Mathematicians From Switzerland

"[Newton] teaches us to take the fluxions, of any given order, of an equation with any given number of variable quantities, which belongs to the differential calculus: but he does not inform us, how to solve the inverse problem; that is to say, he has pointed out no means of resolving differential equations, either immediately, or by the separation of the indeterminate quantities, or by the reduction into series, &c. This theory however had already made very considerable progress in Germany, Holland, and France, as may be concluded from the problems of the catenarian, isochronous, and elastic curves, and particularly by the solution which James Bernoulli had given of the isoperimetrical problem."

"History of calculus"

"Mathematics"

"Frederick, let into the singularity of the man... would not deprive his Academy of a member from whom so much was to be expected. He was therefore admitted with a pension, and pronounced his inaugural oration in the month of January, 1765. Since that period, his Majesty honoured him with frequent and distinguished marks of his esteem; placed him in the financial commission of the Academy, and the architectural department, with the title of Superior Counsellor, at the same time making a considerable addition to his appointment. During these twelve years... Mr. Lambert, in his proper element, devoted his incessant labours to the improvement of science and the public good. He published some excellent performances, and furnished tracks without number, which have been inserted in the Memoires of the Academy, the Astronomical Tables of Berlin, and other collections. All his writings are highly expressive of a universal and original genius."

"We came finally to regard him... as an ingot of pure gold, whose value could not be enhanced by the fashion of the artist."

"Giving himself no manner of uneasiness as to what others might think of him, nor caring either to please or displease, he was uniformly without disguise; and, as he shewed himself on all occasions in the same colours, he at last subdued the prejudice, and forced the admiration of others to identify itself with his own."

"Mean and singular in his dress, he presented himself in a very awkward manner; a stranger to the received usages of society, or careless of conforming to them, he seemed to be occupied with nothing but himself; his philosophic volubility of tongue was unceasing till be found himself alone; and, even then I have seen him, after broaching a subject with some person who was called away, go on and finish it as if he had been speaking all the while to an attentive hearer. Add to this that flashes of self-love, and expressions of the high idea he entertained of his own merit..."

"Mr. Lambert was a man with whom the eye and the ear found it extremely difficult to become familiar."

"The history of Mr. Lambert's intellect during the space of 25 years, the progress of his genius, his rapid advancement in knowledge, and the series of his operations... he noted with equal truth and simplicity, in a sort of journal which is continued from the month of January, 1752, to the month of May, 1777. Such are those fugitive leaves more precious than the leaves of the Sybil. Never were there any which better merited to be preserved; and I request of the academy that they may be printed and annexed to my Eloge, on which they will bestow life and value."

"The reputation of his works is established, and posterity will confirm the decision of the present age."

"The torrent of his ideas, which flowed incessantly and rapidly from his brain, ever brought along with it useful materials for the construction of the system of the world. In these consisted his wealth; and no man could say, with more truth than himself, that all he was worth he carried about with him."

"The works of Mr. Lambert... have been duly appreciated by competent judges, who, by bestowing on them a distinguished reputation, have unalterably fixed the high rank the author has... held in the republic of letters. In the year 1760, he collected the different pieces, still in a fugitive state, of his Novum Organum [Neues Organon]; but which was not published till the year 1764. In the year 1761, he published his Treatise on the Properties of the Orbits of Comets, printed at Ausburg."

"In the month of Sept. 1759, Mr. Lambert was at Ausburg... for the purpose of giving the last touch to his Photometry, and to have it printed under his own eye. At the same period was instituted the Electoral Academy of Sciences at ... they... expressed their desire to have him more particularly attached to them by engaging him to furnish them literary papers, and to assist them with his advice. As a remuneration of his services, he received the title of Honorary Professor, and a pension of 800 florins. ...This connection, however, was of short duration. They accused him of not having the interest of the learned academy sufficiently at heart; and he complained... that they neglected his advice, and were at no pains to reform the abuses which he pointed out to them. They withdrew his pension, and he would not condescend to take any step for its recovery. Mr. Lambert was too much occupied with the abstract principles of science to give his thoughts to things so material; and yet, he was by no means in easy circumstances. He was satisfied if the profit of his works would enable him to lead the life of a philosopher from one publication to another..."

