1.1. Note.

This book is a reprint of the 1840 translation from the German by John Hewlett which contains a translation of Lagrange's Additions and an excerpt from the Memoir of the Life and Character of Euler, by Francis Horner. The 1984 reprint has, in addition, an introduction by Clifford Truesdell. We give below the complete Memoir of the Life and Character of Euler, by Francis Horner as given in the 3rd edition of 2011.

1.2. From the Publisher.

Originally published in 1765, this book is a gentle introduction to algebra by one of history's greatest mathematicians, Leonhard Euler. Starting with basic mathematical concepts such as signs, fractions, powers and roots, logarithms, infinite series, arithmetic and geometric ratios, and the calculation of interest, Euler then discusses how to solve equations of varying degrees, methods of rendering certain formulas rational, and more. In 1771, Joseph-Louis Lagrange included an addendum to the French edition containing topics such as continued fractions and Diophantine equations.

1.3. Memoir of the Life and Character of Euler, by Francis Horner, Esq. M.P.

Leonard Euler was the son of a clergyman in the neighbourhood of Basil, and was born on the 15th of April, 1707. His natural turn for mathematics soon appeared, from the eagerness and facility with which he became master of the elements under the instructions of his father, by whom he was sent to the university of Basil at an early age. There, his abilities and his application were so distinguished, that he attracted the particular notice of John Bernoulli. That excellent mathematician seemed to look forward to the youth's future achievements in science, while his own kind care strengthened the powers by which they were to be accomplished. In order to superintend his studies, which far outstripped the usual routine of the public lecture, he gave him a private lesson regularly once a week; when they conversed together on the acquisitions, which the pupil had been making since their last interview, considered whatever difficulties might have occurred in his progress, and arranged the reading and exercises for the ensuing week.

Under such eminent advantages, the capacity of Euler did not fail to make rapid improvements; and in his seventeenth year, the degree of Master of Arts was conferred on him. On this occasion, he received high applause for his probationary discourse, the subject of which was a comparison between the Cartesian and Newtonian systems.

His father, having all along intended him for his successor, enjoined him now to relinquish his mathematical studies, and to prepare himself by those of theology, and general erudition, for the ministerial functions. After some time, however, had been consumed, this plan was given up. The father, himself a man of learning and liberality, abandoned his own views for those, to which the inclination and talents of his son were of themselves so powerfully directed; persuaded, that in thwarting the propensities of genius, there is a sort of impiety against nature, and that there would be real injustice to mankind in smothering those abilities, which were evidently destined to extend the boundaries of science. Leonard was permitted, therefore, to resume his favourite pursuits; and, at the age of nineteen, transmitting two dissertations to the Academy of Sciences at Paris, one on the masting of ships, and the other on the philosophy of sound, he commenced that splendid career, which continued, for so long a period, the admiration and the glory of Europe.

About the same time, he stood candidate for a vacant professorship in the university of Basil; but having lost the election, he resolved, in consequence of this disappointment, to leave his native country; and in 1727 he set out for Petersburg, where his friends, the young Bernoullis, had settled about two years before, and where he flattered himself with prospects of literary success under the patronage of Catherine I. Those prospects, however, were not immediately realised; nor was it till after he had been frequently and long disappointed, that he obtained any preferment. His first appointment appears to have been to the chair of natural philosophy; and when Daniel Bernoulli removed from Petersburg, Euler succeeded him as professor of mathematics.

In this situation he remained for several years, engaged in the most laborious researches, enriching the academical collections of the continent with papers of the highest value, and producing almost daily improvements in the various branches of physical, and, more particularly, analytical science. In 1741, he complied with a very pressing invitation from Frederic the Great, and resided at Berlin till 1766. Throughout this period, he continued the same literary labours, directed by the same wonderful sagacity and comprehension of intellect. As he advanced with his own discoveries and inventions, the field of knowledge seemed to widen before his view, and new subjects still multiplied on him for further speculation. The toils of intense study, with him, seemed only to invigorate his future exertions. Nor did the energies of Euler's mind give way, even when the organs of the body were overpowered: for in the year 1735, having completed, in three days, certain astronomical calculations, which the academy called for in haste; but which several mathematicians of eminence had declared could not be performed within a shorter period than some months, the intense application threw him into a fever, in which he lost the sight of one eye.

Shortly after his return to Petersburg, in 1766, he became totally blind. His passion for science, however, suffered no decline; the powers of his mind were not impaired, and he continued as indefatigable as ever. Though the distresses of age likewise were now crowding fast upon him, for he had passed his sixtieth year; yet it was in this latter period of his life, under infirmity, bodily pain, and loss of sight, that he produced some of his most valuable works; such as command our astonishment, independently of the situation of the author, from the labour and originality which they display. In fact, his habits of study and composition, his inventions and discoveries, closed only with his life. The very day on which he died, he had been engaged in calculating the orbit of Herschel's planet, and the motions of aerostatic machines. His death happened suddenly in September 1783, from a fit of apoplexy, when he was in the seventy-sixth year of his age.

Such is the short history of this illustrious man. The incidents of his life, like that of most other laborious students, afford very scanty materials for biography; little more than a journal of studies, and a catalogue of publications; but curiosity may find ample compensation in surveying the character of his mind. An object of such magnitude, so far elevated above the ordinary range of human intellect, cannot be approached without reverence, nor nearly inspected, perhaps, without some degree of presumption. Should an apology be necessary, therefore, for attempting the following estimate of Euler's character, let it be considered, that we can neither feel that admiration, nor offer that homage, which is worthy of genius, unless, aiming at something more than the dazzling sensations of mere wonder, we subject it to actual examination, and compare it with the standards of human nature in general.

Whoever is acquainted with the memoirs of those great men, to whom the human race is indebted for the progress of knowledge, must have perceived, that, while mathematical genius is distinct from the other departments of intellectual excellence, it likewise admits in itself of much diversity. The subjects of its speculation are become so extensive and so various, especially in modern times, and present so many interesting aspects, that it is natural for a person, whose talents are of this cast, to devote his principal curiosity and attention to particular views of the science. When this happens, the faculties of the mind acquire a superior facility of operation, with respect to the objects towards which they are most frequently directed, and the invention becomes habitually most active and most acute in that channel of inquiry.

The truth of these observations is strikingly illustrated by the character of Euler. His studies and discoveries lay not among the lines and figures of geometry, those characters, to use an expression of Galileo, in which the great book of the universe is written; nor does he appear to have had a turn for philosophising by experiment, and advancing to discovery through the rules of inductive investigation. The region, in which he delighted to speculate, was that of pure intellect.

He surveyed the properties and affections of quantity under their most abstracted forms. With the same rapidity of perception, as a geometrician ascertains the relative position of portions of extension, Euler ranges through the regions of abstract quantities, unfolding their most involved combinations, and tracing their most intricate proportions. That admirable system of mathematical logic and language, which at once teaches the rules of just inference, and furnishes an instrument for prosecuting deductions, free from the defects, which obscure and often falsify our reasonings on other subjects; the different species of quantity, whether formed in the understanding by its own abstractions, or derived from modifications of the representative system of signs; the investigation of the various properties of these, their laws of genesis, the limits of comparison among the different species, and the method of applying all this to the solution of physical problems; these were the researches on which the mind of Euler delighted to dwell, and in which he never engaged without finding new objects of curiosity, detecting sources of inquiry, which had passed unobserved, and exploring fields of speculation and discovery, which before were unknown.

The subjects, which we have here slightly enumerated, form, when, taken together, what is called the Modern Analysis: a science eminent for the profound discoveries which it has revealed; for the refined artifices that have been devised, in order to bring the most abstruse parts of mathematics within the compass of our reasoning powers, and for applying them to the solution of actual phaenomena, as well as for the remarkable degree of systematic simplicity, with which the various methods of investigation are employed and combined, so as to confirm and throw light on one another. The materials, indeed, had been collecting for years, from about the middle of the seventeenth century; the foundations had been laid by Newton, Leibniz, the elder Bernoullis, and a few others; but Euler raised the superstructure: it was reserved for him to work upon the materials, and to arrange this noble monument of human industry and genius in its present symmetry. Through the whole course of his scientific labours, the ultimate and the constant aim on which he set his mind, was the perfection of Calculus and Analysis. Whatever physical inquiry he began with, this always came in view, and very frequently received more of his attention than that which was professedly the main subject. His ideas ran so naturally in this train, that even in the perusal of Virgil's poetry, he met with images that would recall the associations of his more familiar studies, and lead him back, from the fairy scenes of fiction, to mathematical abstraction, as to the element, most congenial to his nature.

