Mathematicians from Germany

"Every kind of science, if it has only reached a certain degree of maturity, automatically becomes a part of mathematics."

"Wir dürfen nicht denen glauben, die heute mit philosophischer Miene und überlegenem Tone den Kulturuntergang prophezeien und sich in dem Ignorabimus gefallen. Für uns gibt es kein Ignorabimus, und meiner Meinung nach auch für die Naturwissenschaft überhaupt nicht. Statt des törichten Ignorabimus heiße im Gegenteil unsere Losung:"

"The art of doing mathematics consists in finding that special case which contains all the germs of generality."

"One can measure the importance of a scientific work by the number of earlier publications rendered superfluous by it."

"Mathematics knows no races or geographic boundaries; for mathematics, the cultural world is one country."

""Mathematics is a presuppositionless science. To found it I do not need God, as does Kronecker, or the assumption of a special faculty of our understanding attuned to the principle of mathematical induction, as does Poincaré, or the primal intuition of Brouwer, or, finally, as do Russell and Whitehead, axioms of infinity, reducibility, or completeness, which in fact are actual, contentual assumptions that cannot be compensated for by consistency proofs."

"If one were to bring ten of the wisest men in the world together and ask them what was the most stupid thing in existence, they would not be able to discover anything so stupid as astrology."

"If I were to awaken after having slept for a thousand years, my first question would be: Has the Riemann hypothesis been proven?"

"One of the supreme achievements of purely intellectual human activity."

"But he (Galileo) was not an idiot,... Only an idiot could believe that scientific truth needs martyrdom — that may be necessary in religion, but scientific results prove themselves in time."

"Sometimes it happens that a man's circle of horizon becomes smaller and smaller, and as the radius approaches zero it concentrates on one point. And then that becomes his point of view."

"Keep computations to the lowest level of the multiplication table."

"Immer mit den einfachsten Beispielen anfangen."

"I do not see that the sex of the candidate is an argument against her admission as a Privatdozent. After all, the Senate is not a bath-house."

"Good, he did not have enough imagination to become a mathematician."

"History teaches the continuity of the development of science. We know that every age has its own problems, which the following age either solves or casts aside as profitless and replaces by new ones. If we would obtain an idea of the probable development of mathematical knowledge in the immediate future, we must let the unsettled questions pass before our minds and look over the problems which the science of today sets and whose solution we expect from the future. To such a review of problems the present day, lying at the meeting of the centuries, seems to me well adapted. For the close of a great epoch not only invites us to look back into the past but also directs our thoughts to the unknown future."

"A mathematical problem should be difficult in order to entice us, yet not completely inaccessible, lest it mock at our efforts. It should be to us a guide post on the mazy paths to hidden truths, and ultimately a reminder of our pleasure in the successful solution."

"It remains to discuss briefly what general requirements may be justly laid down for the solution of a mathematical problem. I should say first of all, this: that it shall be possible to establish the correctness of the solution by means of a finite number of steps based upon a finite number of hypotheses which are implied in the statement of the problem and which must always be exactly formulated. This requirement of logical deduction by means of a finite number of processes is simply the requirement of rigor in reasoning."

"To new concepts correspond, necessarily, new signs. These we choose in such a way that they remind us of the phenomena which were the occasion for the formation of the new concepts."

"If we do not succeed in solving a mathematical problem, the reason frequently consists in our failure to recognize the more general standpoint from which the problem before us appears only as a single link in a chain of related problems. After finding this standpoint, not only is this problem frequently more accessible to our investigation, but at the same time we come into possession of a method which is applicable also to related problems."

"This conviction of the solvability of every mathematical problem is a powerful incentive to the worker. We hear within us the perpetual call: There is the problem. Seek its solution. You can find it by pure reason, for in mathematics there is no ignorabimus."

"Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts. For with all the variety of mathematical knowledge, we are still clearly conscious of the similarity of the logical devices, the relationship of the ideas in mathematics as a whole and the numerous analogies in its different departments. We also notice that, the farther a mathematical theory is developed, the more harmoniously and uniformly does its construction proceed, and unsuspected relations are disclosed between hitherto separate branches of the science. So it happens that, with the extension of mathematics, its organic character is not lost but only manifests itself the more clearly."

"The organic unity of mathematics is inherent in the nature of this science, for mathematics is the foundation of all exact knowledge of natural phenomena. That it may completely fulfil this high mission, may the new century bring it gifted masters and many zealous and enthusiastic disciples!"

"An old French mathematician said: A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street. This clearness and ease of comprehension, here insisted on for a mathematical theory, I should still more demand for a mathematical problem if it is to be perfect; for what is clear and easily comprehended attracts, the complicated repels us."

"In mathematics, as in any scientific research, we find two tendencies... [T]he tendency toward abstraction seeks to crystallize the logical relations inherent in the maze of material in a systematic and orderly manner. On the other hand, the tendency toward intuitive understanding fosters a more immediate grasp of the objects... a live rapport with them... which stresses the concrete meaning of their relations. ...[I]ntuitive understanding plays a major role in geometry. ...[S]uch concrete intuition is of great value not only for the research worker, but... for anyone who wishes to study and appreciate the results of research in geometry."

"[O]ur purpose is to give a presentation of geometry... in its visual, intuitive aspects. With the aid of visual imagination we can illuminate the manifold facts and problems... beyond this, it is possible... to depict the geometric outline of the methods of investigation and proof, without... entering into the details... In this manner, geometry being as many-faceted as it is and being related to the most diverse branches of mathematics, we may even obtain a summarizing survey of mathematics as a whole, and a valid idea of the variety of problems and the wealth of ideas it contains. Thus a presentation of geometry in large brushstrokes... and based on the approach through visual intuition, should contribute to a more just appreciation of mathematics by a wider range of people than just the specialists."

"[M]athematics is not a popular subject... The reason for this is to be found in the common superstition that [it] is but a continuation... of the fine art of arithmetic, of juggling with numbers. [We] combat that superstition, by offering, instead of formulas, figures that may be looked at and that may easily be supplemented by models which the reader may construct. This book... bring[s] about a greater enjoyment of mathematics, by making it easier... to penetrate the essence of mathematics without... a laborious course of studies."

"The various branches of geometry are all interrelated closely and quite often unexpectedly. This shows up in many places in the book. Even so... it was necessary to make each chapter...self-contained... We hope that... we have rendered each chapter taken by itself... understandable and interesting. We want to take the reader on a leisurely walk... in the big garden that is geometry, so that each may pick for himself a bouquet to his liking."

"Physics is too difficult for physicists!"

"More decisive than any other influence for the young Hilbert at Königsberg was his friendship with Adolf Hurwitz and Minkowski. He got his thorough mathematical training less from lectures, teachers or books, than from conversation."

"David Hilbert—the undisputed, foremost living mathematician in the world and lifelong close friend and collaborator of the by then deceased Minkowski—had already presented to the Göttingen Academy his own version of the same equations a few days earlier [than Einstein]. Although Minkowski and Hilbert accomplished their most important achievements in pure mathematical fields, their respective contributions to relativity should in no sense be seen as merely occasional excursions into the field of theoretical physics. Minkowski and Hilbert were motivated by much more than a desire to apply their exceptional mathematical abilities opportunistically... On the contrary, Minkowski's and Hilbert's contributions to relativity are best understood as an organic part of their overall scientific careers."

"A more thorough study of Euclid's axioms and postulates proved them to be inadequate for the deduction of Euclid's geometry. ...Hilbert and others succeeded in filling the gap by stating explicitly a complete system of postulates for Euclidean and non-Euclidean geometries alike. Among the postulates missing in Euclid's list was the celebrated postulate of Archimedes, according to which, by placing an indefinite number of equal lengths end to end along a line, we should eventually pass any point arbitrarily selected on the line. Hilbert, by denying this postulate, just as Lobatchewski and Riemann had denied Euclid's parallel postulate, succeeded in constructing a new geometry known as non-Archimedean. It was perfectly consistent but much stranger than the classical non-Euclidean varieties. Likewise, it was proved possible to posit a system of postulates which would yield Euclidean or non-Euclidean geometries of any number of dimensions; hence, so far as rational requirements of the mind were concerned, there was no reason to limit geometry to three dimensions."

"Hilbert's problems have the characteristics of any good founding document. Each one is a short essay on its subject, not overly specific, and yet Hilbert makes his intent remarkably clear. He leaves room for change and adjustment. Hilbert's goal was to foster the pursuit of mathematics."

"Hilbert was on the verge of retirement. He was the dignified chairman of the mathematical society's meetings, though he no longer came up with those caustic quips that people would repeat afterwards, imitating as best they could his Baltic accent. It is a pity these were not recorded before it was too late. The samples cited in English translation, in Constance Reid's biography of Hilbert, give only the palest idea of his biting wit."

"quando orientur controversiae, non magis disputatione opus erit inter duos philosophos, quam inter duos computistas. Sufficiet enim calamos in manus sumere sedereque ad abacos, et sibi mutuo (accito si placet amico) dicere: calculemus"

"Languages are the best mirror of the human mind [and] the most ancient monu­ments of peoples."

"Theologus: Amare autem? Philosophus: Felicitate alterius delectari."

"Nam filum labyrintho de compositione continui deque maximo et minimo ac indesignabili at que infinito non nisi geometria praebere potest, ad metaphysicam vero solidam nemo veniet, nisi qui illac transiverit."

"To love is to be delighted by the happiness of someone, or to experience pleasure upon the happiness of another. I define this as true love."

"Omne possibile exigit existere."

"Chaque substance est comme un monde à part, indépendant de toute autre chose, hors de Dieu..."

"As regards the objection that possibles are independent of the decrees of God I grant it of actual decrees (although the Cartesians do not at all agree to this), but I maintain that the possible individual concepts involve certain possible free decrees; for example, if this world was only possible, the individual concept of a particular body in this world would involve certain movements as possible, it would also involve the laws of motion, which are the free decrees of God; but these, also, only as possibilities. Because, as there are an infinity of possible worlds, there are also an infinity of laws, certain ones appropriate to one; others, to another, and each possible individual of any world involves in its concept the laws of its world."

"TO LOVE is to find pleasure in the happiness of others. Thus the habit of loving someone is nothing other than BENEVOLENCE by which we want the good of others, not for the profit that we gain from it, but because it is agreeable to us in itself. CHARITY is a general benevolence. And JUSTICE is charity in accordance with wisdom. … so that one does not do harm to someone without necessity, and that one does as much good as one can, but especially where it is best employed."

"Pour ce qui est des connaissances non-écrites qui se trouvent dispersées parmi les hommes de différents professions, je suis persuadé qu’ils passent de beaucoup tant à l'égard de la multitude que de l'importance, tout ce qui se trouve marqué dans les livres, et que la meilleure partie de notre trésor n'est pas encore enregistrée."

"Il y a jusque dans les exercices des enfants ce qui pourrait arrêter le plus grand Mathématicien."

"When Sir A. Fountaine was at Berlin with Leibnitz in 1701, and at supper with the Queen of Prussia, she asked Leibnitz his opinion of Sir Isaac Newton. Leibnitz said that taking mathematicians from the beginning of the world to the time when Sir Isaac lived, what he had done was much the better half; and added that he had consulted all the learned in Europe upon some difficult points without having any satisfaction, and that when he applied to Sir Isaac, he wrote him in answer by the first post, to do so and so, and then he would find it."

"La nature ne fait jamais des sauts."

"[The consequences of] beliefs that go against the providence of a perfectly good, wise, and just God, or against that immortality of souls which lays them open to the operations of justice.... I even find that somewhat similar opinions, by stealing gradually into the minds of men of high station who rule the rest and on whom affairs depend, and by slithering into fashionable books, are inclining everything toward the universal revolution with which Europe is threatened, and are completing the destruction of what still remains in the world of the generous Greeks and Romans who placed love of country and of the public good, and the welfare of before fortune and even before life."

"I have seen something of the project of M. de St. Pierre, for maintaining a perpetual peace in Europe. I am reminded of a device in a cemetery, with the words: Pax perpetua; for the dead do not fight any longer: but the living are of another humor; and the most powerful do not respect tribunals at all."

"My philosophical views approach somewhat closely those of the late Countess of Conway, and hold a middle position between Plato and Democritus, because I hold that all things take place mechanically as Democritus and Descartes contend against the views of Henry More and his followers, and hold too, nevertheless, that everything takes place according to a living principle and according to final causes — all things are full of life and consciousness, contrary to the views of the Atomists."

"Il y a deux labyrinthes fameux où notre raison s’égare bien souvent : l'un regarde la grande question du libre et du nécessaire, surtout dans la production et dans l'origine du mal ; l'autre consiste dans la discussion de la continuité et des indivisibles qui en paraissent les éléments, et où doit entrer la considération de l'infini."

"Musica est exercitium arithmeticae occultum nescientis se numerare animi."

"It is an extremely useful thing to have knowledge of the true origins of memorable discoveries, especially those that have been found not by accident but by dint of meditation. It is not so much that thereby history may attribute to each man his own discoveries and others should be encouraged to earn like commendation, as that the art of making discoveries should be extended by considering noteworthy examples of it."

"J'ay marqué plus d'une fois, que je tenois l'espace pour quelque chose de purement relatif, comme le temps; pour un ordre des coëxistences, comme le temps est un ordre des successions."

"cur aliquid potius extiterit quam nihil"

"Although the whole of this life were said to be nothing but a dream and the physical world nothing but a phantasm, I should call this dream or phantasm real enough, if, using reason well, we were never deceived by it."

"De quelque manière que Dieu aurait créé le monde, il aurait toujours été régulier et dans un certain ordre général. Mais Dieu a choisi celui qui est le plus parfait, c’est-à-dire celui qui est en même temps le plus simple en hypothèses et le plus riche en phénomènes..."

"Ce miracle de l'Analyse, prodige du monde des idées, objet presque amphibie entre l'Être et le Non-être, que nous appelons racine imaginaire."

"We never have a full demonstration, although there is always an underlying reason for the truth, even if it is only perfectly understood by God, who alone penetrated the infinite series in one stroke of the mind."

"The love of God consists in an ardent desire to procure the general welfare, and reason teaches me that there is nothing which contributes more to the general welfare of mankind than the perfection of reason."

"On est obligé d’ailleurs de confesser que la Perception et ce qui en dépend, est inexplicable par des raisons mécaniques, c’est-à-dire par les figures et par les mouvements. Et feignant qu'il y ait une Machine, dont la structure fasse penser, sentir, avoir perception ; on pourra la concevoir agrandie en conservant les mêmes proportions, en sorte qu’on y puisse entrer, comme dans un moulin. Et cela posé, on ne trouvera en la visitant au dedans, que des pièces, qui poussent les unes les autres, et jamais de quoi expliquer une perception. Ainsi c'est dans la substance simple, et non dans le composé, ou dans la machine qu’il la faut chercher."

"Et comme tout présent état d'une substance simple est naturellement une suite de son état précédent, tellement, que le présent y est gros de l'avenir."

"Il y a aussi deux sortes de vérités, celles de Raisonnement et celle de Fait. Les vérités de Raisonnement sont nécessaires et leur opposé est impossible, et celles de Fait sont contingentes et leur opposé est possible."

"Or, comme il y a une infinité d'univers possibles dans les idées de Dieu, et qu'il n'en peut exister qu'un seul, il faut qu'il y ait une raison suffisante du choix de Dieu qui le détermine à l'un plutôt qu'à l'autre. Et cette raison ne peut se trouver que dans la convenance, dans les degrés de perfection que ces mondes contiennent, chaque possible ayant droit de prétendre à l'existence à mesure de la perfection qu'il enveloppe."

"Thus there is nothing waste, nothing dead in the universe; no chaos, no confusions, save in appearence. We might compare this to the appearence of a pond in the distance, where we can see the confused movement and swarming of the fish, without distinguishing the fish themselves. Thus we are that each living body has a dominante entelechy, which in case of an animal is the soul, but the members of this living body are full of other living things, plants and animals, of which each has in turn ita dominant entelechy or soul."

"Ainsi on peut dire que non seulement l'âme, miroir d'un univers indestructible, est indestructible, mais encore l'animal même, quoique sa machine périsse souvent en partie, et quitte ou prenne des dépouilles organiques."

"Gottfried Leibniz is famous... for his slogan Calculemus, which means "Let us calculate." He envisioned a formal language to reduce reasoning to calculation, and he said that reasonable men, faced with a difficult question of philosophy or policy, would express the question in a precise language and use rules of calculation to carry out precise reasoning. This is the first reduction of reasoning to calculation ever envisioned. ...he actually designed and built a working calculating machine, the Stepped Reckoner ...inspired by the somewhat earlier work of Pascal, who built a machine that could add and subtract. Leibniz's machine could add, subtract, divide, and multiply, and was apparently the first machine with all four arithmetic capabilities."

"[T]he program which Immanuel Kant proposed back in the 1760s... was this: our knowledge of the outside world depends on our modes of perception... Unfortunately, a great revolution took place in or about the year 1768, when he read a paper by Euler which intended to show that space was indeed absolute as Newton had suggested and not relative as Leibnitz suggested. (...in the eighteenth century the question of whether Newton's... or Leibnitz's view of the world was right profoundly affected all philosophy.) After reading Euler's argument... Kant... for the first time proposed that... we must be conscious of [absolute space] a priori. ...Kant died in 1804, long before new ideas about space... had been published... And since one of the things that happened in [our] lifetime has been the substitution of... a Leibnitz universe, the universe of relativity, for Newton's universe... we should think that out again."

"In the interval of 200 years between Kepler and Gauss there arose no great mathematician in Germany excepting Leibniz."

"Leibniz was certainly not alone among great men in presenting in his early work almost all the important mathematical ideas contained in his mature work."

"The main ideas of his philosophy are to be attributed to his mathematical work, and not vice versa."

"The manuscripts of Leibniz... show, perhaps more clearly than his published work, the great importance which Leibniz attached to suitable notation in mathematics and... in logic generally. He was perhaps the earliest to realize fully and correctly the important influence of a calculus [some mindless method of calculation] on discovery. ...There is a frivolous objection... to the effect that such economy of thought is an attempt to substitute unthinking mechanism for living thought. This contention fails... through the simple fact that this economy is only used in certain circumstances. In no science do we try to make subject to a mechanical calculus any trains of reasoning except such that have not been the object of careful thought many times previously. ...this reasoning has been universally recognized as valid, and we do not wish to waste energy of thought in repeating it when so much remains to be discovered by means of this energy. Since the time of Leibniz, this truth has been recognized, explicitly or implicitly, by all the greatest mathematical analysts."

"When one compares the talents one has with those of a Leibniz, one is tempted to throw away one's books and go die quietly in the dark of some forgotten corner."

"Perhaps never has a man read as much, studied as much, meditated more, and written more than Leibniz… What he has composed on the world, God, nature, and the soul is of the most sublime eloquence. If his ideas had been expressed with the flair of Plato, the philosopher of Leipzig would cede nothing to the philosopher of Athens."

"The German idealist philosophical tradition from which Hayek emerged is usually held to begin with Gottfried Leibniz, who wrote mostly during the second half of the seventeenth century. Leibniz put forward the idea of “monads,” a starkly idealist conception. Essentially, “each monad is a soul,” in the words of Bertrand Russell. Leibniz reversed the traditional conception of mind and matter by applying attributes of matter (in terms of sensory experience) to mind. Mind is what it experiences. Every mind or soul becomes an independent attribute of the universe, divinely ordered or arranged. Leibniz’s focus truly was mind. […] Leibniz was born at the end of the Thirty Years’ War. Religious struggles, such as the Thirty Years’ War, are often protracted and intense because they concern fundamental individual beliefs and values to which compromise is not always applicable. Chaos and disorder reigned in the larger society from which Leibniz emerged. It is not surprising that his philosophy moved in the direction of mind from a strictly sociological perspective, for the world was too hard to bear."

"In the History of Mathematics it is generally stated that the higher analysis took its rise in the method of indivisibles of Cavalieri (1635). This assertion... is erroneous. ...Leibniz was led to his invention of the algorithm of the higher analysis by a study of the writings of Pascal, more than by anything else."

"As Leibniz was fond of saying, it is one and the same to be a thing and to be a thing. In other words the “really real” is free from otherness, because what we could ascribe to it as other than what it is would actually be “another being.” For the same reason, being as such is free from change. In a doctrine where to be is to be the same, otherness is the very negation of being. Thus in virtue of its self-identity, which forbids it to change unless indeed it ceased to be, true being is immutable in its own right. This permanency in self-identity is the chief mark of the “really real,” that is, of being."

"As to Leibnitz, he is certainly a good philosopher, but in his Theodicée he goes too far and would have all actions necessary. His foreordained harmony is not the least credible nor feasible. If you can get a book entitled: An Essay on the Origin of Evil, by Dr W. King, you will find a much better solution of the question: 'whence comes evil?' Leibnitz does indeed reconcile it all with the goodness of God, but not so reasonably as Dr King."

"Of all the works of Leibnitz, the "Theodicee" is the one most spoken of in Germany. Yet it is his feeblest production. This book, like several other writings in which Leibnitz expresses his religious sentiments, has obtained for its author an evil reputation, and has caused him to be cruelly misunderstood. His enemies have accused him of maudlin sentimentality and weakness of intellect; his friends, in defending, have proved him an accomplished hypocrite. The character of Leibnitz was for long a subject of controversy amongst us : the most partial critics could not absolve him from the accusation of duplicity; his most eager detractors were the freethinkers and the men of enlightenment. How could they pardon in a philosopher defence of the Trinity, eternal punishment, and the divinity of Christ! Their tolerance did not extend so far as that. But Leibnitz was neither fool nor knave, and by the lofty harmony of his intellect he was well able to defend Christianity in its integrity. I say, in its integrity, for he defended it against semi-Christianity. He established the consistency of the orthodox as opposed to the inconsistency of their adversaries. More than this he never attempted. He thus stood at that point of indifference where diverse systems appear as merely different sides of the same truth."

"When the Eleatic School denied the possibility of motion, Diogenes, as everybody knows, stepped forth as an opponent. He stepped forth literally, for he said not a word, but merely walked several times back and forth, thinking that thereby he had sufficiently refuted those philosophers. Inasmuch as for a long time I have been engaged, at least occasionally, with the problem whether a repetition is possible and what significance it has, whether a thing gains or loses by being repeated, it suddenly occurred to me, "Thou canst take a trip to Berlin, where thou hast been before, and convince thyself now whether a repetition is possible and what significance it may have." At home I had almost been brought to a standstill by the problem. Say what one will, it is sure to play a very important role in modem philosophy; for repetition is a decisive expression for what "recollection" was for the Greeks. Just as they taught that all knowledge is a recollection, so will modem philosophy teach that the whole of life is a repetition. The only modem philosopher who had an inkling of this was Leibnitz ."

"Plus un, moins un, plus un, moins un, etc. En ajoutant les deux premiers termes, les deux suivans, et ainsi du reste, on transforme la suite dans une autre dont chaque terme est zéro. Grandi, jésuite italien, en avait conclu la possibilité de la création; parce que la suite étant toujours égale à ½, il voyait cette fraction naìtre d'une infinité de zéros, ou du néant. Ce fut ainsi que Leibnitz crut voir l'image de la création, dans son arithmétique binaire ou il n'employait que les deux caractères zéro et l'unité. Il imagina que l'unité pouvait représenter Dieu, et zéro, lé néant; et que l'Être Suprême avait tiré du néant, tous les êtres; comme l'unité avec le zéro, exprime tous les nombres dans ce système. Cette idée plut tellement à Leibnitz, qu'il en fit part au jésuite Grimaldi, président du tribunal des mathématiques à la Chine, dans l'espérance que cet emblème de la création convertirait au christianisme, l'empereur d'alors qui aimait particulièrement le sciences. Je ne rapporte ce trait, que pour montrer jusqu'à quel point les préjugés de l'enfance peuvent égarer les plus grands hommes."

"In letters which went between me and that most excellent geometer. G.G. Leibniz, ten years ago, when I signified that I was in the knowledge of a method of determining maxima and minima, of drawing tangents, and the like, and when I concealed it in transposed letters involving this sentence (Data æquatione, etc., above cited) that most distinguished man wrote back that he had also fallen upon a method of the same kind, and communicated his method, which hardly differed from mine, except in his forms of words and symbols."

"Substances do not interact. Every substance is eternal. Bodies are phenomena, not independently real. Choices are determined but free. This is the best possible world. I first encountered Leibniz in an introduction to Modern Philosophy and the image of him as a philosopher so enthralled with his reasoning as to deny the reality in front of him stuck with me for a long time. It wasn't that his arguments were bad, but that their conclusions seemed obviously false. Wouldn't a swift kick in the shin suffice to prove that substances do interact, that bodies are real, and perhaps even that this is not the best possible world? This image of Leibniz as naive and detached from reality was cemented by Voltaire's satirical character Dr Pangloss, who insists over and over again - in the face of the worst suffering and injustice - that this is the best possible world. There is some irony in this image of Leibniz, as Leibniz was the far opposite of an 'ivory tower' philosopher. He consistently pursued positions that would increase his political influence over positions that would increase his leisure for study and reflection. Leibniz claimed the progress of knowledge as his main goal, but he approached this goal from two sides, on one side through his own research and writing while on the other side promoting institutions that would better support, disseminate, and apply knowledge. Today, Leibniz is best known or at least most widely read for his philosophical writings, but philosophy represents only a small part of his life's work. Although this book will focus on explaining Leibniz's philosophy, that philosophy must be approached from within the broader context of his life and time."

"As an interpreter of nature... Leibnitz stands in no comparison with Newton. His general views in physics were vague and unsatisfactory; he had no great value for inductive reasoning; it was not the way of arriving at truth which he was accustomed to take; and hence, to the greatest physical discovery of that age, and that which was established by the most ample induction, the existence of gravity as a fact in which all bodies agree, he was always incredulous, because no proof of it, a priori could be given."

"The principle premisses of Leibniz's philosophy appear to me to be five... : I. Every proposition has a subject and a predicate. II. A subject may have predicates which are qualities existing at various times. (Such a subject is called a substance.) III. True propositions not asserting existence at particular times are necessary and analytic, but such as assert existence at particular times are contingent and synthetic. The latter depend upon final causes. IV. The Ego is a substance. V. Perception yields knowledge of an external world, ie. of existents other than myself and my states. The fundamental objection to Leibniz's philosophy will be found to be the inconsistency of the first premiss with the fourth and fifth; and in this inconsistency we shall find a general objection to Monadism."

"[We] will discuss Soul and Body, the doctrine of God, and Ethics. ...We shall find that Leibniz no longer shows great originality, but tends, with slight alterations of phraseology, to adopt (without acknowledgment) the views of the decried Spinoza. We shall find also many more minor inconsistencies than in the earlier part of [Leibniz's] system, these being due chiefly to the desire to avoid the impieties of the Jewish Atheist, and the still greater impieties to which Leibniz's own logic should have led him."

"Plato, in the Theaetetus, had set to work to refute the identification of knowledge with perception, and from his time onwards almost all philosophers, down to and including Descartes and Leibniz, had taught that much of our most valuable knowledge is not derived from experience."

"In Locke's own day, his chief philosophical opponents were the Cartesians and Leibniz. ...Until the publication of Kant's Critique of Pure Reason in 1781, it might have seemed as if the older philosophical tradition of Descartes, Spinoza, and Leibniz were being definitely overcome by the newer empirical method. The newer method, however, had never prevailed in German universities, and after 1792 it was held responsible for the horrors of the Revolution."

"In Leibniz, a vast edifice of deduction is pyramided upon a pin-point of logical principle. In Leibniz, if the principle is completely true and the deductions are entirely valid, all is well; but the structure in unstable, and the slightest flaw anywhere brings it down in ruins."

"Long before most of these facts were discovered, Leibnitz had conjectured that originally the earth in general, even in the north, enjoyed a much warmer temperature than in the present period of all-ruling and progressive frost; and Buffon and others have established on this idea their hypothesis of a vast central fire in the interior of the earth. The interior parts of the earth and its internal depths are a region totally impervious to the eye of mortal man, and can least of all be approached by those ordinary paths of hypothesis adopted by naturalists and geologists."

"Leibniz's apparent corelessness stands for a fundamental philosophical problem, a quandary that reaches to the foundations of his system of philosophy. In the metaphysics he... presented to the world, Leibniz claimed that the one thing of which we can all be certain is the unity, permanence, immateriality, and absolute immunity to outside influence of the individual mind. In identifying the mind as a "monad"—the Greek word for "unity"—he positioned himself in direct opposition to Spinoza, whose allegedly materialist philosophy of mind he adamantly rejected. And yet the philosopher who made the unity of the individual the fundamental principle of the universe was himself incomparably fragmented, multiplicitous, exposed to the influence of others, and impossible to pin down. How could a monad be so multifarious, not to say nefarious?"

"Leibniz's work lacked the depth and virtuosity of Newton's, but then Leibniz was a librarian, a philosopher, and a diplomat with only a part-time interest in mathematics."

"The lack of early practice in mathematics left its mark on Leibniz's later mathematical style, in which good ideas are sometimes inefficiently developed through lack of technical skill. Often he seemed to lack not only the technique but also the patience to develop the ideas conceived by his wide-ranging imagination."

"Leibnitz, though not propounding any full doctrine on evolution, gave it an impulse by suggesting a view contrary to the sacrosanct belief in the immutability of species... His punishment at the hands of the Church came a few years later, when in 1712, the Jesuits defeated his attempt to found an Academy of Science at Vienna."

"Leibnitz, dominated by ideas of communication, is, in more than one way, the intellectual ancestor of the ideas of this book, for he was also interested in machine computation and in automata. My views in this book are very far from being Leibnitzian, but the problems with which I am concerned are most certainly Leibnitzian. Leibnitz's computing machines were only an offshoot of his interest in a computing language, a reasoning calculus which again was in his mind, merely an extention of his idea of a complete artificial language. Thus, even in his computing machine, Leibnitz's preoccupations were mostly linguistic and communicational."

"But in our opinion truths of this kind should be drawn from notions rather than from notations."

"The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length. … Further, the dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated."

"The enchanting charms of this sublime science reveal themselves in all their beauty only to those who have the courage to go deeply into it. But when a person of that sex, that, because of our mores and our prejudices, has to encounter infinitely more obstacles and difficulties than men in familiarizing herself with these thorny research problems, nevertheless succeeds in surmounting these obstacles and penetrating their most obscure parts, she must without doubt have the noblest courage, quite extraordinary talents and superior genius."

"It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment. [Wahrlich es ist nicht das Wissen, sondern das Lernen, nicht das Besitzen sondern das Erwerben, nicht das Da-Seyn, sondern das Hinkommen, was den grössten Genuss gewährt.] When I have clarified and exhausted a subject, then I turn away from it, in order to go into darkness again. The never-satisfied man is so strange; if he has completed a structure, then it is not in order to dwell in it peacefully, but in order to begin another. I imagine the world conqueror must feel thus, who, after one kingdom is scarcely conquered, stretches out his arms for others."

"In researches in which an infinity of directions of straight lines in space is concerned, it is advantageous to represent these directions by means of those points upon a fixed sphere, which are the end points of the radii drawn parallel to the lines. The centre and the radius of this auxiliary sphere are here quite arbitrary. The radius may be taken equal to unity. This procedure agrees fundamentally with that which is constantly employed in astronomy, where all directions are referred to a fictitious celestial sphere of infinite radius. Spherical trigonometry and certain other theorems, to which the author has added a new one of frequent application, then serve for the solution of the problems which the comparison of the various directions involved can present."

"Less depends upon the choice of words than upon this, that their introduction shall be justified by pregnant theorems."

"Arc, amplitude, and curvature sustain a similar relation to each other as time, motion, and velocity, or as volume, mass, and density."

"I mean the word proof not in the sense of the lawyers, who set two half proofs equal to a whole one, but in the sense of a mathematician, where ½ proof = 0, and it is demanded for proof that every doubt becomes impossible."

"We must admit with humility that, while number is purely a product of our minds, space has a reality outside our minds, so that we cannot completely prescribe its properties a priori."

"To praise it would amount to praising myself. For the entire content of the work … coincides almost exactly with my own meditations which have occupied my mind for the past thirty or thirty-five years."

"I will add that I have recently received from Hungary a little paper on non-Euclidean geometry in which I rediscover all my own ideas and results worked out with great elegance... The writer is a very young Austrian officer, the son of one of my early friends, with whom I often discussed the subject in 1798, although my ideas were at that time far removed from the development and maturity which they have received through the original reflections of this young man. I consider the young geometer J. Bolyai a genius of the first rank."

"Mathematics is the queen of the sciences."

"The function just found cannot, it is true, express rigorously the probabilities of the errors: for since the possible errors are in all cases confined within certain limits, the probability of errors exceeding those limits ought always to be zero, while our formula always gives some value. However, this defect, which every analytical function must, from its nature, labor under, is of no importance in practice, because the value of our function decreases so rapidly... that it can safely be considered as vanishing. Besides, the nature of the subject never admits of assigning with absolute rigor the limits of error."

"There is in this world a joy of the intellect, which finds satisfaction in science, and a joy of the heart, which manifests itself above all in the aid men give one another against the troubles and trials of life. But for the Supreme Being to have created existences, and stationed them in various spheres in order to taste these joys for some 80 or 90 years — that were surely a miserable plan.... Whether the soul were to live for 80 years or for 80 million years, if it were doomed in the end to perish, such an existence would only be a respite. In the end it would drop out of being. We are thus impelled to the conclusion to which so many things point, although they do not amount to a coercive scientific proof, that besides this material world there exists another purely spiritual order of things, with activities as various, as the present, and that this world of spirit we shall one day inherit."