"The tutor and his pupils repaired to Utretcht, and passed a year in Holland; where Mr. Lambert gave to a bookseller of his treatise on the Passage of Light. But in the over ardent pursuit of this object, he found himself in the situation of the astrologer, who fell into a well... In consequence of a habit equally whimsical and invariable in him, he never presented himself but sideways, changed his position as often as the person with him sought to place himself in front, and he retreated in proportion as the other advanced. It was in a situation of this kind, that, making some steps backwards without attending to a stair case which was directly behind him, he fell at once from top to bottom, heels over head. The fall was dreadful; he lay long in a state of absolute insensibility, nor did he return to his senses till the end of twenty-four hours, when he opened his eyes totally black with extravasated blood..."

"Having one day read that Paschal invented a certain arithmetical machine, by a mere effort of his own genius, he could take no rest till he invented one of the same description. He likewise constructed with his own hand a mercurial watch or pendulum, which kept going 27 minutes, and served to ascertain precise portions of time in his physical experiments. His arithmetical scales and a machine for facilitating the art of drawing in perspective are no less worthy of our notice."

"The pupils of Mr. Lambert were the grandsons of the Count and sons of the Podestate of Coire. It was now in instructing his charge, that Mr. Lambert found all those means of instructing himself of which he had hitherto been so much in want. Becoming more and more conscious of the strength of his natural powers, he embraced, without hesitation, physics, astronomy, mathematics, mechanics, nor did he deem himself unequal to the studies of theology, metaphisics, eloquence, and poetry. He composed verses in all the languages he understood, German, French, Latin, Italian; but he would not dare to attempt the versification of the Greeks."

"His mother, in order to prevent his reading when he ought to be asleep, denied him the use of a light. Young Lambert had been at much pains in learning to write a fine hand, which was afterwards of great use to him: he wrote and drew extremely well; he made little designs or drawings, which he sold to his companions for a farthing or a halfpenny according as they contained more or fewer figures; and from this money he supplied himself with candles, which he lighted the moment all those of the family were put out."

"He possessed great powers of invention... Not possessing himself, and being in no condition to obtain the instruments necessary for making observations, or a single machine for the purposes of experimental philosophy, he contrived to supply that deficiency by making them of the most common materials that fell in his way; and the dexterity he came to employ in the management of them made amends for the imperfection of their construction."

"Mr. Lambert was a stranger to the three kingdoms of nature (He was however tolerably conversant in chemistry; he made various experiments on salts... the subject of different papers... in the academy.): he had never given his attention to individuals, nor to facts in that arrangement. All his points of view centered in the starry vault, in a straight line before him, and in the chamber of his brain, where he was continually immured, even when you thought you were with him, and fixed, or at least divided his attention. No divergency in him either to the right or to the left, always in the region of abstractions, objects in the order, of what are called concretes scarcely grazed his sphere."

"He was almost destitute of taste... in spite of his partiality for the muses, he was ever ready to ask as to subjects of taste, What does it prove? ...I was no stranger to his pretensions to wit ...Great men would drive their inferiors to despair if they paid no tribute to humanity."

"Mr. Lambert was upright in every sense of the word. Rectitude of views, rectitude of intentions, rectitude of action. I will not be accused of attributing to him impeccability, more than infallibility. But... Optimism was unquestionably a proper attribute of the deceased."

"Fontenelle, as he concludes his Eloge of Ozanam, informs us, that it used to be a saying of this academician, that it is the prerogative of the mathematician to go to Paradise in a perpendicular line. This, I have no doubt, was Mr. Lambert's route upon quitting the earth; nor had he occasion for a chariot of fire to carry him to heaven, a single ray of light would afford him a vehicle."

"In proportion as his intellectual pursuits were various and complex... the plan of his life was simple and uniform."

"Until late in life he had no access to what is called the great or fine world; but feeling in himself more real beauty and grandeur than he found in those whom he met usually in fashionable circles, he assigned a place to himself, from which it would not have been an easy matter to dislodge him. Such is the effect of the most precious of prerogatives mens conscia recti (A mind conscious of its own rectitude)."

"He had religion, and even devotion... he was still more a Christian than a philosopher, and... all the erroneous flights of a certain false philosophy were utterly unknown to him. He was too great a man to condescend to its acquaintance. His journal takes notice in the month of January 1755, of a composition intituled Oratio de characteribus Christian, ejusque præstantia Præ Philosopho [Prayer of Christian character, and his excellence prior to the Philosopher]. His whole life has been a commentary on this text, and an incontestible proof of it."