That the sources of analysis might be ascertained in their full extent, as well as the various modifications of form and restrictions of rule that become necessary in applying it to different views of nature; he appears to have nearly gone through a complete course of philosophy. The theory of rational mechanics, the whole range of physical astronomy, the vibrations of elastic fluids, as well as the movements of those which are incompressible, naval architecture and tactics, the doctrine of chances, probabilities, and political arithmetic, were successively subjected to the analytical method; and all these sciences received from him fresh confirmation and further improvement.

It cannot be denied that, in general, his attention is more occupied with the analysis itself, than with the subject to which he is applying it; and that he seems more taken up with his instruments, than with the work, which they are to assist him in executing. But this can hardly be made a ground of censure, or regret, since it is the very circumstance to which we owe the present perfection of those instruments; a perfection to which he could never have brought them, but by the unremitted attention and enthusiastic preference which he gave to his favourite object. If he now and then exercised his ingenuity on a physical, or perhaps metaphysical, hypothesis, he must have been aware, as well as any one, that his conclusions would of course perish with that from which they were derived. What he regarded, was the proper means of arriving at those conclusions; - the new views of analysis, which the investigation might open; and the new expedients of calculus, to which it might eventually give birth. This was his uniform pursuit; all other inquiries were prosecuted with reference to it; and in this consisted the peculiar character of his mathematical genius.

The faculties that are subservient to invention he possessed in a very remarkable degree. His memory was at once so retentive and so ready, that he had perfectly at command all those numerous and complex formulae, which enunciate the rules and more important theorems of analysis. As is reported of Leibniz, he could also repeat the Aeneid from beginning to end; and could trust his recollection for the first and last lines in every page of the edition, which he had been accustomed to use. These are instances of a kind of memory, more frequently to be found where the capacity is inferior to the ordinary standard, than accompanying original, scientific genius. But in Euler, they seem to have been not so much the result of natural constitution, as of his most wonderful attention; a faculty, which, if we consider the testimony of Newton sufficient evidence, is the great constituent of inventive power. It is that complete retirement of the mind within itself, during which the senses are locked up; - that intense meditation, on which no extraneous idea can intrude; - that firm, straightforward progress of thought, deviating into no irregular sally, which can alone place mathematical objects in a light sufficiently strong to illuminate them fully, and preserve the perceptions of "the mind's eye" in the same order that it moves along.

Two of Euler's pupils (we are told by M Fuss, a pupil himself) had calculated a converging series as far as the seventeenth term; but found, on comparing the written results, that they differed one unit at the fiftieth figure they communicated this difference to their master, who went over the whole calculation by head, and his decision was found to be the true one. - For the purpose of exercising his little grandson in the extraction of roots, he has been known to form to himself the table of the six first powers of all numbers, from 1 to 100, and to have preserved it actually in his memory.

The dexterity which he had acquired in analysis and calculation, is remarkably exemplified by the manner in which he manages formulae of the greatest length and intricacy. He perceives, almost at a glance, the factors from which they may have been composed; the particular system of factors belonging to the question under present consideration; the various artifices by which that system may be simplified and reduced; and the relation of the several factors to the conditions of the hypothesis. His expertness in this particular probably resulted, in a great measure, from the ease with which he performed mathematical investigations by head. He had always accustomed himself to that exercise; and having practised it with assiduity, even before the loss of sight, which afterwards rendered it a matter of necessity, he is an instance to what an astonishing degree of perfection that talent may be cultivated, and how much it improves the intellectual powers. No other discipline is so effectual in strengthening the faculty of attention; it gives a facility of apprehension, an accuracy and steadiness to the conceptions; and, what is a still more valuable acquisition, it habituates the mind to arrangement in its reasonings and reflections.

If the reader wants a further commentary on its advantages, let him proceed to the work of Euler, of which we here offer a Translation; and if he has any taste for the beauties of method, and of what is properly called composition, we venture to promise him the highest satisfaction and pleasure. The subject is so aptly divided, the order is so luminous, the connected parts seem so truly to grow one out of the other, and are disposed altogether in a manner so suitable to their relative importance, and so conducive to their mutual illustration, that, when added to the precision, as well as clearness with which every thing is explained, and the judicious selection of examples, we do not hesitate to consider it, next to Euclid's Elements, the most perfect model of elementary writing, of which the scientific world is in possession.

When our reader shall have studied so much of these volumes as to relish their admirable style, he will be the better qualified to reflect on the circumstances under which they were composed. They were drawn up soon after our author was deprived of sight, and were dictated to his servant, who had originally been a tailor's apprentice; and, without being distinguished for more than ordinary parts, was completely ignorant of mathematics. But Euler, blind as he was, had a mind to teach his amanuensis, as he went on with the subject. Perhaps, he undertook this task by way of exercise, with the view of conforming the operation of his faculties to the change, which the loss of sight had produced. Whatever was the motive, his Treatise had the advantage of being composed under an immediate experience of the method best adapted to the natural progress of a learner's ideas from the want of which, men of the most profound knowledge are often awkward and unsatisfactory, when they attempt elementary instruction. It is not improbable, that we may be farther indebted to the circumstance of our Author's blindness; for the loss of this sense is generally succeeded by the improvement of other faculties. As the surviving organs, in particular, acquire a degree of sensibility, which they did not previously possess; so the most charming visions of poetical fancy have been the offspring of minds, on which external scenes had long been closed.

And perhaps a philosopher, familiarly acquainted with Euler's writings, might trace some improvement in perspicuity of method, and in the flowing progress of his deductions, after this calamity had befallen him; which, leaving "an universal blank of Nature's works," favours that entire seclusion of the mind, which concentrates attention, and gives liveliness and vigour to the conceptions.

In men devoted to study, we are not to look for those strong, complicated passions, which are contracted amidst the vicissitudes and tumult of public life. To delineate the character of Euler, requires no contrasts of colouring. Sweetness of disposition, moderation in the passions, and simplicity of manners, were his leading features. Susceptible of the domestic affections, he was open to all their amiable impressions, and was remarkably fond of children. His manners were simple, without being singular, and seemed to flow naturally from a heart that could dispense with those habits, by which many must be trained to artificial mildness, and with the forms that are often necessary for concealment. Nor did the equability and calmness of his temper indicate any defect of energy, but the serenity of a soul that overlooked the frivolous provocations, the petulant caprices, and jarring passions of ordinary mortals.

Possessing a mind of such wonderful comprehension, and dispositions so admirably formed to virtue and to happiness, Euler found no difficulty in being a Christian: accordingly, "his faith was unfeigned," and his love "was that of a pure and undefiled heart." The advocates for the truth of revealed religion, therefore, may rejoice to add to the bright catalogue, which already claims a Bacon, a Newton, a Locke, and a Hale, the illustrious name of Euler. But, on this subject, we shall permit one of his learned and grateful pupils* to sum up the character of his venerable master. "His piety was rational and sincere; his devotion "was fervent. He was fully persuaded of the "truth of Christianity; he felt its importance to "the dignity and happiness of human nature; " and looked upon its detractors, and opposers, as "the most pernicious enemies of man."

The length to which this account has been extended may require some apology; but the character of Euler is an object so interesting, that, when reflections are once indulged, it is difficult to prescribe limits to them. One is attracted by a sentiment of admiration, that rises almost to the emotion of sublimity; and curiosity becomes eager to examine what talents and qualities and habits belonged to a mind of such superior power. We hope, therefore, the student will not deem this an improper introduction to the work which he is about to peruse; as we trust he is prepared to enter on it with that temper and disposition, which will open his mind both to the perception of excellence, and to the ambition of emulating what he cannot but admire.

1.4. Review by: Jeremy Gray.
Mathematical Reviews MR0766740 (86e:01076).

This is a facsimile reprint of John Hewlett's 1840 translation of Euler's Algebra and Lagrange's Additions thereto. Most of Euler's contribution is elementary, nothing more advanced than solving quartic equations, but worth having in order to appreciate his leisurely and effective style - would that more great mathematicians wrote so well and to such pedagogic effect. However, one third of the book is his lucid treatment of questions in number theory, and it is this material that drew Lagrange's comments. Here for the first time are the rigorous treatments of continued fractions and "Pell's" equation, and of quadratic forms. The combination of Euler's and Lagrange's tests, of experimental and theoretical research in Weil's description, is justly celebrated by the editors of Euler's Opera omnia, who print the two together, and it is good to see this classic back in print in English. Every library without much Euler should at least have this volume. It is accompanied by an excerpt of Horner's memoir on the life of Euler, and a eulogy by Truesdell, with a useful bibliography.