""It is beyond doubt that the happiness which love can bestow on its chosen souls is the highest that can fall to mortal's lot. But when I imagine myself in the place of the man who, after twenty happy years, now in one moment loses his all, I am moved almost to say that he is the wretchedest of mortals, and that it is better never to have known such happy days. So it is on this miserable earth: 'the purest joy finds its grave in the abyss of time'. What are we without the hope of a better future?"

"May the dream which we call life be for you a happy dream, a foretaste of that true life which we shall inherit in our real home, when the awakened spirit shall labour no longer under the grievous bondage of the flesh, the fetters of space, the whips of earthly pain, and the sting of our paltry needs and desires. Let us carry our burdens to the end, stoutly and uncomplainingly, never losing sight of that higher goal. Glad then shall we be to lay down our weary lives, and to see the dropping of the curtain.""

"Believe me,... the bitterness of life, or at least of mine, which runs through it like a strand of red, and becomes less and less endurable as I grow older, is not compensated in the hundredth part by the joy of life. I will freely admit that these burdens, which to me have been so grievous, would have been lighter to many another; but our temperament is part of ourselves, given to us by the Creator with our very existence, and we have very little power to change it. I find, on the other hand, in this very consciousness of the vanity of life, which nearly all men must confess to as they draw near the end, my strongest assurance of the approach of a more beautiful metamorphosis. In this, my dear friend, let us find comfort, and endeavour to call up calmness to bear life out to the end."

"The perturbations which the motions of planets suffer from the influence of other planets, are so small and so slow that they only become sensible after a long interval of time; within a shorter time, or even within one or several revolutions, according to circumstances, the motion would differ so little from motion exactly described, according to the laws of Kepler, in a perfect ellipse, that observations cannot show the difference. As long as this is true, it not be worth while to undertake prematurely the computation of the perturbations, but it will be sufficient to adapt to the observations what we may call an osculating conic section: but, afterwards, when the planet has been observed for a longer time, the effect of the perturbations will show itself in such a manner, that it will no longer be possible to satisfy exactly all the observations by a purely elliptic motion; then, accordingly, a complete and permanent agreement cannot be obtained, unless the perturbations are properly connected with the elliptic motion."

"The history of the apple is too absurd. Whether the apple fell or not, how can any one believe that such a discovery could in that way be accelerated or retarded? Undoubtedly, the occurrence was something of this sort. There comes to Newton a stupid, importunate man, who asks him how he hit upon his great discovery. When Newton had convinced himself what a noodle he had to do with, and wanted to get rid of the man, he told him that an apple fell on his nose; and this made the matter quite clear to the man, and he went away satisfied."

"The principle that the sum of the squares of the differences between the observed and computed quantities must be a minimum may, in the following manner, be considered independently of the calculus of probabilities. When the number of unknown quantities is equal to the number of the observed quantities depending on them, the former may be so determined as exactly to satisfy the latter. But when the number of the former is less than that of the latter, an absolutely exact agreement cannot be obtained, unless the observations possess absolute accuracy. In this case care must be taken to establish the best possible agreement, or to diminish as far as practicable the differences. This idea, however, from its nature, involves something vague. For, although a system of values for the unknown quantities which makes all the differences respectively less than another system, is without doubt to be preferred to the latter, still the choice between two systems, one of which presents a better agreement in some observations, the other in others, is left in a measure to our judgment, and innumerable different principles can be proposed by which the former condition is satisfied. Denoting the differences between observation and calculation by A, A’, A’’, etc., the first condition will be satisfied not only if AA + A’ A’ + A’’ A’’ + etc., is a minimum (which is our principle) but also if A4 + A’4 + A’’4 + etc., or A6 + A’6 + A’’6 + etc., or in general, if the sum of any of the powers with an even exponent becomes a minimum. But of all these principles ours is the most simple; by the others we should be led into the most complicated calculations."

"It may be true, that men, who are mere mathematicians, have certain specific shortcomings, but that is not the fault of mathematics, for it is equally true of every other exclusive occupation. So there are mere philologists, mere jurists, mere soldiers, mere merchants, etc. To such idle talk it might further be added: that whenever a certain exclusive occupation is coupled with specific shortcomings, it is likewise almost certainly divorced from certain other shortcomings."

"I have had my results for a long time: but I do not yet know how I am to arrive at them."

"If others would but reflect on mathematical truths as deeply and as continuously as I have, they would make my discoveries."

"I confess that Fermat's Theorem as an isolated proposition has very little interest for me, because I could easily lay down a multitude of such propositions, which one could neither prove nor dispose of."

"There are problems to whose solution I would attach an infinitely greater importance than to those of mathematics, for example touching ethics, or our relation to God, or concerning our destiny and our future; but their solution lies wholly beyond us and completely outside the province of science."

"Finally, a few days ago, it has been achieved - but not by my cumbersome search, rather through God’s good grace, I am tempted to say. As the lightning strikes the riddle was solved; I myself would be unable to point to a guiding thread between what I knew before, what I had used in my last attempts, and what made it work."

"I believe you are more believing in the Bible than I. I am not, and, you are much happier than I. I must say that so often in earlier times when I saw people of the lower classes, simple manual laborers who could believe so rightly with their hearts, I always envied them, and now tell me how does one begin this?"

"I scarcely believe that in psychology data are present which can be mathematically evaluated. But one cannot know this with certainty, without having made the experiment. God alone is in possession of the mathematical bases of psychic phenomena."

"You say that faith is a gift; this is perhaps the most correct thing that can be said about it."

"Yes! The world would be nonsense, the whole creation an absurdity without immortality."

"All the measurements in the world do not balance one theorem by which the science of eternal truths is actually advanced."

"Even though much error and hypocrisy may often be mixed in such pietistic tendencies, nevertheless I recognize with all my heart the business of a missionary as a highly honorable one in so far as it leads to civilization the still semisavage part of earth s inhabitants. May my son try it for several years."

"One is forced to the view, for which there is so much evidence even though without rigorous scientific basis, that besides this material world another, second, purely spiritual world order exists, with just as many diversities as that in which we live-—we are to participate in it."

"One day he said: For the soul there is a satisfaction of a higher type; the material is not at all necessary. Whether I apply mathematics to a couple of clods of dirt, which we call planets, or to purely arithmetical problems, it's just the same; the latter have only a higher charm for me."

"A great part of its theories derives an additional charm from the peculiarity that important propositions, with the impress of simplicity on them, are often easily discovered by induction, and yet are of so profound a character that we cannot find the demonstrations till after many vain attempts; and even then, when we do succeed, it is often by some tedious and artificial process, while the simple methods may long remain concealed."

"I am coming more and more to the conviction that the necessity of our geometry cannot be demonstrated, at least neither by, nor for, the human intellect. . . Geometry should be ranked, not with arithmetic, which is purely aprioristic, but with mechanics."

"You know that I write slowly. This is chiefly because I am never satisfied until I have said as much as possible in a few words, and writing briefly takes far more time than writing at length."

"In general the position as regards all such new calculi is this - That one cannot accomplish by them anything that could not be accomplished without them. However, the advantage is, that, provided such a calculus corresponds to the inmost nature of frequent needs, anyone who masters it thoroughly is able - without the unconscious inspiration of genius which no one can command - to solve the respective problems, yea to solve them mechanically in complicated cases in which, without such aid, even genius becomes powerless. Such is the case with the invention of general algebra, with the differential calculus, and in a more limited region with Lagrange's calculus of variations, with my calculus of congruences, and with Mobius's calculus. Such conceptions unite, as it were, into an organic whole countless problems which otherwise would remain isolated and require for their separate solution more or less application of inventive genius."

"The austere sides of life, at least of mine, which move through it like a red thread, and which one faces more and more defenselessly in old age, are not balanced to the hundredth part by the pleasurable. I will gladly admit that the same fates which have been so hard for me to bear, and still are, would have been much easier for many another person, but the mental constitution belongs to our ego, which the Creator of our existence has given us, and we can change little in it."

"I am almost amazed that you consider a professional philosopher capable of no confusion in concepts and definitions. Such things are nowhere more at home than among philosophers who are not mathematicians, and Wolff was no mathematician, even though he made cheap compen- diums. Look around among the philosophers of today, among Schelling, Hegel, Nees von Esenbeck, and their like; doesn t your hair stand on end at their definitions? Read in the history of ancient philosophy what kinds of definitions the men of that day, Plato and others, gave (I except Aristotle). But even in Kant it is often not much better; in my opinion his distinction between analytic and synthetic theorems is such a one that either peters out in a triviality or is false."

"One cannot reduce to concepts the distinction between two systems of three straight lines each (directed lines, of which the one system points forward, upward to the right, the other forward, upward to the left) but one can only demonstrate by holding to actually present spatial things. Two minds cannot reach agreement about it unless their views connect up with one and the same system present in the real world"

"Dark are the paths which a higher hand allows us to traverse here... let us hold fast to the faith that a finer, more sublime solution of the enigmas of earthly life will be present, will become part of us."

"In such apparent accidents which finally produce such a decisive influence on one s whole life, one is inclined to recognize the tools of a higher hand. The great enigma of life never becomes clear to us here below."

"If the object of all human investigation were but to produce in cognition a reflection of the world as it exists, of what value would be all its labor and pains, which could result only in vain repetition, in an imitation within the soul of that which exists without it?"

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

"That this subject [of ] has hitherto been considered from the wrong point of view and surrounded by a mysterious obscurity, is to be attributed largely to an ill-adapted notation. If for instance, +1, -1, √-1 had been called direct, inverse, and lateral units, instead of positive, negative, and imaginary (or even impossible) such an obscurity would have been out of question."

"Gauss's aim was always to give his investigations the form of perfect works of art. He would not rest sooner and never gave a piece of work to the public until he had given it the perfection of form he desired for it. A good building should not show its scaffolding when completed, he used to say. In his demonstrations he used almost entirely the synthetic method, which he had come to prize through his studies of Archimedes and Newton. It is distinguished from the analytic method by its brevity and comprehensiveness. But the road leading to the discovery remains veiled; and indeed it often seems that Gauss frequently and intentionally turned aside from the road that led to mere instruction."

"Abraham Gotthelf Kästner is] the best mathematician among poets and the best poet among mathematicians"

"Pauca sed matura ["few, but ripe"]"

"Procreare jucundum, sed parturire molestum ["to beget is pleasant, but to give birth is painful"]"

"Bei Gegenstdnden mit denen ich mich noch nicht lange beschaftigt habe, bin ich gegen meine eigenen Ansichten, zumal wenn sie einem Laplace widersprechen, misstrauisch und nehme gern die von anderen entgegen. ["I am suspicious of my own views on subjects with which I have not long occupied myself, and gladly accept those of others, especially when my views contradict one of Laplace."]"

"Some of the discoveries of Abel and Jacobi were anticipated by Gauss. In the Disquisitiones Arithmeticæ he observed that the principles which he used in the division of the circle were applicable to many other functions, besides the circular, and particularly to the transcendents dependent on the integral \displaystyle \int \frac{\,dx}{\sqrt{1-x^4}}. From this Jacobi concluded that Gauss had thirty years earlier considered the nature and properties of elliptic functions and had discovered their double periodicity. The papers in the collected works of Gauss confirm this conclusion."

"He is like the fox, who effaces his tracks in the sand with his tail."

"Everything Gauss writes is abomination, as it is so obscure that it is almost impossible to understand it."

"Not only could nobody but Gauss have produced it, but it would never have occurred to anyone but Gauss that such a formula was possible."

"There is no doubt... that mathematicians are generally overzealous about conciseness, and in their passion for brevity indulge in symbols even where these seem no better than a familiar English word or phrase. A faulty judgement has caused mathematicians to equate elegance and conciseness at the cost of intelligibility. Gauss himself wrote elegant, but highly compact, carefully polished papers with no hint of the motivation, meaning, or details of the steps. When criticized, he said that no architect leaves the scaffolding after completing the building. But the fact is that even excellent mathematicians found the reading of Gauss's papers very difficult, and the same is true of many other mathematicians."

"Toward the ends of their lives, Euler, D'Alembert, and Lagrange agreed that the realm of mathematical ideas had been practically exhausted and that no new great minds were appearing on the mathematical horizons. Of course, these men had grown old and their vision was already dimmed, for Laplace, Legendre, and Fourier were in young manhood. In one respect, however, these elder statesmen were correct... their immediate successors continued to explore and polish the very same ideas which the mid-eighteenth century had pursued. But history shows that the human mind is fertile, ingenious, and creative beyond all possible anticipations. ...even the richest vein of thought is ultimately exhausted, and then, indeed, a period of stagnation may ensue. Inevitably, however, there arise new conceptions and new periods of feverish and rewarding research. Euler and his contemporaries failed to reckon with history. ... The man who was to change the course of mathematics was but six years old when Euler and D'Alembert died in 1783... Gauss is commonly ranked with Archimedes and Newton. ...all three of these men were as much devoted to physical research as to mathematics."

"On demandait à Laplace quel était selon lui le plus grand mathématicien de l'Allemagne. C'est Pfaff, répondit-il. - Je croyais, reprit l'interlocuteur, que Gauss lui était supérieur. - Mais, s'écria Laplace, vous me demandez quel est le plus grand mathématicien de l'Allemagne, et Gauss est le plus grand mathématicien de l'Europe."

"It is to Gauss, to the Magnetic Union, and to magnetic observers in general, that we owe our deliverance from that absurd method of estimating forces by a variable standard which prevailed so long even among men of science. It was Gauss who first based the practical measurement of magnetic force (and therefore of every other force) on those long established principles, which, though they are embodied in every dynamical equation, have been so generally set aside, that these very equations, though correctly given... are usually explained... by assuming, in addition to the variable standard of force, a variable, and therefore illegal, standard of mass."

"If explaining minds seems harder than explaining songs, we should remember that sometimes enlarging problems makes them simpler! The theory of the roots of equations seemed hard for centuries within its little world of real numbers, but it suddenly seemed simple once Gauss exposed the larger world of so-called complex numbers. Similarly, music should make more sense once seen through listeners' minds."

"According to his frequently expressed view, Gauss considered the three dimensions of space as specific peculiarities of the human soul; people, which are unable to comprehend this, he designated in his humorous mood by the name Bœotians. We could imagine ourselves, he said, as beings which are conscious of but two dimensions; higher beings might look at us in a like manner, and continuing jokingly, he said that he had laid aside certain problems which, when in a higher state of being, he hoped to investigate geometrically."

"Gauss liked to call [number theory] 'the Queen of Mathematics'. For Gauss, the jewels in the crown were the primes, numbers which had fascinated and teased generations of mathematicians."

"Armed with his prime number tables, Gauss began his quest. As he looked at the proportion of numbers that were prime, he found that when he counted higher and higher a pattern started to emerge. Despite the randomness of these numbers, a stunning regularity seemed to be looming out of the mist."

"The revelation that the graph appears to climb so smoothly, even though the primes themselves are so unpredictable, is one of the most miraculous in mathematics and represents one of the high points in the story of the primes. On the back page of his book of logarithms, Gauss recorded the discovery of his formula for the number of primes up to N in terms of the logarithm function. Yet despite the importance of the discovery, Gauss told no one what he had found. The most the world heard of his revelation were the cryptic words, 'You have no idea how much poetry there is in a table of logarithms.'"

"Maybe we have become so hung up on looking at the primes from Gauss's and Riemann's perspective that what we are missing is simply a different way to understand these enigmatic numbers. Gauss gave an estimate for the number of primes, Riemann predicted that the guess is at worst the square root of N off its mark, Littlewood showed that you can't do better than this. Maybe there is an alternative viewpoint that no one has found because we have become so culturally attached to the house that Gauss built."

"As a boy of six I could understand the proof of a mathematical theorem more readily than that meat had to be cut with one's knife, not one's fork."

"What attracted me so strongly and exclusively to mathematics, apart from the actual content, was particularly the specific nature of the mental processes by which mathematical concepts are handled. This way of deducing and discovering new truths from old ones, and the extraordinary clarity and self-evidence of the theorems, the ingeniousness of the ideas... had an irresistible fascination for me. Beginning from the individual theorems, I grew accustomed to delve more deeply into their relationships and to grasp whole theories as a single entity. That is how I conceived the idea of mathematical beauty..."

"There have been only three epoch-making mathematicians: Archimedes, Newton, and Eisenstein."

"As any reader of Eisenstein must realise, he felt hard pressed for time during the whole of his short mathematical career... His papers, although brilliantly conceived, must have been written by fits and starts, with the details worked out only as the occasion arose; sometimes a development is cut short, only to be taken up again at a later stage."

"As Eisenstein shows, his method for constructing elliptic functions applies beautifully to the simpler case of trigonometric functions. Moreover, this case provides not merely an illuminating introduction to his theory, but also the simplest proofs for a series of results, originally discussed by Euler."

"Looking back from today's vantage, Eisenstein's mathematics appear to us more up to date than ever. It is not so much the harvest of theorems, nor the creation of full-fledged theories, but the way of looking at things which amazes us..."

"It is true that M. Fourier had the opinion that the principal end of mathematics was the public utility and the explanation of natural phenomena; but such a philosopher as he is should have known that the unique end of science is the honor of the human mind, and that from this point of view a question of number is as important as a question of the system of the world."

"Any progress in the theory of partial differential equations must also bring about a progress in Mechanics."

"Wherever Mathematics is mixed up with anything, which is outside its field, you will find attempts to demonstrate these merely conventional propositions a priori, and it will be your task to find out the false deduction in each case."

"History knew a midnight, which we may estimate at about the year 1000 A.D., when the human race lost the arts and sciences even to the memory. The last twilight of paganism was gone, and the new day had not yet begun. Whatever was left of culture in the world was found only in the Saracens, and a Pope eager to learn studied in disguise in their universities, and so became the wonder of the West. At last Christendom, tired of praying to the dead bones of the martyrs, flocked to the tomb of the Saviour Himself, only to find for a second time that the grave was empty and that Christ was risen from the dead. Then mankind too rose from the dead. It returned to the activities and the business of life; there was a feverish revival in the arts and in the crafts. The cities flourished, a new citizenry was founded. Cimabue rediscovered the extinct art of painting; Dante, that of poetry. Then it was, also, that great courageous spirits like Abelard and Saint Thomas Aquinas dared to introduce into Catholicism the concepts of Aristotelian logic, and thus founded scholastic philosophy. But when the Church took the sciences under her wing, she demanded that the forms in which they moved be subjected to the same unconditioned faith in authority as were her own laws. And so it happened that scholasticism, far from freeing the human spirit, enchained it for many centuries to come, until the very possibility of free scientific research came to be doubted. At last, however, here too daylight broke, and mankind, reassured, determined to take advantage of its gifts and to create a knowledge of nature based on independent thought. The dawn of the day in history is know as the Renaissance or the Revival of Learning."

"His [Lagrange's] lectures on differential calculus form the basis of his Theorie des fonctions analytiques which was published in 1797. ...its object is to substitute for the differential calculus a group of theorems based upon the development of algebraic functions in series. A somewhat similar method had been previously used by John Landen in his Residual Analysis... Lagrange believed that he could... get rid of those difficulties, connected with the use of infinitely large and infinitely small quantities, to which philosophers objected in the usual treatment of the differential calculus. ...Another treatise in the same lines was his Leçons sur le calcul des fonctions, issued in 1804. These works may be considered as the starting-point for the researches of Cauchy, Jacobi, and Weierstrass."

"Jacobi's most celebrated investigations are those on elliptic functions, the modern notation in which is substantially due to him, and the theory of which he established simultaneously with Abel, but independently of him. Jacobi's results are given in his treatise on elliptic functions, published in 1829, and in some later papers in Crelle's Journal; they are earlier than Weierstrass's researches... The correspondence between Legendre and Jacobi on elliptic functions has been reprinted in the first volume of Jacobi's collected works. Jacobi, like Abel, recognised that elliptic functions were not merely a group of theorems on integration, but that they were types of a new kind of function, namely, one of double periodicity; hence he paid particular attention to the theory of the theta function."

"I ought also to mention his papers on Abelian transcendants; his investigations on the theory of numbers... his important memoirs on the theory of differential equations, both ordinary and partial; his development of the calculus of variations; and his contributions to the problem of three bodies, and other particular dynamical problems. Most of the results of the researches last named are included in his Vorlesungen über Dynamik."

"The most important of Legendre's works is his Functions elliptiques, issued in two volumes in 1825 and 1826. He took up the subject where Euler, Landen, and Lagrange had left it, and for forty years was the only one to cultivate this new branch of analysis, until at last Jacobi and Abel stepped in with admirable new discoveries."

"The theory of determinants was studied by Hoëné Wronski in Italy and J. Binet in France; but they were forestalled by the great master of this subject, Cauchy. In a paper (Jour. de l'ecole Polyt., IX., 16) Cauchy developed several general theorems. He introduced the name determinant a term previously used by Gauss in the functions considered by him. In 1826 Jacobi began using this calculus, and he gave brilliant proof of its power. In 1841 he wrote extended memoirs on determinants in Crelle's Journal, which rendered the theory easily accessible."

"Cauchy made some researches on the calculus of variations. This subject is now in its essential principles the same as when it came from the hands of Lagrange. Recent studies pertain to the variation of a double integral when the limits are also variable, and to variations of multiple integrals in general. ...In 1837 Jacobi published a memoir, showing that the difficult integrations demanded by the discussion of the second variation, by which the existence of a maximum or minimum can be ascertained, are included in the integrations of the first variation, and thus are superfluous. This important theorem, presented with great brevity by Jacobi, was elucidated and extended by V. A. Lebesgue, C. E. Delaunay, Eisenlohr, S. Spitzer, Hesse, and Clebsch. ...In 1852 G. Mainardi attempted to exhibit a new method of discriminating maxima and minima, and extended Jacobi's theorem to double integrals. Mainardi and F. Brioschi showed the value of determinants in exhibiting the terms of the second variation."

"Gauss' researches on the theory of numbers were the starting-point for a school of writers, among the earliest of whom was Jacobi. The latter contributed to Crelle's Journal an article on cubic residues, giving theorems without proofs. After the publication of Gauss' paper on biquadratic residues, giving the law of biquadratic reciprocity, and his treatment of complex numbers, Jacobi found a similar law for cubic residues. By the theory of elliptical functions, he was led to beautiful theorems on the representation of numbers by 2, 4, 6, and 8 squares."

"The problem of three bodies has been treated in various ways since the time of Lagrange, but no decided advance towards a more complete algebraic solution has been made, and the problem stands substantially where it was left by him. He had made a reduction in the differential equations to the seventh order. This was elegantly accomplished in a different way by Jacobi in 1843."

"Advances in theoretical mechanics, bearing on the integration and the alteration in form of dynamical equations, were made since Lagrange by Poisson, William Rowan Hamilton, Jacobi, Madame Kowalevski, and others. Lagrange had established the "Lagrangian form" of the equations of motion. He had given a theory of the variation of the arbitrary constants which, however, turned out to be less fruitful in results than a theory advanced by Poisson. ...Hamilton's method of integration was freed by Jacobi of an unnecessary complication, and was then applied by him to the determination of a geodetic line on the general ellipsoid. With aid of elliptic coordinates Jacobi integrated the partial differential equation and expressed the equation of the geodetic in form of a relation between two Abelian integrals. Jacobi applied to differential equations of dynamics the theory of the ultimate multiplier. The differential equations of dynamics are only one of the classes of differential equations considered by Jacobi. Dynamic investigations along the lines of Lagrange, Hamilton, and Jacobi were made by Liouville, A. Desboves, Serret, J. C. F. Sturm, Ostrogradsky, J. Bertrand, Donkin, Brioschi, leading up to the development of the theory of a system of canonical integrals."

"Among standard works on mechanics are Jacobi's Vorlesungen über Dynamik, edited by Clebsch, 1866."

"C. J. Jacobi was especially distressed because on several occasions when he came to Gauss to relate some new discoveries the latter pulled out from his desk drawer some papers that contained the very same discoveries. Jacobi resolved to get even. ...Gauss opened his desk drawer and pulled out some papers ...Jacobi then remarked, "It is a pity that you did not publish this work since you have published so may poorer papers.""

"Aside from Cauchy, the greatest contributory to the theory [of determinants] was Carl Gustav Jacob Jacobi. With him the word "determinant" received its final acceptance. He early used the functional determinant which Sylvester has called the Jacobian, and in his famous memoirs in Crelle's Journal for 1841 he considered these forms as well as that class of alternating functions which Sylvester has called alternants."

"In 1829, the year that Abel died, Carl Gustav Jacob Jacobi published his "Fundamenta nova theoriae functionum ellipticarum." Jacobi based his theory of elliptic functions on four functions defined by infinite series and called theta functions. ...The addition theorems of elliptic functions can also be considered as special applications of Abel's theorem on the sum of integrals of algebraic equations. The question now arose whether hyper-elliptic integrals could be inverted in the way elliptic integrals had been inverted to yield elliptic functions. The solution was found by Jacobi in 1832 when he published his result that the inversion could be performed with functions of more than one variable. Thus the theory of Abelian functions of p variables was born, which became an important branch of Nineteenth century mathematics."

"Sylvester has given the name "Jacobian" to the functional determinant in order to pay respect to Jacobi's work on algebra and elimination theory. The best known of Jacobi's papers on this subject is his "De formatione et proprietatibus determinantium" (1841), which made the theory of determinants the common good of the mathematicians."

"The best approach to Jacobi is perhaps through his beautiful lectures on dynamics ("Vorlesungen über Dynamik"), published in 1866 after lecture notes from 1842-43."

"In re mathematica ars proponendi quaestionem pluris facienda est quam solvendi."

"I realize that in this undertaking I place myself in a certain opposition to views widely held concerning the mathematical infinite and to opinions frequently defended on the nature of numbers."

"Had Mittag-Leffler had his way, I should have to wait until the year 1984, which to me seemed too great a demand!"

"The old and oft-repeated proposition "Totum est majus sua parte" [the whole is larger than the part] may be applied without proof only in the case of entities that are based upon whole and part; then and only then is it an undeniable consequence of the concepts "totum" and "pars". Unfortunately, however, this "axiom" is used innumerably often without any basis and in neglect of the necessary distinction between "reality" and "quantity", on the one hand, and "number" and "set", on the other, precisely in the sense in which it is generally false."

"There is no doubt that we cannot do without variable quantities in the sense of the potential infinite. But from this very fact the necessity of the actual infinite can be demonstrated."

"In order for there to be a variable quantity in some mathematical study, the domain of its variability must strictly speaking be known beforehand through a definition. However, this domain cannot itself be something variable, since otherwise each fixed support for the study would collapse. Thus this domain is a definite, actually infinite set of values. Hence each potential infinite, if it is rigorously applicable mathematically, presupposes an actual infinite."

"The potential infinite means nothing other than an undetermined, variable quantity, always remaining finite, which has to assume values that either become smaller than any finite limit no matter how small, or greater than any finite limit no matter how great."

"Er ist aber in Kopenhagen geboren, von israelitischen Eltern, die der dortigen portugisischen Judengemeinde."

"The totality of all alephs cannot be conceived as a determinate, well-defined, and also a finished set. This is the punctum saliens, and I venture to say that this completely certain theorem, provable rigorously from the definition of the totality of all alephs, is the most important and noblest theorem of set theory. One must only understand the expression "finished" correctly. I say of a set that it can be thought of as finished (and call such a set, if it contains infinitely many elements, "transfinite" or "suprafinite") if it is possible without contradiction (as can be done with finite sets) to think of all its elements as existing together, and to think of the set itself as a compounded thing for itself; or (in other words) if it is possible to imagine the set as actually existing with the totality of its elements."

"Every transfinite consistent multiplicity, that is, every transfinite set, must have a definite aleph as its cardinal number."

"I have never proceeded from any Genus supremum of the actual infinite. Quite the contrary, I have rigorously proved that there is absolutely no Genus supremum of the actual infinite. What surpasses all that is finite and transfinite is no Genus; it is the single, completely individual unity in which everything is included, which includes the Absolute, incomprehensible to the human understanding. This is the Actus Purissimus, which by many is called God. I am so in favor of the actual infinite that instead of admitting that Nature abhors it, as is commonly said, I hold that Nature makes frequent use of it everywhere, in order to show more effectively the perfections of its Author. Thus I believe that there is no part of matter which is not — I do not say divisible — but actually divisible; and consequently the least particle ought to be considered as a world full of an infinity of different creatures."

"A set is a Many that allows itself to be thought of as a One."

"The fear of infinity is a form of myopia that destroys the possibility of seeing the actual infinite, even though it in its highest form has created and sustains us, and in its secondary transfinite forms occurs all around us and even inhabits our minds."

"The actual infinite arises in three contexts: first when it is realized in the most complete form, in a fully independent otherworldly being, in Deo, where I call it the Absolute Infinite or simply Absolute; second when it occurs in the contingent, created world; third when the mind grasps it in abstracto as a mathematical magnitude, number or order type."

"That from the outset they expect or even impose all the properties of finite numbers upon the numbers in question, while on the other hand the infinite numbers, if they are to be considered in any form at all, must (in their contrast to the finite numbers) constitute an entirely new kind of number, whose nature is entirely dependent upon the nature of things and is an object of research, but not of our arbitrariness or prejudices."

"This view [of the infinite], which I consider to be the sole correct one, is held by only a few. While possibly I am the very first in history to take this position so explicitly, with all of its logical consequences, I know for sure that I shall not be the last!"

"My theory stands as firm as a rock; every arrow directed against it will return quickly to its archer. How do I know this? Because I have studied it from all sides for many years; because I have examined all objections which have ever been made against the infinite numbers; and above all because I have followed its roots, so to speak, to the first infallible cause of all created things."

"What I assert and believe to have demonstrated in this and earlier works is that following the finite there is a transfinite (which one could also call the supra-finite), that is an unbounded ascending ladder of definite modes, which by their nature are not finite but infinite, but which just like the finite can be determined by well-defined and distinguishable numbers."

"The transfinite numbers are in a certain sense themselves new irrationalities and in fact in my opinion the best method of defining the finite irrational numbers is wholly dissimilar to, and I might even say in principle the same as, my method described above of introducing transfinite numbers. One can say unconditionally: the transfinite numbers stand or fall with the finite irrational numbers; they are like each other in their innermost being; for the former like the latter are definite delimited forms or modifications of the actual infinite."

"I entertain no doubts as to the truths of the transfinites, which I recognized with God’s help and which, in their diversity, I have studied for more than twenty years; every year, and almost every day brings me further in this science."

"As for the mathematical infinite, to the extent that it has found a justified application in science and contributed to its usefulness, it seems to me that it has hitherto appeared principally in the role of a variable quantity, which either grows beyond all bounds or diminishes to any desired minuteness, but always remains finite. I call this the improper infinite [das Uneigentlich-unendliche]."

"Infinity, in its first form (the improper-infinite) presents itself as a variable finite [veranderliches Endliches]; in the other form (which I call the proper infinite [Eigentlich-unendliche]) it appears as a thoroughly determinate [bestimmtes] infinite."

"What I declare and believe to have demonstrated in this work as well as in earlier papers is that following the finite there is a transfinite (transfinitum)--which might also be called supra-finite (suprafinitum), that is, there is an unlimited ascending ladder of modes, which in its nature is not finite but infinite, but which can be determined as can the finite by determinate, well-defined and distinguishable numbers."

"Mathematics, in the development of its ideas, has only to take account of the immanent reality of its concepts and has absolutely no obligation to examine their transient reality."

"Mathematics is in its development entirely free and is only bound in the self-evident respect that its concepts must both be consistent with each other, and also stand in exact relationships, ordered by definitions, to those concepts which have previously been introduced and are already at hand and established. In particular, in the introduction of new numbers, it is only obligated to give definitions of them which will bestow such a determinacy and, in certain circumstances, such a relationship to the other numbers that they can in any given instance be precisely distinguished. As soon as a number satisfies all these conditions, it can and must be regarded in mathematics as existent and real."

"The essence of mathematics lies entirely in its freedom."

"If there is some determinate succession of defined whole real numbers, among which there exists no greatest, on the basis of this second principle of generation a new number is obtained which is regarded as the limit of those numbers, i.e. is defined as the next greater number than all of them."

"[T]here exist no other sets than finite and denumerably infinite sets and continua... [I]n mathematics we can create only finite sequences, further by means of... 'and so on' the order type ω, but only consisting of equal elements... but no other sets. Cantor and his disciples... think they have knowledge of all sorts of further sets; their fundamental principle... comes to about the same as the axiomaticians. ...[T]his principle is unjustified and... we assert that the several paradoxes of the 'Mengenlehre'... have no right to exist... [I]t would have been the duty of Cantorians, immediately to reject a notion which gives rise to contradictions, because it is... not built... mathematically."

"No one shall expel us from the Paradise that Cantor has created."

"In 1874 the German mathematician Georg Cantor made the startling discovery that there are more irrational numbers than rational ones, and more transcendental numbers than algebraic ones. In other words, rather than being oddities, most real numbers are irrational; and among irrational numbers, most are transcendental."

"If we have only to classify a finite number of objects, it is easy to preserve these classifications without change. If the number of objects is indefinite, ...[i.e.,] if we are constantly liable to find new and unforeseen objects springing up, it may happen that the appearance of a new object will oblige us to modify the classification, and it is thus that we are exposed to antinomies. There is no actual infinity. The Cantorians forgot this, and so fell into contradiction. It is true that Cantorism has been useful, but that was when it was applied to a real problem, whose terms were clearly defined, and then it was possible to advance without danger. Like the Cantorians, the logicians have forgotten the fact, and they have met with the same difficulties. ...[B]elief in an actual infinity is essential in the Russellian logic, and this... distinguishes it from the Hilbertian logic. Hilbert takes the... view of extension... to avoid the Cantorian antimonies. Russell takes the... view of comprehension... to regard the infinite as actual. And we have not only infinite classes; when we pass from the genus to the species... the number of conditions is still infinite, for they generally express that the object... is in... relation with all the objects of an infinite class. But all this is ."

