Academics From The United Kingdom

"Hippocrates himself is an example of the concurrent study of the two departments. On the one hand, he was the first of the Greeks who is known to have compiled a book of Elements. This book, we may be sure, contained in particular the most important propositions about the circle included in Euclid, Book III. But a much more important proposition is attributed to Hippocrates; he is said to have been the first to prove that circles are to one another as the squares on their diameters (= Eucl. XII., 2) with the deduction that similar segments of circles are to one another as the squares on their bases. These propositions were used by him in his tract on the squaring of lunes, which was intended to lead up to the squaring of the circle. The latter problem is one which must have exercised practical geometers from time immemorial. Anaxagoras for instance is said to have worked at the problem while in prison."

"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."

"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."

"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"."

"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."

"Almost the whole of Greek science and philosophy begins with Thales."

"In geometry the following theorems are attributed to him [Thales]—and their character shows how the Greeks had to begin at the very beginning of the theory—(1) that a circle is bisected by any diameter (Eucl. I., Def. 17), (2) that the angles at the base of an isosceles triangle are equal (Eucl. I., 5), (3) that, if two straight lines cut one another, the vertically opposite angles are equal (Eucl. I., 15), (4) that, if two triangles have two angles and one side respectively equal, the triangles are equal in all respects (Eucl. I., 26). He is said (5) to have been the first to inscribe a right-angled triangle in a circle: which must mean that he was the first to discover that the angle in a semicircle is a right angle. He also solved two problems in practical geometry: (1) he showed how to measure the distance from the land of a ship at sea (for this he is said to have used the proposition numbered (4) above), and (2) he measured the heights of pyramids by means of the shadow thrown on the ground (this implies the use of similar triangles in the way that the Egyptians had used them in the construction of pyramids)."

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

"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 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."

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

"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 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..."

"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 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."

"In illustration of his entire preoccupation with his studies, we are told that he would forget all about his food and such necessities of life, and would be drawing geometrical figures in the ashes of the fire, or, when anointing himself, in the oil on his body."

"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."

"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."

"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."

"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..."

"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)."

"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."

"The discovery of Hippocrates amounted to the discovery of the fact that from the relation (1)\frac{a}{x} = \frac{x}{y} = \frac{y}{b}it follows that(\frac{a}{x})^3 = [\frac{a}{x} \cdot \frac{x}{y} \cdot \frac{y}{b} =] \frac{a}{b}and if a = 2b, [then (\frac{a}{x})^3 = 2, and]a^3 = 2x^3.The equations (1) are equivalent [by reducing to common denominators or cross multiplication] to the three equations (2)x^2 = ay, y^2 = bx, xy = ab[or equivalently...y = \frac{x^2}{a}, x = \frac{y^2}{b}, y = \frac{ab}{x} ]thumb|Doubling the Cube the 2 solutions of Menaechmusand the solutions of Menaechmus described by Eutocius amount to the determination of a point as the intersection of the curves represented in a rectangular system of Cartesian coordinates by any two of the equations (2). Let AO, BO be straight lines placed so as to form a right angle at O, and of length a, b respectively. Produce BO to x and AO to y. The first solution now consists in drawing a parabola, with vertex O and axis Ox, such that its parameter is equal to BO or b, and a hyperbola with Ox, Oy as asymptotes such that the rectangle under the distances of any point on the curve from Ox, Oy respectively is equal to the rectangle under AO, BO i.e. to ab. If P be the point of intersection of the parabola and hyperbola, and PN, PM be drawn perpendicular to Ox, Oy, i.e. if PN, PM be denoted by y, x, the coordinates of the point P, we shall have \begin{cases}y^2 = b.ON = b.PM = bx\\ and\\ xy = PM.PN = ab\end{cases}whence\frac{a}{x} = \frac{x}{y} = \frac{y}{b}. In the second solution of Menaechmus we are to draw the parabola described in the first solution and also the parabola whose vertex is O, axis Oy and parameter equal to a. The point P where the two parabolas intersect is given by\begin{cases}y^2 = bx\\x^2 = ay\end{cases}whence, as before,\frac{a}{x} = \frac{x}{y} = \frac{y}{b}."

"Once the first principles are disposed of, the body of doctrine contained in the recent textbooks of elementary geometry does not, and from the nature of the case cannot, show any substantial differences from that set forth in the Elements."