1.5. Review of 2006 edition by: Graham Hoare.
The Mathematical Gazette 92 (524) (2008), 373-374.

Written towards the end of his life when he was blind, Euler's Elements reminds me in some ways of the first algebra book that was brought to my attention a few months before embarking on my secondary education. It is full of examples ('questions for practice') and contains very few diagrams. However, there are significant differences. For example, Euler is painstaking in explaining what is going on. By page 9, for example, he is giving a detailed explanation of why $(-b) \times (-d) = +bd$, something that was ducked in my experience at school. Again, arithmetic features strongly to facilitate algebraic understanding. By the end, Euler has reached topics which would stretch good sixth-formers nowadays such as solving third and fourth degree polynomial equations by exact and approximate methods.

The text is divided into four chapters. The first begins with an explanation of signs, progresses to integers and fractions by way of introducing analogous basic algebraic operations including the manipulation of powers. Irrational numbers, represented by fractional exponents are introduced before the section ends with a thorough exposition of logarithms. Since the second chapter extends the first to Compound Quantities, the set of questions for practice on page 26 appears to be misplaced for it asks, for example, to reduce $\Large \frac { x^4 -b^4 }{x^5 -b^2 x^3 }$ to its lowest terms and worse!

As we have hinted, the second chapter mirrors the first. Matters develop smoothly until we arrive at division of compound quantities. In section 2.5, we are soon plunged into the murky waters of infinite series when, for example, 1 is divided by $1-a$. Here Euler wavers for he correctly renders it as $1 + a +...+a^{7}+ a^{8}/(1-a )$ but then gives $1 + a +... +a^{12}$, etc. to infinity. Foreseeing the absurdity, when $a = 2$, that $-1 = 1 + 2 +... + 64$, etc. to infinity, he reverts to the first form. He tries to draw the sting in a footnote for the case $\Large\frac{1}{1+a}\normalsize$ which, for $a = 1$, appears to yield $\large\frac{1}{2}\normalsize = 1-1+1-1+1-1$, etc. to infinity, by observing that 'no infinite series is in reality equal to the fraction from which it is derived, unless the remainder be considered, which, in the present case, is alternately $+\large\frac{1}{2}\normalsize$ and $-\large\frac{1}{2}\normalsize$; that is, $+\large\frac{1}{2}\normalsize$ when the series is 0 and $-\large\frac{1}{2}\normalsize$ when the series is 1, which still gives the same values for the whole expression.' There follows sections involving the extraction of square and cube roots arithmetically and thence to the algebraic equivalents, having established what we now refer to as the binomial theorem. The same caveats apply to such calculations as $^{3}√(c^{3}+ b)$ and $\Large\frac{1}{a+b^{3}\normalsize}$ as we have mentioned above. The word 'convergence' doesn't appear.

The third chapter, Of Ratios and Proportions, deals principally with arithmetic and geometric progressions. The former are illustrated by figurate or polygonal numbers and the latter by the calculation of interest and infinite decimal fractions.

The final chapter, Of Algebraic Equations and of the Resolution of those Equations begins with some interesting remarks as to the purpose of algebra. Euler then considers simple equations, simultaneous equations up to 3 unknowns and problems leading to then. Quadratic equations come next followed by the extraction of the square roots of binomials (e.g. √(5 + 2√6). Some ad hoc techniques are deployed to resolve polynomial equations of the third degree before the methods of Cardan and Scipio Ferreo are considered. As for equations of the fourth degree, Euler considers Bombelli's rule for reducing the problem to resolving cubics before introducing a new method. Finally, we are introduced to 'the method given by Sir Is. Newton at the beginning of his Method of Fluxions to resolve equations by approximation.' This chapter, especially, could have been written by someone in the first half of the twentieth century.

This book is an edited reprint of Part 1 of J Hewlett's 1822 English translation of Euler's Elements of Algebra. Euler's amanuensis was a young servant who acquired something of a mathematical education through his labours. The text, which is handsomely produced, reflects what constitutes elementary mathematics at the time and will be fascinating to historians of mathematics. It is remarkably accessible to the modern reader.

2.1. From the Preface.

Often I have considered the fact that most of the difficulties which block the progress of students trying to learn analysis stem from this: that although they understand little of ordinary algebra, still they attempt this more subtle art.

From this it follows not only that they remain on the fringes, but in addition they entertain strange ideas about the concept of the infinite, which they must try to use. Although analysis does not require an exhaustive knowledge of algebra, even of all the algebraic techniques so far discovered, still there are topics whose consideration prepares a student for a deeper understanding. However, in the ordinary treatise on the elements of algebra, these topics are either completely omitted or are treated carelessly. For this reason, I am certain that the material I have gathered in this book is quite sufficient to remedy that defect.

I have striven to develop more adequately and clearly than is the usual case those things which are absolutely required for analysis. Moreover, I have also unravelled quite a few knotty problems so that the reader gradually and almost imperceptibly becomes acquainted with the idea of the infinite.

There are also many questions which are answered in this work by means of ordinary algebra, although they are usually discussed with the aid of analysis. In this way the interrelationship between the two methods becomes clear.

I have divided this work into two books; in the first of these I have confined myself to those matters concerning pure analysis. In the second book I have explained those things which must be known from geometry, since analysis is ordinarily developed in such a way that its application to geometry is shown. In both parts, however, I have omitted the elementary matters and developed only those things which, in other places, are either completely omitted or only cursorily treated or, finally, follow from new arguments.

Thus, in the first book, since all of analysis is concerned with variable quantities and functions of such variables, I have given a full treatment to functions. I have also treated the transformation of functions and functions as the sum of infinite series. In addition I have developed functions in infinite series.

Many kinds of functions whose characteristic qualities are discovered by higher analysis are classified. First I have distinguished between algebraic and transcendental functions: the former are formed from the ordinary algebraic operations on variable quantities; the latter arise from other procedures or from the infinite repetition of algebraic operations.

The primary subdivision of algebraic functions is into that of non-irrational and irrational. I have shown how the former can not only be simplified, but also factored, and this is very useful in integral calculus. It has been shown to what extent irrational functions can be brought to non-irrational form by means of suitable substitutions. Both types can be developed in infinite series, but this method is usually applied with the greatest usefulness to transcendental functions. It is clear that the theory of infinite series has greatly extended higher analysis. Several chapters have been included in which I have examined the properties and summation of many infinite series; some of these are arranged in such a way that it can be seen that they could hardly be investigated without the aid of analysis. Series of this type are those whose summations are expressed either through logarithms or circular arcs. However, since these are transcendental quantities which can be defined by quadratures of the hyperbola and the circle, for the most part they are usually treated in analysis. After that I shall have progressed from powers of quantities to exponential quantities, which are simply powers whose exponents are variables. From the inverse of these I have arrived at the most natural and fruitful concept of logarithms. Whence not only are very ample uses of these immediately obtained, but also from them it is possible to obtain all those infinite series by which ordinarily these quantities are represented. Then there is produced a method of reasonably simple construction of tables of logarithms. In a like manner I have turned my attention to circular arcs. This type of quantity, although quite different from logarithms, nevertheless, there is such a close mutual relationship that when the latter is viewed as a complex quantity, it is converted into the former. Just as logarithms have their own particular algorithm, which has most useful applications in all of analysis, I have derived algorithms for the trigonometric quantities, so that these calculations can be made as easily as for the logarithmic and algebraic quantities. The extent of the usefulness of this for the solution of very difficult problems becomes clear in several chapters of this book. Indeed, very many other examples from analysis could be offered were they not sufficiently known already, and in fact more are being found almost daily. But this investigation brings the greatest help to the resolution of rational functions into real factors. Since this is so important for integral calculus, I have given this diligent attention. I have investigated those infinite series which arise from the development of this type of function, and are known as recurrent series. For these I have given both summations and general terms and also other important properties. Since the resolution into factors has led to these series, so in turn, I have pondered to what extent the product of several factors, and even infinite products, can be expressed in a series. This business opened the way to knowledge of a myriad of series. Since a series can be expressed as an infinite product, I have found rather convenient numerical expressions with the aid of which the logarithms of sines, cosines, and tangents can easily be computed. Furthermore, from this same source we can derive the solutions of many problems which are concerned with the partition of numbers. Questions of this sort would seem to defeat analysis without this help. Such a diversity of material might easily have grown into several volumes, but I have, as far as possible, expressed everything so succinctly that everywhere the foundation is very clearly explained. The further development is left to the industry of the readers. In this way they will have an opportunity to try their own ingenuity and further develop analysis itself. Nor do I hesitate to proclaim that within this book there are contained many things which are clearly new, but also some sources have been uncovered from which many significant further discoveries can be drawn.