"Why was Cantor so vehemently opposed to infinitesimals? In his valuable essay, "The Metaphysics of the Calculus," Abraham Robinson suggests that Cantor already had enough problems trying to defend transfinite numbers. It seems likely that, consciously or otherwise, Cantor deemed it politically wise to go along with orthodox mathematicians on the question of infinitesimals. Cantor's stance might be compared to that of a pro-marijuana Congressional candidate who advocates harsh penalties for the sale or use of heroin."

"After being relegated to an obscure mid-tier university, blocked from leading journals and openly mocked by his peers, including his former mentor, the late 19th century German mathematician found refuge for his groundbreaking work on infinities in, of all places, the Roman Catholic Church … Catholic theologians welcomed Cantor's ideas, which provided a workable way of understanding mathematical infinities, as evidence that humans could grasp the infinite and could also, therefore, have a greater understanding of God, himself infinite. What a welcome relief this must have been to the chronically depressed Cantor! As John D. Barrow writes in The Infinite Book: A Short Guide to the Boundless, Timeless and Endless, Cantor "started to tell his friends that he had not been the inventor of the ideas about infinity that he had published. He was merely a mouthpiece, inspired by God to communicate parts of the mind of God to everyone else.""

"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."

"Wir Mathematiker sind alle ein bißchen meschugge."

"Please don't read the preface for the teacher."

"I will ask of you only the ability to read English and to think logically—no high school mathematics, and certainly no higher mathematics."

"The multiplication table will not occur in this book, not even the theorem,2 \cdot 2 = 4,but I would recommend, as an exercise, that you define2 = 1 + 1, 4 = (((1 + 1) + 1) + 1)and then prove the theorem."

"My book is written, as befits such easy material, in merciless telegram style ("Axiom," "Definition," "Theorem," "Proof," occasionally "Preliminary Remark")... I hope I have written this book in such a way that a normal student can read it in two days. And then (since he already knows the formal rules from school) he may forget its contents."

"In 1933 Landau was dismissed from his [University of Göttingen] chair on the grounds of his race. An important colleague... Ludwig Bieberbach ...wrote the following lines in a treatise on Personality structure and mathematical creativity: "In this way... the ultimate reason behind the courageous rejection which the students at Göttingen University meted out to a great mathematician, Edmund Landau, was that his un-German style in research and teaching had become intolerable to German sensitivities. A people which has seen how alien desires for dominion are gnawing at its identity, how enemies of the people are working to impose their alien ways on it, must reject teachers of a type alien to it." The English mathematician Godfrey H. Hardy... responded to Bierbach... "There are many of us, many English and many Germans, who said things during the (First) War which we scarcely meant and are sorry to remember now. Anxiety for one's own position, dread of falling behind the rising torrent of folly, determination at all costs not to be outdone, may be natural if not particularly heroic excuses. Prof. Bieberbach's reputation excludes such explanation for his utterances; and I find myself driven to the more uncharitable conclusion that he really believes them true.""

"He possessed an enormous capacity for work - up to 12 or more hours a day. ... He worked to completely rigorous rules. We were once 'working' together in Cambridge, and started immediately after breakfast. I presently said: excuse me for a minute or two. 'Two minutes 47 seconds.' ... He was completely non-musical (as were Klein and Hardy). ... When G. H. Hardy wrote after the First World War to the effect that he had not been a fanatical anti-German, and felt confident that Landau would wish to resume former relations, Landau replied: 'As a matter of fact my opinions were much the same as yours, with trivial changes of sign.'"

"Not only the physical but also the intellectual landscape of German-language mathematics in the early 1930s would be impossible to imagine without German-Jewish mathematicians. Indeed, some fields of mathematics were completely transformed by their contributions. Number theory was transformed by Hermann Minkowski and Edmund Landau, algebra by Ernst Steinitz and Emmy Noether, set theory and general topology by Felix Hausdorff, Abraham Fraenkel and several others—to mention but a few examples."

"The principal events... took place in the early months of 1933... By April the Nazis had almost total control of Germany. One of their first decrees, on April 7, was intended to bring about the dismissal of all Jews from the civil service. ...University professors were civil servants ...Of the five professors teaching mathematics at Götingen, three—Edmund Landau, Richard Courant, and Felix Bernstein—were Jewish. A fourth, Hermann Weyl, had a Jewish wife. ...the April decree did not apply to Landau or Courant, since they fell within the Hindenburg exceptions. ...It did not help that Götingen at large was rather strong for Hitler. This was true of both "town" and "gown." ...(That grand house of which Edmund Landau was so proud had been defaced with a painting of the gallows in 1931.) On April 26 the town newspaper... printed an announcement that six professors were being placed on indefinite leave. ...One holdout was Edmund Landau (the only Götingen math professor... who was a member of the town's synagogue). Relying on the integrity of the law, Landau attempted to resume calculus classes in November... but the Science Student's Council... organized a boycott. Uniformed storm troopers prevented Landau's students from entering the lecture hall. With singular courage, Landau asked the Council leader, a 20-year-old student named Oswald Teichmüller, to write out as a letter his reasons... his reasons were ideological. He... felt it improper that German students should be taught by Jews. We are accustomed to think of Nazis activists as thugs, low-lifes, opportunists and failed-artists... which, indeed, most of them were. ...they also included in their ranks some people of the highest intelligence."

"Hilbert's solution of Waring's Problem was ready to be presented in the joint seminar with Minkowski in the middle of January 1909. After Minkowski's death, Hilbert presented his solution to Göttingen Academy on 6 February 1909, dedicating it to the memory of his friend, who had done so much for number theory. From then on he missed Minkowski, but carried on the seminar with Edmund Landau, Minkowski's successor. In looking for a successor to Minkowski, Klein and Hilbert looked for a young mathematician, whose achievements were still ahead of him. This requirement ruled out Adolf Hurwitz, and the final candidates were Oskar Perron and Edmund Landau. The decision was made by Klein, who said: 'Oh, Perron is such a wonderful person. Everybody loves him. Landau is very disagreeable, very difficult to get along with. But we, being such a group as we are, it is better that we have a man who is not easy'. Landau, though a worthy successor with respect to number theory... showed no interest in geometry and even less in applied mathematics, not to speak of mathematical physics. ...Hilbert knew that in executing his plans concerning physics, he could not count on Landau."

"The thorough analysis of even simple problems in arithmetic may require the application of advanced mathematics. A striking example is that of the distribution of prime numbers. The solution of this problem lies in finding a general formula which tells us the number of primes that lie in any given numerical interval. ...Edmond Landau ...wrote two large volumes analyzing this problem without solving it, using the most advanced mathematics known at the time. Even in the elementary aspects of mathematics we are thus dealing with complex topics which make great demands on our mathematical skills."

"The present work is inspired by Edmund Landau's famous book, Handbuch der Lehre von der Verteilung der Primzahlen, where he posed two extremal questions on cosine polynomials and deduced various estimates on the distribution of primes using known estimates of the extremal quantities. Although since then better theoretical results are available for the error term of the prime number formula, Landau's method is still the best in finding explicit bounds. In particular, Rosser and Schonfeld used the method in their work "Approximate formulas for some functions of prime numbers"."

"It is one thing to reflect, as Bieberbach did, for instance on the relative pedagogical merit of different ways to introduce π in a calculus class: geometrically via the circle, or in Landau's way via the zeroes of the cosine, with this function being defined by the power series. And it is quite a different matter to use such reflections as a basis for the forced removal of a distinguished colleague from teaching. Bieberbach's behaviour came all the more as a shock as nothing in his previous biography seemed to prepare one for it..."

"Landau was the son of a well-to-do Berlin gynecologist (who invented the myomectomy operation)... his mother was from the banking family of Jacoby, and Landau grew up in a Jacoby house amid other Berlin banking families. ...Landau married Marrianne Erlich, daughter of Paul Ehrlich... Ehrlich had been a fellow student with Landau's father. Thus Landau grew up a well-connected and well-to-do person... he was also something of a prodigy. Legend has it that at age three, when his mother forgot her umbrella in a carriage, he replied, "It was number 354," and the umbrella was quickly reacquired. ...Landau was also something of a cynical snob."

"Any book that revisits the foundations of analysis has to reckon with the formidable precedent of Edmund Landau's Grundlagen der Analysis (Foundations of Analysis) of 1930. Indeed, the influence of Landau's book is probably the reason that so few books since 1930 have even been attempted to include the construction of the real numbers in an introduction to analysis. On the other hand, Landau's account is virtually the last word in rigor. The only way to be more rigorous would be to rewrite Landau's proofs in computer-checkable form—which has in fact been done recently. On the other hand Landau's book is almost pathologically reader-unfriendly. ...While memories of Landau still linger, so too does fear of the real numbers. In my opinion, the problem with Landau's book is not so much the rigor (though it is excessive), but the lack of background, history, examples, and explanatory remarks. Also, the fact that he does nothing with the real numbers except construct them. In short, it could be an entirely different story if it were explained that the real numbers are interesting! That is what I have tried to do...."

"We cannot hope to give here a final clarification of the essence of fact, judgement, object, property; this task leads into metaphysical abysses; about these one has to seek advice from men whose name cannot be stated without earning a compassionate smile—e.g. Fichte."

"It seems clear that [set theory] violates against the essence of the continuum, which, by its very nature, cannot at all be battered into a single set of elements. Not the relationship of an element to a set, but of a part to a whole ought to be taken as a basis for the analysis of a continuum."

"Important though the general concepts and propositions may be with which the modern industrious passion for axiomatizing and generalizing has presented us, in algebra perhaps more than anywhere else, nevertheless I am convinced that the special problems in all their complexity constitute the stock and core of mathematics; and to master their difficulties requires on the whole the harder labor."

"Cartan developed a general scheme of infinitesimal geometry in which Klein's notions were applied to the tangent plane and not to the n-dimensional manifold M itself."

"On a certain level of generality A which I call the ground level, you have certain theorems that have been proved and certain unsolved problems P of recognised interest. Suppose you discover a generalisation of one of these theorems and thereby rise to a higher level of generality A. Write it up, but lock it away in a drawer - unless or until it serves to solve one of the problems P on the ground level... But the deeper one drives the spade the harder the digging gets; maybe it has become too hard for us unless we are not given some outside help, be it even by such devilish devices as high-speed computing machines."

"The introduction of numbers as coordinates by reference to the particular division scheme of the open one dimensional continuum is an act of violence whose only practical vindication is the special calculatory manageability of the ordinary number continuum with its four basic operations. The topological skeleton determines the connectivity of the manifold in the large."

"In these days the angel of topology and the devil of abstract algebra fight for the soul of each individual mathematical domain."

"We now come to a decisive step of mathematical abstraction: we forget about what the symbols stand for... [The mathematician] need not be idle; there are many operations which he may carry out with these symbols, without ever having to look at the things they stand for."

"[Physicists and philosophers] stick stubbornly to the principles of a mechanistic interpretation of the world after physics has, in its factual structure, already outgrown the latter. They have the same excuse as the inhabitant of the mainland who for the first time travels on the open sea: he will desperately try to stay in sight of the vanishing coast line, as long as there is no other coast in sight, towards which he steers."

"Symmetry is a vast subject, significant in art and nature. Mathematics lies at its root, and it would be hard to find a better one on which to demonstrate the working of the mathematical intellect."

"This letter, if judged by the novelty and profundity of ideas it contains, is perhaps the most substantial piece of writing in the whole literature of mankind."

"In my work, I have always tried to unite the true with the beautiful; but when I had to choose one or the other, I usually chose the beautiful."

"A new development began for relativity theory after 1925 with its absorption into quantum physics. The first great success was scored by Dirac's quantum mechanical equations of the electron, which introduced a new sort of quantities, the spinors, besides the vectors and tensors into our physical theories. ...But difficulties of the gravest kind turned up when one passed from one electron or photon to the interaction among an indeterminate number of such particles. In spite of several advances a final solution of this problem is not yet in sight and may well require a deep modification of the foundation of quantum mechanics, such as would account in the same basic manner for the elementary electric charge e as relativity theory and our present quantum mechanics account for c and h."

"Einstein's theory of relativity has advanced our ideas of the structure of the cosmos a step further. It is as if a wall which separated us from Truth has collapsed. Wider expanses and greater depths are now exposed to the searching eye of knowledge, regions of which we had not even a presentiment. It has brought us much nearer to grasping the plan that underlies all physical happening."

"It was my wish to present this great subject as an illustration of the itermingling of philosophical, mathematical, and physical thought, a study which is dear to my heart. This could be done only by building up the theory systematically from the foundations, and by restricting attention throughout to the principles. But I have not been able to satisfy these self-imposed requirements: the mathematician predominates at the expense of the philosopher."

"To gaze up from the ruins of the oppressive present towards the stars is to recognise the indestructible world of laws, to strengthen faith in reason, to realise the "harmonia mundi" that transfuses all phenomena, and that never has been, nor will be, disturbed."

"A new theory by the author has been added, which draws the physical inferences consequent on the extension of the foundations of geometry beyond Reimann... and represents an attempt to derive from world-geometry not only gravitational but also electromagnetic phenomena. Even if this theory is still only in its infant stage, I feel convinced that it contains no less truth than Einstein's Theory of Gravitation—whether this amount of truth is unlimited or, what is more probable, is bounded by the Quantum Theory."

"With Mie's view of matter there is contrasted another, according to which matter is a limiting singularity of the field, but charges and masses are force-fluxes in the field. This entails a new and more cautious attitude towards the whole problem of matter."

"Space and time are commonly regarded as the forms of existence of the real world, matter as its substance. A definite portion of matter occupies a definite part of space at a definite moment of time. It is in the composite idea of motion that these three fundamental conceptions enter into intimate relationship."

"The Greeks made Space the subject-matter of a science of supreme simplicity and certainty. and certainty Out of it grew, in the mind of classical antiquity, the idea of pure science. Geometry became one of the most powerful expressions of that sovereignty of the intellect that inspired the thought of those times. At a later epoch, when the intellectual despotism of the Church... had crumbled, and a wave of scepticism threatened to sweep away all that had seemed most fixed, those who believed in Truth clung to Geometry as to a rock, and it was the highest ideal of every scientist to carry on his science "more geometrico". Matter... could be measured as a quantity and... its characteristic expression as a substance was the Law of Conservation of Matter... This, which has hitherto represented our knowledge of space and matter, and which was in many quarters claimed by philosophers as a priori knowledge, absolutely general and necessary, stands to-day a tottering structure."

"First, the physicists in the persons of Faraday and Maxwell, proposed the "electromagnetic field" in contradistinction to matter, as a reality of a different category. Then, during the last century, the mathematicians, … secretly undermined belief in the evidence of Euclidean Geometry. And now, in our time, there has been unloosed a cataclysm which has swept away space, time, and matter hitherto regarded as the firmest pillars of natural science, but only to make place for a view of things of wider scope and entailing a deeper vision. This revolution was promoted essentially by the thought of one man, Albert Einstein."

"The rapid development of science... has, as it were, burst its old shell, now become too narrow."

"Recognition of the subjectivity of the qualities of sense is found in Galilei (and also in Descartes and Hobbes) in a form closely related to the principle underlying the constructive mathematical method of our modern physics which repudiates" qualities"."

"In the field of philosophy Kant was the first to take the next decisive step towards the point of view that not only the qualities revealed by the senses, but also space and spatial characteristics have no objective significance in the absolute sense; in other words, that space, too, is only a form of our perception."

"In the realm of physics it is perhaps only the theory of relativity which has made it quite clear that the two essences, space and time, entering into our intuition, have no place in the world constructed by mathematical physics. Colours are thus "really" not even æther-vibrations, but merely a series of values of mathematical functions in which occur four independent parameters corresponding to the three dimensions of space, and the one of time."

"It is the nature of a real thing to be inexhaustible in content; we can get an ever deeper insight into this content by the continual addition of new experiences, partly in apparent contradiction, by bringing them into harmony with one another. In this interpretation, things of the real world are approximate ideas. From this arises the empirical character of all our knowledge of reality."

"Time is the primitive form of the stream of consciousness. ...If we project ourselves outside the stream of consciousness and represent its content as an object, it becomes an event happening in time, the separate stages of which stand to one another in the relations of earlier and later."

"All characteristics of material things as they are presented to us in the acts of external perception (e.g. colour) are endowed with the separateness of spatial extension, but it is only when we build up a single connected real world out of all our experiences that the spatial extension, which is a constituent of every perception, becomes a part of one and the same all-inclusive space. … every material thing can, without changing content, equally well occupy a position in Space different from its present one. This immediately gives us the property of the homogeneity of space which is the root of the conception, Congruence."

"Consciousness spreads out its web, in the form of time, over reality."

"After Riemann had made known his discoveries, mathematicians busied themselves with working out his system of geometrical ideas formally; chief among these were Christoffel, Ricci, and Levi-Civita. Riemann... clearly left the real development of his ideas in the hands of some subsequent scientist whose genius as a physicist could rise to equal flights with his own as a mathematician. After a lapse of seventy years this mission has been fulfilled by Einstein."

"Above all, the ominous clouds of those phenomena that we are with varying success seeking to explain by means of the quantum of action, are throwing their shadows over the sphere of physical knowledge, threatening no one knows what new revolution."

"The scene of action of reality is not a three-dimensional Euclidean space but rather a four-dimensional world, in which space and time are linked together indissolubly. However deep the chasm may be that separates the intuitive nature of space from that of time in our experience, nothing of this qualitative difference enters into the objective world which physics endeavors to crystallize out of direct experience. It is a four-dimensional continuum, which is neither "time" nor "space". Only the consciousness that passes on in one portion of this world experiences the detached piece which comes to meet it and passes behind it as history, that is, as a process that is going forward in time and takes place in space."

"Hermann Weyl’s 1918 text Das Kontinuum... investigates how much of the mathematical corpus can be retained if we restrict ourselves to predicative definitions and methods of proof. He presents a foundational system in which it is impossible to perform an impredicative definition. ...It is an excellent example of a fully developed non-mainstream foundational system for mathematics."

"Hermann Weyl was both a mathematician and a mathematical physicist. Weyl wrote on mathematics, general relativity... quantum mechanics... art and philosophy. His smaller book on philosophy is entitled The Open World. It is made up of lectures... in 1932 at Yale University. In the philosophy of science, according to Weyl, complexity is essential in understanding the concept of a law of nature. If laws of nature may be arbitrarily complex, he argued, the very concept... becomes vacuous. What difference would remain... if the laws meant to explain them were as complex as the phenomena they are meant to explain? Laws of nature must be simple."

"In the first attempt to introduce gauge theories in physics, Hermann Weyl, around the 1920s, proposed certain scale transformations to be a fundamental symmetry of nature."

"I inquired of Dyson whether Weyl had given an example of his having sacrificed truth for beauty. I learned that the example which Weyl gave was his gauge theory of gravitation which he had worked out in his Raum-Zeit-Materie. Apparently, Weyl became convinced that this theory was not true as a theory of gravitation; but still it was so beautiful that he did not wish to abandon it and so he kept it alive for the sake of beauty. But much later, it did turn out that Weyl's instinct was right after all, when the formalism of gauge invariance was incorporated into quantum electrodynamics."

"Weyl considered an aspect about general relativity... the nonpreservation of direction in a curved space. ...[He] decided to consider the possibility that length was also not preserved. ...To effect this change mathematically, Weyl had to make a slight modification in the structure of general relativity. He assumed that in addition to the usual metric (set of numbers or variables) that described the gravitational field, there was another one related to length. ...amazingly when the result was analyzed Maxwell's equations mysteriously appeared. It almost seemed as if a bit of magic had occurred and scientists quickly became interested in the miracle. ...but with detailed analysis the theory was shown to be flawed. Einstein was the first to put his finger on the flaw. ...Weyl soon acknowledged the flaw and laid his theory to rest. It may have been a failure (actually it was not an entire failure; a similar idea is used today in modern field theory), but it did accomplish something important: it got people interested in the possibility that the electromagnetic and gravitational field could be unified. Einstein soon began working on an alternative theory, as did others."

"Weyl never became a devoted adherent to any single philosophy. He rather was a wanderer through the philosophical fields differing with the changes in his scientific view..."

"He realized that formal mathematics might even have an advantage over immediately insightful ("phenomenological") mathematics, because in its conceptual constitution it was free from the restrictions of the latter. I will call this view the symbolic realism of the "mature" Weyl."

"Quantum theory does not trouble me at all. It is just the way the world works. What eats me, gets me, drives me, pushes me, is to understand how it got that way. What is the deeper foundation underneath it? Where does it come from? So that we won’t see it as something that is unwelcome by friends that we admire—John Bell and many others—it will be something that will make you say, ‘It couldn’t have been otherwise.’ We haven’t gotten to that stage yet, and until we do, we have not met the challenge that is right there. I continue to say that the quantum is the crack in the armor that covers the secret of existence. To me it’s a marvelous stimulus, hope, and driving force. And yet I am afraid that just the word—‘hope’—is what does not eat, or possess, or drive so many of our colleagues in the field. They’re content to take the theory for granted, rather than to find out where it comes from. But you would hardly feel the drive to find out where from if you don’t feel that the theory is utterly right. I have been brought up from ‘childhood’ to feel that it is utterly right. Here I was, reading that book of Weyl’s at the age of eighteen and just crazy about it."

"My goal is to show that the heavenly machine is not a kind of divine living being but similar to a clockwork insofar as all the manifold motions are taken care of by one single absolutely simple magnetic bodily force, as in a clockwork all motion is taken care of by a simple weight. And indeed I also show how this physical representation can be presented by calculation and geometrically."

"Nature uses as little as possible of anything."

"The die is cast; I have written my book; it will be read either in the present age or by posterity, it matters not which; it may well await a reader, since God has waited six thousand years for an interpreter of his words."

"We do not ask for what useful purpose the birds do sing, for song is their pleasure since they were created for singing. Similarly, we ought not to ask why the human mind troubles to fathom the secrets of the heavens. The diversity of the phenomena of nature is so great and the treasures hidden in the heavens so rich precisely in order that the human mind shall never be lacking in fresh nourishment."

"In Terra inest virtus, quae Lunam del."

"There is a force in the earth which causes the moon to move."

"I much prefer the sharpest criticism of a single intelligent man to the thoughtless approval of the masses."

"I used to measure the heavens, now I measure the shadows of Earth. Although my mind was heaven-bound, the shadow of my body lies here."

"Temporis filia veritas; cui me obstetricari non pudet."

"Truth is the daughter of time, and I feel no shame in being her midwife."

"The earth is the sphere, the measure of all; round it describe a dodecahedron; the sphere including this will be Mars. Round Mars describe a tetrahedron; the sphere including this will be Jupiter. Describe a cube round Jupiter; the sphere including this will be Saturn. Now, inscribe in the earth an icosahedron, the sphere inscribed in it will be Venus: inscribe an octahedron in Venus: the circle inscribed in it will be Mercury."

"Either... the moving intelligences of the planets are weakest in those that are farthest from the sun, or... there is one moving intelligence in the sun, the common center, forcing them all round, but those most violently which are nearest, and that it languishes in some sort and grows weaker at the most distant, because of the remoteness and the attenuation of the virtue."

"Geometry has two great treasures: one is the Theorem of Phythagoras, the other the division of a line in extreme and mean ratio. The first we can compare to a mass of gold; the other we may call a precious jewel."

"I propose to show that God, in creating the universe and arranging the spheres, had in view the five regular solids of geometry, and fixed by their dimensions the number, proportions and motions of the spheres. Take the sphere of the earth as a unit and circumscribe it with a regular dodecahedron. The sphere that contains this dodecahedron is the sphere of Mars."

"Vim coeli reserate viri: venit agnita ad usus: Ignotae videas commoda nulla rei."

"Discover the force of the heavens O Men: Once recognised it can be put to use: No use could be seen in unknown things."

"He who will please the crowd and for the sake of the most ephemeral renown will either proclaim those things which nature does not display or even will publish genuine miracles of nature without regard to deeper causes is a spiritually corrupt person… With the best of intentions I publicly speak to the crowd (which is eager for things new) on the subject of what is to come."

"I believe it was an act of Divine Providence that I arrived just at the time when Longomontanus [Tycho Brahe's assistant before Kepler] was occupied with Mars. For Mars alone enables us to penetrate the secrets of astronomy which otherwise would remain forever hidden from us."

"Every corporeal substance, so far forth as it is corporeal, has a natural fitness for resting in every place where it may be situated by itself beyond the sphere of influence of a body cognate with it."

"Gravity is a mutual affection between cognate bodies towards union or conjunction (similar in kind to the magnetic virtue), so that the earth attracts a stone much rather than the stone seeks the earth."

"...wheresoever the earth may be placed, or whithersoever it may be carried by its animal faculty, heavy bodies will always be carried towards it."

"If the earth were not round, heavy bodies would not tend from every side in a straight line towards the center of the earth, but to different points from different sides."

"If two stones were placed... near each other, and beyond the sphere of influence of a third cognate body, these stones, like two magnetic needles, would come together in the intermediate point, each approaching the other by a space proportional to the comparative mass of the other."

"If the moon and earth were not retained in their orbits by their animal force or some other equivalent, the earth would mount to the moon by a fifty-fourth part of their distance, and the moon fall towards the earth through the other fifty-three parts, and they would there meet, assuming, however, that the substance of both is of the same density."

"If the earth should cease to attract its waters to itself all the waters of the sea would be raised and would flow to the body of the moon."

"The sphere of the attractive virtue which is in the moon extends as far as the earth, and entices up the waters; but as the moon flies rapidly across the zenith, and the waters cannot follow so quickly, a flow of the ocean is occasioned in the torrid zone towards the westward."

"If the attractive virtue of the moon extends as far as the earth, it follows with greater reason that the attractive virtue of the earth extends as far as the moon and much farther; and, in short, nothing which consists of earthly substance anyhow constituted although thrown up to any height, can ever escape the powerful operation of this attractive virtue."

"Nothing which consists of corporeal matter is absolutely light, but that is comparatively lighter which is rarer, either by its own nature, or by accidental heat. And it is not to be thought that light bodies are escaping to the surface of the universe while they are carried upwards, or that they are not attracted by the earth. They are attracted, but in a less degree, and so are driven outwards by the heavy bodies; which being done, they stop, and are kept by the earth in their own place."

"But although the attractive virtue of the earth extends upwards, as has been said, so very far, yet if any stone should be at a distance great enough to become sensible compared with the earth’s diameter, it is true that on the motion of the earth such a stone would not follow altogether; its own force of resistance would be combined with the attractive force of the earth, and thus it would extricate itself in some degree from the motion of the earth."

"I was almost driven to madness in considering and calculating this matter. I could not find out why the planet would rather go on an elliptical orbit. Oh, ridiculous me! As the liberation in the diameter could not also be the way to the ellipse. So this notion brought me up short, that the ellipse exists because of the liberation. With reasoning derived from physical principles, agreeing with experience, there is no figure left for the orbit of the planet but a perfect ellipse."

"It is not improbable, I must point out, that there are inhabitants not only on the moon but on Jupiter too, or (as was delightfully remarked at a recent gathering of certain philosophers) that those areas are now being unveiled for the first time. But as soon as somebody demonstrates the art of flying, settlers from our species of man will not be lacking. Who would once have thought that the crossing of the wide ocean was calmer and safer than of the narrow Adriatic Sea, Baltic Sea, or English Channel? Given ships or sails adapted to the breezes of heaven, there will be those who will not shrink from even that vast expanse. Therefore, for the sake of those who, as it were, will presently be on hand to attempt this voyage, let us establish the astronomy, Galileo, you of Jupiter, and me of the moon."

"No operation of addition or subtraction gives rise to diversity, but all are equally related to their pair of Terms, or Elements."

"Geometry is one and eternal shining in the mind of God. That share in it accorded to humans is one of the reasons that humanity is the image of God."

"Since geometry is co-eternal with the divine mind before the birth of things, God himself served as his own model in creating the world (for what is there in God which is not God?), and he with his own image reached down to humanity."

"Now because 18 months ago the first dawn, 3 months ago broad daylight but a very few days ago the full sun of the most highly remarkable spectacle has risen — nothing holds me back. I can give myself up to the sacred frenzy, I can have the insolence to make a full confession to mortal men that I have stolen the golden vessel of the Egyptians to make from them a tabernacle for my God far from the confines of the land of Egypt. If you forgive me I shall rejoice; if you are angry, I shall bear it; I am indeed casting the die and writing the book, either for my contemporaries or for posterity to read, it matters not which: let the book await its reader for a hundred years; God himself has waited six thousand years for his work to be seen."

"If you want the exact moment in time, it was conceived mentally on 8th March in this year one thousand six hundred and eighteen, but submitted to calculation in an unlucky way, and therefore rejected as false, and finally returning on the 15th of May and adopting a new line of attack, stormed the darkness of my mind. So strong was the support from the combination of my labour of seventeen years on the observations of Brahe and the present study, which conspired together, that at first I believed I was dreaming, and assuming my conclusion among my basic premises. But it is absolutely certain and exact that "the proportion between the periodic times of any two planets is precisely the sesquialterate proportion of their mean distances...""

"The Earth sings Mi-Fa-Mi, so we can gather ever from this that Misery and Famine reign in our habitat."

"The heavenly bodies are nothing but a continuous song for several voices (perceived by the intellect, not by the ear); a music which... sets landmarks in the immeasurable flow of time. It is therefore, no longer surprising that man, in imitation of his creator, has at last discovered the art of figured song, which was unknown to the ancients. Man wanted to reproduce the continuity of cosmic time... to obtain a sample test of the delight of the Divine Creator in His works, and to partake of his joy by making music in the imitation of God."

"The soul of the newly born baby is marked for life by the pattern of the stars at the moment it comes into the world, unconsciously remembers it, and remains sensitive to the return of configurations of a similar kind."

"The wisdom of the Lord is infinite as are also His glory and His power. Ye heavens, sing His praises: sun, moon, and planets, glorify Him in your ineffable language! Praise Him, celestial harmonies, and all ye who can comprehend them! And thou, my soul, praise thy Creator! It is by Him and in Him that all exist."

"Just as the eye was made to see colours, and the ear to hear sounds, so the human mind was made to understand, not whatever you please, but quantity."

"[N]either this nor that supposition is worthy of the name of an astronomical hypothesis, but rather that which is implied in both alike."

"Indeed I reply in a single word to the sentiments of the saints on these questions about nature; in theology, to be sure, the force of authorities is to be weighed, in philosophy, however, that of causes. Therefore, a saint is Lactantius, who denied the rotundity of the earth; a saint is Augustine, who, admitting the rotundity, yet denied the antipodes; worthy of sainthood is the dutiful performance of moderns who, admitting the meagreness of the earth, yet deny its motion. But truth is more saintly for me, who demonstrate by philosophy, without violating my due respect for the doctors of the church, that the earth is both round and inhabited at the antipodes, and of the most despicable size, and finally is moved among the stars."

"[W]ithout proper experiments I conclude nothing..."

"I certainly know that I owe it [the Copernican theory] this duty, that as I have attested it as true in my deepest soul, and as I contemplate its beauty with incredible and ravishing delight, I should also publicly defend it to my readers with all the force at my command."

"Wherever there are qualities there are likewise quantities, but not always vice versa."

"There are, in fact, as I began to say above, not a few principles which are the special property of mathematics, such principles as are discovered by the common light of nature, require no demonstration, and which concern quantities primarily; then they are applied to other things, so far as the latter have something in common with quantities. Now there are more of these principles in mathematics than in the other theoretical sciences because of that very characteristic of the human understanding which seems to be such from the law of creation, that nothing can be known completely except quantities or by quantities. And so it happens that the conclusions of mathematics are most certain and indubitable."

"[Quantity is the fundamental feature of things,] the primarium accidens substantiae,' ...prior to the other categories."

"God gives every animal the means of saving its life—why object if he gives astrology to the astronomer?"

"I was merely thinking God's thoughts after Him. Since we astronomers are priests of the highest God in regard to the book of nature, it benefits us to be thoughtful, not of the glory of our minds, but rather, above all else, of the glory of God."

"The laws of nature are but the mathematical thoughts of God."

"Kepler was the first to discover the art of successfully inquiring [into] her laws of nature, since his predecessors merely constructed explanatory concepts which they endeavoured to apply to the course of nature."

"It is not known so generally that Kepler was... a geometrician and algebraist of considerable power, and that he, Desargues, and perhaps Galileo, may be considered as forming a connecting link between the mathematicians of the renaissance and those of modern times. Kepler's work in geometry consists rather in certain general principles enunciated, and illustrated by a few cases, than in any systematic exposition of the subject. In a short chapter on conics inserted in his Paralipomena, published in 1604, he lays down what has been called the principle of continuity, and gives as an example the statement that a parabola is at once the limiting case of an ellipse and of a hyperbola; he illustrates the same doctrine by reference to the foci of conics (the word focus was introduced by him); and he also explains that parallel lines should be regarded as meeting at infinity. He introduced the use of the eccentric angle in discussing properties of the ellipse."

"Kepler's laws were the climax of thousands of years of an empirical geometry of the heavens. They were discovered as the result of about twenty-two years of incessant calculation, without logarithms, one promising guess after another being ruthlessly discarded as it failed to meet the exacting demands of observational accuracy. Only Kepler's Pythagorean faith in a discoverable mathematical harmony in nature sustained him. The story of his persistence in spite of persecution and domestic tragedies that would have broken an ordinary man is one of the most heroic in science."

"After his own fashion, Desargues discussed... Kepler's principle (1604) of continuity, in which a straight line is closed at infinity and parallels meet there..."