"The efforts of a multitude of writers have rather been directed towards producing alternatives for Euclid which shall be more suitable, that is to say, easier, for schoolboys. It is of course not surprising that, in these days of short cuts, there should have arisen a movement to get rid of Euclid and to substitute "a royal road to geometry"; the marvel is that a book which was not written for schoolboys but for grown men (as all internal evidence shows, and in particular the essentially theoretical character of the work and its aloofness from anything of the nature of "practical" geometry) should have held its own as a schoolbook for so long."

"While then for a long time everyone was at a loss, Hippocrates of Chios was the first to observe that, if between two straight lines of which the greater is double of the less it were discovered how to find two mean proportionals in continued proportion, the cube would be doubled; and thus he turned the difficulty in the original problem into another difficulty no less than the former. Afterwards, they say, some Delians attempting, in accordance with an oracle, to double one of the altars fell into the same difficulty. And they sent and begged the geometers who were with Plato in the Academy to find for them the required solution. And while they set themselves energetically to work and sought to find two means between two given straight lines, Archytas of Tarentum is said to have discovered them by means of half-cylinders, and Eudoxus by means of the so-called curved lines. It is, however, characteristic of them all that they indeed gave demonstrations, but were unable to make the actual construction or to reach the point of practical application, except to a small extent Menaechmus and that with difficulty."

"An edition is... still wanted which shall, while in some places adhering... to the original text, at the same time be so entirely remodelled by the aid of accepted modern notation as to be thoroughly readable by any competent mathematician, and this want it is the object of the present work to supply."

"Any satisfactory reproduction of the Conics must fulfil certain essential conditions: (1) it should be Apollonius and nothing but Apollonius, and nothing should be altered either in the substance or in the order of his thought, (2) it should be complete, leaving out nothing of any significance or importance, (3) it should exhibit under different headings the successive divisions of the subject, so that the definite scheme followed by the author may be seen as a whole."

"There is perhaps no question that occupies, comparatively, a larger space in the history of Greek geometry than the problem of the Doubling of the Cube. The tradition concerning its origin is given in a letter from Eratosthenes of Cyrene to King Ptolemy Euergetes quoted by Eutocius... "Eratosthenes to King Ptolemy greeting. "There is a story that one of the old tragedians represented Minos as wishing to erect a tomb for Glaucus and as saying, when he heard that it was a hundred feet every way,Too small thy plan to bound a royal tomb. Let it be double; yet of its fair form Fail not, but haste to double every side.But he was clearly in error; for when the aides are doubled, the area becomes four times as great, and the solid content eight times as great. Geometers also continued to investigate the question in what manner one might double a given solid while it remained in the same form."

"There has been a rush of competitors anxious to be first in the field with a new text-book on the more "practical" lines which now find so much favour. The natural desire of each teacher who writes such a text-book is to give prominence to some special nostrum which he has found successful with pupils. One result is, too often, a loss of a due sense of proportion... It is, perhaps too early yet to prophesy what will be the ultimate outcome of the new order of things; but it would at least seem possible that history will repeat itself and that, when chaos has come again in geometrical teaching, there will be a return to Euclid more or less complete for the purpose of standardising it once more."

"Euclid's work will live long after all the text books of the present day are superseded and forgotten. It is one of the noblest monuments of antiquity; no mathematician worthy of the name can afford not to know Euclid, the real Euclid as distinct from any revised or rewritten versions which will serve for schoolboys or engineers. And, to know Euclid, it is necessary to know his language, and, so far as it can be traced, the history of the "elements" which he collected in his immortal work."

"The researches of the last thirty or forty years into the history of mathematics (I need only mention such names as those of [Carl Anton] Bretschneider, Hankel, Moritz Cantor, [Friedrich] Hultsch, Paul Tannery, Zeuthen, Loria, and Heiberg) have put the whole subject upon a different plane. I have endeavoured in this edition to take account of all the main results of these researches up to the present date. Thus, so far as the geometrical Books are concerned, my notes are intended to form a sort of dictionary of the history of elementary geometry, arranged according to subjects; while the notes on the arithmetical Books VII.-IX. and on Book X follow the same plan."

"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."