"Lambert is dead, and ye live ignorant mortals; ye live enemies of knowledge; ye live an useless burden on the earth, born to consume its good things without the capacity to produce one."

"When I turn my eyes to the place where we were accustomed to see our illustrious colleague, and where we saw him with so much pleasure, and where we used to hear him speak as if he had been inspired, I say to myself, certainly without the smallest intention to detract from the merit of any man: that place, is it filled? or, rather, shall it ever be filled again?"

"Mathematicians are usually regarded as clear and sober thinkers, but some of the men who have been gifted with the most marvelous power of mathematical analysis have not been free from the defects and vagaries of common mortals. Newton was extremely irritable, Laplace was inordinately vain, Monge, the inventor of descriptive geometry, was very forgetful and absent-minded. These men, however, were not fanciful dreamers, and few such are found among great mathematicians. One of these few was Johann Heinrich Lambert, the first man who endeavored to construct a system of the universe."

"Lambert was an 18th century Alsatian scholar, who is today regarded as a physicist, geometer, statistician, astronomer and philosopher and a representative of German rationalism. ...Among the achievements of Lambert... are the discovery of ; the formulation of laws governing light absorption, and thereby the establishment of photometry; the formulation of a law of motion of comets or planets. He is among the first to appreciate the nature of the Milky Way; he established several theorems of non-Euclidean geometry, developed De Moivre's theorems on the trigonometry of complex variables and introduced the hyperbolic sine and cosine functions. He proved the irrationality of π and π2, created a general theory of errors and finally, was the first to express Newton's second law of motion in the notation of the differential calculus."

"The famous Lambert, another Leibnitz, because of the universality and thoroughness of his knowledge, deserves a place among those mathematicians who had preserved a knowledge and taste for geometry at a time when the wonders of analysis concerned all, and who made the most glorious applications of it."

"This is all to which weak and limited beings can pretend, beings who occupy a point, and last but a moment in this mighty edifice built for eternity."

"In order to supply the defects of experience, we will have recourse to the probable conjectures of analogy, conclusions which we will bequeath to our posterity to be ascertained by new observations, which, if we augur rightly, will serve to establish our theory and to carry it gradually nearer to absolute certainty."

"We will next proceed by the lamp of experience, consulting with care the observations deposited in the records of astronomy."

"We will found our hypothesis in the general laws of motion, whose effects are every where the same, and whose influence extends to the utmost limits of matter."

"We suppose the existence of a wise and beneficent Being who presided over the formation of the World, and who is pleased to display his infinite perfections on this illustrious theatre."

"But, are the faculties of our nature equal to this? and what are the principles which ought to guide us in these researches?"

"We would wish to discover the Plan of the Universe, and the means employed by the Eternal Architect in the execution of his magnificent design. We will first contemplate the System of which we make a part, and of which our Sun is the center. Thence we will ascend towards those Suns and those innumerable Worlds which are scattered through the immensity of space."

"If in two ellipses having a common major axis we take two such arcs that their chords are equal, and that also the sums of the radii vectores, drawn respectively from the foci to the extremities of these arcs, are equal to each other, then the sectors formed in each ellipse by the arc and the two radii vectores are to each other as the square roots of the parameters of the ellipses."

"I shall now address you on the subject of the present situation in research in the foundations of mathematics. Since there remain open questions in this field, I am not in a position to paint a definitive picture of it for you. But it must be pointed out that the situation is not so critical as one could think from listening to those who speak of a foundational crisis. From certain points of view, this expression can be justified; but it could give rise to the opinion that mathematical science is shaken at its roots."

"Enlightened humanity has sought in rational definiteness its liberating refuge from the dominating influence of the merely authoritative. At the present time, however, this has for a large part been lost to consciousness, and to many people scientific validity that has to be acknowledged appears as an oppressing authority."

"Bernays's publications extend over the most diverse fields of mathematics … and are all marked by thoroughness and reliability … He is distinguished by a deep-seated love for science as well as a trustworthy character and nobility of thought, and is highly valued by everyone. In all matters concerning fundamental questions in mathematics, he is the most knowledgeable expert and, especially for me, the most valuable and productive colleague."