"In his curious tract on Stereometry, published in 1615, Kepler made some advances in the doctrine of infinitesimals. Prompted to the task by a dispute with the seller of some casks of wine, he studied the measurement of solids formed by the revolution of a curve round any line whatever. In solving some of the simplest of these problems, he conceived a circle to be formed of an infinite number of triangles having all their vertices in the centre, and their infinitely small bases in the circumference of the circle, and by thus rendering familiar the idea of quantities infinitely great and infinitely small, he gave an impulse to this branch of mathematics. The failure of Kepler, too, in solving some of the more difficult of the problems which he himself proposed roused the attention of geometers, and seems particularly to have attracted the notice of Cavaleri."

"When Gilbert of Colchester, in his “New Philosophy,” founded on his researches in magnetism, was dealing with tides, he did not suggest that the moon attracted the water, but that “subterranean spirits and humors, rising in sympathy with the moon, cause the sea also to rise and flow to the shores and up rivers”. It appears that an idea, presented in some such way as this, was more readily received than a plain statement. This so-called philosophical method was, in fact, very generally applied, and Kepler, who shared Galileo’s admiration for Gilbert’s work, adopted it in his own attempt to extend the idea of magnetic attraction to the planets."

"Now, if the Earth move, it is a Planet, and shines to them in the Moone, and to the other Planetary inhabitants, as the Moone and they doe vs upon the Earth: but shine she doth, as Galilie, Kepler, and others prove, and then they per consequens, the rest of the Planets are inhabited, as well as the Moone, which he grants in his dissertation with Galilies Nuncius Siderius, that there be Joiviall and Saturnine Inhabitants, &tc. and that those severall Planets, have their severall Moones about them, as the Earth hath hers, as Galileus hath already evinced by his glasses... yet Kepler, the Emperours Mathematitian, confirms out of his experience, that he saw as much, by the same helpe. Then (I say) the Earth and they be Planets alike, inhabited alike, moved about by the Sunne, the common center of the World alike, and it may be those two greene children... that fell from Heaven, came from thence. We may likewise insert with Campanella and Brunus, that which Melissus, Democritus, Leucipus maintained in their ages, there be infinite Worlds, and infinite Earths, or systemes, because infinite starres and planets, like unto this of ours. Kepler betwixtiest and earnest in his Perspectives, Lunar Geography, dissertat cum nunc:syder seemes in part to agree with this, and partly to contradict; for the Planets he yeelds them to be inhabited, he doubts of the Starres: and so doth Tycho in his Astronomicall Epistles, out of consideration of their variety and greatnesse... that he will never beleeve those great and huge Bodies were made to no other use, then this that we perceave, to illuminate the Earth, a point insensible, in respect of the whole. But who shall dwell in these vast Bodies, Earths, Worlds, if they be inhabited? rational creatures, as Kepler demands? Or have they soules to be saved? Or do they inhabit a better part of the World then we doe? Are we or they Lords of the World? ...this only he proves, that we are in the best place, best World, nearest the Heart of the Sun. Thomas Campanella... subscribes to this of Keplerus, that they are inhabited hee certainly supposeth... and that there are infinite worlds, having made an Apologie for Galileus..."

"Kepler's achievements in mathematics would alone have been sufficient to win for him enduring fame; he first enunciated clearly the principle of continuity in mathematics, treating the parabola as at once the limiting case of the ellipse and the hyperbola, and showing that parallel lines can be regarded as meeting at infinity; he introduced the word 'focus' into geometry; while in his Stereometria Dolorum, published 1615, he applied the conception to the solution of certain volumes and areas by the use of infinitesimals, thus preparing the way for Desargues, Cavalieri, Barrow, and the developed calculus of Newton and Leibniz."

"The Neo-Platonic background, which furnished the metaphysical justification for much of this mathematical development (at least as regards its bearing on astronomy) awoke Kepler's full conviction and sympathy. Especially did the aesthetic satisfactions gained by this conception of the universe as a simple, mathematical harmony, appeal vigorously to his artistic nature."

"Founder of exact modern science though he was, Kepler combined with his exact methods and indeed found his motivation for them in certain long discredited superstitions, including what it is not unfair to describe as sunworship."

"The sun, according to Kepler, is God the Father, the sphere of the fixed stars is God the Son, the intervening ethereal medium, through which the power of the sun is communicated to impel the planets around their orbits, is the Holy Ghost."

"Kepler in the first thirty years of the seventeenth century "reduced to order the chaos of data" left by , and added to them just the thing that was needed—mathematical genius. Like Copernicus he created another world-system which, since it did not ultimately prevail, merely remains as a strange monument of colossal intellectual power working on insufficient materials; and even more than Copernicus he was driven by semi-religious fervour—a passion to uncover the magic of mere numbers and to demonstrate the music of the spheres. ...He has to his credit a collection of discoveries and conclusions—some of them more ingenious than useful—from which we today can pick out three that have a permanent importance in the history of astronomy."

"Johannes Kepler... imbibed Copernican principles while at the University of Tubingen. His pursuit of science was repeatedly interrupted by war, religious persecution, pecuniary embarrassments, frequent changes of residence, and family troubles. In 1600 he became for one year assistant to... ... His first attempt to explain the solar system was made in 1596, when he thought he had discovered a curious relation between the five regular solids and the number and distance of the planets. The publication of this pseudo-discovery brought him much fame. At one time he tried to represent the orbit of Mars by the oval curve which we now write in polar coördinates, \rho = 2r cos^3\theta. Maturer reflection and intercourse with Tycho Brahe and Galileo led him to investigations and results worthy of his genius—"Kepler's laws." He enriched pure mathematics as well as astronomy. It is not strange that he was interested in the mathematical science which had done him so much service; for "if the Greeks had not cultivated s, Kepler could not have superseded Ptolemy." The Greeks never dreamed that these curves would ever be of practical use; Aristaeus and Apollonius studied them merely to satisfy their intellectual cravings after the ideal; yet the conic sections assisted Kepler in tracing the march of the planets in their elliptic orbits. Kepler made also extended use of logarithms and decimal fractions, and was enthusiastic in diffusing a knowledge of them. At one time, while purchasing wine, he was struck by the inaccuracy of the ordinary modes of determining the contents of kegs. This led him to the study of the volumes of solids of revolution and to the publication of the Stereometria Doliorum [Vinariorum] in 1615. In it he deals first with the solids known to Archimedes and then takes up others. Kepler made wide application of an old but neglected idea, that of infinitely great and infinitely small quantities. Greek mathematicians usually shunned this notion, but with it modern mathematicians completely revolutionized the science. In comparing rectilinear figures, the method of superposition was employed by the ancients, but in comparing rectilinear and curvilinear figures with each other, this method failed because no addition or subtraction of rectilinear figures could ever produce curvilinear ones. To meet this case, they devised the , which was long and difficult; it was purely synthetical, and in general required that the conclusion should be known at the outset. The new notion of infinity led gradually to the invention of methods immeasurably more powerful. Kepler conceived the circle to be composed of an infinite number of triangles having their common vertices at the centre, and their bases in the circumference; and the sphere to consist of an infinite number of pyramids. He applied conceptions of this kind to the determination of the areas and volumes of figures generated by curves revolving about any line as axis, but succeeded in solving only a few of the simplest out of the 84 problems which he proposed for investigation in his Stereometria. Other points of mathematical interest in Kepler's works are (1) the assertion that the circumference of an ellipse, whose axes are 2a and 2b, is nearly π (a + b); (2) a passage from which it has been inferred that Kepler knew the variation of a function near its maximum value to disappear; (3) the assumption of the principle of continuity (which differentiates modern from ancient geometry), when he shows that a has a focus at infinity, that lines radiating from this "cæcus focus" are parallel and have no other point at infinity. The Stereometria led Cavalieri... to the consideration of infinitely small quantities."

"As I have stated the most remarkable aspect of Kepler's pursuit of science is the constancy with which he applied himself to his chosen quest. To use a phrase of Shelley's his 'was a character superior in singleness'."

"A law explains a set of observations; a theory explains a set of laws. The quintessential illustration of this jump in level is the way in which Newton’s theory of mechanics explained Kepler’s law of planetary motion. Basically, a law applies to observed phenomena in one domain (e.g., planetary bodies and their movements), while a theory is intended to unify phenomena in many domains. Thus, Newton’s theory of mechanics explained not only Kepler’s laws, but also Galileo’s findings about the motion of balls rolling down an inclined plane, as well as the pattern of oceanic tides. Unlike laws, theories often postulate unobservable objects as part of their explanatory mechanism. So, for instance, Freud’s theory of mind relies upon the unobservable ego, superego, and id, and in modern physics we have theories of elementary particles that postulate various types of quarks, all of which have yet to be observed."

"In his 1619 book The Harmony of the World he tells us that he discovered a harmonic law while delivering a lecture on astronomy to his students. Kepler found that for each planet, the cube of the average distance from the sun is proportional to the square of the period of revolution. Kepler later found a similar law for the satellites of Jupiter. Today we know that such a law holds for any system of bodies that circulates around a central parent body. There are many applications of Kepler's law; for instance, half a century later it gave Isaac Newton the clue to his discovery of the law of universal gravitation."

"More than two hundred years before Poncelet, the important concept of a occurred independently to... Johann Kepler... and the French architect Girard Desargues... Kepler (in his Paralipomena in Vitellionem, 1604) declared that a parabola has two foci, one of which is infinitely distant in two opposite directions, and that any point on the curve is joined to this "blind focus" by a line parallel to the axis."

"The effective inventor of the telescope and compound microscope was Galileo... Galileo's account of the path of the rays through the concave eye-piece and convex objective which he used was not satisfactory and was considerably improved by Kepler, who suggested the use of two convex lenses which became the basis of later instruments. Kepler had already written an important optical treatise in the form of a commentary on Witelo's Perspectiva... His improvements to the telescope may be regarded as what he had learned from the thirteenth-century writer."

"With the discovery of the law of inertia and the subsequent downfall of the Aristotelian theory of motion on which Kepler had based his work, his physical theories soon became outmoded and were then rendered obsolete by Newton's work. Yet Kepler's laws of planetary motion remained, so that Edmond Halley could write in his review of Newton's Principia that the first eleven propositions were found to agree with the phenomena of celestial motions, as discovered by the great sagacity and diligence of Kepler."

"Although the concept of heavenly harmony was a theme mentioned in the literature of the time... Kepler's world harmony had little influence on his contemporaries. ...With the rise of the experimental science advocated by Francis Bacon and greatly facilitated by the invention and development of scientific instruments, the general trend of the seventeenth century was towards a mechanical natural philosophy in which metaphysical speculation would play little part. Another factor... may possibly be recognized in the nature of developments that had taken place in mathematics during the sixteenth century, for the advances in algebra and the introduction of symbolism favored a nominalist view of mathematics in contrast to the realist Platonic view of geometry that Kepler adopted as a foundation for his theory of a world harmony."

"When he discovered the polyhedral hypothesis soon after being sent to teach mathematics in Graz, he changed his mind [about becoming a Lutheran minister] , indicating... that he now saw his work in astronomy as an exercise of a priestly vocation. ...he claimed that, in the Harmonice mundi, he offered to the world nothing less than the plan of creation, which God himself had waited six thousand years for someone to comprehend."

"Kepler is the first who ventured here [into] an exact mathematical treatment of the problems (of astronomical science), the first to establish natural laws in the specific sense of the new science."

"Dumbleton was one of the first to express functional relationships in graphical form. ...Dumbleton also gave a proof of the Merton mean-speed rule... stating that "the latitude of a uniformly difform movement corresponds to the degree of the midpoint." He used the method in the Suma [Suma logicæ et philosophiæ naturalis] to study the problem of the variation in the strength of light as a function of the distance from its source. ...He realized that that the decrease in intensity of illumination was not linearly proportional to the distance... But he did not succeed in finding the exact quantitative relationship, which is that the intensity of illumination due to a luminous source is inversely proportional to the square of the distance, a law discovered by Johannes Kepler in 1604."

"I esteem myself happy to have as great an ally as you in my search for truth. I will read your work … all the more willingly because I have for many years been a partisan of the Copernican view because it reveals to me the causes of many natural phenomena that are entirely incomprehensible in the light of the generally accepted hypothesis. To refute the latter I have collected many proofs, but I do not publish them, because I am deterred by the fate of our teacher Copernicus who, although he had won immortal fame with a few, was ridiculed and condemned by countless people (for very great is the number of the stupid)."

"I have as yet read nothing beyond the preface of your book, from which, however, I catch a glimpse of your meaning, and feel great joy on meeting with so powerful an associate in the pursuit of truth, and consequently, such a friend to truth itself; for it is deplorable that there should be so few who care about truth, and who do not persist in their perverse mode of philosophising. But as this is not the fit time for lamenting the melancholy condition of our times, but for congratulating you on your elegant discoveries in confirmation of the truth, I shall only add a promise to peruse your book dispassionately, and with the conviction that I shall find in it much to admire. This I shall do the more willingly because many years ago I became a convert to the opinions of Copernicus, and by his theory have succeeded in explaining many phenomena which on the contrary hypothesis are altogether inexplicable. I have arranged many arguments and confutations of the opposite opinions, which, however, I have not yet dared to publish, fearing the fate of our master, Copernicus, who, although he has earned immortal fame among a few, yet by an infinite number (for so only can the number of fools be measured) is hissed and derided. If there were many such as you I would venture to publish my speculations, but since that is not so I shall take time to consider of it."

"I thank you because you are the first one, and practically the only one, to have complete faith in my assertions."

"To say... that the motion of the Earth meeting with the motion of the Lunar Orb, the concurrence of them occasioneth the Ebbing and Flowing [of the seas], is an absolute vanity, not onely be­cause it is not exprest, nor seen how it should so happen, but the falsity is obvious, for that the Revolution of the Earth is not con­trary to the motion of the Moon, but is towards the same way. So that all that hath been hitherto said, and imagined by others, is, in my judgment, altogether invalid. But amongst all the famous men that have philosophated upon this admirable effect of Nature, I more wonder at Kepler than any of the rest, who being of a free and piercing wit, and having the motion ascri­bed to the Earth, before him, hath for all that given his ear and assent to the Moons predominancy over the Water, and to oc­cult properties, and such like trifles."

"J. Kepler was the first (that I know of) that discover'd the true cause of the Tide, and he explains it largely in his Introduction to the Physics of the Heavens, given in his Commentaries to the Motion of the Planet Mars, where after he has shewn the Gravity or Gravitation of all Bodies towards another, he thus writes: "The Orb of the attracting Power, which is in the Moon is extended as far as the Earth, and draws the Waters under the Torrid Zone, acting upon places where it is vertical, insensibly on included Seas, but sensibly on the Ocean, whose Beds are large, and the Waters have the liberty of reciprocation, that is, of rising and falling"; and in the 70th Page of his Lunar Astronomy,—"But the cause of the Tides of the Sea appear to be the Bodies of the Sun and Moon drawing the Waters of the Sea.""

"Afterwards that incomparable Philosopher Sir Isaac Newton, improv'd the hint, and wrote so amply upon this Subject as to make the Theory of the Tides his own, by shewing that the Waters of the Sea rise under the Moon and the Place opposite to it: For Kepler believ'd "that the Impetus occasion'd by the presence of the Moon, by the absence of the Moon, occasions another Impetus; till the Moon returning, stops and moderates the Force of that Impetus, and carries it round with its motion." Therefore this Spheroidical Figure which stands out above the Sphere (like two Mountains, the one under the Moon and the other in the place opposite to it) together with the Moon (which it follows) is carried by the Diurnal Motion, (or rather, according to the truth of the matter, as the Earth turns towards the East it leaves those Eminencies of Water, which being carried by their own motion slowly towards the East, are as it were unmov'd) in its journey makes the Water swell twice and sink twice in the space of 25 Hours, in which time the Moon being gone from the Meridian of any Place, returns to it again."

"Galileo argued that nature, God's second book, is written in mathematical letters... Kepler is even more explicit in his work on world harmony; he says: God created the world in accordance with his ideas of creation. These ideas are the pure archetypal forms which Plato termed Ideas, and they can be understood by man as mathematical constructs. They can be understood by Man, because Man was created in the spiritual image of God. Physics is reflection on the divine Ideas of Creation, therefore physics is divine service."

"One wonders how many modern scientists faced by a similar situation in their work would fail to be impressed by such remarkable numerical coincidences."

"If Kepler had been a mathematician of the twentieth century, he would have stopped his laborious observational inductions after noting his first law, and deduced the other two analytically."

"Copernicus, Kepler and Galileo were ‘revisionists’ in rejecting the geocentric system of Ptolemy (which held sway for some 1500 years) and, against an oppressive and repressive mainstream opinion (and officialdom), reinstated—with improvements—the heliocentric system of Aristarchos of Samos (3rd cent BCE)."

"Kepler (and Desargues) regarded the two "ends" of the ["straight"] line as meeting at "infinity" so that the line has the structure of a circle. In fact, Kepler actually thought of a line as a circle with its center at infinity."

"Over and above the specific theorems created by men such as Desargues, Pascal and La Hire, several new ideas and outlooks were beginning to appear. The first is the idea of continuous change of a mathematical entity from one state to another... [i.e., of a] a geometrical figure. It was Kepler, in his Astronomiae Optica of 1604, who first seemed to grasp the fact that parabola, ellipse, hyperbola, circle, and the degenerate conic consisting of a pair of lines are continuously derivable from each other. ...The notion of a continuous change in a figure was also employed by Pascal. He allowed two consecutive vertices of his hexagon to approach each other so that the figure became a pentagon. In the same manner he passed from pentagons to quadrilaterals. The second idea to emerge from the work of the projective geometers is that of transformation and invariance."

"The Pythagorean dream of musical harmony governing the motion of the stars never lost its mysterious impact, its power to call forth responses from the depth of the unconscious mind. ...But, one might ask, was the "Harmony of the Spheres" a poetic conceit or a scientific concept. A working hypothesis or a dream dreamt through a mystic's ear? ...Even Aristotle laughed "harmony, heavenly harmony" out of the courts of earnest, exact science. Yet... Johannes Kepler became enamoured with the Pythagorean dream, and on this foundation of fantasy, by methods of reasoning equally unsound, built the solid edifice of modern astronomy. It is one of the most astonishing episodes in the history of thought, and an antidote to the pious belief that the Progress of Science is governed by logic."

"The Harmony of the World is the continuation of the Cosmic Mystery, and the climax of his lifelong obsession. What Kepler attempted here is, simply, to bare the ultimate secret of the universe in an all-embracing synthesis of geometry, music, astrology, astronomy and epistemology. It was the first attempt of this kind since Plato, and it is the last to our day. After Kepler, fragmentation of experience sets in again, science is divorced from religion, religion from art, substance from form, matter from wind."

"Kepler made free use if indivisibles in both astronomical work and a treatise on measuring volumes of wine casks. He went far beyond the practical needs... and wrote an extensive tract on indivisible methods. Two illustrative examples are his approaches to the areas of a circle and an ellipse."

"But to return to Kepler, his great sagacity, and continual meditation on the planetary motions, suggested to him some views of the true principles from which these motions flow. In his preface to the commentaries concerning the planet Mars, he speaks of gravity as of a power that was mutual betwixt bodies, and tells us that the earth and moon tend towards each other, and would meet in a point so many times nearer to the earth than to the moon, as the earth is greater than the moon, if their motions did not hinder it. He adds that the tides arise from the gravity of the waters towards the moon. But not having just enough notions of the laws of motion, he does not seem to have been able to make the best use of these thoughts; nor does he appear to have adhered to them steadily, since in his epitome of astronomy, published eleven years after, he proposes a physical account of the planetary motions, derived from different principles."

"He [Kepler] supposes, in that treatise [epitome of astronomy], that the motion of the sun on his axis is preserved by some inherent vital principle; that a certain virtue, or immaterial image of the sun, is diffused with his rays into the ambient spaces, and, revolving with the body of the sun on his axis, takes hold of the planets and carries them along with it in the same direction; as a load-stone turned round in the neighborhood of a magnetic needle makes it turn round at the same time. The planet, according to him, by its inertia endeavors to continue in its place, and the action of the sun's image and this inertia are in a perpetual struggle. He adds, that this action of the sun, like to his light, decreases as the distance increases; and therefore moves the same planet with greater celerity when nearer the sun, than at a greater distance. To account for the planet's approaching towards the sun as it descends from the aphelium to the perihelium, and receding from the sun while it ascends to the aphelium again, he supposes that the sun attracts one part of each planet, and repels the opposite part; and that the part which is attracted is turned towards the sun in the descent, and that the other part is towards the sun in the ascent. By suppositions of this kind he endeavored to account for all the other varieties of the celestial motions."

"Luckily, Napier came on the scene with his logarithms just when Johannes Kepler, the discoverer of the laws of planetary motion, was deeply immersed in mind-numbing, tedious calculations, filling hundreds of folio pages with lengthy arithmetic operations, in his construction of the orbit of Mars from the observational data of Tycho Brahe. To Kepler, this discovery was a gift from heaven, for logarithms reduced considerably the time he had to spend just doing arithmetic calculations, a task which he detested."

"As living bodies have hair, so does the earth have grass and trees, the cicadas being its dandruff; as living creatures secrete urine in a bladder, so do the mountains make springs; sulphur and volcanic products correspond to excrement, metals and rainwater to blood and sweat; the sea water is the earth's nourishment … At the same time the anima terrae [soul of the earth] is also a formative power (facultas formatrix) in the earth's interior and expresses, for example, the five regular bodies in precious stones and fossils ..... It is important that in Kepler's view the anima terrae is responsible for the weather and also for meteoric phenomena. Too much rain, for instance, is an illness of the earth."

"Kepler also thought of the Inverse Square Law; he thought of it first. ...Kepler regarded gravitational attraction as analogous to propagation of light... Consider now the intensity of light falling on a planet P at a distance R from the Sun. Let S be the total amount if light emitted by the Sun. ...the intensity will be the same at all points distance R from the Sun. But these points constitute a spherical sheet (with center the Sun) whose radius is R and whose surface area, therefore, is 4πR2. Consequently, intensity of radiation =\frac {S}{4\pi}\cdot\frac {1}{R^2}i.e., the intensity is inversely proportional to the square of the distance between the planet P and the Sun. ...Kepler thought carefully about the possibility, but was dubious... to his credit; he mistrusted the idea for a very good reason. ...although during a solar eclipse the Moon blocks the Sun's radiation to part of the Earth, there is no discontinuity in the Earth's motion. If gravitational attraction were radiated as light is radiated, this too would be temporarily blocked by the Moon, so that during the eclipse it would discontinue its eliptical orbit..."

"Kepler was a brilliant thinker and a lucid writer, but he was a disaster as a classroom teacher. He mumbled. He digressed. He was at times utterly incomprehensible. He drew only a handful of students his first year at Graz; the next year there were none. He was distracted by an incessant interior clamour of associations and speculations vying for his attention. And one pleasant summer afternoon, deep in the interstices of one of his interminable lectures, he was visited by a revelation that was to alter radically the future of astronomy. Perhaps he stopped in mid-sentence. His inattentive students, longing for the end of the day, took little notice, I suspect, of the historic moment."

"Kepler's project in was to give 'true and perfect reasons for the numbers, quantities, and periodic motions of celestial orbits.' The perfect reasons must be based on the simple mathematical principles, which had been discovered by Kepler in the solar system, by using geometric demonstrations. The general scheme of his model was borrowed... from Plato's 'Timaeus', but the mathematical relations for the s (pyramid, cube, , , ) were taken by Kepler from the works of Euclid and Ptolemy. Kepler followed Proclus and believed that 'the main goal of Euclid was to build a geometric theory of the so-called Platonic solids.' Kepler was fascinated by Proclus and often quotes him calling him a 'Pythagorean'."

"Nearly three thousand years ago, the ancient Egyptians knew that a glass lens can make an object look bigger. Nero... is said to have looked through an emerald to watch his gladiators fighting... By the ninth century, people were using 'reading stones' to assist their failing eyesight. These were polished lumps of clear glass, rounded on one side and flat on the other; you sat them on top of the document you were trying to read... The first true spectacles were almost certainly invented in Italy between 1280 and 1300. They acted like a magnifying glass and corrected long-sightedness; it would be another 300 years before lenses able to correct short-sightedness would be developed, in part because these were much harder to make. Johannes Kepler (astronomer, astrologer and mathematician) was the first to explain how convex and concave lenses corrected eyesight. ...lenses were (and still are) made by grinding glass using various types of abrasive material, which in Kepler's time were already being used by jewellers."

"Kepler states expressly that he gave the name Foci to certain points related to the conic sections which had previously "no name." With their new name he associated his new views about the points themselves, and his doctrines of Continuity (under the name Analogy) and Parallelism, which would soon have become known, and would after a time have been taken up by competent mathematicians.... A letter of Henry Briggs to Kepler [dated Mar 10, 1625] suggest improvements in the Paralipomena ad Vitellionem. In this letter Briggs... comprehended and accepted Kepler's way of looking at parallels as lines to or from a point at infinity in one direction or its opposite."

"It was only the third new set of planetary tables in European history. And whereas Copernicus's and Ptolemy's tables were more or less equally accurate, Kepler's were some 50 times more so. Within a few years, it was possible to pinpoint the time of transit of Mercury across the face of the sun so that it was possible to observe it in transit for the first time in human history. Of course, Kepler's theories were more difficult, especially since he had incorporated logarithms, which had only been invented a few years earlier. Much of the book, therefore, was made up of explanatory text that told the reader how to use the tables. ...The printing ...was finished on time in September 1627 ...but he was not optimistic ...noting, "There will be few purchasers, as is always the case with mathematical works, especially in the present chaos.""

"Every perception of colour is an illusion.. ..we do not see colours as they really are. In our perception they alter one another."

"The concern of the artist is with the discrepancy between physical fact and psychological effect."

"I want color and form to have contradictorily functions."

"For me, abstraction is real, probably more real than nature. I'll go further and say that abstraction is nearer my heart. I prefer to see with closed eyes."

"In order to use color effectively it is necessary to recognize that color deceives continually."

"In visual perception a color is almost never seen as it really is — as it physically is. This fact makes color the most relative medium in art."

"If one says 'Red' and there are 50 people listening, it can be expected that there will be 50 reds in their minds. And one can be sure that all these reds will be very different."

"Anxiety is dead."

"A painter works to formulate with or in colors.. .My paintings follow the second option."

"THE ORIGIN OF ART: The discrepancy between physical fact and psychic effect. THE CONTENT OF ART: Visual information of our reaction to life. THE MEASURE OF ART: The ratio of effort to effect. THE AIM OF ART: Revelation and evocation of vision."

"Seeing several of these paintings [his series paintings 'Hommage to the square', Josef Albers painted in 1963-64] next to each other makes it obvious that each painting is an instrumentation on its own."

"But besides relatedness and influence I should like to see that my colors remain, as much as possible, a 'face' –their own 'face', as it was achieved – uniquely — and I believe consciously - in Pompeian wall-paintings - by admitting coexistence of such polarities as being dependent and independent — being dividual and individual."

"I helped my father who was a house painter and decorative painter. He made stage sets, he made glass paintings, he made everything. I was in the workshop and watched him. So as a child so-called art was not my view. That was, in my opinion, my father's job. But I liked to watch him; he comes, as my mother also, from a very craftsman's background. My father's parents were carpenters. They were also builders partly. They were painters. And several of them were very, active in the theater and all such nonsense, you know. On my mother's side there was much more heavy craft. They were blacksmiths. They made a specialty horse shoes and nails for them.. .So, as a child, my main fun was to watch others working. I loved to walk to the neighboring carpenter's place and up to the neighboring shoemaker in my home town."

"And I learned very early [Josef Albers was then ten years old] how to make imitation of wood grain. This is something I have in common with Georges Braque. Braque also learned very early from his father how to imitate marble or wood grain. So I could easily make the appearance of oak or walnut on pine. That is very easy; a very simple technique. And I learned how to imitate marble. I never made such a good joke as Braque did. When he was in the Mediterranean he fooled his friends. He painted a rowboat that had wood on one side and marble on the other side. You see, when he'd row out of the city it looked as if he were in a boat of a different material than when he came back, you see, one side was imitation wood and the other side was imitation marble."

"I discovered soon that teaching has the handicap of retrospection. And that I don't believe in. So I started [at the w:Bauhaus in Dessau] instead a method of handling material with the material itself. So that was my main change. Whereas Itten before [Itten left the Bauhaus in 1923 and Albers followed him as art-teacher] had only spoken about the appearance, 'matiere' - (the French word) and I said I would turn from 'matiere' - the outside - to the inside, to the capacity of the material, before the appearance. And that changed the attitude basically I think."

"I made my examination in Berlin in 1915. And I must say also that Berlin was for me in another way very important. At the time there were all these new movements – 'Die Brücke'... ...the 'Der Blaue Reiter', Walden of the 'Storm' Gallery. Then Kassierer who bought the Chagalls, the first Chagall's that were ever seen in Europe were there. And there was 'Die Brücke'. Rottluff, Heckel, and Kirchner. You know we saw all that. Which was good. You see, Kassierer was then the man who bought the modern French painters. He had particularly Degas who I consider still today a very good painter, one of the best. But, anyway, in spite of my teaching my art was my concern. On the little money I had collected I lived in Berlin very cheaply, ate very cheaply. And already in 1920 I saved the first salaries I received to go to Munich.. .So for the first time I saw the old masters, Rubens and all at the Alte Pinakothek."

"I had to go to the Bauhaus to the basic course that was given by Itten. And I submitted to that although I was a little older than Itten. But I have not the best memories of my studies there. So when that course was over everyone had to exhibit his work and then it was decided whether or not one could continue. I was accepted to continue. But I wanted to go into a workshop and I wanted to make stained glass. That was my old dream. Glass pictures. But Itten thought I was not ready for that. Certainly to delay my study in glass, Itten said, 'Glass painting is a branch of wall painting and you should go first to our wall painting workshop,' And I said, 'That's nonsense. Wall painting has to do with reflected light and glass painting with direct light.' So I said 'Sorry, I'll do my own stuff on my own.' I had no money. Just a Rucksack and a hammer. And I started these assemblages. That was in 1921, But in all books on assemblages these things are not mentioned."

"This is what has Gropius the director made the Bauhaus famous. Not its lamps or its furniture. They are all out of fashion already. But the way of approaching formal problems or material as such, that has made it famous. And the emphasis on material, especially its capacity is my contribution. That was never cleared between us teachers. Kandinsky did what he thought should be done. Klee developed an absolutely different method. Schlemmer developed absolutely something else. Klee was my so-called form master. In the workshops there they had a crafts master and a form master. The crafts master had to direct the practical work, the mechanics of the workshop. And the form master had to develop the, formal qualities. Klee was my form master in the glass workshop. He came to me and never criticized anything. He talked about something else. Never asked about any form problem with the windows I was working on. Never a word. He was too respectful. He was the nicest master I could ask for. He talked about exhibitions. He thought I should exhibit. That's another story. We had a good relationship because we never dealt with the same problems. He didn’t attack our problems. He never brought up a problem."

"When we are honest – that's my saying – if we are honest then we will reveal ourselves. But we do not have to make an effort to be individualistic, different from others. You see that is the nonsense of the last 15, 20 years [Albers refers here critically to American Abstract Expressionism ]. What is wrong there is that everyone wants to be different from the already different ones. And then they ended up all alike. And we are tired of that. And the youngsters feel that now. And they don't continue, you see. They see this will not last. These exaggerated performers always speak in the highest dramatic voice. And in order to achieve it get always drunk before you come to action. Sick. It's over. So I'm quite critical against many of my colleagues. It is not their self-expression. What makes me to be more than my neighbor only when I think I have to say something more than he can. That is self-disclosure. I once gave a talk in Chicago and right in the beginning I said – a lady came to me and said, 'You are against self-expression. And I am mad against you now.' 'And I'll stand upside down to demonstrate that, I said, 'Stop the sentence. You are self-disclosing; you are not self-expressing.'"

"Art is not to be looked at. Art is looking at us. What is art to others is not necessarily art to me. Nor for the same reason and vice versa. What was art to me or was not some time ago might have lost that value or gained it in the meantime and maybe again though art is not an object but experience. To be able to perceive it we need to be receptive. Therefore art is there where art meets us now. The content of art is visual formulation of our relation to life. The measure of art, the ratio of effort to effect, the aim of art revelation and evocation of vision."

"Yes it was 1949. How I came to that. That's like how one gets to know a human being. It so happens that I've always had a preference – as everyone has prejudices and preferences – for the square as a shape in preference to the circle as a shape. And I have known for a long time that a circle always fools me by not telling me whether it's standing still or not. And if a circle circulates you don't see it. The outer curve looks the same whether it moves or does not move. So the square is much more honest and tells me that it is sitting on one line of the four, usually a horizontal one, as a basis. And I have also come to the conclusion that the square is a human invention, which makes it sympathetic to me. Because you don't see it in nature. As we do not see squares in nature, I thought that it is man-made. But I have corrected myself. Because squares exist in salt crystals, our daily salt. We know this because we can see it in the microscope. On the other hand, we believe we see circles in nature. But rarely precise ones. Nature, it seems, is not a mathematician. Probably there are no straight lines either. Particularly not since Einstein says in his theory of relativity that there is no straight line, rod knows whether there are or not, I don't. I still like to believe that the square is a human invention. And that tickles me. So when I have a preference for it then I can only say excuse me."

