Mathematicians From England

"Step by step men of science are coming to recognise that mechanism is not at the bottom of phenomena, but is only the conceptual shorthand by aid of which they can briefly describe and resume phenomena."

"The whole early history... is... so intimately connected with the names Galilei, Hooke, Mariotte and Leibniz that I have introduced some account of their work. The labours of Lagrange and Riccati also required some recognition. ...These early writers form the basis... not without interest, whether judged from the special standpoint of the elastician or from the wider footing of... the growth of human ideas. With a similar aim I have introduced throughout the volume... memoirs having purely historical value which had escaped Dr Hunter's notice. Another class of memoirs which I have inserted are... of mathematical value, omitted apparently by pure accident. For example all the memoirs of F. E. Neumann, the second memoir of Duhamel, those of [P. H.] Blanchet etc. I cannot hope that the work is complete in this respect even now, but I trust that nothing of equal importance has escaped..."

"The original work as planned by Clifford was to have been entitled The First Principles of the Mathematical Sciences Explained to the Non-Mathematical and to have contained six chapters on Number, Space, Quantity, Position, Motion, and Mass respectively. ...Shortly before his death he expressed a wish that the book should only be published after very careful revision and that its title should be changed to [our title]. Upon Clifford's death the labour of revision and completion was entrusted to Mr. R. C. Rowe... On the sad death of Professor Rowe... I was requested... to take up the... editing thus left incomplete. ...For the latter half of Chapter III and for the whole of Chapter IV... I am alone responsible. Yet whatever there is in them of value I owe to Clifford; whatever is feeble or obscure is my own. ...With Chapter V. my task has been by no means light. ... I felt it impossible to rewrite the whole without depriving the work of its right to be called Clifford's, and yet at the same time it was absolutely necessary to make considerable changes. ...Without any notice of mass or force it seemed impossible to close a discussion on motion; something I felt must be added. I have accordingly introduced a few pages on the laws of motion [and] since found that Clifford intended to write a concluding chapter on mass. How to express the laws of motion in a form of which Clifford would have approved was indeed an insoluble riddle... because I was unaware of his having written on the subject. I have accordingly expressed... my own views on the subject [i.e.,] a strong desire to see the terms matter and force, together with the ideas associated with them, entirely removed from scientific terminology—to reduce, in fact, all dynamic to kinematic. I should hardly have ventured to put forward these views had I not recently discovered that they have... the weighty authority of Professor Mach... But since writing these pages I have also been referred to a discourse delivered by Clifford at the Royal Institution in 1873, some... of which appeared in Nature June 10, 1880 [pp. 122-123.] Therein it is stated that 'no mathematician can give any meaning to the language about matter, force, inertia used in current text-books of mechanics.' This fragmentary account of the discourse undoubtedly proves that Clifford held on the categories of matter and force as clear and original ideas as on all subjects of which he has treated; only, alas! they have not been preserved. Footnote: Mr. R. Tucker who... searched Clifford's note books... sends me... the following... in Clifford's handwriting: 'Force is not a fact at all, but an idea embodying what is approximately the fact.'"

"Even the fathers of statistical science forgot that a random series is bound to exhibit some pattern... Professor Pearson was among the first scholars interested in creating artificial random number generators, tables one could use as inputs for... simulations (precursors of our Monte Carlo simulator). ...[T]hey did not want these tables to exhibit... regularity. Yet real randomness does not look random!...A single random run is bound to exhibit some pattern ..."

"The great danger... not content with our real knowledge of the relation of the finite to the infinite, they slur over our vast ignorance by the help of the imagination. Myth supplies the place of true knowledge where we are ignorant of the connection between finite and infinite. Hence... most concrete systems of religion present us with a certain amount of knowledge but a great deal of myth."

"The scientific conception of chance is that of a measure based on experience; a knowledge of the average results of many events is used to replace ignorance of the result of any individual event. ...The judgment which Science gives in this case is decisive; judged by the so called "permanences," or runs of colour, Monte Carlo roulette is no true worship of the goddess at all."

"[W]e are forced to accept... that the random spinning of a roulette manufactured and daily readjusted with extraordinary care is not obedient to the laws of chance, but is chaotic in its manifestations."

"[B]ear these points in mind, the association of Death and Chance, the notion of both as chaotic in their action, and their embodiment in a great artistic ideal—the Dance of Death—which gave so much colouring to mediaeval thought and life. We find this sombre notion everywhere—on the church walls, on the bridges, in the engravings and s, but as well in the sermons, the poetry, and the very turn of folk-sentiment."