I have used the same arrangement in the second volume, where I have treated those topics which are commonly called higher geometry. Before I discuss conic sections, which in other treatments almost always come first, I have proposed a theory of curves with enough generality that it can advantageously be applied to an examination of the nature of any curve whatsoever. I use only an equation by which the nature of every curve is expressed, and I show how to derive from this both the shape and its primary characteristics. It has seemed to me that this is most especially advantageous in the case of conic sections. Until this time they have ordinarily been treated only from the geometric viewpoint, or if by analysis, in an awkward and unnatural way. I have first explained their general properties from the general equation for a second order curve. Then I have subdivided them into genera and species, considering whether they have branches going to infinity or the whole curve is included in a bounded region. In the former case something else has to be considered, that is, how many branches extend to infinity and of what nature each of these is; namely, whether or not the branch has a straight line asymptote. In this way I have obtained the customary three types of conic section. The first is the ellipse, totally contained in a bounded region; the second is the hyperbola, which has four infinite branches asymptotic to two straight lines; the third is the parabola with two infinite branches without asymptotes. In a like manner I have described third order curves, which I have divided into sixteen kinds, after considering their general properties. Indeed, to these kinds I have reduced the seventy-two of Newton's classification. I have described this method with such clarity that for curves of any higher order whatsoever, a classification may easily be made. I have made a test of the method in the case of fourth order curves. Having explained whatever concerns the order of a curve, I returned to uncovering the general properties of all curves. Thus I have explained a method for defining tangents to curves, their normals, and curvature, which is ordinarily measured by the radius of the osculating circle. Although all of these nowadays are ordinarily accomplished by means of differential calculus, nevertheless, I have here presented them using only ordinary algebra, in order that the transition from finite analysis to analysis of the infinite might be rendered easier. I have also investigated points of inflection of curves, cusps, double points, and multiple points. The definitions of these kinds of points follow without difficulty from the equations. At the same time I readily admit that these matters can be much more easily worked out by differential calculus. I have touched on the controversy over a second order cusp, where both arcs which come together in the cusp curve in the same direction. It seems to me that I have settled this question in such a way that there can remain no doubt. Finally, I have adjoined several chapters in which I have explained how to find curves with certain stated properties. I give the solution to a number of problems concerned with circles. Since there are some topic from geometry which seem to offer strong support for learning analysis, I have added an appendix in which I have presented, using calculus, the theory of solids and the surfaces of solids. I have shown, insofar as it is possible to do so through equations in three variables, the nature of these surfaces. Then having divided surfaces into orders, as was done in the case of curves, according to the power of the variables in the equation, I have shown that the only surface of the first order is the plane. I have divided surfaces of the second order, by considering their parts which extend to infinity, into six types. In a similar way a division can be made for surfaces of other orders. I have also considered the intersections of two surfaces. These, insofar as they can be understood through equations, I have shown generally to be curves which do not lie in a single plane.

For the rest, since many things here will be met which have already been treated by others I ought to ask pardon, since I shall not have given credit in every place to all who have toiled herein. I have endeavoured to develop everything as briefly as possible. If the history of each problem had been discussed, that would have increased the size of this work beyond reasonable bounds. Many of these problems, whose solutions can be found elsewhere, in this work have solutions which arise from different arguments. For this reason I would seem in no little part to be exonerated. I do hope that both these things, but also especially those which are entirely new, will be acceptable to most of those who enjoy this work.

2.2. Review by: Paul J Campbell.
Mathematics Magazine 62 (4) (1989), 283.

This first English translation was occasioned by a talk by André Weil in which he remarked that students of mathematics would profit more from a study of Euler than from modern textbooks. The work treats infinite series, infinite products, and continued fractions-topics Euler regarded as suitable for learning before calculus (but after trigonometry, logarithms, and exponentials). Today's students would benefit by seeing this book after calculus, as they would then be able to appreciate better its spirit of undaunted calculation.

2.3. Review by: P Shiu.
The Mathematical Gazette 73 (466) (1989), 361-362.

In 1748 the great mathematician Euler wrote a famous textbook called Introductio in Analysis Infinitorum which was then translated into French, German and Russian. As Professor André Weil has suggested that our students would profit more from a study of the Introductio rather than of the available modern textbooks, this admirable English translation cannot be described as too late, especially when there is such a dearth of work by great mathematicians that is suitable for the undergraduates let alone the sixth-formers. I myself occasionally recommend to my students G H Hardy's Pure mathematics, which was described by J E Littlewood as the work of a missionary among cannibals. One is tempted to speculate whether British mathematics, between the time of Newton and Hardy, would still have been so isolated from the continent had there been an English version of Euler's Introductio. In any case I now have another book to recommend to my students for extra reading.

The 380 articles are divided into 18 chapters which include the transformation of functions by substitution, the development of functions in infinite series and thorough discussions on the exponential, the logarithm and the trigonometric functions. Much of these depends on what we now call the method of indeterminate coefficients. On this Euler was, to quote Professor Weil, "bold to the point of rashness ... but fortune favours the bold". For example, although Gauss was the first to use complex numbers confidently and in a scientific manner, nevertheless it was Euler who found the formula $e^{i\theta} = \cos \theta + i \sin \theta$. Many of us first saw beauty in mathematics when we came across this formula and here is the chance to read all about it from the man himself. The last four chapters are particularly interesting and instructive. According to H Davenport, analytic number theory began with the work of Dirichlet, and in particular with Dirichlet's memoir of 1837 in which he proved that there are always infinitely many primes $p$ in an arithmetic progression $p \equiv a(\bmod b)$, where $a$ and $b$ are coprime, by proving that the series $\sum_{p \equiv a(\bmod b) } \large\frac{1}{p}$ diverges. The subject was brought into prominence by Riemann in 1859 with his introduction of the Riemann zeta function $\zeta(s) = \sum _{n>0} \Large\frac{1}{n^{s}\normalsize}$ together with the notorious Riemann Hypothesis. Here in Chapter 15 we find the path was cleared by Euler for the foundations to be laid by these worthy successors; for example, there is an explanation of his justly famous proof that $\sum \large\frac{1}{p}\normalsize$ diverges. This is followed by a chapter on the partition of numbers, a subject created by Euler for the later arrival of Hardy, Ramanujan and Littlewood. The last chapter is on continued fractions, a topic the foundation of which was given by Lagrange who always acknowledged his indebtedness to Euler.

The author suggests in the preface that "most of the difficulties that block the progress of students trying to learn analysis stem from (the lack of) ordinary algebra". The subject of analysis had, of course, quite a different meaning in his days; and by ordinary algebra he meant what can nowadays be loosely described as manipulative skill. Teachers still say more or less the same of present day students! If it was not beneath the great Euler to get his hands dirty and mess around with equations and numbers, it must be good for all of us. In fact the text includes a profusion of examples to illustrate the ideas being discussed, and many of the examples involved such prodigious numerical calculations that there is no doubt that the author took much pleasure in doing them. More to the point is that many a famous result was first discovered by numerical calculations. Thus the values for $\zeta(2)$ was calculated to 20 decimal places before he discovered that its exact value is $\large\frac{1}{6}\normalsize \pi^{2}$, and we should bear in mind that he did not have even an abacus!

The instructive style is nicely preserved by an excellent translation. The exposition is masterly and encouraging; one often feels the enthusiasm of the author and his wish to share his knowledge. As one would expect from the writings of Euler there is a complete lack of pomposity let alone arrogance; indeed only after putting the book down does one suddenly recall that most of the results were the author's own discoveries and creations. On this point I should also recommend the chapter on Euler in André Weil's Number theory: An approach through history from Hammurapi to Legendre which gives the background to the discoveries of many of the famous results, in particular his success in the evaluation of $\zeta(2k)$ as rational multiples of $\pi^{2k}$ for the first 8 values of $k$.

2.4. Review of the 1885 German translation by: C J Scriba.
Mathematical Reviews MR0715928 (85d:01030).

On 4 July 1744 Euler wrote to Christian Goldbach: "I have meanwhile finished a new book entitled 'Introductio in analysin infinitorum', wherein I treat the higher parts of algebra and geometry. There I solve a large number of difficult problems without applying infinitesimal calculus - almost nothing of this can be found anywhere else. After I had made a plan for a complete treatise on infinitesimal analysis I noticed that many things ought to be presented first which do not actually belong there and which are not dealt with anywhere; from these topics the present book has arisen as a preparation for the study of infinitesimal analysis."

