Mathematicians From Switzerland

"The Elements has been considered as offering models of scrupulously rigorous demonstrations. It is certainly true that in point of rigour it compares favourably with its modern rivals; but when examined in the light of strict mathematical logic, it has been pronounced by C.S. Peirce to be "riddled with fallacies." The results are correct only because the writer's experience keeps him on his guard."

"Comparatively few of the propositions and proofs in the Elements are his [Euclid's] own discoveries. In fact, the proof of the "Theorem of Pythagoras" is the only one directly ascribed to him."

"It is a remarkable fact in the history of geometry, that the Elements of Euclid, written two thousand years ago, are still regarded by many as the best introduction to the mathematical sciences."

"The history of mathematics is important also as a valuable contribution to the history of civilization. Human progress is closely identified with scientific thought. Mathematical and physical researches are a reliable record of intellectual progress."

"The history of mathematics may be instructive as well as agreeable ; it may not only remind us of what we have, but may also teach us to increase our store. Says De Morgan, "The early history of the mind of men with regards to mathematics leads us to point out our own errors; and in this respect it is well to pay attention to the history of mathematics." It warns us against hasty conclusions; it points out the importance of a good notation upon the progress of the science; it discourages excessive specialization on the part of the investigator, by showing how apparently distinct branches have been found to possess unexpected connecting links; it saves the student from wasting time and energy upon problems which were, perhaps, solved long since; it discourages him from attacking an unsolved problem by the same method which has led other mathematicians to failure; it teaches that fortifications can be taken by other ways than by direct attack, that when repulsed from a direct assault it is well to reconnoitre and occupy the surrounding ground and to discover the secret paths by which the apparently unconquerable position can be taken."

"As regards algebra, the early Arabs failed to adopt either the Diophantine or the Hindu notations. An examination of [the algebra of Al-Khwarizmi] shows that the exposition was altogether rhetorical, i.e., devoid of all symbolism."

"My quotations from Newton suggest the motive which induced him to take a stand against the use of hypotheses, namely, the danger of becoming involved in disagreeable controversies. ...Newton could no more dispense with hypotheses in his own cogitations than an eagle can dispense with flight. Nor did Newton succeed in avoiding controversy."

"The opinion is widely prevalent that even if the subjects are totally forgotten, a valuable mental discipline is acquired by the efforts made to master them. While the Conference admits that, considered in itself this discipline has a certain value, it feels that such a discipline is greatly inferior to that which may be gained by a different class of exercises, and bears the same relation to a really improving discipline that lifting exercises in an ill-ventilated room bear to games in the open air. The movements of a race horse afford a better model of improving exercise than those of the ox in a tread-mill."

"Our so-called "Arabic" notation owes its excellence to the application of the principle of local value and the use of a symbol for zero. It is now conclusively established that the principle of local value was used by the ns much earlier than by the Hindus and that the Maya of Central America used the principle and symbols for zero in a well-developed numeral system of their own. The notation of Babylonia used the scale of 60, that of the Maya, the scale 20 (except in one step). It follows, therefore, that the present controversy on the origin of our numerals does not involve the question of the first use of local value and symbols for zero; it concerns itself only with the time and place of the first application of local value to the decimal scale and with the origin of the forms or shapes of our ten numerals. ... Hurt by the alleged arrogance of certain Greek scholars, Sebokht praises the science of the Hindus and speaks of "their valuable methods of computation. . . . I wish only to say that this computation is done by means of nine signs." Unfortunately, he leaves it to us to guess whether or not he used the zero. The passage, written about 662 A.D., is the earliest reference that has been found outside of India to our numerals. ...The form of the symbols with the zero, used in India, differed so widely from the old forms without the zero used there, that the former seem to have had an independent origin and to have been imported into India. ...The following are outstanding facts: 1. The earliest reliable record of the use of our numerals with zero is an inscription of 867 A.D. in India. 2. The validity of the testimony of early Arabic writers ascribing to India the numerals with zero is shaken, but not destroyed. 3. There is not a scintilla of evidence in the form of old manuscripts or numeral inscriptions to support the Greek origin of our numerals. 4. At present the hypothesis of the Hindu origin of our numerals stands without serious rival. But this hypothesis is by no means firmly established."

"Professor Sylvester's first high class at the new university Johns Hopkins consisted of only one student, G. B. Halsted, who had persisted in urging Sylvester to lecture on the modern algebra. The attempt to lecture on this subject led him into new investigations in quantics."