"Diophantos lived in a period when the Greek mathematicians of great original power had been succeeded by a number of learned commentators, who confined their investigations within the limits already reached, without attempting to further the development of the science. To this general rule there are two most striking exceptions, in different branches of mathematics, Diophantos and Pappos. These two mathematicians, who would have been an ornament to any age, were destined by fate to live and labour at a time when their work could not check the decay of mathematical learning. There is scarcely a passage in any Greek writer where either of the two is so much as mentioned. The neglect of their works by their countrymen and contemporaries can be explained only by the fact that they were not appreciated or understood. The reason why Diophantos was the earliest of the Greek mathematicians to be forgotten is also probably the reason why he was the last to be re-discovered after the Revival of Learning. The oblivion, in fact, into which his writings and methods fell is due to the circumstance that they were not understood. That being so, we are able to understand why there is so much obscurity concerning his personality and the time at which he lived. Indeed, when we consider how little he was understood, and in consequence how little esteemed, we can only congratulate ourselves that so much of his work has survived to the present day."

"The most probable view is that adopted by Nesselmann, that the works which we know under the three titles formed part of one arithmetical work, which was, according to the author's own words, to consist of thirteen Books. The proportion of the lost parts to the whole is probably less than it might be supposed to be. The Porisms form the part the loss of which is most to be regretted, for from the references to them it is clear that they contained propositions in the Theory of Numbers most wonderful for the time."

"It may be in some measure due to the defects of notation in his time that Diophantos will have in his solutions no numbers whatever except rational numbers, in [the non-numbers of] which, in addition to surds and imaginary quantities, he includes negative quantities. ...Such equations then as lead to surd, imaginary, or negative roots he regards as useless for his purpose: the solution is in these cases ὰδοπος, impossible. So we find him describing the equation 4=4x+20 as ᾰτοπος because it would give x=-4. Diophantos makes it throughout his object to obtain solutions in rational numbers, and we find him frequently giving, as a preliminary, conditions which must be satisfied, which are the conditions of a result rational in Diophantos' sense. In the great majority of cases when Diophantos arrives in the course of a solution at an equation which would give an irrational result he retraces his steps and finds out how his equation has arisen, and how he may by altering the previous work substitute for it another which shall give a rational result. This gives rise, in general, to a subsidiary problem the solution of which ensures a rational result for the problem itself. Though, however, Diophantos has no notation for a surd, and does not admit surd results, it is scarcely true to say that he makes no use of quadratic equations which lead to such results. Thus, for example, in v. 33 he solves such an equation so far as to be able to see to what integers the solution would approximate most nearly."

"Nesselmann observes that we can, as regards the form of exposition of algebraic operations and equations, distinguish three historical stages of development... 1. ...Rhetoric Algebra, or "reckoning by complete words." ...the absolute want of all symbols, the whole of the calculation being carried on by means of complete words, and forming... continuous prose. ...2. ...Syncopated Algebra... is essentially rhetorical and therein like the first in its treatment of questions, but we now find for often-recurring operations and quantities certain abbreviational symbols. ...3. ...Symbolic Algebra ...uses a complete system of notation by signs having no visible connection with the words or things which they represent, a complete language of symbols, which supplants entirely the rhetorical system, it being possible to work out a solution without using a single word of the ordinary written language, with the exception (for clearness' sake) of a conjunction here and there, and so on. Neither is it the Europeans posterior to the middle of the seventeenth century who were the first to use Symbolic forms of Algebra. In this they were anticipated many centuries by the Indians."

"The tragedy of Karbala decided not only the fate of the caliphate, but of the Mohammedan kingdoms long after the Caliphate had waned and disappeared."

"The sword of Mahomet and the Coran are the most fatal enemies of civilization, liberty, and truth which the world has yet known."

"Neither their (the Muslims') tenets nor their practice will in any respect bear to come into competition with Christian, or even with Jewish morality. ... For instance, we call the Muslims chaste because they abstained from indiscriminate profligacy, and kept carefully within the bounds prescribed as licit by their Prophet. But those bounds besides the utmost freedom of divorce and change of wives, admitted an illimitable licence of cohabitation with 'all that the right hand of the believer might possess,' or, in other words, with any possible number of damsels he might choose to purchase, or receive in gift, or take captive in war."

"But of this genera of writing, William Muir's The Life of Mahomet, first published in 1861 in four volumes, was the best. It was a pioneering study and it has not been improved upon since then. William Muir had strong Christian views but he was also a painstaking and conscientious researcher, and he exhausted most of the sources on the Prophet's life, which were not many. The basic material on Muhammad is limited and new biographies could not really be new except in details, treatment and emphases."

"The Western societies have had hundreds of years of reformation behind them. Islam has never had its reformation, and that is part of the problem. If you look back to the 16th and 17th centuries when men were killing one another in the name of religion throughout Europe, that's where we're at more or less, historically speaking, in terms of the level of debate and discourse. The Quran is considered an untouchable text, not a historical document. ... This is the literal word of God, and it is very dangerous to play with that in the Middle East today."