Euler divided this Introductio into two parts. The first one (whose German translation is the reprint under review) contains what Euler considered to be prerequisites for the study of infinitesimal calculus. (The second part deals with the theory of curves and surfaces, as preparation to the applications of the calculus to geometry. Maser's German translation comprises only the first part.) Divided into 18 chapters, Part I of the Introductio presents the following topics: functions, their classification, transformations and development into infinite series; exponential functions, logarithms, and circular (trigonometric) functions and their series; infinite products, summation, recurrent series; angular sections; applications to the partition of numbers; continued fractions.

Thus it is the concept of functions (here still defined as analytical expression) that with Euler attains the central position in higher mathematics. The rules of common algebra are extended to infinite series and infinite products, problems of convergence are hardly mentioned, and where we would apply a limiting process, Euler is operating with infinitely small or infinitely large numbers. But although these foundational problems were only settled satisfactorily during the 19th century, the Introductio is a fascinating book to read. One of its highlights is the derivation of $\sum _{1}^{∞}\large\frac{1}{n^{2}\normalsize}$ = $\large\frac{1}{6}\normalsize \pi^{2}$ and, more generally, the determination of the values of the function $\zeta(n)$ for even values of $n$; another is the introduction of generating functions into additive number theory in connection with problems of the partition of numbers.

3.1. Review by: P Shiu.
The Mathematical Gazette 74 (470) (1990), 392-393.

This is the second half of the English translation of the famous textbook Introductio in analysis infinitorum by the great mathematician Euler. Book I, which contains analysis and higher arithmetic, was reviewed in Volume 73 (Number 466) of the Gazette. The subject matter of Book II is the theory of curves, and the content is divided into 22 chapters, together with 6 appendices on surfaces.

Many teachers lament the absence of geometry, especially the conics, from school syllabuses, and the appearance of such a textbook by Euler must be welcome. However, I find this second book much less enjoyable. For a start, any sensible reading of the text will require good accompanying figures and diagrams. Although the figures are nicely produced from the original it is a major irritation to find that they are all placed together at the end of the book. It has also to be said that the contributions from the author on topics in Book II are much less profound and certainly not as important. Thus, whereas many of Euler's results in arithmetic or analysis in Book I seem to possess an air of finality about them, most of the contributions here leave much room for drastic improvement, and they certainly lack the air of permanence. My guess is that the book is of interest only to those who wish to see what was known on the subject some 250 years ago. Take, for example, the last chapter which is on "Problems pertaining to the circle". The author demonstrated again his love for calculations by first explaining how to obtain the angle $57°12' 44'' 48''' 22'''' 29''''' 21''''''$ corresponding to one radian. Examples of the "pertaining" problems are the search for the solutions to $\cos \theta = \theta$ and $\sin 2\theta = \theta$, which were done in laborious details and high accuracy using the false position method. It is explained in the last paragraph that such problems were posed "in order that the nature of the circle might be penetrated more deeply, since attempts at these quadrature problems have been unsuccessful by all previous methods. If it should happen in the solution of one or another problem that the arc might be commensurable with the whole circumference ...". It was, of course, not even known that π was irrational, and the transcendence of π was established only in 1882.

3.2. Review by: Doru Stefanescu.
Mathematical Reviews MR1025504 (91i:01143).

One of the chief works of Leonhard Euler is his Introductio in Analysin Infinitorum. It was published in two Latin volumes in 1748 at Lausanne and it had a tremendous impact on the development of mathematics. The first book is devoted to mathematical analysis and was recently translated into German [1983] and into English [1988]. The present volume is the first English translation of the second book of Introductio in Analysin Infinitorum. It is divided into two distinct sections, both referring to geometrical subjects.

Geometry is one of the branches of mathematics to which Euler made substantial contributions. After 1728 Euler published several papers on analytic geometry. They referred to both plane and solid analytic geometry and included the use of parametric representations of curves and the consideration of coordinates in the study of quadrics. Many of the results and methods described in these papers were included in the second volume of the Introductio. The first section is a treatise about plane curves. It contains 22 chapters. Euler systematically uses coordinates. He makes the distinction between algebraic and transcendental curves. The algebraic curves are classified with respect to their degree (called "order" by Euler). There are detailed studies of the curves of degree two, three or four. Special chapters are devoted to the study of the branches that go to infinity, of the asymptotes of a curve, of the curvature, the intersection of curves, the similarities and affinities of curves. We mention the elimination methods used in the study of the intersection of curves of arbitrary degree and the use of polar coordinates. In a special chapter some transcendental curves are studied and the intercendental curves first considered by Leibniz are also mentioned. In the final chapter Euler expounds the method of "false position" for obtaining approximate solutions to problems that involve transcendental equations.

The second part of Euler's book is an appendix on surfaces. Spatial coordinates and change of coordinates are used systematically. In the six chapters of the appendix the standard classification of the three-dimensional quadrics is obtained and the intersection of a surface with a plane and the intersection of two surfaces are investigated.

The second book of Introductio in Analysin Infinitorum is the first important treatise of analytic geometry published after Descartes' Geometry (1937). Together with G Cramer's Introduction à l'analyse des lignes courbes algébriques (1750) it reflects the progress that analytic geometry realised in the 18th century. The English version of Euler's book respects the spirit of the original Latin edition. The quarto is replaced by the more fashionable octavo, but the figures are grouped at the end of the volume as was usual in the mathematical books published before the middle of the 19th century. The translator includes as a preface the pertinent parts of Euler's introduction that referred to both volumes.

This beautiful edition brings back to the mathematical community a fundamental work that marked the evolution of our science.

4.1. From the Publisher.

What differential calculus, and, in general, analysis of the infinite, might be can hardly be explained to those innocent of any knowledge of it. Nor can we here offer a definition at the beginning of this dissertation as is sometimes done in other disciplines. It is not that there is no clear definition of this calculus; rather, the fact is that in order to understand the definition there are concepts that must first be understood. Besides those ideas in common usage, there are also others from finite analysis that are much less common and are usually explained in the course of the development of the differential calculus. For this reason, it is not possible to understand a definition before its principles are sufficiently clearly seen. In the first place, this calculus is concerned with variable quantities. Although every quantity can naturally be increased or decreased without limit, still, since calculus is directed to a certain purpose, we think of some quantities as being constantly the same magnitude, while others change through all the stages of increasing and decreasing. We note this distinction and call the former constant quantities and the latter variables. This characteristic difference is not required by the nature of things, but rather because of the special question addressed by the calculus.

4.2. From the Preface.

What differential calculus, and, in general, analysis of the infinite, might be can hardly be explained to those innocent of any knowledge of it. Nor can we here offer a definition at the beginning of this dissertation as is sometimes done in other disciplines. It is not that there is no clear definition of this calculus; rather, the fact is that in order to understand the definition there are concepts that must first be understood. Besides those ideas in common usage, there are also others from finite analysis that are much less common and are usually explained in the course of the development of the differential calculus. For this reason, it is not possible to understand a definition before its principles are sufficiently clearly seen.

In the first place, this calculus is concerned with variable quantities. Although every quantity can naturally be increased or decreased without limit, still, since calculus is directed to a certain purpose, we think of some quantities as being constantly the same magnitude, while others change through all the stages of increasing and decreasing. We note this distinction and call the former constant quantities and the latter variables. This characteristic difference is not required by the nature of things, but rather because of the special question addressed by the calculus.

In order that this difference between constant quantities and variables might be clearly illustrated, let us consider a shot fired from a cannon with a charge of gunpowder. This example seems to be especially appropriate to clarify this matter. There are many quantities involved here: First, there is the quantity of gunpowder; then, the angle of elevation of the cannon above the horizon; third, the distance travelled by the shot; and, fourth, the length of time the shot is in the air. Unless the same cannon is used throughout the experiment, we must also bring into our calculations the length of the barrel and the weight of the shot. Here, we will not consider variations in the cannon or the shot, lest we become entailed in very complicated questions. Hence, if we always keep the same quantity of powder, the elevation of the barrel will vary continuously with the distance travelled and the shot's duration of time in the air. In this case, the amount of powder, or the force of the explosion, will be the constant quantity. The elevation of the barrel, the distance travelled, and the time in the air should be the variable quantities. If for each degree of elevation we were to define these things, so that they may be noted for future reference, the changes in distance and duration of the flight arise from all of the different elevations. There is another question: Suppose the elevation of the barrel is kept the same, but the quantity of powder is continuously changed. Then the changes that occur in the flight need to be defined. In this case, the elevation will be the constant, while the quantity of powder, the distance, and duration are the variable quantities. Hence, it is clear that when the question is changed, the quantities that are constant and those that are variables need to be noted. At the same time, it must be understood from this that in this business the thing that requires the most attention is how the variable quantities depend on each other. When one variable changes, the others necessarily are changed. For example, in the former case considered, the quantity of powder remains the same, and the elevation is changed; then the distance and duration of the flight are changed. Hence, the distance and duration are variables that depend on the elevation; if this changes, then the others also change at the same time. In the latter case, the distance and duration depend on the quantity of charge of powder, so that a change in the charge must result in certain changes in the other variables.