"There science is dealing with physical facts, in art we are dealing with psychic effects. With this I come to my first statement: 'The source of art – that is, where it comes from – is the discrepancy between physical fact and psychic effect'. That's what I'm talking about. When I want to speak about why I am doing the same thing now, which is squares, for – how long? – 19 years. Because there is no final solution in any visual formulation. Although this may be just a belief on my part, I have some assurances that that is not the most stupid thing to do, through Cezanne whom I consider as one of the greatest painters. From Cézanne we have, so the historians tell us – 250 paintings of Mont St. Victoire. But we know that Cézanne has left in the fields often more than he took home because he was disappointed with his work. So we may conclude he did many more than 250 of the same problem."

"Duplicity in events: What happens here as new, happens somewhere else just the same way. That's so exciting. That is one of the secrets of life. Why did I sometimes build a lamp in the Bauhaus and somebody comes from Holland and says, 'Oh, somebody in Holland makes just the same lamp.' Such duplicity shows that the time is ripe for a problem and thus it is in the air, and will be solved here – and there. With this we are finding the 'creative process', for which somebody is coming to ask me about. I would say, 'I paint because I have no time not to paint.' That's my creative process."

"I have received a question I have expected, 'Don't you deal with accidents?' Yes, I deal with accidents, just as Arp admits it all the time. And I admit it, too. But I like to have them under my command and not sign them because they are accidents. If it remains only accident then sign it 'accident' or 'fate' or 'the Lord', whatever you prefer. It's not you because you have not visioned it. You see visual formulation deals with vision, visual information and visual reaction. So I speak differently from all those who deliver themselves to uncontrolled accidents."

"I would say, 'My things have the look of icons.' Unconsciously they look at you not as my face is now – you see me in profile – icons are only this way. And so are my paintings."

"I say all the time, if I sell that to you, you pay me for 3 colors. And I sell you 4, I betray you. Not to cheat you, but to pet you. You see I betray you in a positive way. I make you see more than there is. And that's in all my art that way. Absolutely something else. And that's what my book is about. You never see what you see. I lead you to see something else. And therefore I direct you. That's help."

"I have not built any theory. I have only tried to build up sensitive eyes, as my book says. And I have tried to achieve that by aiming at very distinct color relationships again – like how do they influence each other? Change each other in light and in intensity, in transparency, opacity? How do they change each other in all different directions? That we make all the students aware, through experience, that color is the most relative medium in art, and that we never really see what we see. All neighboring means which occur every minute different, not only in changing light but also by our changing moods. And in the end, the study of color again is a study of ourselves."

"That question is so big that there is no end [Albers refers here to the relationship between colors and the source of light]. You see, they have just you cannot participate with it if you have not lubricated your eyes very thoroughly to see the little changes produced in our eye that has another action than any optical apparatus like photography. We must know that we have two ways of seeing. For instance, when we are indoors another part of the retina is engaged compared with when we are outdoors. If we are in warm light or cool light, in higher light or lower light."

"I think it's true, as many say, I have dealt for many years with the problems that w:Op art so-called, is dealing with. For many years I have studied the logic and magic of color. And so I know what's involved when it comes to the interaction of colors, more than many who refuse to study it. But I found a way to study it, I think, that's all. And besides I refuse to be the father of a new bandwagon."

"I made true the first English sentence [Albers came from Germany] that I uttered (better stuttered) on our arrival at Black Mountain College in November 1933 [right after the closing of the Bauhaus art school in Dessau Germany, by the Nazi's] When a student asked me what I was going to teach I said: 'to open eyes'. And this has become the motto of all my teaching [famous pupils of Josef Albers at Black Mountain College were for instance the young generation starting American artists, like Robert Rauschenberg, Helen Frankenthaler, Cy Twombly, Ray Johnson and Susan Weil."

"In 1923, when I had been a student at the 'Bauhaus' in Weimar.. ..Gropius [the director of Bauhaus] asked me to teach the basic course 'Werklehhre'. He wanted me to introduce newcomers to the principles of handicrafts. He knew that I came from that background and had appropriate practice and knowledge."

"I did not teach painting but seeing. I concentrated on the basic courses for beginners. I taught drawing (purposely without nudes), color (without any painting as such) and design (as 'structural organization'). And so the graduate students came 'down' to the basic courses for beginners."

"Amateurism is an emptiness and I accept it because it has no preconceived ideas or rules to be applied. This is for me [as art teacher] a most welcome situation and I like to keep my students amateurs and dilettantes."

"I have taught – until 10 years ago – for nearly 40 years, that is almost half of my life. And when I think that over – now afterwards -, I come to a surprising conclusion, namely that I did not teach arts as such, but philosophy and psychology of art."

"But I've noticed something with other artists who do use the whole range of forms of colours and black - in Albers, for instance, who experiments with yellow, red, blue, the whole scale. Of course I love his colour paintings, but when I see a black-and-white such as 'The Homage to a Black and White Square' [Josef Albers painted large series with this title], I like that best, you know. I think it has something to do with deciding just exactly what you really like best. There is always that wonderful element of doubt."

"[Josef] Albers was a beautiful teacher [at the Black Mountain college ] and an impossible person. He wasn't easy to talk to, and I found his criticism so excruciating and so devastating that I never asked for it. Years later, though, I'm still learning what he taught me, because what he taught me had to do with the entire visual world. He didn't teach you how to 'do painting'. The focus was always on your personal sense of looking.. .I consider Albers the most important teacher I've ever had, and I'm sure that he considers me one of his poorest students."

"I don't think he [Joseph Albers] ever realized that it was his discipline that I came for [on Black Mountain College, were Josef Albers was then a leading teacher]. Besides, my response to what I learned from him was just the opposite of what he intended.. .I was very hesitant about arbitrarily designing forms and selecting colors that would achieve some predetermined result, because I didn't have any ideas to support that sort of thing – I didn't want color to serve me, in other words."

"It is my own personal psychosis that it is only by the background that you can see what is in front of you. Only be accepting all that surrounds you can you be totally self-visualized. And at the same time, your self-visualization is a reflection of your surroundings. Albers was right about that."

"[Josef] Albers' rule is to make order. As for me, I consider myself successful when I do something that resembles the lack of order I sense."

"When asked later in life about his working methods for the ['Homage to the 'Square' paintings, Albers would often explain that he always began with the center square because his father, who, among other things, painted houses, had instructed him as a young man that when you paint a door you start in the middle and work outwards. [Albers:] 'That way you catch the drips, and don’t get your cuffs dirty'."

"The German artist [Albers was from German origine] whose lifelong exploration of shape and color theories influenced a generation of artists and challenged audiences worldwide in addition of being one of the leading artists of the 20th century."

"Es handelt sich um meinen liebsten Jugendtraum, nämlich um den Nachweis, dass die Abel’schen Gleichungen mit Quadratwurzeln rationaler Zahlen durch die Transformations-Gleichungen elliptischer Functionen mit singularen Moduln grade so erschöpft werden, wie die ganzzahligen Abel’schen Gleichungen durch die Kreisteilungsgleichungen."

"Now it is time for us to realize that, in his Grundzüge, Kronecker did not merely intend to give his own treatment of the basic problems of ideal-theory which form the main subject of Dedekind's life-work. His aim was a higher own. He was, in fact, attempting to describe and to initiate a new branch of mathematics, which would contain both number-theory and algebraic geometry as special cases. This grandiose conception has been allowed to fade out of our sight, partly because of the intrinsic difficulties of carrying it out, partly owing to historical accidents and to the temporary successes of the partisans of purity and of Dedekind. It will be the main purpose of this lecture to try to rescue it from oblivion, to revive it, and to describe the few modern results which may be considered as belonging to the Kroneckerian program."

"What is immobile must suffer violence. The light-winged bird will easily escape the huge dragon, but the firmly rooted big tree must remain where it is and may have to give up its leaves, fruit, perhaps even its life."

"On the chessboard, lies and hypocrisy do not survive long. The creative combination lays bare the presumption of a lie; the merciless fact, culminating in the checkmate, contradicts the hypocrite."

"Education in Chess has to be an education in independent thinking and judgement. Chess must not be memorized, simply because it is not important enough. If you load your memory you should know why. Memory is too valuable to be stocked with trifles. Of my fifty-seven years I have applied at least thirty to forgetting most of what I had learned or read, and since I succeeded in this I have acquired a certain ease and cheer which I should never again like to be without. If need be, I can increase my skill in Chess, if need be I can do that of which I have no idea at present. I have stored little in my memory, but I can apply that little, and it is of good use in many and varied emergencies. I keep it in order, but resist every attempt to increase its dead weight."

"You should keep in mind no names, nor numbers, nor isolated incidents, not even results, but only methods. The method is plastic. It is applicable in every situation."

"He who wants to educate himself in Chess must evade what is dead in Chess — artificial theories, supported by few instances and upheld by an excess of human wit; the habit of playing with inferior opponents; the custom of avoiding difficult tasks; the weakness of uncritically taking over variations or rules discovered by others; the vanity which is self-sufficient; the incapacity for admitting mistakes; in brief, everything that leads to a standstill or to anarchy."

"Dr. Tarrasch is a thinker, fond of deep and complex speculation. He will accept the efficacy and usefulness of a move if at the same time he considers it beautiful and theoretically right. But I accept that sort of beauty only if and when it happens to be useful. He admires an idea for its depth, I admire it for its efficacy. My opponent believes in beauty, I believe in strength. I think that by being strong, a move is beautiful too."

"Without error there can be no brilliancy."

"Put two players against each other who both have perfect technique, who both avoid weaknesses, and what is left?—a sorry caricature of chess.{{cite book"

"Although the adage "If you find a good move, look for a better one" is often attributed to Lasker, it actually dates earlier."

"It is no easy matter to reply correctly to Lasker's bad moves."

"Emanuel Lasker was undoubtedly one of the most interesting people I came to know in my later life."

"I rather liked Lasker's stubborn intellectual independence, a most rare quality in a generation whose intellectuals are almost invariably mere camp-followers."

"What he really yearned for was some scientific understanding and that beauty peculiar to the process of logical creation, a beauty from whose magic spell no one can escape who has ever felt even its slightest influence."

"Spinoza's material life and economic independence were based on the grinding of lenses; in Lasker’s life chess played a similar part. But Spinoza was luckier, for his business was such as to leave his mind free and independent; whereas master-chess grips its exponent, shackling the mind and brain, so that the inner freedom and independence of even the strongest character cannot remain unaffected."

"That Lasker was a great fighter is an observation which is common to all studies of his play. Nobody can estimate what enormous will-power went into Lasker's fighting ability; and yet at the core of this quality was his belief that each position is unique, that it has some hidden aspect which the skeptic, the man of resource, will finally unearth."

"Lasker also remarked with his detached, penetratingly ironic insight that Dawid Janowski took so much pleasure in a won position that he could not bear to part with it and wind it up to a victorious conclusion."

"Tarrasch teaches knowledge, Lasker teaches wisdom."

"Steinitz always looked for the objectively right move. Tarrasch always claimed to have found the objectively right move. Lasker did nothing of the kind. He never bothered about what might or might not be the objectively right move; all he cared for was to find whatever move was likely to be most embarrassing for the specific person sitting on the other side of the board."

"He hated chess as much as he loved it, using it chielfy as a means of livelihood while he devoted himself to problems of philosophy and mathematics."

"As I pored over the games of the great masters, two styles appealed to me above all others: Lasker and Steinitz. In Lasker I saw, above all, the supreme tactical genius. Whether a game was won or lost mattered little to him; he fought on to get the most out of every position."

"My chess hero."

"The greatest of the champions was, of course, Emanuel Lasker."

"The twentieth century return to Middle Age scholastics taught us a lot about formalisms. Probably it is time to look outside again. Meaning is what really matters."

"Software engineering is the part of computer science which is too difficult for the computer scientist."

"[Software engineering is the] establishment and use of sound engineering principles to obtain economically software that is reliable and works on real machines efficiently."

"... es ist wahr, ein Mathematiker, der nicht etwas Poet ist, wird nimmer ein vollkommener Mathematiker sein."

"Objections... inspired Kronecker and others to attack Weierstrass' "sequential" definition of irrationals. Nevertheless, right or wrong, Weierstrass and his school made the theory work. The most useful results they obtained have not yet been questioned, at least on the ground of their great utility in mathematical analysis and its implications, by any competent judge in his right mind. This does not mean that objections cannot be well taken: it merely calls attention to the fact that in mathematics, as in everything else, this earth is not yet to be confused with the Kingdom of Heaven, that perfection is a chimaera, and that, in the words of Crelle, we can only hope for closer and closer approximations to mathematical truth—whatever that may be, if anything—precisely as in the Weierstrassian theory of convergent sequences of rationals defining irrationals."

"The arithmetization of mathematics... which began with Weierstrass... had for its object the separation of purely mathematical concepts, such as number and correspondence and aggregate, from intuitional ideas, which mathematics had acquired from long association with geometry and mechanics. These latter, in the opinion of the formalists, are so firmly entrenched in mathematical thought that in spite of the most careful circumspection in the choice of words, the meaning concealed behind these words, may influence our reasoning. For the trouble with human words is that they possess content, whereas the purpose of mathematics is to construct pure thought. But how can we avoid the use of human language? The... symbol. Only by using a symbolic language not yet usurped by those vague ideas of space, time, continuity which have their origin in intuition and tend to obscure pure reason—only thus may we hope to build mathematics on the solid foundation of logic."

"[Up to that time] one would have said that a continuous function is essentially capable of being represented by a curve, and that a curve has always a tangent. Such reasoning has no mathematical value whatever; it is founded on intuition, or rather on a visible representation. But such representation is crude and misleading. We think we can figure to ourselves a curve without thickness; but we only figure a stroke of small thickness. In like manner we see the tangent as a straight band of small thickness, and when we say that it touches the curve, we wish merely to say that these two bands coincide without crossing. If that is what we call a curve and a tangent, it is clear that every curve has a tangent; but this has nothing to do with the theory of functions. We see to what error we are led by a foolish confidence in what we take to be visual evidence. By the discovery of this striking example Weierstrass has accordingly given us a useful reminder, and has taught us better to appreciate the faultless and purely arithmetical methods with which he more than any one has enriched our science."

"If the task of philosophy is to break the domination of words over the human mind [...], then my concept notation, being developed for these purposes, can be a useful instrument for philosophers [...] I believe the cause of logic has been advanced already by the invention of this concept notation."

"This ideography is a "formula language", that is, a lingua characterica, a language written with special symbols, "for pure thought", that is, free from rhetorical embellishments, "modeled upon that of arithmetic", that is, constructed from specific symbols that are manipulated according to definite rules."

"I hope I may claim in the present work to have made it probable that the laws of arithmetic are analytic judgments and consequently a priori. Arithmetic thus becomes simply a development of logic, and every proposition of arithmetic a law of logic, albeit a derivative one. To apply arithmetic in the physical sciences is to bring logic to bear on observed facts; calculation becomes deduction."

"Nur im Zusammenhange eines Satzes bedeuten die Wörter etwas. Es wird also darauf ankommen, den Sinn eines Satzes zu erklären, in dem ein Zahlwort vorkommt."

"Your discovery of the contradiction caused me the greatest surprise and, I would almost say, consternation, since it has shaken the basis on which I intended to build arithmetic."

"'Facts, facts, facts,' cries the scientist if he wants to emphasize the necessity of a firm foundation for science. What is a fact? A fact is a thought that is true. But the scientist will surely not recognize something which depends on men's varying states of mind to be the firm foundation of science."

"It really is worth the trouble to invent a new symbol if we can thus remove not a few logical difficulties and ensure the rigour of the proofs. But many mathematicians seem to have so little feeling for logical purity and accuracy that they will use a word to mean three or four different things, sooner than make the frightful decision to invent a new word."

"If I compare arithmetic with a tree that unfolds upward into a multitude of techniques and theorems while its root drives into the depths, then it seems to me that the impetus of the root."

"Every good mathematician is at least half a philosopher, and every good philosopher is at least half a mathematician."

"Ein Philosoph, der keine Beziehung zur Geometrie hat, ist nur ein halber Philosoph, und ein Mathematiker, der keine philosophische Ader hat, ist nur ein halber Mathematiker."

"A judgment, for me is not the mere grasping of a thought, but the admission of its truth."

"Equality gives rise to challenging questions which are not altogether easy to answer... a = a and a = b are obviously statements of differing cognitive value; a = a holds a priori and, according to Kant, is to be labeled analytic, while statements of the form a = b often contain very valuable extensions of our knowledge and cannot always be established a priori. The discovery that the rising sun is not new every morning, but always the same, was one of the most fertile astronomical discoveries. Even to-day the identification of a small planet or a comet is not always a matter of course. Now if we were to regard equality as a relation between that which the names 'a' and 'b' designate, it would seem that a = b could not differ from a = a (i.e. provided a = b is true). A relation would thereby be expressed of a thing to itself, and indeed one in which each thing stands to itself but to no other thing."

"Without some affinity in human ideas art would certainly be impossible; but it can never be exactly determined how far the intentions of the poet are realized."

"Being true is different from being taken as true, whether by one or by many or everybody, and in no case is it to be reduced to it. There is no contradiction in something's being true which everybody takes to be false. I understand by 'laws of logic' not psychological laws of takings-to-be-true, but laws of truth. ...If being true is thus independent of being acknowledged by somebody or other, then the laws of truth are not psychological laws: they are boundary stones set in an eternal foundation, which our thought can overflow, but never displace. It is because of this that they have authority for our thought if it would attain truth. They do not bear the relation to thought that the laws of grammar bear to language; they do not make explicit the nature of our human thinking and change as it changes."

"The ideal of strictly scientific method in mathematics which I have tried to realise here, and which perhaps might be named after Euclid I should like to describe in the following way... The novelty of this book does not lie in the content of the theorems but in the development of the proofs and the foundations on which they are based... With this book I accomplish an object which I had in view in my Begriffsschrift of 1879 and which I announced in my Grundlagen der Arithmetik. I am here trying to prove the opinion on the concept of number that I expressed in the book last mentioned."

"A scientist can hardly meet with anything more undesirable than to have the foundations give way just as the work is finished. I was put in this position by a letter from Mr. Bertrand Russell when the work was nearly through the press."

"Is it always permissible to speak of the extension of a concept, of a class? And if not, how do we recognize the exceptional cases? Can we always infer from the extension of one concept's coinciding with that of a second, that every object which falls under the first concept also falls under the second?"

"The historical approach, with its aim of detecting how things began and arriving from these origins at a knowledge of their nature, is certainly perfectly legitimate; but it also has its limitations. If everything were in continual flux, and nothing maintained itself fixed for all time, there would no longer be any possibility of getting to know about the world, and everything would be plunged into confusion."

"We suppose, it would seem, that concepts grow in the individual mind like leaves on a tree, and we think to discover their nature by studying their growth; we seek to define them psychologically, in terms of the human mind. But this account makes everything subjective, and if we follow it through to the end, does away with truth. What is known as the history of concepts is really a history either of our knowledge of concepts or of the meanings of words."

"Often it is only after immense intellectual effort, which may have continued over centuries, that humanity at last succeeds in achieving knowledge of a concept in its pure form, by stripping off the irrelevant accretions which veil it from the eye of the mind."

"Gottlob Frege created modern logic including "for all," "there exists," and rules of proof. Leibniz and Boole had dealt only with what we now call "propositional logic" (that is, no "for all" or "there exists"). They also did not concern themselves with rules of proof, since their aim was to reach truth by pure calculation with symbols for the propositions. Frege took the opposite track: instead of trying to reduce logic to calculation, he tried to reduce mathematics to logic, including the concept of number."

"Bertrand Russell found Frege's famous error: Frege had overlooked what is now known as the Russell paradox. Namely, Frege's rules allowed one to define the class of x such that P(x) is true for any "concept" P. Frege's idea was that such a class was an object itself, the class of objects "falling under the concept P." Russell used this principle to define the class R of concepts that do not fall under themselves. This concept leads to a contradiction... argument: (1) if R falls under itself then it does not fall under itself; (2) this contradiction shows that it does not fall under itself; (3) therefore by definition it does fall under itself after all."

"From the medieval development of Aristotle's logic through Leibniz's Characteristica Universalis through Frege and Russell and up to the present development of symbolic logic, it could be argued that exactly the reverse [of Jacques Derrida's argument] is the case; that by emphasizing logic and rationality, philosophers have tended to emphasize written language as the more perspicuous vehicle of logical relations. Indeed, as far as the present era in philosophy is concerned, it wasn't until the 1950s that serious claims were made on behalf of the ordinary spoken vernacular languages, against the written ideal symbolic languages of mathematical logic. When Derrida makes sweeping claims about "the history of the world during an entire epoch," the effect is not so much apocalyptic as simply misinformed."

"Our representative absolutist is Gottlob Frege, whose writings did as much as anything to revive the 'mathematizing' approaches of the Platonist tradition around 1900, and did so—quite explicitly—as a means of protecting philosophy from subordination to the facts of history and psychology. ...The Platonist strand in Descartes' philosophy was revived... by... Frege, who promulgated the original programme of 'conceptual analysis' in his Foundations of Arithmetic. ...Frege ...was rebelling ...against the tendency to telescope formal and prescriptive 'laws of thought', which were the proper concern of logic, with the empirical and descriptive 'laws of thinking', which were the business of cognitive psychologists... [W]e should ignore all merely empirical discoveries, whether about the development of understanding in the individual mind or about the historical evolution of our communal understanding. ...Philosophers must concern themselves with 'concepts' only as timeless, intellectual ideals, towards which the human mind struggles, at best, painfully and little by little. ...[A]ctual conceptions current in any existing community are philosophically significant only as an approximation to the eternal system of ideal 'concepts'. ...[A]ny actual, historical set of conceptions has a legitimate intellectual claim on us, only to the extent that it approximates that ideal."

"Logic is an old subject, and since 1879 it has been a great one."

"Most often it is the case that people know that something big can be manipulated with it [i.e., the screw], but not how and in what way it is connected to time, and that untold time, and finally such force of machines, wheels, and shafts is necessary as cannot be produced nor be had."

"I had not only opportunity of seeing how different things have been made, but also manual work made me strong."

"[His work is addressed]... not to the learned and experienced mathematicians who are already, or should be, better acquainted with them... [and most of whom] have studied mechanics more as a subject of curiosity and a hobby, than with any view of service to the public. The people we had in mind were rather the mechanic, handicraftsman and the like, who, without education or knowledge of foreign languages have no access to many sources of information..."

""Theatrum machiuamm universale," &c. by Jacob Leupold, Leipsic, seven volumes, folio, 1724, 1727,1774. This is the greatest and most complete work of this kind that ever was published. The first volume is little more than an introduction to the work; the second and third volumes contain a description of hydraulic machines; the next two volumes relate to machines for raising weights, the theory of levelling, and other subjects; and the sixth treats principally on machines connected with the construction of bridges; the seventh volume is entitled, "Theatre arithmetico geometrique," where the author treats of all instruments employed in these two sciences This work would have been much more considerable if its author had lived to complete the immense task he had undertaken."

"In the Histoire de l'Academie for the year 1725, p. 78, it is stated that when M. du Fay was at Strasbourg, M. Jacob Leupold had a pump which threw water in a continuous stream, using only one piston, and that he made a great mystery of it; but that M. du Fay immediately stated the reason of it."

"Jacob Leupold (1674-1727) German engineer who collected, for the first time in print, the basic principles of mechanical engineering."

"Leupold is also credited as an early inventor of air pumps. He designed his first pump in 1705, and in 1707 he published a book Antlia pneumatica illustrata. In 1711 following an advice of its president Wilhelm Leibniz, Prussian Academy of Sciences acquired Leupold's pump. In 1720 Leupold started to work on the manuscript of his prominent encyclopedie Theatrium machinarum, a nine-volume series on machine design and technology, published between 1724 and 1739 . It was the first systematic analysis of mechanical engineering in the world."

"For scholars and laymen alike it is not philosophy but active experience in mathematics itself that can alone answer the question: What is mathematics?"

"It becomes the urgent duty of mathematicians, therefore, to meditate about the essence of mathematics, its motivations and goals and the ideas that must bind divergent interests together."

"Mathematics as an expression of the human mind reflects the active will, the contemplative reason, and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality. Though different traditions may emphasize different aspects, it is only the interplay of these antithetic forces and the struggle for their synthesis that constitute the life, usefulness, and supreme value of mathematical science."

"Empirical evidence can never establish mathematical existence--nor can the mathematician's demand for existence be dismissed by the physicist as useless rigor. Only a mathematical existence proof can ensure that the mathematical description of a physical phenomenon is meaningful."

"Mathematical techniques to achieve numerical solutions for began to appear about the turn of the century. The first definitive work was carried out by Richardson, who in a paper delivered to the in London in 1910 introduced a finite-difference technique for numerical solution of . Called a "relaxation technique," that approach is still used today to obtain numerical solutions for so-called {[w|elliptic partial differential equation}}s (the equations that govern inviscid subsonic flows are such equations). However, modern numerical analysis is usually considered to have begun in 1928, when Courant, Friedrichs, and Lewy published a definitive paper on the numerical solution of so-called s (the equations that govern inviscid compressible flow are such equations)."

"The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth, space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality."

"The whole world appears resolved into such world-lines. And I should like to say beforehand that, according to my opinion, it would be possible for the physical laws to find their fullest expression as correlations of these world-lines."

"The word postulate of relativity... appears to me very stale... I should rather like to give this statement the name Postulate of the absolute world (or briefly, world-postulate)."

"The rigid electron is in my view a monster in relation to Maxwell's equations, whose innermost harmony is the principle of relativity."

"Oh, that Einstein, always cutting lectures... I really would not believe him capable of it."

"It came as a tremendous surprise, for in his student days Einstein had been a lazy dog... He never bothered about mathematics at all."

"[Zurich,] where the students, even the most capable among them, ... are accustomed to get everything spoon-fed."

"H. A. Lorentz has found out the Relativity theorem and has created the Relativity postulate as a hypothesis that electrons and matter suffer contractions in consequence of their motion according to a certain law. A. Einstein has brought out the point very clearly, that this postulate is not an artificial hypothesis but is rather a new way of comprehending the time-concept, which is forced upon us by observation of natural phenomena."

"The assumption of the contraction of the electron in Lorentz's theory must be introduced at an earlier stage than Lorentz has actually done."

"By laying down the relativity postulate from the outset, sufficient means have been created for deducing henceforth the complete series of Laws of Mechanics from the principle of conservation of energy (and statements concerning the form of the energy) alone."

"It would be very unsatisfactory if the new way of looking at the time-concept, which permits a Lorentz transformation, were to be confined to a single part of Physics."

"Many authors say that classical mechanics stand in opposition to the relativity postulate, which is taken to be the basic of the new Electro-dynamics"

"Not only the physical but also the intellectual landscape of German-language mathematics in the early 1930s would be impossible to imagine without Gernan-Jewish mathematicians. Indeed, some fields of mathematics were completely transformed by their contributions. Number theory was transformed by Hermann Minkowski and Edmund Landau, algebra by Ernst Steinitz and Emmy Noether, set theory and general topology by Felix Hausdorff, Abraham Fraenkel and several others—to mention but a few examples."

"David Hilbert—the undisputed, foremost living mathematician in the world and lifelong close friend and collaborator of the by then deceased Minkowski—had already presented to the Göttingen Academy his own version of the same equations a few days earlier [than Einstein]. Although Minkowski and Hilbert accomplished their most important achievements in pure mathematical fields, their respective contributions to relativity should in no sense be seen as merely occasional excursions into the field of theoretical physics. Minkowski and Hilbert were motivated by much more than a desire to apply their exceptional mathematical abilities opportunistically... On the contrary, Minkowski's and Hilbert's contributions to relativity are best understood as an organic part of their overall scientific careers."

"With the rejection of such classical absolutes as length and duration, our ability to conceive of an objective impersonal world, independent of the presence of an observer, seems to be imperiled. The great merit of Minkowski was to show that an absolute world could nevertheless be imagined, although it was a far different world from that of classical physics. In Minkowski's world the absolute which supersedes the absolute length and duration of classical physics is the Einsteinian interval. ... Thus suppose that, as measured in our Galilean frame of reference, two flashes occur at points A and B, situated at a distance l apart, and suppose the flashes are separated in time by an interval t. If we change our frame of reference, both l and t will change in value, becoming l and t respectively, exhibiting by their changes the relativity of length and duration. In Minkowski's words, "Henceforth space and time themselves are mere shadows." On the other hand, the mathematical construct l^2 - c^2t^2 will remain invariant, and so we shall have l^2 - c^2t^2 = l'^2 - c^2t'^2. It is this invariant expression, which involves both length and duration, or both space and time, which constitutes the Einsteinian interval; and the objective world which it cannotes is the world of four-dimensional space-time. The Einsteinian interval... remains the same for all observers, just as distance alone or duration alone were mistakenly believed to remain the same for all observers in classical physics. ...the Einsteinian interval still remains an invariant as measured for all frames of reference, whether accelerated or not. In the case of accelerated frames, however, we must restrict our attention to Einsteinan intervals of infinitesimal magnitude, and then add up the intervals when finite magnitudes are involved."

"A four dimensional continuum described by the co ordinates x1, x2, x3, x4, was called "world" by Minkowski, who also termed a point-event a "world point." From a "happening" in three-dimensional space, physics becomes, as it were, an "existence" in the four-dimensional world. This four dimensional "world" bears a close similarity to the three-dimensional "space" of Euclidean analytical geometry. ...We can regard Minkowski's "world" in a formal manner as a four-dimensional Euclidean space (with imaginary time co-ordinate); the Lorentz transformation corresponds to a "rotation" of the co-ordinate system in the four-dimensional world."

"The discovery of Minkowski... is to be found... in the fact of his recognition that the four-dimensional space-time continuum of the theory of relativity, in its most essential formal properties, shows a pronounced relationship to the three-dimensional continuum of Euclidean geometrical space. In order to give due prominence to this relationship, however, we must replace the usual time co-ordinate t by an imaginary magnitude, \sqrt -1\cdot ct, proportional to it. Under these conditions, the natural laws satisfying the demands of the (special) theory of relativity assume mathematical forms, in which the time co-ordinate plays exactly the same role as the three space-coordinates. Formally, these four co-ordinates correspond exactly to the three space co-ordinates in Euclidean geometry. ...These inadequate remarks can give the reader only a vague notion of the important idea contributed by Minkowski. Without it the general theory of relativity... would perhaps have got no farther than its long clothes."

"The geometrical theory of numbers... first gained prominence when Hermann Minkowski (1793-1909), who served as professor of mathematics at several universities, published his Geometrie der Zahlen (1896)."

"Minkowski's idea and the solution of the twin paradox can best be explained by means of an analogy between space and spacetime... Time as a fourth dimension rests vertically on the other three—just as in space the vertical juts out of the two-dimensional plane as a third dimension. Distances through spacetime comprise four dimensions, just as space has three. The more you go in one direction, the less is left for the others. When a rigid body is at rest and does not move in any of the three dimensions, all of its motion takes place on the time axis. It simply grows older. ...The faster he moves away from his frame of reference... and covers more distance in the three dimensions of space, the less of his motion through spacetime as a whole is left over for the dimension of time. ...Whatever goes into space is deducted from time. ...In comparison with the distances light travels, all distances in the dimensions of space, even those involving airplane travel, are so very small that we essentially move only along the time axis, and we age continually. Only if we are able to move away from our frame of reference very quickly, like the traveling twin... would the elapsed time shrink to near zero, as it approached the speed of light. Light itself... covers its entire distance through spacetime only in the three dimensions of space... Nothing remains for the additional dimension... the dimension of time... Because light particles do not move in time, but with time, it can be said that they do not age. For them "now" means the same thing as "forever." They always "live" in the moment. Since for all practical purposes we do not move in the dimensions of space, but are at rest in space, we move only along the time axis. This is precisely the reason we feel the passage of time. Time virtually attaches to us."

"When he enrolled as a student in the Polytechnic, Einstein developed a know-it-all attitude, and he paid little attention to Minkowski's lectures and skipped many. Minkowski described him as a "lazy dog," and years later, upon the publication of the theory of relativity, he commented, "I really would not have believed him capable of it.""

"Ever since Hermann Minkowski's now infamous comments in 1908 concerning the proper way to view space-time, the debate has raged as to whether or not the universe should be viewed as a four-dimensional, unified whole wherein the past, present, and future are regarded as equally real or whether the views espoused by the possibilists, historicists, and presentests regarding the unreality of the future (and, for presentests, the past) are more accurate. Now, a century after Minkowski's proposed block universe first sparked debate, we present a new, more conclusive argument in favor of eternalism."

"In a Newtonian view, space and time are separate and different. Symmetries of the laws of physics are combinations of rigid motions of space and an independent shift in time. But... these transformations do not leave Maxwell's equations invariant. Pondering this, the mathematicians Henri Poincaré and Hermann Minkowski were led to a new view of the symmetries of space and time, on a purely mathematical level. If they had described these symmetries in physical terms, they would have beaten Einstein to relativity, but they avoided physical speculations. They did understand that symmetries in the laws of electromagnetism do not affect space and time independently but mix them up. The mathematical scheme describing these intertwined changes is known as the Lorentz group, after the physicist, Hendrik Lorentz."

"Minkowski, building on Einstein's work, had now discovered that the Universe is made of a four-dimensional "spacetime" fabric that is absolute, not relative."

"Minkowski calls a spatial point existing at a temporal point a world point. These coordinates are now called 'space-time coordinates'. The collection of all imaginable value systems or the set of space-time coordinates Minkowski called the world. This is now called the manifold. The manifold is four-dimensional and each of its space-time points represents an event."