"The Saudi government has been pumping money in a quiet kind of revolution to shape Islam in its own images since 1973, [with] oil price rises. It wasn't a noisy revolution like the Iranian revolution was. It didn't have so much hubbub and noise associated with it and all. But it was quietly done [with] Saudi influence, using money, and the building of [madrassas] -- that is, religious schools and mosques all across the world. ... The very particular kind of Islam associated with Saudi Arabia ... is an upstart. It was created in the 18th century. It was constrained and confined entirely to the Arabian Peninsula right through to the late 1960s. All of a sudden, this [Wahhabi] Islam -- which is espoused by these young men, which considers even a Muslim like myself, because of my Shiite background, to be dirty or not a real Muslim ... [is] probably the dominant form [of Islam] in the United States. It spreads from one end of the world to the next. It's been a quiet, silent revolution that's been happening, and suddenly exploded on the scene with Sept. 11."

"Perhaps the most dangerous element that was picked out of the Muslim tradition and changed and transformed in the hands of these young men who perpetrated Sept. 11 is this idea of committing suicide. They call it martyrdom, of course. Suicide is firmly rejected in Islam as an act of worship. In the tradition, generally, to die in battle for a larger purpose -- that is, for the sake of the community at large -- is a noble thing to do. Self-sacrifice yourself as you defend the community -- that is a traditional thing, and that has a traditional meaning of "jihad." But what is non-traditional, what is new is this idea that jihad is almost like an act of private worship. You become closer to God by blowing yourself up in such a way. You, privately, irrespective of what effect it has on everyone else. ... For these young men, that is the new idea of jihad. This idea of jihad allows you to lose all the old distinctions between combatants and non-combatants, between just and unjust wars, between the rules of engagement of different types. All of that is gone, because now the act of martyrdom is an act of worship ... in and of itself. It's like going on the pilgrimage. It's like paying your alms, which every Muslim has to do. It's like praying in the direction of Mecca, and so on and so forth. It is an individual act of worship. That's terrifying, and that's new. That's an entirely new idea, which these young men have taken out, developed."

"The defensiveness of Islam is its crucial feature today. It's what, by the way, is in such contrast to the most interesting period of Islamic history, when Islam was an open, absorbing religion, constantly taking in outside influences, as opposed to its current hedgehog-like posture, prickly to the outside ... always looking backward. This is not how it was in the creative moment, in the first four, six, eight centuries of Islam, where it was constantly seeking out and absorbing."

"Evil is something that, when you see it, when you know it, it's intimate. It's almost sensual. That is why people who have been tortured know it by instinct. They don't need to be told what it is, and they may have a very hard time putting it into words. ... That's the nature of the phenomenon. It's hard to put into words."

"Suggesting evil is human doesn't mean we can always understand it, or doesn't mean there's only one way of understanding it. It's sort of like a great work of art. You can never fully absorb it. It's got many dimensions. It lives on through time, in different ways."

"Bin Laden's real audience is the Middle East, his other Muslims. I think he thought that, by this act, he would win large numbers of converts to his cause ... [to] bring Arab regimes down. He would perhaps even take power in this or that country, preferably Saudi Arabia. That is where he is looking to; that is who is the audience. That is who his symbols are directed towards. So this is unlike anything else in the history of Islam. Early Muslims, when they left the Arabian Peninsula and entered the [Fertile Crescent], were conquerors. They converted peoples, and they gave them time to convert. So they didn't force them sometimes, and they were perfectly happy ruling over them. They were setting up a state, and then people converted over time. Syria remained Christian for hundreds of years after the Muslim conquest. So something different is going on here. The obvious sense in which the United States is evil is in the cultural icons that are seen everywhere. They are seemingly trivial things, the influence of the America culture, which is everywhere: TV, how women dress, the lack of importance of religion. So these are the senses in which they are rejecting the United States. But you're right; they don't see Americans as people. ... They block that out. They only see as people the Muslims they want to convert to their side, and that's terrifying."

"I have always thought there were dark ... corners in religion. I took that for granted. That's not the surprising thing for me. ... The frightening thing is rather that, in the Arab world, we have let the darkness of religion flourish."

"People say to me, "Kanan, this is ridiculous, democracy in Iraq, a complete pipe dream," ... That's realism. You know, in a way, the realists are right, they are always right. Even when they are morally wrong."