"Does not the beauty of the artist's work lie for us in the accuracy with which his symbols resume innumerable facts of our past emotional experience? ... [A]esthetic judgment... how exactly parallel it is to the scientific judgment."

"[R]andom spinning being assumed, the distribution of chance in the game depends upon the mechanical perfection of the teetotum; it must be equally likely to fall on all its thirty-seven sides, i.e. the frequency of all the numbers must in the long run be very nearly the same."

"I just passed two cars in West London driving with Palestinian flags flying from each window, bouncing up and down in their cars, seemingly celebrating like they were having a party. Make no mistake, this is a dangerous and terrifying time for all Jews around the world."

"When you tell people that you are studying maths at uni, they are like, “Oh …”. Especially a blonde Essex girl."

"Men are not as scared to lose and they’ve got a lot more time to devote to, not exactly pointless things, but to being good at things like Countdown."

"This culture has developed, with those who’ve created it, doing so in the name of today's incarnation of Labour. There’s nothing kinder nor gentler about it."

"I’m really interested in male and female brains and whether female brains or male brains are better at maths. You sit men and women down and give them a maths test and they will do fairly equally. Then you set up the same test, but with different people, and make them tick a box to say whether they are a man or a woman, and the women do significantly worse in the maths test than they did previously in a group set."

"More than other subjects there’s a myth that you have to be an absolute genius to be good at maths and to enjoy it, so I think it’s less accessible for people. Even the word “maths” makes people screw their face up."

"I see the maths face quite a lot. It’s the blind panic that they have to do maths in front of people. It’s just fear and dread. There’s definitely a maths face – try it on someone."

"Acton, half an hour ago. Popped into a cafe for some baklava with the kids and our Ukrainian friends. People have been brutally murdered, kidnapped and there are people in London dancing."

"Algebra is the offer made by the devil to the mathematician. The devil says: `I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvellous machine.'"

"My own supervisor, William Hodge, the creator of the fertile theory of harmonic forms, was not a genius like Ramanujan but resembled Lefschetz."

"This 'Hodge conjecture' has by now achieved a considerable status, almost on a par with the Riemann hypothesis or the Poincaré conjecture."

"I always want to try to understand why things work. I’m not interested in getting a formula without knowing what it means. I always try to dig behind the scenes, so if I have a formula, I understand why it’s there. And understanding is a very difficult notion. People think mathematics begins when you write down a theorem followed by a proof. That’s not the beginning, that’s the end. For me the creative place in mathematics comes before you start to put things down on paper, before you try to write a formula. You picture various things, you turn them over in your mind. You’re trying to create, just as a musician is trying to create music, or a poet. There are no rules laid down. You have to do it your own way. But at the end, just as a composer has to put it down on paper, you have to write things down. But the most important stage is understanding. A proof by itself doesn’t give you understanding. You can have a long proof and no idea at the end of why it works. But to understand why it works, you have to have a kind of gut reaction to the thing. You’ve got to feel it."

"Between the time of the gift of the Portsmouth Papers and the 1930s... there was as yet no real discipline of the history of science and of mathematics. The number of individuals producing lasting historical contributions in the history of science and mathematics was small, including such heroic figures as J. L Heiberg, G. Eneström, Thomas Little Heath, and Paul Tannery."

"The only one of the works of Aristarchus which has been preserved, is the very interesting short treatise "On the distances of sun and moon". It is a great merit of Thomas Heath that he called attention to the mathematical value of this treatise and that he published a translation with an excellent historical astronomical commentary."

"It would be inconvenient to interrupt the account of Menaechmus's solution of the problem of the two mean proportionals in order to consider the way in which he may have discovered the conic sections and their fundamental properties. It seems to me much better to give the complete story of the origin and development of the geometry of the conic sections in one place, and this has been done in the chapter on conic sections associated with the name of Apollonius of Perga. Similarly a chapter has been devoted to algebra (in connexion with Diophantus) and another to trigonometry (under Hipparchus, Menelaus and Ptolemy)."

"Dr. James Gow did a great service by the publication in 1884 of his Short History of Greek Mathematics, a scholarly and useful work which has held its own and has been quoted with respect and appreciation by authorities on the history of mathematics in all parts of the world. At the date when he wrote, however, Dr. Gow had necessarily to rely upon the works of the pioneers Bretschneider, Hankel, Allman, and (first edition). Since then the subject has been very greatly advanced... scholars and mathematicians... have thrown light on many obscure points. It is therefore high time for the complete story to be rewritten."