"In 1908 the famous mathematician Minkowski made a remarkable discovery concerning the Lorentz formulae. He showed that, although each observer has his own private space and private time, a public concept which is the same for all observers can be formed by combining space and time as a kind of 'distance' by multiplying it by the velocity of light, c; in other words, with any time interval we can associate a definite spatial interval, namely the distance which light can travel in empty space in that period. If, according to a particular observer, the difference in time between any two events is T, this associated spatial interval is cT. Then, if R is the space-distance between these two events, Minkowski showed that the difference of the squares of cT and R has the same value for all observers in uniform relative motion. The square root of this quantity is called the space-time interval between two events. Hence, although time and three-dimensional space depend on the observer, this new concept of space-time is the same for all observers."

"According to the special theory there is a finite limit to the speed of causal chains, whereas classical causality allowed arbitrarily fast signals. Foundational studies... soon revealed that this departure from classical causality in the special theory is intimately related to its most dramatic consequences: the relativity of simultaneity, time dilation, and length contraction. By now it had become clear that these kinematical effects are best seen as consequences of Minkowski space-time, which in turn incorporates a nonclassical theory of causal structure. However, it has not widely been recognized that the converse of this proposition is also true: the causal structure of Minkowski space-time contains within itself the entire geometry (topoligical and metrical structure) of Minkowski space-time. ...The problem of the independence of topological and metrical structures of space-time was clearly recognized by early writers on relativity such as Russell (1954) and, of course, Eddington..."

"It is less than four years since cohomological methods (i.e. methods of Homological Algebra) were introduced into Algebraic Geometry in Serre's fundamental paper[11], and it seems certain that they are to overflow the part of mathematics in the coming years, from the foundations up to the most advanced parts. ... [11] Serre, J. P. Faisceaux algébriques cohérents. Ann. Math. (2), 6, 197–278 (1955)."

"The introduction of the digit 0 or the group concept was general nonsense too, and mathematics was more or less stagnating for thousands of years because nobody was around to take such childish steps..."

"The question you raise “how can such a formulation lead to computations” doesn’t bother me in the least! Throughout my whole life as a mathematician, the possibility of making explicit, elegant computations has always come out by itself, as a byproduct of a thorough conceptual understanding of what was going on. Thus I never bothered about whether what would come out would be suitable for this or that, but just tried to understand – and it always turned out that understanding was all that mattered."

"I can illustrate the ... approach with the ... image of a nut to be opened. The first analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months — when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado! A different image came to me a few weeks ago. The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration ... the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it ... yet finally it surrounds the resistant substance."

"We should ask our fellow physicists to invent a principle of anti-interference, which would bring light out of two obscurities (Leray and Grothendieck)."

"Alexandre Grothendieck was very different from Weil in the way he approached mathematics: Grothendieck was not just a mathematician who could understand the discipline and prove important results—he was a man who could create mathematics. And he did it alone."

"No one but Grothendieck could have taken on algebraic geometry in the full generality he adopted and seen it through to success. It required courage, even daring, total self confidence and immense powers of concentration and hard work. Grothendieck was a phenomenon."

"Many people who knew Grothendieck during his time at I.H.E.S. speak of his kindness, his openness to any kind of question, his gentle humor. He was often barefoot. He fasted once a week in opposition to the war in Vietnam. Mazur recalled that Grothendieck had met a family at the local train station with nowhere to stay, and he invited them to live in the basement apartment of his home. He had a machine installed that helped make taramasalata—a fish-roe spread—so that they could sell prepared food at the market."

"He really never worked on examples, I only understand things through examples and then gradually make them more abstract. I don’t think it helped Grothendieck in the least to look at an example. He really got control of the situation by thinking of it in absolutely the most abstract possible way. It’s just very strange. That’s the way his mind worked."

"Weil's new mathematical language, algebraic geometry, had enabled him to articulate subtleties about solutions to equations that hitherto had been impossible. But if there was any hope of extending Weil's ideas to prove the Riemann Hypothesis, it was clear they would need to be developed beyond the foundations he had laid in his prison cell in Rouen. It would be another mathematician from Paris who would bring the bones of Weil's new language to life. The master architect who performed this task was one of the strangest and most revolutionary mathematicians of the twentieth century - Alexandre Grothendieck."

"Applications in arithmetic geometry (such as Weil conjectures, Ramanujan conjecture, Mordell conjecture, Shafarevich conjecture, Tate conjectures) are unthinkable in the classical style, these really need Grothendieck's foundations of algebraic geometry."

"Many mathematicians are rather childlike, unworldly in some sense, but Grothendieck more than most. He just seemed like an innocent—not very sophisticated, no pretense, no sham. He thought very clearly and explained things very patiently, without any sense of superiority. He wasn’t contaminated by civilization or power or one-up-manship."

"In Récoltes et Semailles, Grothendieck counts his twelve disciples. The central character is Pierre Deligne, who combines in this tale the features of John, "the disciple whom Jesus loved”", and Judas the betrayer. The weight of symbols!"

"Jean Dieudonné and Laurent Schwartz were able to discipline Grothendieck just enough to prevent him from running off in all directions, and to restrain his excessive attraction to extreme generality."

"Grothendieck’s undertaking throve thanks to unexpected synergies: the immense capacity for synthesis and for work of Dieudonné, promoted to the rank of scribe, the rigorous, rationalist and well-informed spirit of Serre, the practical know-how in geometry and algebra of Zariski’s students, the juvenile freshness of the great disciple Pierre Deligne, all acted as counterweights to the adventurous, visionary and wildly ambitious spirit of Grothendieck."

"[My aim is] to design logic as a calculating discipline, especially to give access to the exact handling of relative concepts, and, from then on, by emancipation from the routine claims of natural language, to withdraw any fertile soil from "cliché" in the field of philosophy as well. This should prepare the ground for a scientific universal language that, widely differing from linguistic efforts like Volapük [a universal language like Esperanto, very popular in Germany at the time], looks more like a sign language than like a sound language."

"When I started to trace the later development of logic, the first thing I did was to look at Schröder's Vorlesungen über die Algebra der Logik, ...[whose] third volume is on the logic of relations (Algebra und Logik der Relative, 1895). The three volumes immediately became the best-known advanced logic text, and embody what any mathematician interested in the study of logic should have known, or at least have been acquainted with, in the 1890s. While, to my knowledge, no one except Frege ever published a single paper in Frege's notation, many famous logicians adopted Peirce-Schröder notation, and famous results and systems were published in it. Löwenheim stated and proved the Löwenheim theorem (later reproved and strengthened by Thoralf Skolem, whose name became attached to it together with Löwenheim's) in Peircian notation. In fact, there is no reference in Löwenheim's paper to any logic other than Peirce's. To cite another example, Zermelo presented his axioms for set theory in Peirce-Schröder notation, and not, as one might have expected, in Russell-Whitehead notation."

"Nevertheless, it remains conceivable that the measure relations of space in the infinitely small are not in accordance with the assumptions of our geometry [Euclidean geometry], and, in fact, we should have to assume that they are not if, by doing so, we should ever be enabled to explain phenomena in a more simple way."

"As is known, scientific physics dates its existence from the discovery of the differential calculus. Only when it was learned how to follow continuously the course of natural events, attempts, to construct by means of abstract conceptions the connection between phenomena, met with success. To do this two things are necessary: First, simple fundamental concepts with which to construct; second, some method by which to deduce, from the simple fundamental laws of the construction which relate to instants of time and points in space, laws for finite intervals and distances, which alone are accessible to observation (can be compared with experience)."

"Magnitude-notions are only possible where there is an antecedent general notion which admits of different specialisations. According as there exists among these specialisations a continuous path from one to another or not, they form a continuous or discrete manifoldness; the individual specialisations are called in the first case points, in the second case elements, of the manifoldness."

"Definite portions of a manifoldness, distinguished by a mark or by a boundary, are called Quanta. Their comparison with regard to quantity is accomplished in the case of discrete magnitudes by counting, in the case of continuous magnitudes by measuring. Measure consists in the superposition of the magnitudes to be compared; it therefore requires a means of using one magnitude as the standard for another. In the absence of this, two magnitudes can only be compared when one is a part of the other; in which case also we can only determine the more or less and not the how much. The researches which can in this case be instituted about them form a general division of the science of magnitude in which magnitudes are regarded not as existing independently of position and not as expressible in terms of a unit, but as regions in a manifoldness."

"If in the case of a notion whose specialisations form a continuous manifoldness, one passes from a certain specialisation in a definite way to another, the specialisations passed over form a simply extended manifoldness, whose true character is that in it a continuous progress from a point is possible only on two sides, forwards or backwards. If one now supposes that this manifoldness in its turn passes over into another entirely different, and again in a definite way, namely so that each point passes over into a definite point of the other, then all the specialisations so obtained form a doubly extended manifoldness. In a similar manner one obtains a triply extended manifoldness, if one imagines a doubly extended one passing over in a definite way to another entirely different; and it is easy to see how this construction may be continued. If one regards the variable object instead of the determinable notion of it, this construction may be described as a composition of a variability of n + 1 dimensions out of a variability of n dimensions and a variability of one dimension."

"Measure-relations can only be studied in abstract notions of quantity, and their dependence on one another can only be represented by formulæ. On certain assumptions, however, they are decomposable into relations which, taken separately, are capable of geometric representation; and thus it becomes possible to express geometrically the calculated results. In this way, to come to solid ground, we cannot, it is true, avoid abstract considerations in our formulæ, but at least the results of calculation may subsequently be presented in a geometric form. The foundations of these two parts of the question are established in the celebrated memoir of Gauss, Disqusitiones generales circa superficies curvas."

"For Space, when the position of points is expressed by rectilinear co-ordinates, ds = \sqrt{ \sum (dx)^2 }; Space is therefore included in this simplest case. The next case in simplicity includes those manifoldnesses in which the line-element may be expressed as the fourth root of a quartic differential expression. ...I restrict myself... to those manifoldnesses in which the line element is expressed as the square root of a quadric differential expression. ...Manifoldnesses in which, as in the Plane and in Space, the line-element may be reduced to the form \sqrt{ \sum (dx)^2 }, are... only a particular case of the manifoldnesses to be here investigated; they require a special name, and therefore these manifoldnesses... I will call flat. In order now to review the true varieties of all the continua which may be represented in the assumed form, it is necessary to get rid of difficulties arising from the mode of representation, which is accomplished by choosing the variables in accordance with a certain principle."

"Let us imagine that from any given point the system of shortest lines going out from it is constructed; the position of an arbitrary point may then be determined by the initial direction of the geodesic in which it lies, and by its distance measured along that line from the origin. It can therefore be expressed in terms of the ratios dx0 of the quantities dx in this geodesic, and of the length s of this line. ...the square of the line-element is \sum (dx)^2 for infinitesimal values of the x, but the term of next order in it is equal to a homogeneous function of the second order... an infinitesimal, therefore, of the fourth order; so that we obtain a finite quantity on dividing this by the square of the infinitesimal triangle, whose vertices are (0,0,0,...), (x1, x2, x3,...), (dx1, dx2, dx3,...). This quantity retains the same value so long as... the two geodesics from 0 to x and from 0 to dx remain in the same surface-element; it depends therefore only on place and direction. It is obviously zero when the manifold represented is flat, i.e., when the squared line-element is reducible to \sum (dx)^2, and may therefore be regarded as the measure of the deviation of the manifoldness from flatness at the given point in the given surface-direction. Multiplied by -¾ it becomes equal to the quantity which Privy Councillor Gauss has called the total curvature of a surface. ...The measure-relations of a manifoldness in which the line-element is the square root of a quadric differential may be expressed in a manner wholly independent of the choice of independent variables. A method entirely similar may for this purpose be applied also to the manifoldness in which the line-element has a less simple expression, e.g., the fourth root of a quartic differential. In this case the line-element, generally speaking, is no longer reducible to the form of the square root of a sum of squares, and therefore the deviation from flatness in the squared line-element is an infinitesimal of the second order, while in those manifoldnesses it was of the fourth order. This property of the last-named continua may thus be called flatness of the smallest parts. The most important property of these continua for our present purpose, for whose sake alone they are here investigated, is that the relations of the twofold ones may be geometrically represented by surfaces, and of the morefold ones may be reduced to those of the surfaces included in them..."

"With every simple act of thinking, something permanent, substantial, enters our soul. This substantial somewhat appears to us as a unit but (in so far as it is the expression of something extended in space and time) it seems to contain an inner manifoldness; I therefore name it "mind-mass." All thinking is, accordingly, formation of new mind masses."

"Mind-masses entering the soul appear to us as ideas, the quality of the latter depending on the inner state of the former."

"Forming mind-masses amalgamate, combine or compound themselves in definite degree, partly with each other, partly with older mind-masses. The manner and strength of these combinations depend on conditions which are but imperfectly recognised by Herbart... They depend chiefly upon the inner relationship of the mind-masses."

"The soul is a compact of mind-masses combined in a most intimate and manifold manner. It grows constantly by accession of mind-masses, and upon these depend its development."

"Mind-masses, once formed, are imperishable, their combinations are indissoluble; only the relative strength of these combinations is altered by the incoming of new mind masses."

"Every entering mind-mass excites all related mind-masses and this excitation is the more powerful the more insignificant the diversity of their inner states (quality)."

"If... part of the related mind-masses hang together among themselves, then these are not only immediately excited but also mediately and consequently in proportion more powerfully than the rest."

"The simplest and most common manifestation of the activity of older mind-masses is Reproduction, which consists in the striving of the active mind-mass to engender one similar to itself."

"Every mind-mass strives to produce a like formed mind-mass and accordingly strives to produce that form of motion of the matter by which it was formed."

"We now apply these laws of mental processes, to which the explanation of our own inner consciousness leads, to explain the order and adaptation observed on the earth, i.e., to explain Being and historical development."

"An immediate consequence of these principles of explanation is that the souls of organic beings, i.e., the compacts of mind-masses, arisen during life, continue to exist after death. (Their isolated persistence is not sufficient). But in order to explain the orderly development of organic nature in which the earlier collected experiences obviously serve as basis for the later creations, it is necessary to assume that these mind-masses enter into a greater compact of mind-masses, the Earth-Soul, and that these serve a higher soul-life according to the same laws as the mind-masses engendered in our nerve-processes observe in their service of our own soul-life."

"The substratum of mental activity must be sought only in ponderable matter."

"There remains only the assumption that the ponderable masses within the rigid earth-crust are supporters of the soul-life of the earth."

"It is absurd to assume that upon the rigid earth-crust the organic originated from the inorganic. In order to explain the nascence of the lowest organisms on the earth-crust, one must assume an already existing organising principle or a thought-process, under conditions that would render organic combinations impossible. We must accordingly assume that these conditions are valid only for the life-process in the actual state of the earth's surface, and only so far as we can explain them may we estimate the possibility of life-processes under other conditions."

"Zend-Avesta, a truly life giving word creating new life in knowledge as in faith! ...As Fechner in his Nanna sought to show that plants have souls, so the point of departure of his contemplations in the Zend-Avesta is the doctrine that the stars have souls. The method he employs is not that of the abstraction of general laws by induction and the application and testing of these in the explanation of nature, it is analogy. He compares the earth with our own organism, which we know to be endowed with a soul. He searches out not merely in a one-sided way the similarities, but does equal justice to the dissimilarities, too, and so arrives at the conclusion that all the former show the earth to be a being with a soul, and that all the latter indicate that it is a being with a soul far higher than our own."

"The adaptation observed in men, animals and plants... one part of this adaptation is explained from a thought-process in the interior of these bodies... another part, however, the adaptation of the organism, by a thought-process in a greater whole."

"There is within the limits of our experience no reason to seek the causes of these adaptations in a greater whole. All organisms are designed only for life upon the earth. The state of the earth's crust accordingly contains all (external) reasons of its arrangement. ...They are peculiar (individual). According to all that experience teaches we must assume that they are not repeated on other heavenly bodies."

"From the standpoint of exact natural science, of the explanation of nature from causes, the assumption of an earth-soul is... an hypothesis for the explanation of Being and of the historical development of the organic world."

"The souls of perished creatures shall... form the elements of the soul-life of the earth."

"The different thought-processes seem to differ chiefly in respect to their temporal rhythm. If plants have souls, then hours and days must be for them what seconds are for us. The corresponding period for the earth-soul, at least for its outward activity, possibly embraces many thousands of years."

"Thesis. Finite, Representable. Antithesis. Infinite, System of Notions lying at the limit of the representable."

"I. Thesis. Finite elements of Space and Time. Antithesis. Continuity."

"II. Thesis. Freedom, i.e., not the power absolutely to originate, but to pass judgement between two or more possibilities. Antithesis. Determinism."

"II. Thesis. In order that decision by arbitrary power may be possible in spite of completely definite laws of the action of ideas, one must assume that the psychic mechanism itself has, or at least in its development acquires, the peculiar property of inducing the necessity of these laws. Antithesis. No one can, in case of affairs, abandon the conviction that the future is co-determined by his transactions."

"III. Thesis. A God working in Time. (Government of the world). Antithesis. A timeless, personal, omniscient, al-mighty, all-benevolent God (Providence)."

"IV. Thesis. Immortality. Antithesis. A thing in and by itself endowed with transcendental freedom, radical evil, intelligible character and lying at the basis of our temporal appearance."

"IV. Thesis. Freedom is very well compatible with sound lawfulness of the course of nature. But the concept of a timeless God is then untenable. But the restriction which omnipotence and omniscience suffer through freedom of the creature in the sense above determined, must be eliminated by the assumption of a temporally acting God, of a ruler of the hearts and destinies of men; the concept of Providence must be supplemented and in part replaced by the notion of government of the world. [No Antithesis indicated.]"

"The method applied by Newton to the grounding of the Infinitesimal Calculus, and which since the beginning of this century has been recognised by the best mathematicians as the only one that furnishes sure results, is the method of limits. The method consists in this, viz., instead of considering a continuous transition from one value of a quantity to another, from one position to another, or, speaking generally, from one determination of a concept to another, one considers in the first place a transition through a finite number of intervals and then allows the number of these intervals to increase so that the distances of two successive points of division all decrease infinitely."

"The concept-systems of antithesis are concepts that are indeed thoroughly determined by negative predicates but are not positively representable. Just because a precise and complete representation of these concept-systems is impossible, they are inaccessible to direct investigation and elaboration by our reflection. They may, however, be regarded as lying at the limit of the representable, i.e., we can form a concept-system lying within the representable, which passes over into the given system by simple change of magnitude ratio. By abstracting from the ratios of the quantities, the concept-system remains unchanged in case of transition to the limit. At the limit itself, however, some of the correlative concepts of the system lose their susceptibility of being represented, and those, indeed, that mediate the relation between other concepts."

"Natural science is the attempt to comprehend nature by precise concepts. According to the concepts by which we comprehend nature not only are observations completed at every instant but also future observations are pre-determined as necessary, or, in so far as the concept-system is not quite adequate therefor, they are predetermined as probable; these concepts determine what is "possible" (accordingly also what is "necessary," or the opposite of which is impossible), and the degree of the possibility (the "probability") of every separate event that is possible according to them, can be mathematically determined, if the event is sufficiently precise. If what is necessary or probable according to these concepts occurs, then the latter are thereby confirmed and upon this confirmation by experience rests our confidence in them. If, however, something happens which according to them is not expected and which is therefore according to them impossible or improbable, then arises the problem so to complete them, or if necessary, to transform them, that according to the completed or ameliorated concept-system, what is observed ceases to be impossible or improbable. The completion or amelioration of the concept-system forms the "explanation" of the unexpected observation. By this process our comprehension of nature becomes gradually always more complete and assured, but at the same time recedes even farther behind the surface of phenomena."

"There are no degrees of being, a gradual difference of only states or relations being thinkable. Accordingly, if an agent strives to preserve or to restore itself, then it must be a state or relation."

"Kant has rightly observed that by the resolution of the concept of a thing we can find neither that it exists nor that it is the cause of something else, and accordingly that the concepts of being and causality are not analytical but can be derived only from experience. When however he later feels himself obliged to assume that the notion of causality originates in a pre-experiential property of the cognising subject and therefore stamps it a mere rule of time-series, by which, in experience, with each observation as cause any other could be connected as effect, then is the child thrown out with the bath. (Indeed, we must derive the relations of causality from experience; but we must not fail to correct and to complete our conception of these facts of experience by reflection.)"

"The word hypothesis has now a somewhat different significance from that given it by Newton. We are now accustomed to understand by hypothesis all thoughts connected with the phenomena. Newton was far from the crude thought that explanation of phenomena could be attained by abstraction."

"Riemann has shown that as there are different kinds of lines and surfaces, so there are different kinds of space of three dimensions; and that we can only find out by experience to which of these kinds the space in which we live belongs. In particular, the axioms of plane geometry are true within the limits of experiment on the surface of a sheet of paper, and yet we know that the sheet is really covered with a number of small ridges and furrows, upon which (the total curvature not being zero) these axioms are not true. Similarly, he says although the axioms of solid geometry are true within the limits of experiment for finite portions of our space, yet we have no reason to conclude that they are true for very small portions; and if any help can be got thereby for the explanation of physical phenomena, we may have reason to conclude that they are not true for very small portions of space."

"Riemann was the first; I read his inaugural dissertation and his major work on Abelian functions. Starting out thus was a stroke of luck for which I have always been grateful. These are not hard to read, as long as one realizes that every word is loaded with meaning: there is perhaps no other mathematician whose writing matches Riemann's for density."

"In... 1859 Bernhard Riemann... presented a paper to the [Berlin] Academy... "On the Number of Prime Numbers Less Than a Given Quanitity." ...Riemann tackled the problem with the most sophisticated mathematics of his time... inventing for his purposes a mathematical object of great power and subtlety. ...[H]e made a guess about that object, and then remarked:One would of course like to have a rigorous proof of this, but I have put aside the search for such a proof after some fleeting vain attempts because it is not necessary for the immediate objective...That ... guess lay almost unnoticed for decades. Then... gradually seized... imaginations... until it attained the status of an overwhelming obsession. ...The Riemann Hypothesis... remained an obsession all through the twentieth century and remains one today, having resisted every attempt at proof and disproof. [It is] now the great white whale of mathematical research."

"Riemann's style is extremely difficult. His tragically brief life was too occupied with mathematical creativity for him to devote himself to elegant exposition or to... polished presentation... Riemann's best ideas have been incorporated in later, more readable works. Nonetheless... No secondary source can duplicate Riemann's insight. Riemann was so far ahead of his time that it was 30 years before anyone could begin really to grasp his ideas... [T]he results which Siegel found in private papers were a major contribution to the field... in 1932, seventy years after Riemann discovered them. Any simplification, paraphrasing, or reworking... runs the risk of missing an important idea, of obscuring a point of view which was a source of Riemann's insight, or of introducing new technicalities or side issues which are not of real concern. There is no mathematician... whom I would trust to revise his work."

"The profound purely scientific significance of Riemann's work in pure mathematics and mathematical physics has been long since recognised, and time is more and more disclosing the great philosophical import of portions of that same work, as for example, of the famous Habilitationschrift on "The Hypotheses that Form the Foundations of Geometry." Riemann was indeed distinctly a philosophical mathematician. Grundlage (the foundations of things), more than anything else, fascinated his marvellous genius, and his greatest work was exploration among the roots of knowledge... Some of his profoundest ideas have certainly not been duly exploited. In their German dress, they are to many people practically inaccessible. ... Like Riemann's purely philosophical ideas, which were influenced by Fechner, his psychology will, at least in its terminology, be found to be at variance in many points with the views and tendencies of the time. They have, however, a quite independent significance as throwing light upon Riemann's own intellectual development, and thus, apart from whatever intrinsic merit they may possess, form a valuable page in the history of the development of thought."

"Einstein independently discovered Riemann's original program, to give a purely geometric explanation to the concept of "force." …To Riemann, the bending and warping of space causes the appearance of a force. Thus forces do not really exist; what is actually happening is that space itself is being bent out of shape. The problem with Riemann's approach... was that he had no idea specifically how gravity or electricity and magnetism caused the warping of space. ...Here Einstein succeeded where Riemann failed."

"At the University of Göttingen, the shy and gifted Riemann studied mathematics under Gauss and physics under W. Weber, Gauss's collaborator in the invention of the telegraph. Like Gauss, Riemann was deeply interested in physical science, and from that source drew the inspiration for his mathematical investigations."

"In the field of non-Euclidean geometry, Riemann... began by calling attention to a distinction that seems obvious once it is pointed out: the distinction between an unbounded straight line and an infinite line. The distinction between unboundedness and infiniteness is readily illustrated. A circle is an unbounded figure in that it never comes to an end, and yet it is of finite length. On the other hand, the usual Euclidean concept of a straight line is also unbounded in that it never reaches an end but is of infinite length. ...he proposed to replace the infiniteness of the Euclidean straight line by the condition that it is merely unbounded. He also proposed to adopt a new parallel axiom... In brief, there are no parallel lines. This ... had been tried... in conjunction with the infiniteness of the straight line and had led to contradictions. However... Riemann found that he could construct another consistent non-Euclidean geometry."

"Bernhard Riemann (1826–1866) was known as “the mathematician from Göttingen.” (Of course so were Gauss and Hilbert.) He provided one answer to the question, Where do functions live?, in a time in which the more general question was, Where are people to live and grow and thrive? Here, think of the First Industrial Revolution of water and steam power, the factory, and the rise of urbanization; the novels of Charles Dickens or Émile Zola; the reconstruction of Paris under Napoleon III and Haussmann; and the soon-to-come Second Industrial Revolution of electricity, chemistry, and the further transformation of agriculture. Consequently, in the city all sorts of things were now mixed together: classes, values, roles. Those things and statuses might be separated out into a simpler less mixed-together world. Technically, such a mixture of what is not to be mixed together is called pollution. And the mathematicians and the city planners aimed to purify and make sense where there was once pollution and disorder."

"In a memoir presented to the Royal Society of Göttingen in 1858, but afterwards withdrawn, and only published in Poggendorff's Annalen in 1867, after the death of the author, Bernhard Riemann]deduces the phenomena of the induction of electric currents from a modified form of \frac{\mathrm{d^2}V}{\mathrm{d}x^2} + \frac{\mathrm{d^2}V}{\mathrm{d}y^2} + \frac{\mathrm{d^2}V}{\mathrm{d}z^2} + 4 \pi \rho= \frac{1}{a^2} \frac{\mathrm{d^2}V}{\mathrm{d}t^2}where V is the electrostatic potential and a velocity. This equation is of the same form as those which express the propagation of waves and other disturbances in elastic media. The author, however, seems to avoid making explicit mention of any medium through which the propagation takes place. The mathematical investigation given by Riemann has been examined by Clausius, who does not admit the soundness of the mathematical processes, and shews that the hypothesis that potential is propogated like light does not lead either to the formula of Weber, or to the known laws of electrodynamics."

"Riemann bloomed relatively late and did not live to see his fortieth birthday, but he revolutionized virtually everything he touched."

"Riemann's insight followed his discovery of a mathematical looking-glass through which he could gaze at the primes. Alice's world was turned upside down when she stepped through her looking-glass. In contrast, in the strange mathematical world beyond Riemann's glass, the chaos of the primes seemed to be transformed into an ordered pattern as strong as any mathematician could hope for. He conjectured that this order would be maintained however far one stared into the never-ending world beyond the glass. His prediction of an inner harmony on the far side of the mirror would explain why outwardly the primes look so chaotic. The metamorphosis provided by Riemann's mirror, where chaos turns to order, is one which most mathematicians find almost miraculous. The challenge that Riemann left the mathematical world was to prove that the order he thought he could discern was really there."

""As is known, scientific physics dates its existence from the discovery of the differential calculus. Only when it was learned how to follow continuously the course of natural events, attempts, to construct by means of abstract conceptions the connection between phenomena, met with success. To do this two things are necessary: First, simple fundamental concepts with which to construct; second, some method by which to deduce, from the simple fundamental laws of the construction which relate to instants of time and points in space, laws for finite intervals and distances, which alone are accessible to observation (can be compared with experience)." [Riemann.] The first of the two problems here indicated by Riemann consists in setting up the differential equation, based upon physical facts and hypotheses. The second is the integration of this differential equation and its application to each separate concrete case, this is the task of mathematics."

"After Riemann had made known his discoveries, mathematicians busied themselves with working out his system of geometrical ideas formally; chief among these were Christoffel, Ricci, and Levi-Civita. Riemann... clearly left the real development of his ideas in the hands of some subsequent scientist whose genius as a physicist could rise to equal flights with his own as a mathematician. After a lapse of seventy years this mission has been fulfilled by Einstein."

"It was long accepted as a fact that a metrical character could be described by means of a quadratic differential form, but the fact was not clearly understood. Riemann many years ago pointed out that the metrical groundform might, with equal right, essentially, be a homogeneous function of the fourth order in the differentials, or even a function built up in some other way. and that it need not even depend rationally on the differentials. But we dare not stop even at that point."

"The essay of Bernhard Riemann, "On the Hypotheses which lie at the Base of Geometry," owes its great celebrity to the fact that he was a mathematical analyst of the first order, one of the favorite pupils of Gauss, under the inspiration of whose teachings, if not at his suggestion, the essay was written—by whom, in fact, it was presented, in 1854, shortly before his (Gauss's) death to the philosophical faculty of Goettingen, and by whom its cardinal propositions were expressly indorsed as an exposition of his own speculative opinions. Every intelligent reader of this essay will agree... that its intrinsic merit is not at all commensurate with the attention with which it was received and the interest with which it is still generally considered. Not only are its statements, both of the problem and of the proposed methods of solution, crude and confused, but they bear the impress throughout of Riemann's very imperfect acquaintance with the nature of logical processes and even with the import of logical terms. It is apparent... that its author was an utter stranger to the discussions respecting the nature of space which have been so vigorously carried on by the best thinkers of our time ever since the days of Kant, and that he was so little familiar with the history of logic as to be without the faintest suspicion of the manifold ambiguity of such terms as "concept" and "quantity," and of the necessity of their exact definition preliminary to an inquiry respecting the very foundations of human knowledge."

"Riemann himself modestly apologizes for the philosophical shortcomings of his essay on the ground of his inexperience in philosophical matters. But the crudeness of his speculations affords a very striking illustration... of the well-known fact that exclusive devotion to the labors of the mathematical analyst has a tendency to develop certain special powers of the intellect at the expense of its general grasp and strength. Although Sir William Hamilton, no doubt, overstated the case against the mathematicians, I believe that his suggestions are not wholly unworthy of attention, and that there is force in the words of D'Alembert (referred to by Sir William Hamilton)... We have here five distinct propositions, which... may be stated in distinct form as follows: 1. That the nature of space is to be deduced from its concept. 2. That the concept of space can be formed and determined only by its subsumption under a higher concept. 3. That our space is a "triply extended Multiple or Aggregate," the higher concept under which its concept is to be subsumed being that of an "n-fold extended Multiple" or a "multiply extended Aggregate" (eine n-fach ausgedehnte Mannigfaltigkeit), and that—translating Riemann's phraseology into its plain logical import—the (logical) extension of this higher concept determines the number of the possible kinds of space. 4. That the conceptual possibility of space is coextensive with its empirical possibility, though not with its empirical reality. 5. That continuous quantities are coördinate with discrete quantities, i.e., are species of the same genus, both being in their nature multiples or aggregates."

"To imagine that conclusions respecting the nature of space and the origin of its concept can be drawn from the mere fact that space is a function of three variables, and may thus in a manner be classified with similar functions, is a mockery of all reasoning from which an old scholastic would have turned with the scornful reminder that coördination and subsumption, for the purpose of effectually aiding in the formation of a particular concept, must not only be under a genus, but under a genus proximum."

"Riemann's third proposition that space is an "n-fold extended multiple" or a "multiply extended aggregate"... The term "Mannigfaltigkeit," as here employed, is a standing puzzle to the readers of Riemann's essay. ...Riemann adopted the term from Gauss, who was probably the originator of its employment for the designation of "space in general" (as distinguished from "flat space," in the metageometrical sense). Gauss, in turn, took the expression no doubt from Herbart... whose philosophy is to a great extent a sort of reproduction of the old Eleatic quandaries about "The One and the Many." Herbart, in fine, had obtained it from Kant, whose disciple he was, or believed himself to be, and whose phrase 'Mannigfaltigkeiten der Empfindung'is variously found... in... his followers. ...space is not a "multiple" or "aggregate" at all, but... its very essence is continuity. This... follows from its conceptual nature as well as from its relativity. ...space itself is not in any intelligible sense a quantity."

"Riemann's fourth proposition is founded on a confusion between conceptual possibility and real or empirical possibility. Conceptual possibility is determined solely by the consistency or inconsistency of the elements of the concept to be formed—it is tested simply by the logical law of non-contradiction; while empirical possibility depends upon the consistency... with the various conditions of sensible reality or... laws of nature. ...Upon this distinction depend the utility and scope of the artifice not unfre quently resorted to in certain analytical investigations of supposing the existence of a fourth spatial dimension for the purpose of reducing certain functions to a symmetrical form and this distinction too is the basis of an observation made by Boole... "Space is presented to us, in perception, as possessing the three dimensions of length, breadth, and depth. But in a large class of problems relating to the properties of curved surfaces, the rotation of solid bodies around axes, the vibration of elastic media, etc., this limitation appears in the analytical investigation to be of an arbitrary character, and, if attention were paid to the processes of solution alone, no reason could be discovered why space should not exist in four, or in any greater number of, dimensions. The intellectual procedure in the imaginary world thus suggested can be apprehended by the clearest light of analogy." Upon the same ground... Hermann Grassmann, who is sometimes referred to as one of the founders of transcendental geometry, has developed the theory of extension in its general application to an indefinite number of dimensions, although he certainly did not cherish the delusion (as seems to be supposed by Victor Schlegel) that this could be the source of inferences respecting the number of actual or empirically possible dimensions of space. On this subject we have Grassmann's own explicit declaration: "It is clear," he says, "that the concept of space can in no wise be generated by thought. ...Whoever maintains the contrary must undertake to derive the dimensions of space from the pure laws of thought—a problem which is at once seen to be impossible of solution.""