Those quantities that depend on others in this way, namely, those that undergo a change when others change, are called functions of these quantities. This definition applies rather widely and includes all ways in which one quantity can be determined by others. Hence, if x designates the variable quantity, all other quantities that in any way depend on $x$ or are determined by it are called its functions. Examples are $x^{2}$, the square of $x$, or any other powers of $x$, and indeed, even quantities that are composed with these powers in any way, even transcendentals, in general, whatever depends on $x$ in such a way that when $x$ increases or decreases, the function changes. From this fact there arises a question; namely, if the quantity $x$ is increased or decreased, by how much is the function changed, whether it increases or decreases? For the more simple cases, this question is easily answered. If the quantity $x$ is increased by the quantity $\omega$, its square $x^{2}$ receives an increase of $2x\omega + \omega^{2}$. Hence, the increase in $x$ is to the increase of $x^{2}$ as $\omega$ is to $2x\omega + \omega^{2}$, that is, as 1 is to $2x + \omega$. In a similar way, we consider the ratio of the increase of $x$ to the increase or decrease that any function of $x$ receives. Indeed, the investigation of this kind of ratio of increments is not only very important, but it is in fact the foundation of the whole of analysis of the infinite. In order that this may become even clearer, let us take up again the example of the square $x^{2}$ with its increment of $2x\omega + \omega^{2}$, which it receives when $x$ itself is increased by $\omega$. We have seen that the ratio here is $2x + \omega$ to 1. From this it should be perfectly clear that the smaller the increment is taken to be, the closer this ratio comes to the ratio of $2x$ to 1. However, it does not arrive at this ratio before the increment itself, ω, completely vanishes. From this we understand that if the increment of the variable goes to zero, then the increment of $x^{2}$ also vanishes. However, the ratio holds as $2x$ to 1. What we have said here about the square is to be understood of all other functions of $x$; that is, when their increments vanish as the increment of $x$ vanishes, they have a certain and determinable ratio. In this way, we are led to a definition of differential calculus: It is a method for determining the ratio of the vanishing increments that any functions take on when the variable, of which they are functions, is given a vanishing increment. It is clearly manifest to those who are not strangers to this subject that the true character of differential calculus is contained in this definition and can be adequately deduced from it.

Therefore, differential calculus is concerned not so much with vanishing increments, which indeed are nothing, but with the ratio and mutual proportion. Since these ratios are expressed as finite quantities, we must think of calculus as being concerned with finite quantities. Although the values seem to be popularly discussed as defined by these vanishing increments, still from a higher point of view, it is always from their ratio that conclusions are deduced. In a similar way, the idea of integral calculus can most conveniently be defined to be a method for finding those functions from the knowledge of the ratio of their vanishing increments.

In order that these ratios might be more easily gathered together and represented in calculations, the vanishing increments themselves, although they are really nothing, are still usually represented by certain symbols. Along with these symbols, there is no reason not to give them a certain name. They are called differentials, and since they are without quantity, they are also said to be infinitely small. Hence, by their nature they are to be so interpreted as absolutely nothing, or they are considered to be equal to nothing. Thus, if the quantity $x$ is given an increment ω, so that it becomes $x + \omega$, its square $x^{2}$ becomes $x^{2} + 2x\omega + \omega^{2}$, and it takes the increment $2x\omega + \omega^{2}$. Hence, the increment of $x$ itself, which is $\omega$, has the ratio to the increment of the square, which is$2x\omega + \omega^{2}$, as 1 to $2x + \omega$. This ratio reduces to 1 to $2x$, at least when $\omega$ vanishes. Let $\omega = 0$, and the ratio of these vanishing increments, which is the main concern of differential calculus, is as 1 to $2x$. On the other hand, this ratio would not be true unless that increment $\omega$ vanishes and becomes absolutely equal to zero. Hence, if this nothing that is indicated by $\omega$ refers to the increment of the quantity $x$, since this has the ratio to the increment of the square $x^{2}$ as 1 to $2x$, the increment of the square $x^{2}$ is equal to $2x\omega$ and for this reason is also equal to zero. Although both of these increments vanish simultaneously, this is no obstacle to their ratios being determined as 1 to $2x$. With respect to this nothing that so far has been represented by the letter $\omega$, in differential calculus we use the symbol $dx$ and call it the differential of $x$, since it is the increment of the quantity $x$. When we put $dx$ for $\omega$, the differential of $x^{2}$ becomes $2x dx$. In a similar way, it is shown that the differential of the cube $x^{3}$ will be equal to $3x^{2}dx$. In general, the differential of any quantity $x^{n}$ will be equal to $nx^{n-1}dx$. No matter what other functions of $x$ might be proposed, differential calculus gives rules for finding its differential. Nevertheless, we must constantly keep in mind that since these differentials are absolutely nothing, we can conclude nothing from them except that their mutual ratios reduce to finite quantities. Thus, it is in this way that the principles of differential calculus, which are in agreement with proper reasoning, are established, and all of the objections that are wont to be brought against it crumble spontaneously; but these arguments retain their full rigour if the differentials, that is, the infinitely small, are not completely annihilated.

To many who have discussed the rules of differential calculus, it has seemed that there is a distinction between absolutely nothing and a special order of quantities infinitely small, which do not quite vanish completely but retain a certain quantity that is indeed less than any assignable quantity. Concerning these, it is correctly objected that geometric rigour has been neglected. Because these infinitely small quantities have been neglected, the conclusions that have been drawn are rightly suspected. Although these infinitely small quantities are conceived to be few in number, when even a few, or many, or even an innumerable number of these are neglected, an enormous error may result. There is an attempt wrongfully to refute this objection with examples of this kind, whereby conclusions are drawn from differential calculus in the same way as from elementary geometry. Indeed, if these infinitely small quantities, which are neglected in calculus, are not quite nothing, then necessarily an error must result that will be the greater the more these quantities are heaped up. If it should happen that the error is less, this must be attributed to a fault in the calculation whereby certain errors are compensated by other errors, rather than freeing the calculation from suspicion of error. In order that there be no compensating one error by a new one, let me fix firmly the point I want to make with clear examples. Those quantities that shall be neglected must surely be held to be absolutely nothing. Nor can the infinitely small that is discussed in differential calculus differ in any way from nothing. Even less should this business be ended when the infinitely small is described by some with the example wherein the tiniest mote of dust is compared to a huge mountain or even to the whole terrestrial globe. If someone undertakes to calculate the magnitude of the whole terrestrial globe, it is the custom easily to grant him an error not only of a single grain of dust, but of even many thousands of these. However, geometric rigour shrinks from even so small an error, and this objection would be simply too great were any force granted to it. Then it is difficult to say what possible advantage might be hoped for in distinguishing the infinitely small from absolutely nothing. Perhaps they fear that if they vanish completely, then will be taken away their ratio, to which they feel this whole business leads. It is avowed that it is impossible to conceive how two absolutely nothings can be compared. They think that some magnitude must be left for them that can be compared. They are forced to admit that this magnitude is so small that it is seen as if it were nothing and can be neglected in calculations without error. Neither do they dare to assign any certain and definite magnitude, even though incomprehensibly small. Even if they were assumed to be two or three times smaller, the comparisons are always made in the same way. From this it is clear that this magnitude gives nothing necessary for undertaking a comparison, and so the comparison is not taken away even though that magnitude vanishes completely.

Now, from what has been said above, it is clear that that comparison, which is the concern of differential calculus, would not be valid unless the increments vanish completely. The increment of the quantity $x$, which we have been symbolising by $\omega$, has a ratio to the increment of the square $x^{2}$, which is $2x\omega + \omega^{2}$, as 1 to $2x + \omega$. But this always differs from the ratio of 1 to $2x$ unless $\omega = 0$, and if we do require that $\omega = 0$, then we can truly say that this ratio is exactly as 1 to $2x$. In the meantime, it must be understood that the smaller the increment $\omega$ becomes, the closer this ratio is approached. It follows that not only is it valid, but quite natural, that these increments be at first considered to be finite and even in drawings, if it is necessary to give illustrations, that they be finitely represented. However, then these increments must be conceived to become continuously smaller, and in this way, their ratio is represented as continuously approaching a certain limit, which is finally attained when the increment becomes absolutely nothing. This limit, which is, as it were, the final ratio of those increments, is the true object of differential calculus. Hence, this ratio must be considered to have laid the very foundation of differential calculus for anyone who has a mind to contemplate these final ratios to which the increments of the variable quantities, as they continuously are more and more diminished, approach and at which they finally arrive.