"Professor Florian Cajori died August 14, 1934. In May of the following year I was invited by the University of California Press to edit this work. ...this is a revision of Motte's translation of the Principia. From many conversations with Professor Cajori, I know that he had long cherished the idea of revising Newton's immortal work by rendering certain parts into modern phraseology (e.g., to change the reading of "reciprocally in the subduplicate ratio of " to "inversely as the square root of") and to append historical and critical notes which would provide instruction to some readers and interest to all. This is his last work; one of the most fitting to crown a life devoted to investigation and to the history of the sciences in his chosen field."

"As a mathematician Dean Cajori has achieved a name which very few in this world can equal, a name which is respected all over the globe. His text books and his writings have been published all over the world. We are proud of all the achievements of our "Caj", of course, but we are especially proud of what he has done for us here, and it is for this reason that we shall always hold him in our memory. As a friend and as an instructor he has been more to us than we can ever measure, and we shall always look back upon the days when we had "Caj"."

"They [the students] looked upon "Caj" as one of their best friends in all the College, and thought of him as the one responsible for their later success. He has had a personal appeal to a great many students... It was the appeal of the one with a human interest in what somebody else is doing, the appeal of the true friend and the hearty well-wisher. It was the appeal of "Caj"."

"Of no little importance are Euler's labors in analytical mechanics. ...He worked out the theory of the rotation of a body around a fixed point, established the general equations of motion of a free body, and the general equation of hydrodynamics. He solved an immense number and variety of mechanical problems, which arose in his mind on all occasions. Thus on reading Virgil's lines. "The anchor drops, the rushing keel is staid," he could not help inquiring what would be the ship's motion in such a case. About the same time as Daniel Bernoulli he published the Principle of the Conservation of Areas and defended the principle of "least action," advanced by P. Maupertius. He wrote also on tides and on sound."

"The most influential mathematics textbook of ancient times is easily named, for the Elements of Euclid has set the pattern in elementary geometry ever since. The most effective textbook of the medieval age is less easily designated; but a good case can be made out for the Al-jabr of Al-Khwarizmi, from which algebra arose and took its name. Is it possible to indicate a modern textbook of comparable influence and prestige? Some would mention the Géométrie of Descartes or the Principia of Newton or the Disquisitiones of Gauss; but in pedagogical significance these classics fell short of a work by Euler titled Introductio in analysin infinitorum."

"He calculated without any apparent effort, just as men breathe, as eagles sustain themselves in the air."

"Quanquam nobis in intima naturae mysteria penetrare, indeque veras caussas Phaenomenorum agnoscere neutiquam est concessum: tamen evenire potest, ut hypothesis quaedam ficta pluribus phaenomenis explicandis aeque satisfaciat, ac si vera caussa nobis esset perspecta."

"A function of a variable quantity is an analytic expression composed in any way whatsoever of the variable quantity and numbers or constant quantities."

"Till now the mathematicians tried in vain to discover some order in the sequence of the prime numbers and we have every reason to believe that there is some mystery which the human mind shall never penetrate. To convince oneself, one has only to glance at the tables of primes which some people took the trouble to compute beyond a hundred thousand, and one perceives that there is no order and no rule. This is so much more surprising as the arithmetic gives us definite rules with the help of which we can continue the sequence of the primes as far as we please, without noticing, however, the least trace of order."

"It will seem a little paradoxical to ascribe a great importance to observations even in that part of the mathematical sciences which is usually called Pure Mathematics, since the current opinion is that observations are restricted to physical objects that make impression on the senses. As we must refer the numbers to the pure intellect alone, we can hardly understand how observations and quasi-experiments can be of use in investigating the nature of numbers. Yet, in fact, as I shall show here with very good reasons, the properties of the numbers known today have been mostly discovered by observation, and discovered long before their truth has been confirmed by rigid demonstrations. There are many properties of the numbers with which we are well acquainted, but which we are not yet able to prove; only observations have led us to their knowledge. Hence we see that in the theory of numbers, which is still very imperfect, we can place our highest hopes in observations; they will lead us continually to new properties which we shall endeavor to prove afterwards. The kind of knowledge which is supported only by observations and is not yet proved must be carefully distinguished from the truth; it is gained by induction, as we usually say. Yet we have seen cases in which mere induction led to error. Therefore, we should take great care not to accept as true such properties of the numbers which we have discovered by observation and which are supported by induction alone. Indeed, we should use such discovery as an opportunity to investigate more exactly the properties discovered and to prove or disprove them; in both cases we may learn something useful."

"La construction d'une machine propre à exprimer tous les sons de nos paroles , avec toutes les articulations , seroit sans-doute une découverte bien importante. … La chose ne me paroît pas impossible."