"Riemann's fifth proposition... This pernicious fallacy is one of the traditional errors current among mathematicians, and has been prolific of innumerable delusions. It is this error which has stood in the way of the formation of a rational, intelligible, and consistent theory of irrational and imaginary quantities, so called, and has shrouded the true principles of the doctrine of "complex numbers" and of the calculus of quaternions in an impenetrable haze. ... There are no "discrete quantities" except those which are dealt with in special (common) and general arithmetic, that is to say numbers. ...a number is not a quantity at all, nor a measure of quantity, but simply an intellectual vehicle of quantities—a purely subjective instrumentality for their comparison and admeasurement. ...quantities have been first divided into extensive quantities (space) and intensive quantities (forces, colors, sounds, and all subjective affections), and the extensive quantities have then been subdivided into continuous and discrete. Now, the fact is that all objects of apprehension, including all data of sense, are in themselves, i.e., within the act of apprehension, essentially continuous. They become discrete only by being subjected, arbitrarily or necessarily, to several acts of apprehension, and by thus being severed into parts, or coördinated with other objects similarly apprehended into wholes. To say that a datum of sensation or of subjective feeling is in itself discrete is to assert that it is absolute, and to deny that quantity is essentially relative. And to maintain (with those who speak of positive, negative, fractional, irrational, imaginary, complex, linear, or directional numbers) that number may be continuous is to ignore the plainest and most unmistakable fact in all our intellectual operations, and to misinterpret all the teachings of the history of mathematics. ...It is not to be expected... that mathematicians will cease, at this late day, to speak of arithmetical or algebraic symbols as "quantities;" but there may be a little hope... that they might return to the old phrase "geometrical (and other) magnitudes." The mischief lies, not so much in the use of a particular word, as in the employment of the same word for the denotation of objects differing from each other toto genera."

"The foregoing discussion has brought us to the point where the reader is in a condition.. to realize the great fundamental absurdity of Riemann's endeavor to draw inferences respecting the nature of space and the extension of its concepts from algebraic representations of "multiplicities." An algebraic multiple and a spatial magnitude are totally disparate. That no conclusions about forms of extension or spatial magnitudes are derivable from the forms of algebraic functions is evident upon the most elementary considerations."

"If Riemann's argument were fundamentally valid, it could be presented in very succinct and simple form. It would be nothing more than a suggestion that, because algebraic quantities of the first, second, and third degrees denote geometrical magnitudes of one, two, and three dimensions respectively, there must be geometrical magnitudes of four, five, six, etc., dimensions corresponding to algebraic quantities of the fourth, fifth, sixth, etc., degree. ...the analytical argument in favor of the existence, or possibility, of transcendental space is another flagrant instance of the reification of concepts."

"Einstein's theory, more especially the second part (the general theory), is intimately connected with the discoveries of the non-Euclidean geometricians, Riemann in particular. Indeed, had it not been for Riemann's work, and for the considerable extension it has conferred upon our understanding of the problem of space, Einstein's general theory could never have arisen."

"The essence of Riemann's discoveries consists in having shown that there exist a vast number of possible types of spaces, all of them perfectly self-consistent. When, therefore, it comes to deciding which one of these possible spaces real space will turn out to be, we cannot prejudge... Experiment and observation alone can yield us a clue. To a first approximation, experiment and observation prove space to be Euclidean, and this accounts for our natural belief... merely by force of habit. But experiment is necessarily inaccurate, and we cannot foretell whether our opinions will not have to be modified when our experiments are conducted with greater accuracy. Riemann's views thus place the problem of space on an empirical basis excluding all a priori assertions on the subject. ...the relativity theory is very intimately connected with this empirical philosophy; for... Einstein is compelled to appeal to a varying non-Euclideanism of four-dimensional space-time in order to account with extreme simplicity for gravitation. ...had the extension of the universe been restricted on a priori grounds... to three dimensional Euclidean space, Einstein's theory would have been rejected on first principles. ...as soon as we recognise that the fundamental continuum of the universe and its geometry cannot be posited a priori... a vast number of possibilities are thrown open. Among these the four-dimensional space-time of relativity, with its varying degrees of non-Euclideanism, finds a ready place."

"Prior to Riemann's discoveries it was thought that the absence of a boundary would necessitate the infiniteness of space. To-day we know that this belief is unjustified, for space can be finite and yet unbounded."

"With the new views advocated by Riemann... the texture, structure or geometry of space is defined by the metrical field, itself produced by the distribution of matter. Any non-homogeneous distribution of matter would then entail a variable structure of geometry for space from place to place. ... Riemann's exceedingly speculative ideas on the subject of the metrical field were practically ignored in his day, save by the English mathematician Clifford, who translated Riemann's works, prefacing them to his own discovery of the non-Euclidean Clifford space. Clifford realised the potential importance of the new ideas and suggested that matter itself might be accounted for in terms of these local variations of the non-Euclidean space, thus inverting in a certain sense Riemann's ideas. But in Clifford's day, this belief was mathematically untenable. Furthermore, the physical exploration of space seemed to yield unvarying Euclideanism. ...it was reserved for the theoretical investigator Einstein, by a stupendous effort of rational thought, based on a few flimsy empirical clues, to unravel the mystery and to lead Riemann's ideas to victory. (In all fairness to Einstein... he does not appear to have been influenced directly by Riemann.) Nor were Clifford's hopes disappointed, for the varying non-Euclideanism of the continuum was to reveal the mysterious secret of gravitation, and perhaps also of matter, motion, and electricity. ... Einstein had been led to recognize that space of itself was not fundamental. The fundamental continuum whose non-Euclideanism was fundamental was... one of Space-Time... possessing a four-dimensional metrical field governed by the matter distribution. Einstein accordingly applied Riemann's ideas to space-time instead of to space... He discovered that the moment we substitute space-time for space (and not otherwise), and assume that free bodies and rays of light follow geodesics no longer in space but in space-time, the long-sought-for local variations in geometry become apparent. They are all around us, in our immediate vicinity... We had called their effects gravitational effects... never suspecting that they were the result of those very local variations in the geometry for which our search had been in vain....the theory of relativity is the theory of the space-time metrical field."

"Let us revert to the metrical field, as defining the space-time structure. Although Riemann had attributed the existence of the structure, or metrical field, of space to the binding forces of matter, there is not the slightest indication in Einstein's special theory that any such view is going to be developed later on; in fact, it does not appear that Einstein was influenced in the slightest degree by Riemann's ideas. ...in the special theory, the problem of determining whence the structure, or field, arises, what it is, what causes it, is not even discussed in a tentative manner. Space-time, with its flat structure, is assumed to be given or posited by the Creator. But in the general theory the entire situation changes when Einstein accounts for gravitation, hence for a varying lay of the metrical field, in terms of a varying non-Euclidean structure of space-time around matter. We are then compelled to recognise not only that the metrical field regulates the behaviour of material bodies and clocks, as was also the case in the special theory, but, furthermore, that a reciprocal action takes place and that matter and energy in turn must affect the lay of the metrical field. But we are still a long way from Riemann's view that the field is not alone affected but brought into existence by matter; and it is only when we consider the cosmological part of Einstein's theory that this idea of Riemann's may possibly be vindicated. And here we come to a parting of the ways with de Sitter and Eddington on one side, Einstein and Thirring on the other, and Weyl somewhere in between the two extremes."

"My memoirs "On the Mechanical Theory of Heat" are of different kinds. Some are devoted to the development of the general theory and to the application thereof to those properties of bodies which are usually treated of in the doctrine of heat. Others have reference to the application of the mechanical theory of heat to electricity. ...Other memoirs... have reference to the conceptions I have formed of the molecular motions which we call heat. These conceptions, however, have no necessary connexion with the general theory, the latter being based solely on certain principles which may be accepted without adopting any particular view as to the nature of molecular motions. I have therefore kept the consideration of molecular motions quite distinct from the exposition of the general theory."

"The steam-engine having furnished us with a means of converting heat into a motive power, and our thoughts being thereby led to regard a certain quantity of work as an equivalent for the amount of heat expended in its production, the idea of establishing theoretically some fixed relation between a quantity of heat and the quantity of work which it can possibly produce, from which relation conclusions regarding the nature of heat itself might be deduced, naturally presents itself. Already, indeed, have many successful efforts been made with this view; I believe, however, that they have not exhausted the subject, but that, on the contrary, it merits the continued attention of physicists... The most important investigation in connexion with this subject is that of S. Carnot. Later... represented analytically... by Clapeyron"

"Carnot proves that whenever work is produced by heat... a... quantity of heat passes from a warm body to... cold... [e.g.,] the vapour... generated in the of a steam-engine... passes... to the condenser where it is precipitated... This transmission Carnot regards as the change of heat corresponding to the work... He says... no heat is lost in the process, that... [its] quantity remains unchanged; and he adds, "This is a fact... never... disputed... confirmed by various calorimetric experiments. To deny it, would be to reject the entire theory of heat, of which it forms the principal foundation.""

"The careful experiments of Joule, who developed heat... by the application of mechanical force, establish... not only the possibility of increasing the quantity of heat, but also the fact that the newly-produced heat is proportional to the work expended in its production.<!--p. 15->"

"[M]any facts.. lately transpired... tend to overthrow the hypothesis that heat is... a body, and to prove that it consists in a motion of the... particles of bodies. If... so... principles of mechanics may be applied to heat; this motion may be converted into work, the loss of '... being proportional to... work produced.<!--p. 15->"

"[M]any facts have lately transpired which tend to overthrow the hypothesis that heat is itself a body, and to prove that it consists in a motion of the ultimate particles of bodies. If this be so, the general principles of mechanics may be applied to heat; this motion may be converted into work, the loss of ' in each particular case being proportional to the quantity of work produced. These circumstances, of which Carnot was also well aware, and the importance of which he expressly admitted, pressingly demand a comparison between heat and work, to be undertaken with reference to the divergent assumption that the production of work is not only due to an alteration in the distribution of heat, but to an actual consumption thereof; and inversely, that by the expenditure of work, heat may be produced."

"[T]he new theory is opposed, not to the real fundamental principle of Carnot, but to the addition "no heat is lost;" for it is... possible that in the production of work... a certain portion of heat may be consumed, and a further portion transmitted from a warm body to a cold one; and both portions may stand in a certain definite relation to the quantity of work produced.<!--p. 17->"

"I. Deductions from the principle of the equivalence of heat and work. We shall forbear entering... on the nature of the motion... supposed... within a body, and shall assume... a motion of the particles... and that heat is the measure of their '. Or... more generally... lay down one maxim... founded on the above assumption: — In all cases where work is produced by heat, a quantity of heat proportional to the work done is consumed; and inversely, by the expenditure of a like quantity of work, the same amount of heat may be produced."

"[T]he entire quantity of heat, Q, absorbed by the gas during a change of volume and temperature may be decomposed into two portions. One of these, U, which comprises the sensible heat and the heat necessary for interior work, if... present... determined by the state of the gas at the beginning and at the end of the alteration; while the other portion... the heat expended on exterior work, depends, not only upon the state of the gas at these two limits but also upon the manner in which the alterations have been effected..."

"In my memoir "On the Moving Force of Heat, &c." I have shown that the theorem of the equivalence of heat and work, and Carnot's theorem, are not mutually exclusive, but that, by a small modification... they can be brought into accordance."

"Theorem of the equivalence of Heat and Work. ...Mechanical work may be transformed into heat, and conversely heat into work, the magnitude of the of the one being always proportional to that of the other."

"The s... may be divided into two classes: those which the atoms of a body exert upon each other... which depend... upon the nature of the body, and those which arise from the foreign influences to which the body may be exposed. According to these two classes of forces... I have divided the work done by heat into interior and exterior work."

"Let Q... be the quantity of heat which must be imparted to a body during its passage... from one condition to another, any heat withdrawn from the body being... [a] negative quantity... Q may be divided into three parts... the first... in increasing the heat... in the body, the second in producing the interior [work]... the third in producing the exterior work. ...[T]he second ...[and] first... together... represented by... function U... [are] completely determined by the initial and final states of the body. The third part... the equivalent of exterior work, can, like this work itself, only be determined when the precise manner in which the changes of condition took place is known. If W be the quantity of exterior work, and A the equivalent of heat for the unit of work, the value of the third part will be A · W, and the first fundamental theorem will be..."

"When the several changes are... such... that... the body returns to its original condition... these changes form a cyclical process, we haveand..."

"[W]e may consider the as well as the whole condition of the body... as determined so soon as its t and v are given. We... make these two magnitudes... independent variables, and... consider the pressure p as well as the quantity U... as functions of these. If... t and v receive the increments dt and dv, the corresponding quantity of exterior work done... during an increment of volume dv will be pdv. Hence... and... we obtain"

"Carnot's theorem... brought into agreement with the first fundamental theorem, expresses a relation between... the transformation of heat into work, and the passage of heat from a warmer to a colder body... regarded as... heat at a higher, into heat at a lower temperature. The theorem... may be enunciated... as:—In all cases where a quantity of heat is converted into work, and where the body effecting this transformation... returns to its original condition, another quantity of heat must necessarily be transferred from a warmer to a colder body; and the magnitude of the last quantity of heat, in relation to the first, depends only upon the temperatures of the bodies between which heat passes, and not upon the nature of the body effecting the transformation."

"This principle, upon which the whole of the following development rests, is... Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time. Everything we know the interchange of heat between two bodies of different temperatures confirms this; for heat everywhere manifests a tendency to equalize existing differences of temperature..."

"[W]e... consider the conversion of work into heat and... the passage of heat from a higher to a lower temperature as positive transformations."

"[T]he equivalence-value of the transformation of work into the quantity of heat Q, of the temperature t, may be represented... wherein f(t) is...[the same] function of the temperature... for all cases. When Q is negative... it will indicate that the quantity... transformed... from heat into work."

"In a similar manner... passage of the quantity of heat Q, from... temperature t1 to the temperature t2 must be proportional to the quantity Q, and... can only depend upon the two temperatures.... expressed bywherein F(t1,t2) is...[the same] function of... temperatures... for all cases, and... must change... sign when... temperatures are interchanged; so..."

"[I]n every reversible cyclical process... the two transformations... must be equal in magnitude, but opposite in sign; so that their algebraical sum must be zero."

"so that the temperatures t and t′ being arbitrary, the function of two temperatures which applies to the second kind of transformation is reduced... to the function of one temperature which applies to the first kind."

"[A] simpler symbol for the last function, or rather for its reciprocal... will... be... more convenient... Let...so that T is now the unknown function of the temperature... T1, T2, &c. shall represent... values of this function, corresponding to... t1, t2, &c."

"[T]he second fundamental theorem in the mechanical theory of heat... appropriately... called the theorem of the equivalence of transformations..: If two transformations which, without necessitating any other permanent change, can mutually replace one another, be called equivalent, then the generation of the quantity of heat Q of the temperature t from work, has the equivalence-value'and the passage of the quantity of heat Q from the temperature t1 to the temperature t2, has the equivalence-value'wherein T is a function of the temperature, independent of the nature of the process by which the transformation is effected."

"If to the last expression we give the form...the passage of the quantity of heat Q, from the temperature t1 to the temperature t2, has the same equivalence value as... the transformation of the quantity Q from heat at the temperature t1 into work, and from work into heat at the temperature t2."

"We proceed now to the consideration of non-reversible cyclical processes. ...[W]e obtain the following theorem, which applies generally to all cyclical processes, those that are reversible forming the limit:—The algebraical sum of all [non-reversible] transformations occurring in a cyclical process can only be positive."

"[T]he equationis the analytical expression, for all reversible cyclical processes, of the second fundamental theorem in the mechanical theory of heat."

"[T]he function T... hitherto... undetermined; ...by means of a very probable hypothesis it will be possible ...to do. I refer to... my former memoir... that a permanent gas, when it expands at a constant temperature, absorbs only so much heat as is consumed by the exterior work thereby performed. This assumption has been verified by... experiments of Regnault, and in... probability is accurate for all gases to the same degree as Mariotte and Gay-Lussac's law, so that for an ', for which the latter law is perfectly accurate, the above assumption will also be perfectly accurate."

"[A]ccording to Mariotte and Gay-Lussac's law,{{center|1=p = \frac{a+t}{v} \cdot \text{const}.}}"

"T is nothing more than the temperature counted from a°, or about 273° C. below the freezing-point; and, considering the point... as the absolute zero... T is simply the absolute temperature. ...For this reason I introduced, at the commencement, ...T for the reciprocal value of the function f(t)."

"[T]he mechanical theory of heat, had... origin in the well-known fact that heat may be employed for producing mechanical work... [W]e may naturally anticipate that the theory... will in its turn help to place this application... in a clearer light."

"Besides these reasons, which apply to all thermo-dynamic machines, there are others, applicable... particularly to the... steam-engine... [W]ith respect to vapour at a maximum ... this new theory has led... to laws which differ... from those formerly accepted as true..."

"[A] fact proved by Rankine and myself... when a quantity of vapour, at its maximum density... enclosed by a surface impenetrable to heat, expands and thereby displaces... e, g. a , with its full force of expansion, a part of the vapour must undergo condensation; whereas in most works on the steam-engine, amongst others in the excellent work of De Pambour, Watt's theorem, that under these circumstances the vapour remains... at... maximum density, is assumed... fundamental..."

"[I]t was formerly assumed, in determining the volumes of the unit of weight of saturated vapour at different temperatures, that vapour even at... maximum density... obeys Mariotte's and GayLussac's laws. ...I have ...shown in my first memoir... the volumes in question can be calculated... under the assumption, that a permanent gas when it expands at a constant temperature only absorbs so much heat as is consumed in the external work thereby performed, and that these calculations lead to values which, at least at high temperatures, differ considerably from Mariotte's and Gay-Lussac's laws."

"William Thomson... in March 1851... regarded this result as a proof of the improbability of the above assumption which I had employed. Since then, however, he and J. P. Joule have together undertaken to test experimentally the accuracy... [and] have... shown... with... permanent gases, atmospheric air and hydrogen, the assumption is so nearly true... deviations from exactitude may be disregarded. With [non-permanent gas,] ... deviations were greater... in... accordance with...[my] remark... that the latter would probably be... be accurate for each gas in the same measure as Mariotte's and Gay-Lussac's laws were applicable... Thomson now calculates the volumes of saturated vapours in the same manner as myself."

"[T]he mechanical theory of heat... render[s] a new investigation of... [the former theory of steam-engines] necessary."

"In the present memoir I... develope... principles of the calculation of the work of the steam-engine. I have... limited myself to the steam-engines now in use, without... consideration of... recent... interesting attempts to employ vapour in a superheated state."

"The expression "a machine is driven by heat" is not... strictly accurate. ...[I]n consequence of the changes produced by heat upon ...matter in the machine, the parts ...are set in motion. ...[T]his matter ...[is] that which manifests the action of heat."

"[T]he matter... must at... regularly-recurring periods be present in the machine in equal quantity, and in the same state."

"[W]e may apply the theorems concerning cyclical processes to all thermo-dynamic machines, and thereby arrive at conclusions... independent of the nature of the processes executed by the several machines."

"If for the entire universe we conceive the same magnitude to be determined, consistently and with due regard to all circumstances, which for a single body I have called entropy, and if at the same time we introduce the other and simpler conception of energy, we may express in the following manner the fundamental laws of the universe which correspond to the two fundamental theorems of the mechanical theory of heat. 1. The energy of the universe is constant. 2. The entropy of the universe tends to a maximum."

"In The Kind of Motion We Call Heat, Clausius had shown how to relate the temperature and pressure of a volume of gas to the motion of the atoms, and was able to deduce their average speed. ...That calculation drew a quick response from the Dutch meteorologist Christopher Buys Ballot. ...It atoms were really flying through the air at hundreds of meters per second, shouldn't the fragrant vapors of a hot dinner race through the room...? In figuring out the answer... Clausias added a fundamentally new innovation to gas theory. Atoms... banged into each other a good deal. ...battling through all the other atoms ...What mattered was the average distance between collisions. This turned out to be an all-important quantity... and Clausius gave it the name mean free path."

"In their calculations, Clausius (and Waterston, for that matter) had imagined all atoms in a gas moving at the same speed. They knew this wasn't true... but they didn't have the mathematical sophistication to tackle the full problem. Maxwell... defined a mathematical function called the distribution of velocities, which kept track of how many atoms were moving at any particular speed relative to the average, and by dealing in terms of this distribution... was able to give his calculations a precision that those of Clausius lacked."

"The views of Joule, Mayer, and others were assimilated into the theory of heat engines by Kelvin at Glasgow and Rudolph Clausius at Berlin. They noted that when gases and vapours expanded against an opposing force and performed mechanical work they lost heat. ...the law was put forward as a general principle by Clausius and Kelvin in 1851. Whilst the amount of heat decreased during the cycle of operations of the Carnot heat engine, it was seen that there was a quantity which remained constant throughout the cycle. The amount of heat given out was smaller than that taken in by the heat engine, but the quantity of heat taken in divided by the temperature of the heat source had quantitatively the same value as the amount of heat given out divided by the temperature of the heat sink. Clausius in 1865 termed this quotient, the entropy. Clausius pointed out that Carnot's perfect heat engine was rather an abstraction...The entropy... tended to increase in spontaneous natural processes, not to remain constant as in the perfect heat engine."

"The equation of Clausius to which I must now call your attention is of the following form:pV=\frac{2}{3}T-\frac{2}{3}\sum\sum(\frac{1}{2}Rr).Here p denotes the pressure of a fluid, and V the volume of the vessel which contains it. The product pV, in the case of gases at constant temperature, remains, as Boyle's Law tells us, nearly constant for different volumes and pressures. ...The other member of the equation consists of two terms, the first depending on the motion of the particles, and the second on the forces with which they act on each other. The quantity T is the kinetic energy of the system... that part of the energy which is due to the motion of the parts of the system. ...In the second term, r is the distance between any two particles, and R is the attraction between them. ...The quantity &frac12;Rr or half the product of the attraction into the distance across which the attraction is exerted is defined by Clausius as the virial of the attraction. &sum;&sum;(&frac12;Rr)... indicates that the value of &frac12;Rr is to be found for every pair of particles and the results added together. Clausius has established this equation by a very simple mathematical process... it indicates two causes which may affect the pressure of the fluid on the vessel which contains it... We may therefore attribute the pressure of a fluid either to the motion of its particles or to a repulsion between them."

"To him we are indebted for the conception of the mean length of the path of a molecule of a gas between its successive encounters with other molecules. As soon as it was seen how each molecule, after describing an exceedingly short path, encounters another, and then describes a new path in a quite different direction, it became evident that the rate of diffusion of gases depends not merely on the velocity of the molecules, but on the distance they travel between each encounter."

"He opened up a new field of mathematical physics by shewing how to deal mathematically with moving systems of innumerable molecules."

"Carnot's annunciation of his theory was defective in that it took no notice of the fact that the hot body gives out more heat than the cold one receives from it, and that it regarded as equal the amount of heat received upon one isothermal side of a cycle and that emitted from the other side; a principle that may hold good for infinitely small cycles, but not for larger ones, in which a difference exists between the thermic quantities proportioned to the size of the cycle. This error and the true condition as pointed out by Clausius are defined by Prof. Rankine, who says, in his paper "On the Economy of Heat in Expansive Machines": "Carnot was the first to assert the law that the ratio of the maximum mechanical effect to the whole heat expended in an expansive machine is a function solely of the two temperatures at which the heat is respectively received and emitted, and is independent of the nature of the working substance. But his investigations, not being based on the principle of the dynamic convertibility of heat, involve the fallacy that power can be produced out of nothing. The merit of combining Carnot's law, as it is termed, with that of the convertibility of heat and power, belongs to Mr. Clausius and Prof. William Thomson; and, in the shape in which they have brought it, it may be stated thus: The maximum proportion of heat converted into expansive power by any machine is a function solely of the temperatures at which heat is received and emitted by the working substance, which function for each pair of temperatures is the same for all substances in nature." The law as thus modified and newly expressed might, as M. Langlois remarks, be designated as the equation of Clausius. But Clausius himself, acknowledging the influence which the Frenchman's ideas had exercised upon him, called it the theorem of Carnot."

"Sadi’s pamphlet finds its way into the hands of... Rudolf Clausius. It is he who grasps the fundamental issue at stake, formulating a law that was destined to become famous: if nothing else around it changes, heat cannot pass from a cold body to a hot one. ...[A] ball may fall, but it can also come back up, by rebounding... Heat cannot. This is the only basic law of physics that distinguishes the past from the future. None of the others do. Not Newton's laws governing... mechanics... not the equations for electricity and magnatism... by Maxwell. Not Einstein's on relativistic gravity, nor those of quantum mechanics... by Heisenberg, Schrödinger, and Dirac. Not those for elementary particles... by twentieth-century physicists. Not one of these distinguishes... past from... future. If a sequence of events is allowed by these equations, so is the same sequence run backward in time."

"The essential feature of Maxwell's work was showing that the properties of gases made sense not if gas molecules all flew around at a similar "average" velocity, as Clausius had surmised, but only if they moved at all sorts of speeds, most near the average, but some substantially faster or slower, and a few very fast or slow. ...Just as Quetelet's average man was fictitious, and key insights into society came from analyzing the spread of features around the average, understanding gases meant figuring out the range and distribution of molecular velocities around the average. And that distribution, Maxwell calculated, matched the bell-shaped curve describing the range of measurement errors."

"There is no doubt that Clausius with this paper created classical thermodynamics. Compared with his work here, all preceding except Carnot's is of small moment. Clausius exhibits here the quality of a great discoverer: to retain from his predecessors major and minor—in this case, from LaPlace, Poisson, Carnot, Mayer, Holtzmann, Helmholtz, and Kelvin—what is sound while frankly discarding the rest, to unite previously disparate theories, and by one simple if drastic change to construct a complete theory that is new yet firmly based upon previous partial success. ...By no means disregarding the results of experiment, Clausius was the first theorist of thermodynamics who was not enslaved to them... those which to him seemed dubious were to be rejected... Clausius had another handle... his kinetic theory of gases... Both Rankine's model and Clausius' model... led to a theory... "dynamical"... [F]aith... gave... Rankine and Clausius... confidence... while Kelvin, not yet an atomist, wavered. ...[I]n the molecular theory Clausius was not only the wiser man but also the better physicist."

"The name and fame of Professor Clausius stand as high in this country as in his own. ...his writings ...fell into my hands at a time when I knew but little of the Mechanical Theory of Heat. In those days their author was my teacher; and in many respects I am proud to acknowledge him as my teacher still."

"As I was reading the extract from your paper in the geometric sum and difference... I was struck by the marvelous similarity between your results and those discoveries which I made even as early as 1832... I conceived the first idea of the geometric sum and difference of two or more lines and also of the geometric product of two or three lines in that year (1832). This idea is in all ways identical to that presented in your paper. But since I was for a long time occupied with entirely different pursuits, I could not develop this idea. It was only in 1839 that I was led back to that idea and pursued this geometrical analysis up to the point where it ought to be applicable to all mechanics. It was possible for me to apply this method of analysis to the theory of tides, and in this I was astounded by the simplicity of the calculations resulting from this method."

"From the imputation of confounding axioms with assumed concepts Euclid himself, however, is free. Euclid incorporated the former among his postulates while he separated the latter as common concepts—a proceeding which even on the part of his commentators was no longer understood, and likewise with modern mathematicians, unfortunately for science, has met with little imitation. As a matter of fact, the abstract methods of mathematical science know no axioms at all."

"Geometry can in no way be viewed... as a branch of mathematics; instead, geometry relates to something already given in nature, namely, space. I... realized that there must be a branch of mathematics which yields in a purely abstract way laws similar to geometry."

"It is clear... that the concept of space can in no wise be generated by thought. ...Whoever maintains the contrary must undertake to derive the dimensions of space from the pure laws of thought—a problem which is at once seen to be impossible of solution."

"The first impulse came from the consideration of negatives in geometry; I was accustomed to viewing the distances AB and BA as opposite magnitudes. Arising from this idea was the conclusion that if A, B, and C are points of a straight line, then in all cases AB + BC = AC, this being true whether AB and BC are directed in the same direction or in opposite directions (where C lies between A and B). In the latter case AB and BC were not viewed as merely lengths, but simultaneously their considered since they were oppositely directed, Thus dawned the distinction between the sum of lengths and the sum of distances which were fixed in direction. From this resulted the requirement for establishing this latter concept of sum, not simply for the case where the distances were directed in the same or opposite directions, but also for any other case. This could be done in the most simple manner, since the law that AB + BC = AC remains valid when A, B, and C do not lie on a straight line. This then was the first step which led to a new branch of mathematics... I did not however realize how fruitful and how rich was the field that I had opened up; rather that result seemed scarcely worthy of note until it was combined with a related idea."

"While I was pursuing the concept of geometrical product, as this idea was established by my father... I concluded that not only rectangles, but also parallelograms, may be viewed as products of two adjacent sides, provided that the sides are viewed not merely as lengths, but rather as directed magnitudes. When I joined this concept of geometrical product with the previously established idea of geometrical sum the most striking harmony resulted. Thus when I multiplied the sum of two vectors by a third coplaner vector, the result coincided (and must always coincide) with the result obtained by multiplying separately each of the two original vectors by the third... and adding together (with due attention to positive and negative values) the two products. [Thus A(B + C) = AB + AC.] From this harmony I came to see a whole new area of analysis was opening up which could lead to important results."

"A work on tidal theory... led me to Lagrange's Mécanique analytique and thereby I returned to those ideas of analysis. All the developments in that work were transformed through the principles of the new analysis in such a simple way that the calculations often came out more than ten times shorter than in Lagrange's work."

"The concept of rotation led to geometrical exponential magnitudes, to the analysis of angles and of trigonometric functions, etc. I was delighted how thorough the analysis thus formed and extended, not only the often very complex and unsymmetric formulae which are fundamental in tidal theory, but also the technique of development parallels the concept."

"I feel entitled to hope that I have found in this new analysis the only natural method according to which mathematics should be applied to nature, and according to which geometry may also be treated, whenever it leads to general and to fruitful results."

"The concept of centroid as sum led me to examine Möbius' Barycentrische Calcul, a work of which until then I knew only the title; and I was not little pleased to find here the same concept of the summation of points to which I had been led in the course of the development. This was the first, and... the only point of contact which my new system of analysis had with the one that was already known."

"I define as a unit any magnitude that can serve for the numerical derivation of a series of magnitudes, and in particular I call such a unit an original unit if it is not derivable from another unit. The unit of numbers, that is one, I call the absolute unit, all others relative. Zero can never be a unit."

"It was natural that Grassmann chose to introduce his system, not by means of a paper, but rather by means of a long and complicated book. ...such ideas as Grassmann's form of the scaler (dot) and vector (cross) products... have counterparts in modern vector analysis."

"One may say without great exaggeration that Grassmann invented linear algebra and, with none at all, that he showed how properly to apply it to geometry. ...He ...anticipated in its most important aspects Peano's treatment of the natural numbers, published 28 years later. ...A feature of Grassmann's work, far in advance of the times, is the tendency towards the use of the implicit definition. ...The definition of a linear space (or vector space) came into mathematics, in the sense of becoming widely known, around 1920, when Hermann Weyl and others published formal definitions. ...Grassmann did not put down a formal definition—again, the language was not available—but there is no doubt that he had the concept."

"The history of geometry may be conveniently divided into five periods. The first extends from the origin of the science to about A. D. 550, followed by a period of about 1,000 years during which it made no advance, and in Europe was enshrouded in the darkness of the middle ages; the second began about 1550, with the revival of the ancient geometry; the third in the first half of the 17th century, with the invention by Descartes of analytical or modern geometry; the fourth in 1684, with the invention of the differential calculus; the fifth with the invention of descriptive geometry by Monge in 1795. The quaternions of Sir William Rowan Hamilton the Ausdehnungslehre of Dr. Hermann Grassmann, and various other publications, indicate the dawn of a new period. Whether they are destined to remain merely monuments of the ingenuity and acuteness of their authors, or are to become mighty instruments in the investigation of old and the discovery of new truths, it is perhaps impossible to predict."

"The wonderful and comprehensive system of Multiple Algebra invented by Hermann Grassmann, and called by him the Ausdehnungslehre or Theory of Extension, though long neglected by the mathematicians even of Germany, is at the present time coming to be more and more appreciated and studied. In order that this system, with its intrinsic naturalness, and adaptability to all the purposes of Geometry and Mechanics, should be generally introduced to the knowledge of the coming generation of English-speaking mathematicians, it is very necessary that a text-book should be provided, suitable for use in colleges and universities, through which students may become acquainted with the principles of the subject and its applications."

"As the great generality of Grassmann's processes—all results being obtained for n-dimensional space—has been one of the main hindrances to the general cultivation of his system, it has been thought best to restrict the discussion to space of two and three dimensions."

"Grassmann's first publication of his new system was made in 1844 in a book entitled "Die Lineale Ausdehnungslehre Ein Neuer Zweig der Mathematik." His novel and fruitful ideas were however presented in a somewhat abstruse and unusual form, with the result, as the author himself states in the preface to the second edition issued in 1878, that scarcely any notice was taken of the book by Mathematicians. He was finally convinced that it would be necessary to treat the subject in an entirely different manner in order to gain the attention of the mathematical world. Accordingly in 1862 he published "Die Ausdehnungslehre vollständig und in strenger Form bearbeitet," in which the treatment is algebraic... Since that time his great work has been more fully appreciated, but not even yet, in the opinion of the writer, at its real value."

"The exchange theorem... is sometimes called the Steinitz exchange theorem after Ernst Steinitz... The result was first proved Hermann Günther Graßmann..."