We find among some ancient authors some trace of these ideas, so that we cannot deny to them at least some conception of the analysis of the infinite. Then gradually this knowledge grew, but it was not all of a sudden that it has arrived at the summit to which it has now come. Even now, there is more that remains obscure than what we see clearly. As differential calculus is extended to all kinds of functions, no matter how they are produced, it is not immediately known what method is to be used to compare the vanishing increments of absolutely all kinds of functions. Gradually this discovery has progressed to more and more complicated functions. For example, for the rational functions, the ultimate ratio that the vanishing increments attain could be assigned long before the time of Newton and Leibniz, so that the differential calculus applied to only these rational functions must be held to have been invented long before that time. However, there is no doubt that Newton must be given credit for that part of differential calculus concerned with irrational functions. This was nicely deduced from his wonderful theorem concerning the general evolution of powers of a binomial. By this outstanding discovery, the limits of differential calculus have been marvellously extended. We are no less indebted to Leibniz insofar as this calculus at that time was viewed as individual tricks, while he put it into the form of a discipline, collected its rules into a system, and gave a crystal-clear explanation. From this there followed great aids in the further development of this calculus, and some of the open questions whose answers were sought were pursued through certain definite principles. Soon, through the studies of both Leibniz and the Bernoullis, the bounds of differential calculus were extended even to transcendental functions, which had in part already been discussed. Then, too, the foundations of integral calculus were firmly established. Those who followed in the elaboration of this field continued to make progress. It was Newton who gave very complete papers in integral calculus, but as to its first discovery, which can hardly be separated from the beginnings of differential calculus, it cannot with absolute certainty be attributed to him. Since the greater part has yet to be developed, it is not possible to say at this time that this calculus has absolutely been discovered. Rather, let us with a grateful mind acknowledge each one according to his efforts toward its completion. This is my judgment as to the attribution of glory for the discovery of this calculus, about which there has been such heated controversy.

No matter what name the mathematicians of different nations are wont to give to this calculus, it all comes to this, that they all agree on this outstanding definition. Whether they call the vanishing increments whose ratios are under consideration by the name differentials or fluxions, these are always understood to be equal to zero, and this must be the true notion of the infinitely small. From this it follows that everything that has been debated about differentials of the second and higher orders, and this has been more out of curiosity then of usefulness, comes back to something very clear, namely, that when everything vanishes together we must consider the mutual ratio rather than the individual quantities. Since the ratio between the vanishing increments of the functions is itself expressed by some function, and if the vanishing increment of this function is compared with others, the result must be considered as the second differential. In this way, we must understand the development of differentials of higher orders, in such a way that they always are seen to be truly finite quantities and that this is the only proper way for them to be represented. At first sight, this description of analysis of the infinite may seem, for the most part, both shallow and extremely sterile, although that obscure notion of the infinitely small hardly offers more. In truth, if the ratios that connect the vanishing increments of any functions are clearly known, then this knowledge very often is of the utmost importance and frequently is so important in extremely arduous investigations that without it almost nothing can be clearly understood. For instance, if the question concerns the motion of a shot fired from a cannon, the air resistance must be known in order to know what the motion will be through a finite distance, as well as both the direction of the path at the beginning and also the velocity, on which the resistance depends. But this changes with time. However, the less distance the shot travels, the less the variation, so that it is possible more easily to come to knowledge of the true relationship. In fact, if we let the distance vanish, since in that case both the difference in direction and change in velocity also are removed, the effect of resistance produced at a single point in time, as well as the change in the path, can be defined exactly. When we know these instantaneous changes or, rather, since these are actually nothing, their mutual relationship, we have gained a great deal. Furthermore, the work of integral calculus is to study changing motion in a finite space. It is my opinion that it is hardly necessary to show further the uses of differential calculus and analysis of the infinite, since it is now sufficiently noted, if even a cursory investigation is made. If we want to study more carefully the motion of either solids or fluids, it cannot be accomplished without analysis of the infinite. Indeed, this science has frequently not been sufficiently cultivated in order that the matter can be accurately explained. Throughout all the branches of mathematics, this higher analysis has penetrated to such an extent that anything that can be explained without its intervention must be esteemed as next to nothing.

I have established in this book the whole of differential calculus, deriving it from true principles and developing it copiously in such a way that nothing pertaining to it that has been discovered so far has been omitted. The work is divided into two parts. In the first part, after laying the foundations of differential calculus, I have presented the method for differentiating every kind of function, for finding not only differentials of the first order, but also those of higher order, and those for functions of a single variable as well as those involving two or more variables. In the second part, I have developed very fully applications of this calculus both in finite analysis and the study of series. In that part, I have also given a very clear explanation of the theorem concerning maxima and minima. As to the application of this calculus to the geometry of plane curves, I have nothing new to offer, and this is all the less to be required, since in other works I have treated this subject so fully. Even with the greatest care, the first principles of differential calculus are hardly sufficiently developed that I should bring them, as it were drawn from geometry, to this science. Here, everything is kept within the bounds of pure analysis, so that in the explanation of the rules of this calculus there is no need for any geometric figures.

4.3. Review by: Gerhard Wanner
SIAM Review 43 (1) (2001), 230-231.

A book written nearly 250 years ago is perhaps not a typical item within this collection of book reviews. But this is one of the greatest books in the history of mathematics, translated from Latin into English for the first time. Among the great mathematicians in his- tory, only Euler made genuine pedagogical efforts and had an absolutely clear writing style. Compared to him, Newton, Leibniz, and Weierstrass were mystery-mongers; Gauss and Cauchy, antiteachers; and Lagrange, a pedantic author of often intricate and complicated writings. Euler made the effort to produce excellent text-books, wonderfully organised and written as clearly as possible. They served as a basis for the education of the young mathematical geniuses of the late 18th and early 19th centuries (for example, Laplace, Abel, and Jacobi) and thereby shaped today's mathematics enormously, with the effect that Euler's notation appears remarkably modern today. In particular, he wrote Introduction to the Analysis of Infinites from 1748; Institutionem Calculi Differentialis from 1755, the book under review; Institutionem Calculi Integralis from 1768, 1769; and Complete Guide to Algebra (1770, Opera Onmia, Vol. 1). Euler's pedagogical concern is also confirmed by the satisfaction that he got when his Algebra, written for this purpose in German, could be understood by his cook, who had no mathematical education at all.

Johann Bernoulli, Euler's teacher, was known to be an enthusiastic advocate of the then "modern" calculus, and was bored when he had to explain some of the "ancient stuff." Later, Euler recognised that "most of the difficulties which block the progress of students trying to learn analysis stem from this: that ... they understand little of ordinary algebra ... ." This can be seen as a sort of anti-Bourbaki movement in the 18th century and led to the publication of the famous Introductio (1748; English translation by J D Blanton), which should precede, in Euler's (and this reviewer's) opinion, the study of differential calculus.

Only later, in 1755, did Euler write the Institutiones Calculi Differentialis, whose translation is the book under review. Even here, Euler does not start off with "full Bourbaki," since the rules and theorems about derivatives only begin on page 77, and they are preceded by long and careful (but also sometimes boring) discussions of finite differences, series, the infinite, and the infinitely small. It is marvellous to ad-mire how one of the principal architects explains this science, which "throughout all the branches of mathematics... has penetrated to such an extent that anything that can be explained without its intervention must be esteemed as next to nothing." While the formulas and the general plan could already be seen from the Latin original, only the English translation shows with what care Euler expresses himself in order to make his ideas as clear as possible. Especially interesting is his opinion (on page x of the Preface) about the "heated controversy" concerning the priority dispute between Newton, Leibniz, and the Bernoullis. Another interesting part is the long discussion about such incredible series as

          $1 + 2 + 4 + 8 + 16 + ... = \Large\frac{1}{1-2}\normalsize = -1$

(on pages 57-61) and the eagerness with which Euler defends them, although seeing their "absurd results." Euler's dream, that "we can keep the usefulness of divergent series and preserve their reputation," came partly true later, when the worst of the Cauchy-Abel criticism was over.