"To those who ask what the infinitely small quantity in mathematics is, we answer that it is actually zero. Hence there are not so many mysteries hidden in this concept as they are usually believed to be."

"All the greatest mathematicians have long since recognized that the method presented in this book is not only extremely useful in analysis, but that it also contributes greatly to the solution of physical problems. For since the fabric of the universe is most perfect, and is the work of a most wise Creator, nothing whatsoever takes place in the universe in which some relation of maximum and minimum does not appear. Wherefore there is absolutely no doubt that every effect in the universe can be explained as satisfactorily from final causes, by the aid of the method of maxima and minima, as it can from the effective causes themselves. Now there exist on every hand such notable instances of this fact, that, in order to prove its truth, we have no need at all of a number of examples; nay rather one's task should be this, namely, in any field of Natural Science whatsoever to study that quantity which takes on a maximum or a minimum value, an occupation that seems to belong to philosophy rather than to mathematics. Since, therefore, two methods of studying effects in Nature lie open to us, one by means of effective causes, which is commonly called the direct method, the other by means of final causes, the mathematician uses each with equal success. Of course, when the effective causes are too obscure, but the final causes are more readily ascertained, the problem is commonly solved by the indirect method; on the contrary, however, the direct method is employed whenever it is possible to determine the effect from the effective causes. But one ought to make a special effort to see that both ways of approach to the solution of the problem be laid open; for thus not only is one solution greatly strengthened by the other, but, more than that, from the agreement between the two solutions we secure the very highest satisfaction."

"Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate."

"Now I will have less distraction."

"As analysis was more cultivated, it gained a predominancy over geometry; being found to be a far more powerful instrument for obtaining results; and possessing a beauty and an evidence, which, though different from those of geometry, had great attractions for minds to which they became familiar. The person who did most to give to analysis the generality and symmetry which are now its pride, was also the person who made Mechanics analytical; I mean Euler."

"Perhaps the most surprising thing about mathematics is that it is so surprising. The rules which we make up at the beginning seem ordinary and inevitable, but it is impossible to foresee their consequences. These have only been found out by long study, extending over many centuries. Much of our knowledge is due to a comparatively few great mathematicians such as Newton, Euler, Gauss, or Riemann; few careers can have been more satisfying than theirs. They have contributed something to human thought even more lasting than great literature, since it is independent of language."

"I discovered the works of Euler and my perception of the nature of mathematics underwent a dramatic transformation. I was de-Bourbakized, stopped believing in sets, and was expelled from the Cantorian paradise. I still believe in abstraction, but now I know that one ends with abstraction, not starts with it. I learned that one has to adapt abstractions to reality and not the other way around. Mathematics stopped being a science of theories but reappeared to me as a science of numbers and shapes."

"In 1736, during his first stay in St. Petersburg, Euler tackled the now famous problem of the seven bridges of Königsberg. His contribution to this problem is often cited as the birth of graph theory and topology."

"It is the invaluable merit of the great Basle mathematician Leonard Euler, to have freed the analytical calculus from all geometric bounds, and thus to have established analysis as an independent science, which from his time on has maintained an unchallenged leadership in the field of mathematics."

"Euler's step was daring. In strict logic, it was an outright fallacy... Yet it was justified by analogy, by the analogy of the most successful achievements of a rising science that he called... "Analysis of the Infinite." Other mathematicians, before Euler, passed from finite differences to infinitely small differences, from sums with a finite number of terms to sums with an infinity of terms, from finite products to infinite products. And so Euler passed from equations of a finite degree (algebraic equations) to equations of infinite degree, applying the rules made for the finite... This analogy... is beset with pitfalls. How did Euler avoid them? ...Euler's reasons are not demonstrative. Euler does not reexamine the grounds for his conjecture... only its consequences. ...He examines also the consequences of closely related analogous conjectures... Euler's reasons are, in fact, inductive."

"Euler and Ramanujan are mathematicians of the greatest importance in the history of constants (and of course in the history of Mathematics ...)"

"If we compared the Bernoullis to the Bach family, then Leonhard Euler is unquestionably the Mozart of mathematics, a man whose immense output... is estimated to fill at least seventy volumes. Euler left hardly an area of mathematics untouched, putting his mark on such diverse fields as analysis, number theory, mechanics and hydrodynamics, cartography, topology, and the theory of lunar motion. ...Moreover, we owe to Euler many of the mathematical symbols in use today, among them i, π, e, and f(x). And as if that were not enough, he was a great popularizer of science..."

"He was later to write that he had made some of his best discoveries while holding a baby in his arms surrounded by playing children."

"Read Euler: he is our master in everything."