"The outstanding personalities of Euclid and Archimedes demand chapters to themselves. Euclid, the author of the incomparable Elements, wrote on almost all the other branches of mathematics known in his day. Archimedes's work, all original and set forth in treatises which are models of scientific exposition, perfect in form and style, was even wider in its range of subjects. The imperishable and unique monuments of the genius of these two men must be detached from their surroundings and seen as a whole if we would appreciate to the full the pre-eminent place which they occupy, and will hold for all time, in the history of science."

"It is true that in recent years a number of attractive histories of mathematics have been published in England and America, but these have only dealt with Greek mathematics as part of the larger subject, and in consequence the writers have been precluded... from presenting the work of the Greeks in suflicient detail. The same remark applies to the German histories of mathematics, even to the great work of Moritz Cantor..."

"If one would understand the Greek genius fully, it would be a good plan to begin with their geometry."

"Aristotle would... by no means admit that mathematics was divorced from aesthetic; he could conceive, he said, of nothing more beautiful than the objects of mathematics."

"The best history of Greek mathematics which exists at present is undoubtedly that of Gino Loria under the title Le scienze esatte nell' antica Grecia (second edition 1914...) ...the arrangement is chronological ...they raise the question whether in a history of this kind it is best to follow chronological order or to arrange the material according to subjects... I have adopted a new arrangement, mainly according to subjects..."

"It is a defect in the existing histories that, while they state generally the contents of, and the main propositions proved in, the great treatises of Archimedes and Apollonius, they make little attempt to describe the procedure by which the results are obtained. I have therefore taken pains, in the most significant cases, to show the course of the argument in sufficient detail to enable a competent mathematician to grasp the method used and to apply it, if he will, to other similar investigations."

"Eudoxes... not only based the method [of exhaustion] on rigorous demonstration... but he actually applied the method to find the volumes (1) of any pyramid, (2) of the cone, proving (1) that any pyramid is one third part of the prism which has the same base and equal height, and (2) that any cone is one third part of the cylinder which has the same base and equal height. Archimedes, however, tells us the remarkable fact that these two theorems were first discovered by Democritus, though he was not able to prove them (which no doubt means, not that he gave no sort of proof, but that he was not able to establish the propositions by the rigorous methods of Eudoxes. Archimedes adds that we must give no small share of the credit for these theorems to Democritus... another testimony to the marvellous powers, in mathematics as well as in other subjects, of the great man who, in the words of Aristotle, "seems to have thought of everything". ...Democritus wrote on irrationals; he is also said to have discussed the question of two parallel sections of a cone (which were evidently supposed to be indefinitely close together), asking whether we are to regard them as equal or unequal... Democritus was already close on the track of infinitesimals."

"It is... the author's confident hope that this book will give a fresh interest to the story of Greek mathematics in the eyes both of mathematicians and of classical scholars."

"For the mathematician the important consideration is that the foundations of mathematics and a great portion of its content are Greek. The Greeks laid down the first principles, invented the methods ab initio, and fixed the terminology. Mathematics in short is a Greek science, whatever new developments modern analysis has brought or may bring."

"Greek mathematics reveals an important aspect of the Greek genius of which the student of Greek culture is apt to lose sight."

"The work Was begun in 1913, but the bulk of it was written, as a distraction, during the first three years of the war, the hideous course of which seemed day by day to enforce the profound truth conveyed in the answer of Plato to the Delians. When they consulted him on the problem set them by the Oracle, namely that of duplicating the cube, he replied, 'It must be supposed, not that the god specially wished this problem solved, but that he would have the Greeks desist from war and wickedness and cultivate the Muses, so that, their passions being assuaged by philosophy and mathematics, they might live in innocent and mutually helpful intercourse with one another'. Truly,Greece and her foundations are Built below the tide of war, Based on the crystàlline sea Of thought and its eternity."

"Take the case of a famous problem which plays a great part in the history of Greek geometry, the doubling of the cube, or its equivalent, the finding of two mean proportionals in continued proportion between two given straight lines. ...if all the recorded solutions are collected together, it is much easier to see the relations, amounting in some cases to substantial identity, between them, and to get a comprehensive view of the history of the problem. I have therefore dealt with this problem in a separate section of the chapter devoted to 'Special Problems,' and I have followed the same course with the other famous problems of squaring the circle and trisecting any angle."

"The trisection of an angle was effected by means of a curve discovered by Hippias of Elis, the sophist, a contemporary of Hippocrates as well as of Democritus and Socrates. The curve was called the quadratrix because it also served (in the hands, as we are told, of Dinostratus, brother of Menæchmus, and of Nicomedes) for squaring the circle. It was theoretically constructed as the locus of the point of intersection of two straight lines moving at uniform speeds and in the same time, one motion being angular and the other rectilinear."

