices, is furnished in the circumstances resulting from the manner in which it was prepared for the press, as related by the German editors of the later editions.

A

The object of the celebrated author was to compose an elementary treatise, by which a beginner, without any other assistance might make himself complete master of Algebra. The loss of sight had suggested this idea to him, and his activity of mind did not suffer him to defer the execution of it. For this purpose M. Euler pitched on a young man, whom he had engaged as a servant on his departure from Berlin, suffi ciently master of arithmetic, but in other respects without the least knowledge of mathematics. He had learned the trade of a tailor, and with regard to his capacity was not above mediocrity. This young man, however, has not only retained what his illustrious master taught and dictated to him, but in a short time was able to perform the most difficult algebraic calculations, and to resolve with readiness whatever analytic questions were proposed to him. p. xxv.'

In this work almost every part is excellent, but none more so than that which relates to indeterminate and diophantine problems. The whole of this branch of analysis is most ably methodised, and cast into the best possible form for those persons to read, who wish to study it without the assistance of a master. The substance of the investigations of Kersey, Fermat, Ozanam and others, is thrown into an orderly system; and many highly useful theorems are exhibited in the course of this masterly discussion.

The French translation of this work received the addition of several notes by the translator; and was more especially enriched by some valuable papers from the hand of the celebrated Lagrange, on the subjects of continued fractions, quadratic equations of two unknown quantities in rational numbers, double and triple equalities, the division of alge

braic functions into fractions, &c. These additions are all inserted in the English translation: and in the edition of 1797 some notes were added by the translator, who seems, however, to have been but moderately skilled in mathematics, since his translation contained most of the errors of the French edition, and full two hundred new ones. This, of: course, rendered the book extremely faulty, and made it in a manner unsafe to put into the hands of any one who was extending his knowledge of algebra without the advantage of a tutor. The present editor, (who dates his preface from the Royal Military Academy, but has not given his name,) we are happy to say, has carefully purged it from these inaccuracies; and has, besides, made some important additions and alterations.

Such of the former notes as are retained (among which are all those of the French translator) have been placed at the bottom of the pages to which they refer, and several others have been added where they were thought necessary, beside those subjoined in the second volume; in the latter of which are demonstrated all the numerical propositions that the author has referred to, but not investigated in the body of the work. These notes, as far as could be done, are so arranged as to form a concise abstract of the theory of numbers; which, being a subject that has not much engaged the attention of English mathematicians, it is presumed, that those who have not an opportunity of consulting foreign writers on this branch of analysis, may there find some useful information. A few of these notes are new; the others have been chiefly derived from the works of Waring, Gauss and Legendre.'

It is principally on account of these notes, that we speak at all of the present edition; and we may safely recommend them to students of algebra, especially at colleges and public institutions where young men wish to penetrate a little below the surface. They exhibit neat and satisfactory demonstrations of several curious propositions, not often met with in English books, though they have been discussed at great length by several foreign authors: such, for example as that, if n be a prime number, and prime to n, then 1-1, is divisible by n that the square root of any number hot a square, cannot be expressed by a rational fraction: that a tractional root of an equation cannot produce integral coefficients and an integral absolute terms: that the equation 1xx3=y2, is impossible, in integers, except in the case when 2: that every prime number of the form 4n+1, is the sum of two squares that all prime numbers of the forms 8n+1, and 8n+5, are exclusively of all others, contained in the formula 2+2 that all prime numbers of the forms 8n+1, and 8n+3, are exclusively of all others, contained in the formula y2+2x2: that all prime numbers of the forms 8n+1, and