"Some of the groundbreaking work in the treatment of n-dimensional geometry—was carried out by Hermann Günther Grassmann. ...Grassmann was responsible for the creation of an abstract science of "spaces," inside which the usual geometry was only a special case. Grassmann published his pioneering ideas (originating a branch of mathematics known as linear algebra) in 1844, in a book commonly known as Ausdehnungslehre... Grassmann's suggestion that BA = -AB violates one of the sacrosanct laws of arithmetic... Grassmann faced up squarely to this disturbing possibility and invented a new consistent algebra (known as exterior algebra) that allowed for several processes of multiplication and at the same time could handle geometry in any number of dimensions."

"As professor in the Polytechnic School [autumn of 1858] in Zurich I found myself for the first time obliged to lecture upon the elements of the differential calculus and felt, more keenly than ever before, the lack of a really scientific foundation for arithmetic. In discussing the notion of the approach of a variable magnitude to a fixed limiting value, and especially in proving the theorem that every magnitude which grows continually, but not beyond all limits, must certainly approach a limiting value, I had recourse to geometric evidences. Even now such resort to geometric intuition in a first presentation of the differential calculus, I regard as exceedingly useful, from the didactic standpoint, and indeed indispensable, if one does not wish to lose too much time. But that this form of introduction into the differential calculus can make no claim to being scientific, no one will deny. For myself this feeling of dissatisfaction was so overpowering that I made the fixed resolve to keep meditating on the question till I should find a purely arithmetic and perfectly rigorous foundation for the principles of infinitesimal analysis."

"The statement is so frequently made that the differential calculus deals with continuous magnitude, and yet an explanation of this continuity is nowhere given; even the most rigorous expositions of the differential calculus do not base their proofs upon continuity but, with more or less consciousness of the fact, they either appeal to geometric notions or those suggested by geometry, or depend upon theorems which are never established in a purely arithmetic manner. Among these, for example, belongs the above mentioned theorem, and a more careful investigation convinced me that this theorem, or any one equivalent to it, can be regarded in some way as a sufficient basis for infinitesimal analysis. It then only remained to discover its true origin in the elements of arithmetic and thus at the same time to secure a real definition of the essence of continuity. I succeeded Nov. 24, 1858."

"What advantage will be gained by even a purely abstract definition of real numbers of a higher type, I am as yet unable to see, conceiving as I do of the domain of real numbers as complete in itself."

"I regard the whole of arithmetic as a necessary, or at least natural, consequence of the simplest arithmetic act, that of counting, and counting itself as nothing else than the successive creation of the infinite series of positive integers in which each individual is defined by the one immediately preceding; the simplest act is the passing from an already-formed individual to the consecutive new one to be formed. The chain of these numbers forms in itself an exceedingly useful instrument for the human mind; it presents an inexhaustible wealth of remarkable laws obtained by the introduction of the four fundamental operations of arithmetic."

"Addition is the combination of any arbitrary repetitions of the above-mentioned simplest act into a single act; from it in a similar way arises multiplication. While the performance of these two operations is always possible, that of the inverse operations, subtraction and division, proves to be limited. Whatever the immediate occasion may have been, whatever comparisons or analogies with experience, or intuition, may have led thereto; it is certainly true that just this limitation in performing the indirect operations has in each case been the real motive for a new creative act; thus negative and fractional numbers have been created by the human mind; and in the system of all rational numbers there has been gained an instrument of infinitely greater perfection. This system, which I shall denote by R, possesses first of all a completeness and self-containedness which I have designated... as characteristic of a body of numbers [Zahlkőrper] and which consists in this, that the four fundamental operations are always performable with any two individuals in R, i.e., the result is always an individual of R, the single case of division by the number zero being excepted."

"The system R forms a well-arranged domain of one dimension extending to infinity on two opposite sides. What is meant by this is sufficiently indicated by my use of expressions borrowed from geometric ideas; but just for this reason it will be necessary to bring out clearly the corresponding purely arithmetic properties in order to avoid even the appearance as if arithmetic were in need of ideas foreign to it."

"If a, c are two different numbers, there are infinitely many different numbers lying between a, c."

"If a is any definite number, then all numbers of the system R fall into two classes, A1 and A2, each of which contains infinitely many individuals; the first class A1 comprises all numbers a1 that are < a, the second class A2 comprises all numbers a2 that are > a; the number a itself may be assigned at pleasure to the first or second class, being respectively the greatest number of the first class or the least of the second. In every case the separation of the system R into the two classes A1, A2 is such that every number of the first class A1 is less than every number of the second class A2."

"The way in which the irrational numbers are usually introduced is based directly upon the conception of extensive magnitudes—which itself is nowhere carefully defined—and explains number as the result of measuring such a magnitude by another of the same kind. Instead of this I demand that arithmetic shall be developed out of itself."

"That such comparisons with non-arithmetic notions have furnished the immediate occasion for the extension of the number-concept may, in a general way, be granted (though this was certainly not the case in the introduction of complex numbers); but this surely is no sufficient ground for introducing these foreign notions into arithmetic, the science of numbers."

"Just as negative and fractional rational numbers are formed by a new creation, and as the laws of operating with these numbers must and can be reduced to the laws of operating with positive integers, so we must endeavor completely to define irrational numbers by means of the rational numbers alone. The question only remains how to do this."

"The above comparison of the domain R of rational numbers with a straight line has led to the recognition of the existence of gaps, of a certain incompleteness or discontinuity of the former, while we ascribe to the straight line completeness, absence of gaps, or continuity. In what then does this continuity consist? Everything must depend on the answer to this question, and only through it shall we obtain a scientific basis for the investigation of all continuous domains."

"In the preceding section attention was called to the fact that every point p of the straight line produces a separation of the same into two portions such that every point of one portion lies to the left of every point of the other. I find the essence of continuity in the converse, i.e., in the following principle: "If all points of the straight line fall into two classes such that every point of the first class lies to the left of every point of the second class, then there exists one and only one point which produces this division of all points into two classes, this severing of the straight line into two portions." ...every one will at once grant the truth of this statement; the majority of my readers will be very much disappointed in learning that by this commonplace remark the secret of continuity is to be revealed."

"Although the real theory might have been less useful than the complex in obtaining properties of special functions, its significance for the development of mathematics as a whole has been incomparably greater. It was in the real variable that the necessity for a rigorous theory of the number system of analysis was first recognized. ...the reconstruction of the real number system by Weierstrass in the 1860's and by Dedekind and Cantor in the 1870's led in the last three decades of the nineteenth century, to a profound reconsideration of the nature of all mathematical reasoning. This in turn initiated some of the most searching examinations of all deductive reasoning since the days or Aristotle. Thus the theory of the functions of a real variable since the 1870's has increasingly acquired more than merely a local interest: its problems, solved and unsolved, are significant in fields far distant from technical mathematics."

"In discussing the notion of the approach of a variable magnitude to a fixed limiting value, [Dedekind] had recourse, as had Cauchy before him, to the evidence of the geometry of continuous magnitude. ...Dedekind's approach was somewhat different from that of Weierstrass, Méray, Heine, and Cantor in that, instead of considering in what manner the irrationals are to be defined so as to avoid the vicious circle of Cauchy, he asked himself... what is the nature of continuity? ...The philosophy and mathematics of Leibniz had led him to agree with Galileo that continuity was a property concerning conjunctive aggregation, rather than a unity or coincidence of parts. Leibniz had regarded a set as forming a continuum if between any two elements there was always another element of the set. ...Ernst Mach likewise regarded this property of denseness of an assemblage as constituting its continuity, but... rational numbers... possess the property of denseness and yet do not constitute a continuum. Dedekind...found the essence of... continuity, not by a vague hang-togetherness, but in the nature of the division of the line by a point. ...in any division of the points into two classes such that every point of the one is to the left of every point of the other, there is one and only one point which produces this division. This is not true of the ordered system of rational numbers."

"The modern theory of functions of one real variable was first worked out by H. Hankel, Dedekind, G. Cantor, Dini, and Heine, and then carried further, principally, by Weierstrass, Schwarz, Du Bois-Reymond, Thomae, and Darboux. Hankel established the principle of the condensation of singularities; Dedekind and Cantor gave definitions for irrational numbers..."

"The tacit assumption on which analytic geometry operated was that it was possible to represent the points on a line... by means of numbers. This assumption is... equivalent to the assertion that a perfect correspondence can be established... The great success of analytic geometry... gave this assumption an irresistible pragmatic force. It was essential to include this principle... But how? Under such circumstances mathematics proceeds by fiat. It bridges the chasm between intuition and reason by a convenient postulate. ...The very vagueness of all intuition renders such a substitution... highly acceptable. ... On the one hand there was the logically consistent concept of a real number and its aggregate, the arithmetic continuum; on the other hand, the vague notions of the point and its aggregate, the linear continuum. All that was necessary was to declare the identity of the two... to assert that: It is possible to assign to any point on a line a unique real number, and, conversely, any real number can be represented in a unique manner by a point on a line. This is the famous Dedekind-Cantor axiom."

"Dedekind's language in introducing irrational numbers leaves a little to be desired. He introduces the irrational &alpha; as corresponding to the cut and defined by the cut. But he is not too clear of where &alpha; comes from. He should say that... &alpha; is no more than the cut. ...Heinrich Weber told Dedekind this, and in a letter of 1888 Dedekind replied that... &alpha; is not the cut itself but something distinct, which corresponds to the cut and brings about the cut. Likewise, while the rational numbers generate cuts, they are not the same as the cuts. He says we have the mental power to create such concepts."

"Julius Wilhelm Richard Dedekind stands out as one of the most prominent contributors of the 19th century to the theory of algebraic numbers. He wrote various important memoirs on the binomial equation and on the theory of modular and Abelian functions, but is best known for his treatises Was sind und was sollen die Zahlen? (1888) and Stetigkeit und irrationale Zahlen (1872). In the latter work he set forth his idea of the Schnitt (cut) in relation to irrational numbers,—an idea he had in mind as early as 1858."

"The beauty of [ Eudoxus' ] theory of proportions [ expounded in Book V of Euclid's Elements ] was its adaptability to this new climate. ...The length \sqrt2 is determined by the two sets of positive rationalsL_\sqrt2 = \{r: r^2 < 2\}, \qquad U_\sqrt2 = \{r: r^2 > 2\}Dedekind... decided to let \sqrt2 be this pair of sets! In general, let any partition of the positive rationals into sets L, U such that any member of L is less than any member of U be a positive real number. This idea, now known as the Dedekind cut, is more than just a twist of Eudoxus; it gives a complete and uniform construction of all real numbers, or points on a line, using just the discrete, finally resolving the fundamental conflict in Greek mathematics."

"Literametrics' may become to philology, what 'Biometrics' has already become to the biological sciences."

"Every one knows that the fine phrase "God geometrizes" is attributed to Plato, but few know where this famous passage is found, or the exact words in which it was first expressed. Those who, like the author, have spent hours and even days in the search of the exact statements, or the exact references, of similar famous passages, will not question the timeliness and usefulness of a book whose distinct purpose it is to bring together into a single volume exact quotations, with their exact references, bearing on one of the most time-honored, and even today the most active and most fruitful of all the sciences, the queen-mother of all the sciences, that is, mathematics."

"Es ist eine Mannigfaltigkeit und in derselben eine Transformationsgruppe gegeben; man soll die Mannigfaltigkeit angehören Gebilde hinsichtlich solcher Eigenschaften untersuchen; die durch die Transformationen der Gruppe nicht geändert werden. (Given a manifold with its associated transformation group, one should investigate those structures of the manifold that have properties which are invariant under the transformation group.)"

"The theory of binary forms and the projective geometry of systems of points on a conic are one and the same, i.e., to every proposition concerning binary forms corresponds a proposition concerning such systems of points, and vice versa. ... Elementary plane geometry and the projective investigation of a quadric surface with reference to one of its pointa are one and the same."

"In ordinary geometry a surface is conceived as a locus of points; in Lie's geometry it appears as the totality of all the spheres having contact with the surface."

"It has been the final aim of Lie from the beginning to make progress in the theory of differential equations ..."

"As regards quartic surfaces, Rohn has investigated an enormous number of special cases; but a complete enumeration he has not reached. Among the special surfaces of the fourth order the Kummer surface with 16 conical points is one of the most important. The models constructed by Plücker in connection with his theory of complexes of lines all represent special cases of the Kummer surface."

"Next to the elementary transcental functions the elliptic functions are usually regarded as the most important. There is, however, another class for which at least equal importance must be claimed on account of their numerous applications in astronomy and mathematical physics; these are the hypergeometric functions, so called owing to their connecton with Gauss's hypergeometric series."

"The proof that π is a transcental number will forerver mark an epoch in mathematical science. It gives the final answer to the problem of squaring the circle and settles this vexed question once for all. This problem requires to derive the number π by a finite number of elementary geometrical processes, i.e. with the use of the ruler and compasses alone."

"With Klein, even politics has been introduced into the question: he asserts that “It would seem as if a strong naive space intuition were an attribute of the Teutonic race, while the critical, purely logical sense is more developed in the Latin and Hebrew races.” That such an assertion is not in agreement with facts will appear clearly when we come to examples. It is hardly doubtful that, in stating it, Klein implicitly considers intuition, with its mysterious character, as being superior to the prosaic way of logic and is evidently happy to claim that superiority for his countrymen. We have heard recently of that special kind of ethnography with Nazism: we see that there was already something of this kind in 1893."

"Ehrenfest had always emphasized the importance of Klein's lectures to his students, and we read many of those that circulated in lithograph form. They are full of sweeping insights that reveal the interconnections between different mathematical fields: geometry, function theory, number theory, mechanics, and the internal dialectics of mathematics that manifest themselves through the concept of a group. During my stay in G6ttingen, Courant invited me to help prepare Klein's lectures on the history of nineteenth and early twentieth century mathematics for publication, which I did. These first appeared in Springer's well-known "yellow series," and they remain, with all their personal recollections, the most vivid account of the mathematics of this period."

"From outside Germany, Klein epitomized the cultured German elite. Self-assured, handsome, highly educated, and married to Hegel's granddaughter, he had all the perquisites of a German professor with a devoted cadre of students. Within Germany, however, there was a split between the school of analysis typified by the great and influential German mathematician Karl Weierstrass, and the proponents of more geometric methods associated with Riemann. Klein had identified himself, and his students, with the latter, and thereby contributed to widening the rift—for Klein's enthusiasm was the sort that divides as much as it unifies."

"Let θ be an algebraic integer and assume that all conjugates of θ, except θ itself, have an absolute value less than 1. Then –θ also has this property; on the other hand, θ is real. Without loss of generality, we may therefore suppose θ ≥ 0. Since the norm of θ is a rational integer, we have θ ≥ 1, except for the trivial case θ = 0. Recently, R. Salem ... discovered the interesting theorem that the set S of all θ is closed and that θ = 1 is an isolated point of S. Consequently there exists a smallest θ = θ1 > 1. We shall prove that θ1 is the positive zero of x3 – x – 1 and that also θ1 is isolated in S. Moreover we shall prove that the next number of S, namely the smallest θ = θ2 > θ1, is the positive zero of x4 – x3 – 1 and that θ2 is again an isolated point of S. Since θ1 = 1.324..., θ2 = 1.380..., both numbers are less than 2½; therefore our statements are contained in the following: . Let θ be an algebraic integer whose conjugates lie in the interior of the unit circle; if ±θ ≠ 0, 1, θ1, θ2, then θ2 > 2."

"I am afraid that mathematics will perish by the end of this century if the present trend for senseless abstraction — as I call it: theory of the empty set — cannot be blocked up."

"Ours, according to Leibnitz, is the best of all possible worlds, and the laws of nature can therefore be described in terms of extremal principles. Thus, arising from corresponding variational problems, the differential equations of mechanics have invariance properties relative to certain groups of coordinate transformations."

"The theory of functions of several variables turns out to be essentially more difficult than the theory of one variable because of the existence of points of indeterminacy. In the case n > 1, a mere glance at the poles already indicates a behavior which is completely different from that in the case n = 1. The reason is that, in case n > 1, the poles are not isolated and, in general, there does not exist a Laurent expansion. In a neighborhood of a nonregular point we are forced to view meromorphic functions as quotients of power series."

"One of the many importants ideas introduced by Minkowski into the study of convex bodies was that of gauge function. Roughly, the gauge function is the equation of a convex body. Minkowski showed that the gauge function could be defined in a purely geometric way and that it must have certain properties analogous to those possessed by the distance of a point from the origin. He also showed that conversely given any function possessing these properties, there exists a convex body with the given function as its gauge function."

"Ein Bourgeois, wer noch Algebra treibt! Es lebe die unbeschrankte Individualitat der transzendenten Zahlen! ["It's a bourgeois, who still does algebra! Long live the unrestricted individuality of transcendental numbers!"]"

"... Siegel kept working well into his eighties, after he had returned to Göttingen."

"... when Carl Ludwig Siegel announced that he would hold class on a University holiday, his students left the room empty on the appointed day, hiding nearby to see what he did. “Sure enough, Siegel got up front in the empty room, started in with the beautiful lecture as though he had a full room,” said Merrill Flood *35. After he had continued for a while, “we sheepishly trooped in, and listened to his lecture.”"

"In 1966 Siegel oversaw the publication of his collected works in three volumes. He spent the rest of his life editing (and writing) a fourth volume. As the story goes, he burned everything else, fearing that a historian—as he himself had done with Riemann—would get into his papers."

"It is difficult to compare a differential geometer with a function theorist, or those working on ordinary and partial differential equations with numerical analysts. Christoffel not only contributed to all these fields, but his interests extended to orthogonal polynomials and continued fractions, and the applications of his work to the foundations of tensor analysis, to geodetical science, to the theory of shock waves, to the dispersion of light. Nevertheless, it is widely recognised, at least in the German speaking countries of Europe, that Riemann was the best mathematician of the 19th century, behind Gauss and ahead of Weierstrass. In our opinion Christoffel's teacher Dirichlet, belongs to the next most important group of mathematicians which includes (in chronological order of birth) Jacobi, Kummer, Kronecker, Dedekind, Cantor and Klein. Christoffel himself should be placed in a second group following these. This second group, which may partly overlap with the former, would include such illustrious names as Möbius, von Staudt, Plücker, Heine, Du Bois-Reymond, Carl Neumann, Lipschitz, Fuchs, Schwarz, Hurwitz and Minkowski."

"Since an algebraic function w(z) is defined implicitly by an equation of the form f(z,w) = 0, where f is a polynomial, it is understandable that the study of such functions should be possible by algebraic methods. Such methods also have the advantage that the theory can be developed in the most general setting, viz. over an arbitrary field, and not only over the field of complex numbers (the classical case)."

"... H. Hasse perceived the connection between complex multiplication and the Riemann hypothesis for congruence zeta functions, which was later proved by A. Weil in a fully general form. This observation led M. Deuring to establish a purely algebraic treatment of complex multiplication of elliptic curves. He could, moreover, along the same line of ideas, determine the zeta functions of elliptic curves with complex multiplication. The definition of zeta function of an algebraic curve defined over an algebraic number field is originally due to Hasse; and Weil is the first contributor to this subject."

"... the theory of valuations may be viewed as a branch of topological algebra. In fact, historically speaking, it represents the first invasion of topology, more precisely, of early metric topology, into the domains of algebra. The introduction of metric methods into algebra has been so fruitful that today many of the deeper algebraic theories carry their mark. In this regard, one should distinguish between the classical use in algebra of the natural metric of the real or complex number fields, such as in proving the "fundamental theorem of algebra," and the much more recent use of the far less evident metrics which are derived from arithmetic notions of divisibility and which constitute the principal notion of valuation theory. Such a metric occurs for the first time in Hensel's construction of the p-adic numbers ..."

"Let F be a field of characteristic 0, and let F be a finite dimensional vector space over F. Let E denote the algebra of all endomorphisms of V, and let L be any Lie subalgebra of E. Among the algebraic Lie algebras contained in E and containing L, there is one that is contained in all of them, and this is called the algebraic hull of L in E. Here, an algebraic Lie algebra is defined as the Lie algebra of an algebraic group. It is an easy consequence of the definitions that if A and B are algebraic groups of automorphisms of V such that A⊂B then the Lie algebra of A is contained in the Lie algebra of B. Hence the existence of the algebraic hull of L is an immediate consequence of the following basic result: let G be the intersection of all algebraic groups of automorphisms of V whose Lie algebras contain L."

"A Lie algebra is said to be algebraic if it is isomorphic with the Lie algebra of an affine algebraic group. In view of the fact that entirely unrelated affine algebraic groups (typically, vector groups and toroidal groups) may have isomorphic Lie algebras, this notion of algebraic Lie algebra calls for some clarification. The most relevant result in this direction is due to M. Goto. It says that a finite- dimensional Lie algebra L over a field of characteristic 0 is algebraic if and only if the image of L under the adjoint representation is the Lie algebra of an algebraic subgroup of the group of automorphisms of L ..."

"... Lie algebras have a significance reaching beyond the domain of algebra, because they play such an important role in the theory of Lie groups. Thus, classical Lie algebra theory is strongly dominated by the fact that the finite-dimensional analytic representations of a simply connected analytic group are identifiable with the finite-dimensional representations of its Lie algebra. In the theory of infinite-dimensional representations, the connection with Lie algebra representations is somewhat tenuous, but it is nevertheless at the core of the major advances made in that theory during the last 30 years."

"The investigations of Siegel on discrete groups of motions of the with a fundamental region of finite volume ... make possible a simple characterization of groups conjugate to the modular group by a minimal condition."

"... in 1949, a paper appeared by a German mathematician Hans Maass, which raised some rather interesting problems. You see, earlier the automorphic functions and forms — one had essentially thought of functions that were often called holomorphic, analytical. And Maass started studying functions that were not of that nature, but were instead solutions of a certain eigenvalue problem, which had a certain type of behavior with respect to the discrete group which corresponds to the modular forms. Maass also worked essentially just on the modular group and its subgroups, not on general groups."

"Das ist keine Mathematik; das ist Theologie."

"Ich habe mich davon überzeugt, daß die Theologie auch nützlich sein kann."

"Hilberts Doktor-Vater Ferdinand Lindemann nannte diesen Existenzbeweis „unheimlich“ und Paul Gordan meinte (zitiert nach Otto Blumenthal ‚ Lebensgeschichte‘ in Hilberts ‚Gesammelten Abhandlungen‘, Band 3 (Berlin 1935), S. 388–429, dort S. 394): „Das ist keine Mathematik; das ist Theologie.“ Etwas später milderte Gordan seinen Ausspruch etwas ab und meinte: „Ich habe mich davon überzeugt, daß die Theologie auch nützlich sein kann“. Aber es gab auch viele Mathematiker, die sich nicht überzeugen ließen. Oskar Becker (1889–1964) beispielsweise reagierte sehr heftig und bezeichnete den Hilbert’schen Beweis des Basis-Satzes als „Schleichweg einer Schein-Konstruktion“ (O. Becker in: ‚Mathematische Existenz‘, 1927, op. cit., S. 471)."

"If now a far-reaching theory has grown . . . I attribute this result primarily to Professor Gordan. I am not here referring to his trenchant and profound labours, which shall be fully reported upon hereafter. In this place I must report what cannot be expressed in quotations or references, namely, that Professor Gordan has spurred me on when I flagged in my labours, and that he has helped me . . . over many difficulties which I should never have overcome alone."

"Der Beweis des Hilbertschen Satzes und anderer Sätze ist sehr abstrakt, aber an sich ganz einfach und darum logisch zwingend. Eben darum leitet diese Arbeit von Hilbert eine neue Epoche der algebraischen Geometrie ein. Ebenso einfach ist dann auch die Anwendung auf die Invariantentheorie, die ich hier noch weniger zergliedern kann. Die ganze Frage der Endlichkeit der Invarianten, welche Gordan seinerzeit nur mit umfangreichen Rechnungen für binäre Formen hatte erledigen können (vgl. oben S. 308), wird hier mit einem Schlage für Formen mit beliebig vielen Veränderlichen gelöst. Ihrer Eigenart entsprechend wurde diese Arbeit zunächst mit sehr verschiedener Stimmung aufgenommen. Mich hat sie damals bestimmt, Hilbert bei nächster Gelegenheit nach Göttingen zu ziehen. Gordan war anfangs ablehnend: „Das ist nicht Mathematik, das ist Theologie.“ Später sagte er dann wohl: „Ich habe mich überzeugt, daß auch die Theologie ihre Vorzüge hat.“ In der Tat hat er den Beweis des Hilbertschen Grundtheorems selbst später sehr vereinfacht (Münchener Naturforscherversammlung 1899)."

"The proof of this theorem of Hilbert's and of others is very abstract, but in itself quite simple and hence logically compelling. And for just this reason this work of Hilbert's ushered in a new epoch of algebraic geometry. But application to invariant theory is just as simple, but I can analyze it here even less. The whole question of the finiteness of the invariants, which Gordan had been able to solve for binary forms only by means of comprehensive calculations (see p. 290), is here solved, with one stroke, for forms with arbitrarily many variables. Because of its uniqueness, this work was first received with very diverse reactions. I had then resolved to draw Hilbert to Goettingen at the earliest opportunity. Gordan at first declined, saying, "It is not mathematics, it is theology" But later he said: "I have convinced myself that even theology has its merits". In fact, Gordan himself later on much simplified Hilbert's basic theorem (Muenchener Naturforscherversammlung 1899)."

"A queer fellow, impulsive and one-sided. A great walker and talker - he liked that kind of walk to which frequent stops at a beer-garden or a cafe belong. Either with friends, and then accompanying his discussions with violent gesticulations, completely irrespective of his surroundings; or alone, and then murmuring to himself and pondering over mathematical problems; or if in an idler mood, carrying out long numerical calculations by heart. There always remained something of the eternal "Bursche" of the 1848 type about him – an air of dressing gown, beer and tobacco, relieved however by a keen sense of humor and a strong dash of wit. When he had to listen to others, in classrooms or at meetings, he was always half asleep."

"His strength rested on the invention and calculative execution of formal processes. There exist papers of his where twenty pages of formulas are not interrupted by a single text word; it is told that in all his papers he himself wrote the formulas only, the text being added by his friends."

"Gordan - anfänglich diesen begrifflichen Deduktionen gegenüber mehr ablehnend: „das ist keine Mathematik, das ist Theologie!" - ist dann zweimal (53), (69) dem diesem Beweise zugrunde liegenden Hilbertschen Endlichkeitssatze nähergetreten, indem er die gegebenen Formen F nach verschiedenen Kriterien in eine Reihe anordnete, die das Bilden eines endlichen Moduls aus ihnen deutlich machte; das erstemal in komplizierterer Weise speziell für die Invariantenformen, das zweitemal allgemein und einfach."

"38 Jahre von 1874 an hat Gordan in Erlangen verbracht. Sie sind für ihn gleichmäßig verlaufen: täglich Vorlesungen, Arbeit, und die unentbehrlichen Spaziergänge entweder mit Mitarbeitern … in drastisch lebhaften Zwiegesprächen, unbekümmert um alle Umgebung, oder allein in tiefem Nachdenken und seine Gedanken im Kopfe so fertig verarbeitend, dass er seine Rechnungen zuhause fast ohne Striche [Streichungen] ausführen konnte."

"In seiner eigenen Wissenschaft war es weniger ein Vertiefen in fremde Arbeiten -- denn solche las er sehr wenig -, als ein Überblick über die inneren Zusammenhänge und ein instinktives Gefühl für die Wege und Ziele der mathematischen Bestrebungen, was ihn schon aus kleinen Andeutungen Wertvolles von Minderem scheiden lieb. Aber den auf die Grundlagen gehenden Begriffsentwicklungen ist Gordan nie gerecht geworden: auch in seinen Vorlesungen hat er alle Grunddefinitionen begrifflicher Art, selbst die der Grenze, vollständig gemieden. Sein Vorlesungsprogramm hat sich nur auf die Vorlesungen allgemeiner Art, gelegentlich auch auf binäre Formentheorie, erstreckt; die Übungen waren mit Vorliebe der Algebra entnommen. Über Jacobisches, so über Funktionaldeterminanten, trug er gern vor, nie über Funktionentheoretisches, höhere Geometrie oder Mechanik; auch ließ er keine Seminarvorträge halten. Die Vorlesungen wirkten wesentlich durch die Lebhaftigkeit der Ausdrucksweise und durch eine zum Selbststudium anregende Kraft, eher als durch Systematik und Strenge."

"Gordan, eine in sich geschlossene Individualität, war kräftig und einheitlich im Leben und in der Arbeit. Kein Neuerer in der Wissenschaft: er griff nur an, was seiner Art gemäß war; aber was er angriff, führte er nnermüdlich durch bis zu Ende. Aus dem Stoffe selbst heraus neue kombinatorische Methoden zu schaffen und seine Instrumente kräftig zu handhaben, das war sein mächtiges Können: er war Algorithmiker."

"The University of Berlin had been greatly influenced by the successful French research institutes, such as the Ecole Polytechnique, that had been founded by Napoleon. It had, after all, been founded during the French occupation. One of the key mathematical ambassadors was a brilliant mathematician by the name of Peter Gustav Lejeune Dirichlet. Although he was born in Germany in 1805, Dirichlet's family was of French origin. A return to his roots took him to Paris in 1822, where he spent five years soaking up the intellectual activity that was bubbling out of the academies. Wilhelm von Humboldt's brother Alexander, an amateur scientist, met Dirichlet on his travels and was so impressed that he secured him a position back in Germany. Dirichlet was something of a rebel. Perhaps the atmosphere on the streets in Paris had given him a taste for challenging authority. In Berlin, he was quite happy to ignore some of the antiquated traditions demanded by the rather stuffy university authorities, and often flouted their demands to demonstrate his command of the Latin language."

"Georg Hamel was born in 1877 in Düren, Germany, and died in 1954 in Landshut, Germany. In 1897, Hamel went to the University of Berlin, where he was taught by Hermann Schwarz and Max Planck, to name two. Subsequently, he went to Göttingen University, where he studied with Felix Klein and David Hilbert. He was awarded a doctorate under the supervision of Hilbert in 1901. The subject of his dissertation was Hilbert’s fourth problem. In 1905, he went to Brno. It was during the period of his work in Brno that his 1905 paper on Hamel bases was written."

"I was amazed at how this outstanding connoisseur of Indian knowledge in the field of exact sciences showed himself to be so captive to Cantor’s authority."

"Dirichlet was not satisfied to study Gauss' "Disquisitiones arithmeticae" once or several times, but continued throughout life to keep in close touch with the wealth of deep mathematical thoughts which it contains by perusing it again and again. For this reason the book was never placed on the shelf but had an abiding place on the table at which he worked....Dirichlet was the first one, who not only fully understood this work, but made it also accessible to others."

"A peculiar beauty reigns in the realm of mathematics, a beauty which resembles not so much the beauty of art as the beauty of nature and which affects the reflective mind, which has acquired an appreciation of it, very much like the latter."

"It is true that mathematics, owing to the fact that its whole content is built up by means of purely logical deduction from a small number of universally comprehended principles, has not unfittingly been designated as the science of the self-evident [Selbstverständlichen]. Experience however, shows that for the majority of the cultured, even of scientists, mathematics remains the science of the incomprehensible [Unverständlichen]."

"Mathematical knowledge, therefore, appears to us of value not only in so far as it serves as means to other ends, but for its own sake as well, and we behold, both in its systematic external and internal development, the most complete and purest logical mind-activity, the embodiment of the highest intellect-esthetics."

"Out of the interaction of form and content in mathematics grows an acquaintance with methods which enable the student to produce independently within certain though moderate limits, and to extend his knowledge through his own reflection. The deepening of the consciousness of the intellectual powers connected with this kind of activity, and the gradual awakening of the feeling of intellectual self-reliance may well be considered as the most beautiful and highest result of mathematical training."

"I have come to the conclusion that the exertion, without which a knowledge of mathematics cannot be acquired, is not materially increased by logical rigor in the method of instruction."

"It may be asserted without exaggeration that the domain of mathematical knowledge is the only one of which our otherwise omniscient journalism has not yet possessed itself."

"If in Germany the goddess Justitia had not the unfortunate habit of depositing the ministerial portfolios only in the cradles of her own progeny, who knows how many a German mathematician might not also have made an excellent minister."

"The true mathematician is always a good deal of an artist, an architect, yes, of a poet. Beyond the real world, though perceptibly connected with it, mathematicians have intellectually created an ideal world, which they attempt to develop into the most perfect of all worlds, and which is being explored in every direction. None has the faintest conception of this world, except he who knows it."

"Just as the musician is able to form an acoustic image of a composition which he has never heard played by merely looking at its score, so the equation of a curve, which he has never seen, furnishes the mathematician with a complete picture of its course. Yea, even more: as the score frequently reveals to the musician niceties which would escape his ear because of the complication and rapid change of the auditory impressions, so the insight which the mathematician gains from the equation of a curve is much deeper than that which is brought about by a mere inspection of the curve."

"The domain, over which the language of analysis extends its sway, is, indeed, relatively limited, but within this domain it so infinitely excels ordinary language that its attempt to follow the former must be given up after a few steps. The mathematician, who knows how to think in this marvelously condensed language, is as different from the mechanical computer as heaven from earth."

"The analysis which is based upon the conception of function discloses to the astronomer and physicist not merely the formulae for the computation of whatever desired distances, times, velocities, physical constants; it moreover gives him insight into the laws of the processes of motion, teaches him to predict future occurrences from past experiences and supplies him with means to a scientific knowledge of nature, i.e. it enables him to trace back whole groups of various, sometimes extremely heterogeneous, phenomena to a minimum of simple fundamental laws."