The translator has done a marvellous job, and, with the exception of half a page of "translator's introduction" and an index, has spared us comments, well-intentioned explanations, improvements, and the like, so that the book is pure Euler at its best.

It is beyond question that this book must be in any library and should be seen by any teacher, but the question remains what today's student could learn from a book written two and a half centuries ago, and whether there have been no better books on the market in the meantime.

Well, my personal experience is that, during my youth, I was sitting in Gröbner's office with Euler's Opera Omnia standing on a shelf, an impressive, four-meters-broad, 29-centimetres-high collection of volumes. Never dared I look into one of these; I'd no idea where to start, and, anyway, it was all in Latin. Only later did I recognise that everything I tried to understand in the various later literature was always explained best and in the clearest way in the original works of Euler. So, to be introduced to these is an important event for every mathematician, independent of a possible interest in history. It is extremely valuable to have a translation of this book, which must be seen to be a part of the cultural heritage of humanity, on the same level as Bach's B-minor Mass, Michelangelo's Pietà, or Shakespeare's Hamlet.

4.4. Review by: John Hannah.
Newsletter of the New Zealand Mathematical Society 82 (2001).

This book is a translation of the first part of Leonhard Euler's 1755 textbook Institutiones Calculi Differentialis, and it offers us a fascinating snapshot both of Euler's understanding of differential calculus, and of his expository or teaching style.

From our modern standpoint, Euler's grasp of the foundations of calculus seems (on the surface) to be decidedly primitive. Here is someone who is happy to talk about the infinite and the infinitely small, someone who introduces differentiation with virtually no mention of limits, and someone who is happy to entertain the idea that

         $1 + 2 + 4 + 8 + 16 + ... = -1$

But, as we might expect from someone who was also enormously successful at applying calculus to a wide variety of problems, there are lots of worthwhile ideas lurking behind this caricature of Euler's thoughts. In the end, even limits seem to be there, just below the surface. Consider, for example, what Euler has to say about the above equation. His extended discussion comes at the end of Chapter 3 (On the Infinite and Infinitely Small). Here calculus is treated as an experimental science, and readers are invited to join Euler in his exploration, his conjectures and refutations, and his (possibly tentative) conclusion. He thinks aloud, as it were, wondering what sense he can make of such calculations.

For Euler, the expansion of the fraction

          $\Large\frac{1}{1-x}\normalsize$ = $1 + x + x^{2} + x^{3} + x^{4} + x^{5} + ...$

is an algebraic calculation derived by repeated long division. Letting $x = 1$ and = 2 gives the equations

          $1 + 1 + 1 + 1 + 1 + ... = ∞$

          $1 + 2 + 4 + 8 + 16 + ... = -1$

There is clearly something puzzling going on here: a sum of positive terms has given a negative answer, and a term by term comparison of the left-hand sides might even suggest that negative numbers are greater than infinity. But this idea does not make sense since, arguing in the same way for the series

          $\Large\frac{1}{(1-x)^{2}\normalsize}$ = $1 + 2x + 3x^{2} + 4x^{3} + 5x^{3} + ...$,

he gets two equations

          $1 + 2 + 3 + 4 + 5 + ... = ∞$

          $1 + 4 + 12 + 32 + 180 + ... = 1$

which would then suggest that 1 is bigger than ∞.

At this stage, Euler recalls the division process which gave rise to the series, and he observes that the remainders involved actually grow larger the longer we continue dividing. So the remainders cannot be ignored. After analysing the remainders for the geometric series in more detail, Euler observes that "the sum of a series ought to be a limit the closer to which the partial sums should approach, the more terms are added."

But, he laments, these divergent series allow him to "discover many excellent results." He suggests that the difficulty lies in the word sum, and that he should really refer to $\Large\frac{1}{1-x}\normalsize$ as merely "a finite expression" from which the series $1 + x + x^{2} + x^{3} + x^{4} + x^{5} + ...$ "can be derived." But now comes an interesting twist, typical perhaps of a mathematician. Euler decides to give the word sum a different meaning! "Let us say that the sum of any infinite series is a finite expression from which the series can be derived. ... Since divergent series do not have a sum, properly speaking, there is no real difficulty which arises from this new meaning. Finally, with the aid of this definition we can keep the usefulness of divergent series and preserve their reputations."

We do not get to see how Euler will put this resolution into practice in this volume. He takes up the topic again in Chapter 1 of the next Part of Institutiones Calculi Differentialis, but for an English version we will have to wait for the completion of the future project hinted at by Blanton in his Introduction. In the meantime, perhaps we should probably bear in mind Euler's strategy (redefining the troublesome word) when we read his explanation of differentials in the Preface and see his use of them throughout the present volume. What did he really mean by those infinitely small quantities?

Euler's book also makes interesting reading from the educational point of view. Textbooks have of course changed hugely since 1755, and two changes jump out of the pages as soon as you open this book: there are no pictures and no exercises.

The first omission is a natural consequence of Euler's approach to the foundations of differential calculus (and is not caused, for example, by the difficulty of including illustrations). As he explains at the end of his introduction, he has discussed the geometry of plane curves elsewhere, and in this volume "everything is kept within the bounds of pure analysis, so that in the explanation of the rules of this calculus there is no need for any geometric figures."

What about exercises? Euler certainly expected his readers to do some. In Chapter 7, after a long discussion about the equality of various mixed partial derivatives, he advises his readers that they "must not only meditate on these properties with great care and examine their truth, but also work through many examples." The book itself is full of worked examples, but Euler expects a certain maturity of his audience. Once he has given a reasonable variety of examples, he encourages his readers to try similar exercises, but does feel any need to lead them further by the hand (page 91).

My present-day students would be horrified at the thought of having to make up their own exercises, and uppermost in their minds would be the problem of not knowing what the right answers were. Euler says nothing explicit about this problem, but there are perhaps some clues to his attitude in other, more subtle, features of the exposition. Euler delights in showing his readers different ways of obtaining the same result. Sometimes it is a simple observation: $(a+x)(b-x)$ can be differentiated as a product, or can be multiplied out and then differentiated term by term. But some examples are more intricate. The function $\arcsin x$ is differentiated firstly by using its expression in terms of a complex logarithm, and secondly by a kind of "first principles" differentiation of $x = \sin y$. So students can check the correctness of their answers for themselves. (Present-day students can be asked to do this too and, after the expected initial resistance, many comment on the pleasant glow of reassurance which they experience, especially during exams.)

These alternative solutions serve another purpose, showing the connections between different parts of the subject (as in the arcsin example above). Making connections like this has not been a strong feature of late 20th century mathematics texts, perhaps because of the influence of the axiomatic approach once you have one proof of a result, why would you need another? One recent author who bucks this trend is Gilbert Strang, but I have found that students do not like his books because of this very feature: his enthusiasm to show the interconnectedness of mathematics makes it very difficult for a student to find what he has to say on a particular topic! On the other hand, I think we all complain about the inability of students to transfer knowledge from one (section of a) course to another. It is interesting that making connections is one of the key ideas behind the numeracy projects currently under way in the primary sector. Perhaps we need to make it a more explicit part of our own courses too.

My Latin is a rusting hulk, abandoned during high school, but as far as I can tell Blanton's translation stays faithful to its eighteenth century origins, even to the point of retaining that era's wonderfully verbose (and occasionally rather obscure) style. Who nowadays, for example, would say that a particular form of an algebraic expression "is most accommodated to the expeditious differentiation of any rational function" (page 90)? Although he has modernised a few of Euler's notations (like using $x^{2}$ instead of $xx$), Blanton has retained the authentic $√(-1)$, reversing his decision to use $i$ in his earlier translation of Euler's Introduction to Analysis of the Infinite (Springer-Verlag, 1988). Similarly, in Chapter 2, Blanton follows Euler and consistently uses the single word series where a modern mathematician would probably want to draw a distinction between a sequence of terms and the series obtained by adding all these terms.

There are one or two misprints, but most of these are trivial. One, which made me reach for a piece of paper to check the calculations, involved a misreading of Euler's equation $y = A sv x$ for the arc (or inverse) of the versed sine (page 112). Perhaps because this function has fallen into disuse (at least in mathematics courses), Blanton opted for a verbal equivalent but forgot the inverse part, translating the equation as: $y$ is equal to the versed sine of $x$.

So overall this is a very interesting book. My only serious complaint is that Blanton stopped at the end of Euler's first Part, and that I'll have to wait a while before I see Euler's applications of differential calculus. Either that, or I'll have to brush up on my Latin!