"The actual writers of Elements of whom we hear were the following. Leon, a little younger than Eudoxus, was the author of a collection of propositions more numerous and more serviceable than those collected by Hippocrates. Theudius of Magnesia, a contemporary of Menæchmus and Dinostratus, "put together the elements admirably, making many partial or limited propositions more general". Theudius's book was no doubt the geometrical text-book of the Academy and that used by Aristotle."

"Theodorus of Cyrene and Theaetetus generalised the theory of irrationals, and we may safely conclude that a great part of the substance of Euclid's Book X. (on irrationals) was due to Theætetus. Theætetus also wrote on the five regular solids, and Euclid was therefore no doubt equally indebted to Theætetus for the contents of his Book XIII. In the matter of Book XII. Eudoxus was the pioneer. These facts are confirmed by the remark of Proclus that Euclid, in compiling his Elements, collected many of the theorems of Eudoxus, perfected many others by Theætetus, and brought to irrefragable demonstration the propositions which had only been somewhat loosely proved by his predecessors."

"Menæchmus, a pupil of Eudoxus, and a contemporary of Plato, found the two mean proportionals by means of conic sections, in two ways, (α) by the intersection of two parabolas, the equations of which in Cartesian co-ordinates would be x2=ay, y2=bx, and (β) by the intersection of a parabola and a rectangular hyperbola, the corresponding equations being x2=ay, and xy=ab respectively. It would appear that it was in the effort to solve this problem that Menæchmus discovered the conic sections, which are called, in an epigram by Eratosthenes, "the triads of Menæchmus"."

"Hippocrates also attacked the problem of doubling the cube. ...Hippocrates did not, indeed, solve the problem, but he succeeded in reducing it to another, namely, the problem of finding two mean proportionals in continued proportion between two given straight lines, i.e. finding x, y such that a:x=x:y=y:b, where a, b are the two given straight lines. It is easy to see that, if a:x=x:y=y:b, then b/a = (x/a)3, and, as a particular case, if b=2a, x3=2a3, so that the side of the cube which is double of the cube of side a is found."

"The problem of doubling the cube was henceforth tried exclusively in the form of the problem of the two mean proportionals."

"Archytas of Tarentum found the two mean proportionals by a very striking construction in three dimensions, which shows that solid geometry, in the hands of Archytas at least, was already well advanced. The construction was usually called mechanical, which it no doubt was in form, though in reality it was in the highest degree theoretical. It consisted in determining a point in space as the intersection of three surfaces: (a) a cylinder, (b) a cone, (c) an "anchor-ring" with internal radius = 0."

"Eudoxus was perhaps the greatest of all Archimedes's predecessors, and it is his achievements, especially the discovery of the method of exhaustion, which interest us in connexion with Archimedes."

"The Pythagoreans discovered the existence of incommensurable lines, or of irrationals. This was, doubtless, first discovered with reference to the diagonal of a square which is incommensurable with the side, being in the ratio to it of √2 to 1. The Pythagorean proof of this particular case survives in Aristotle and in a proposition interpolated in Euclid's Book X.; it is by a reductio ad absurdum proving that, if the diagonal is commensurable with the side, the same number must be both odd and even. This discovery of the incommensurable... showed that the theory of proportion invented by Pythagoras was not of universal application and therefore that propositions proved by means of it were not really established. ...The fatal flaw thus revealed in the body of geometry was not removed till Eudoxus discovered the great theory of proportion (expounded in Euclid's Book V.), which is applicable to incommensurable as well as to commensurable magnitudes."

"It is to be feared that few who are not experts in the history of mathematics have any acquaintance with the details of the original discoveries in mathematics of the greatest mathematician of antiquity, perhaps the greatest mathematical genius that the world has ever seen."

"By the time of Hippocrates of Chios the scope of Greek geometry was no longer even limited to the Elements; certain special problems were also attacked which were beyond the power of the geometry of the straight line and circle, and which were destined to play a great part in determining the direction taken by Greek geometry in its highest flights. The main problems in question were three: (1) the doubling of the cube, (2) the trisection of any angle, (3) the squaring of the circle; and from the time of Hippocrates onwards the investigation of these problems proceeded pari passu with the completion of the body of the Elements."

"Archimedes is said to have requested his friends and relatives to place upon his tomb a representation of a cylinder circumscribing a sphere within it, together with the inscription giving the ratio (3/2) which the cylinder bears to the sphere; from which we may infer that he himself regarded the discovery of this ration as his greatest achievement."