"The truth is that the mathematical sciences are growing in complete security and harmony. The ideas of Dedekind, Poincare, and Hilbert have been systematically developed with great success, without any conflict in the results. It is only from the philosophical point of view that objections have been raised. They bear on certain ways of reasoning peculiar to analysis and set theory. These modes of reasoning were first systematically applied in giving a rigorous form to the methods of the calculus. [According to them,] the objects of a theory are viewed as elements of a totality such that one can reason as follows: For each property expressible using the notions of the theory, it is [an] objectively determinate [fact] whether there is or there is not an element of the totality which possesses this property. Similarly, it follows from this point of view that either all the elements of a set possess a given property, or there is at least one element which does not possess it"

"P. Bernays has pointed out on several occasions that, since the consistency of a system cannot be proved using means of proof weaker than those of the system itself, it is necessary to go beyond the framework of what is, in Hilbert’s sense, finitary mathematics if one wants to prove the consistency of classical mathematics, or even that of classical number theory. Consequently, since finitary mathematics is defined as the mathematics in which evidence rests on what is intuitive, certain abstract notions are required for the proof of the consistency of number theory.... In the absence of a precise notion of what it means to be evident, either in the intuitive or in the abstract realm, we have no strict proof of Bernays’ assertion; practically speaking, however, there can be no doubt that it is correct..."

"What is perhaps the greatest blow that has ever come to the student body of Colorado College came last Friday when it was announced that Dean Florian Cajori, for about thirty years the best-known and best-liked professor in the College, had resigned and will not be back with us next year. It was not only on account of the value of his service as an instructor that the students felt such a sense of loss at the announcement, but more on account of the friendship and intimate relationship which he has shown to us. "Caj"… has been closer to this student body than any other one man. It was usually "Caj" who made the speech at the Barbecue, it was "Caj" who talked upcoming events in chapel, it was "Caj" who was always out there at the picnic or the Festival or the ball game. No form of student activity has seemed entirely complete unless our "Caj" has been there or has had something to do with it."

"The grandest achievement of the Hindus and the one which, of all mathematical inventions, has contributed most to the general progress of intelligence, is the invention of the principle of position in writing numbers. Generally we speak of our notation as the “Arabic” notation, but it should be called the “Hindu” notation, for the Arabs borrowed it from the Hindus. That the invention of this notation was not so easy as we might suppose at first thought, may be inferred from the fact that, of other nations, not even the keen-minded Greeks possessed one like it."

"J. J. Sylvester was an enthusiastic supporter of reform [in the teaching of geometry]. The difference in attitude on this question between the two foremost British mathematicians, J. J. Sylvester, the algebraist, and Arthur Cayley, the algebraist and geometer, was grotesque. Sylvester wished to bury Euclid "deeper than e'er plummet sounded" out of the schoolboy's reach; Cayley, an ardent admirer of Euclid, desired the retention of Simson's Euclid. When reminded that this treatise was a mixture of Euclid and Simson, Cayley suggested striking out Simson's additions and keeping strictly to the original treatise."

"Most of his [Euler's] memoirs are contained in the transactions of the Academy of Sciences at St. Petersburg, and in those of the Academy at Berlin. From 1728 to 1783 a large portion of the Petropolitan transactions were filled by his writings. He had engaged to furnish the Petersburg Academy with memoirs in sufficient number to enrich its acts for twenty years a promise more than fulfilled, for down to 1818 [Euler died in 1793] the volumes usually contained one or more papers of his. It has been said that an edition of Euler's complete works would fill 16,000 quarto pages."

"In 1735 the solving of an astronomical problem, proposed by the Academy, for which several eminent mathematicians had demanded several months' time, was achieved in three days by Euler with aid of improved methods of his own... With still superior methods this same problem was solved by the illustrious Gauss in one hour."

"Fermat died with the belief that he had found a long-sought-for law of prime numbers in the formula 2^{2^n} + 1 = a prime, but he admitted that he was unable to prove it rigorously. The law is not true, as was pointed out by Euler in the example 2^{2^5} + 1 = 4,294,967,297 = 6,700,417 times 641. The American lightning calculator Zerah Colburn, when a boy, readily found the factors but was unable to explain the method by which he made his marvellous mental computation."

"The miraculous powers of modern calculation are due to three inventions : the Arabic Notation, Decimal Fractions and Logarithms."