"To the reader of today much in the conception and mode of expression of that time appears strange and unusual. Between us and the mathematicians of the late seventeenth century stands Leonhard Euler... He is the real founder of our modern conception. However non-rigorous he may be in details: he ends and conquers the previous epoch of direct geometric infinitesimal considerations and introduces the period of mathematical analysis according to form and content. Whatever was written after him on the logarithmic series is necessarily based no longer on the already obscured predecessors in the receding mathematical Renaissance, but on Euler's Introductio in analysin infinitorum... in which the entire seventh chapter [De Quantitabus exponentialibus ac Logarithmis] treats of logarithms."

"Who has studied the works of such men as Euler, Lagrange, Cauchy, Riemann, , and Weierstrass, can doubt that a great mathematician is a great artist? The faculties possessed by such men, varying greatly in kind and degree with the individual, are analogous with those requisite for constructive art. Not every mathematician possesses in a specially high degree that critical faculty which finds its employment in the perfection of form, hi conformity with the ideal of logical completeness; but every great mathematician possesses the rarer faculty of constructive imagination."

"Galileo does not attempt any theory to account for the flexure of the beam. This theory, supplied by , was applied by Mariotte, Leibnitz, De Lahire, and Varignon, but they neglect compression of the fibres, and so place the neutral in the lower face of Galileo's beam. The true position of the neutral plane was assigned by James Bernoulli 1695, who in his investigation of the simplest case of bent beam, was led to the consideration of the curve called the "elastica." This "elastica" curve speedily attracted the attention of the great Euler (1744), and must be considered to have directed his attention to the s. Probably the extraordinary divination which led Euler to the formula connecting the sum of two elliptic integrals, thus giving the fundamental theorem of the addition equation of s, was due to mechanical considerations concerning the "elastica" curve; a good illustration of the general principle that the pure mathematician will find the best materials for his work in the problems presented to him by natural and physical questions."

"Euler published so much and in so many different fields that an edited volume is probably the only way (at least at this time) to do him something like justice, since no one person will know enough to span all of his work."

"Following a suggestion by Daniel Bernoulli, Euler gave the first treatment of elastic lines by means of the in the Additamentum I to his Methodus inveniendi (1744...) which carries the title De curvis elasticus. Euler characterized the equilibrium position of an elastic line by the following variational principle: Among all curves of equal length, joining two points where they have prescribed tangents, to determine that which minimizes the value of the expression \int ds/\rho^2 [where \rho is the radius of curvature]. In other words, Euler interpreted an elastic line as an inextensible curve \boldsymbol\zeta with a "" of \int \kappa^2 ds, \kappa [i.e., 1/\rho] being the function of \boldsymbol\zeta, whose positions of (stable) equilibrium are characterized by the minima of the potential energy, i.e., by 's principle of . Thus the problem of the elastic line leads to the isoperimetric problem\int_{\boldsymbol \zeta} \kappa^2 ds \to min \qquad with \int_{\boldsymbol\zeta} ds = L..."

"It is customary to consider Chebyshev, Gauss, Jacobi, and Legendre as the main creators of the theory of orthogonal polynomials. However, their contributions were directly influenced by Brouncker and Wallis who, in March of 1655, made discoveries which influenced the development of analysis for the next hundred years. Namely, Wallis found an infinite product of rational numbers converging to 4/π and Brouncker gave a remarkable continued fraction for this quantity. ...The only mathematician who understood the importance of these discoveries was Euler. ...he felt that the recovery of the original Brouncker's proof could open up new perspectives for analysis. As usual, Euler was right."

"The study of Euler's works will remain the best school for the different fields of mathematics and nothing else can replace it."

"Madam, I have come from a country where people are hanged if they talk."

"Euler lacked only one thing to make him a perfect genius: He failed to be incomprehensible."

"Euler calculated the force of the wheels necessary to raise the water in a reservoir … My mill was carried out geometrically and could not raise a drop of water fifty yards from the reservoir. Vanity of vanities! Vanity of geometry!"

"Somebody said "Talent is doing what others find difficult. Genius is doing easily what others find impossible." ...by that definition, Euler was a genius. He could do the seemingly impossible, and he did it throughout his long and illustrious life. ...Way to Go, Uncle Leonhard!"

"The Introductio does not boast an impressive number of editions, yet its influence was pervasive. In originality and in the richness of its scope it ranks among the greatest of textbooks; but it is outstanding also for clarity of exposition. Published two hundred and two years ago, it nevertheless possesses a remarkable modernity of terminology and notation, as well as of viewpoint. Imitation is indeed the sincerest form of flattery."