Non Fiction Authors From England

"As medieval Christian man could only conceive of religion in terms of a trinity, so his modern descendant can only conceive of politics in terms of a theology or, as we now say, ideology, of left-wing and right-wing forces and factions."

"To the modern Western mind, it is not conceivable that men would fight and die in such numbers over mere differences of religion; there have to be some other “genuine” reasons underneath the religious veil. We are prepared to allow religiously defined conflicts to accredited eccentrics like the Northern Irish, but to admit that an entire civilization can have religion as its primary loyalty is too much. Even to suggest such a thing is regarded as offensive by liberal opinion, always ready to take protective umbrage on behalf of those whom it regards as its wards. This is reflected in the present inability, political, journalistic, and scholarly alike, to recognize the importance of the factor of religion in the current affairs of the Muslim world and in the consequent recourse to the language of left-wing and right-wing, progressive and conservative, and the rest of the Western terminology, the use of which in explaining Muslim political phenomena is about as accurate and as enlightening as an account of a cricket match by a baseball correspondent."

"Islam is one of the world's great religions. Let me be explicit about what I, as a historian of Islam who is not a Muslim, mean by that. Islam has brought comfort and peace of mind to countless millions of men and women. It has given dignity and meaning to drab and impoverished lives. It has taught people of different races to live in brotherhood and people of different creeds to live side by side in reasonable tolerance. It inspired a great civilization in which others besides Muslims lived creative and useful lives and which, by its achievement, enriched the whole world. But Islam, like other religions, has also known periods when it inspired in some of its followers a mood of hatred and violence. It is our misfortune that part, though by no means all or even most, of the Muslim world is now going through such a period, and that much, though again not all, of that hatred is directed against us."

"The three major Middle Eastern religions are significantly different in their relations with the state and their attitudes to political power. Judaism was associated with the state and was then disentangled from it; its new encounter with the state at the present time raises problems which are still unresolved. Christianity, during the first formative centuries of its existence, was separate from and indeed antagonistic to the state with which it only later became involved. Islam from the lifetime of its founder was the state, and the identity of religion and government is indelibly stamped on the memories and awareness of the faithful from their own sacred writings, history, and experience."

"You know the expression "My country, right or wrong." Well, these days, Americans are apt to think, "My country, wrong.""

"The attitude to black Africans remains on the whole negative. Some Muslim authors give balanced and factual accounts, based on personal knowledge, of the black kingdoms; a few even write pious treatises to defend the dark peoples against their detractors. Such defense was clearly felt to be necessary, because of the survival of old prejudices. Even the great geographer Idrisi, in concluding his account of the first climate (geographical zone) with some general remarks on its inhabitants, repeats the old cliches about furrowed feet and stinking sweat and ascribes "lack of knowledge and defective minds" to the black peoples. Their ignorance, he says, is notorious; men of learning and distinction are almost unknown among them, and their kings only acquire what they know about government and justice from the instruction of learned visitors from farther north. The thirteenth-century Persian writer Nasir al-Din Tusi remarks that the Zanj differ from animals only in that "their two hands are lifted above the ground" and continues, "Many have observed that the ape is more teachable and more intelligent than the Zanji.""

"The evidence for the growth of anti-black prejudice comes in the main from two groups of sources. The first of these is literary, especially poetry and anecdote. Several Arabic poets, of the pre-Islamic and early Islamic periods, are described as "black" and are known collectively, to the literary tradition, as aghribat al-'Arab-"the crows of the Arabs."' Some of them-mostly preIslamic-were Arabs of swarthy complexion; others were of mixed Arab and African parentage. For the latter, and still more for the pure Africans, blackness was an affliction. In many verses and narratives, they are quoted as suffering from insult and discrimination, as showing resentment at this, and yet in some way as accepting the inferior status resulting from their African ancestry. One such was the poet Suhaym (d. 660), born a slave and of African origin. His name, obviously a nickname, might be translated as "little black man." In one poem he laments: 'If my color were pink, women would love me. But the lord has marred me with blackness.'"

"On 8 June 632, according to the traditional biography, the Prophet died after a short illness. He had achieved a great deal. To the pagan peoples of western Arabia he had brought a new religion which, with its monotheism and its ethical doctrines, stood on an incomparably higher level than the paganism it replaced. He had provided that religion with a revelation which was to become in the centuries to follow the guide to thought and count of countless millions of Believers. But he had done more than that; he had established a community and a well organized and armed state, the power and prestige of which made it a dominant factor in Arabia. What then is the final significance of the career of the Arabian Prophet? For the traditional Muslims the question scarcely arises. Muhammad was the last and greatest of the Apostles of God, sent as the Seal of Prophecy to bring the final revelation of god's word to mankind. His career and success were fore-ordained and inevitable and needed no the pious fantasy of later generations of believers clothed the dim figure of the Prophet with a rich and multi-coloured fabric of fable, legend, and miracle, not realizing that by diminishing his essential historic humanity they were robbing him of one of his most attractive qualities."

"For a long time now there has been a rising tide of rebellion against this Western paramountcy, and a desire to reassert Muslim values and restore Muslim greatness. The Muslim has suffered successive stages of defeat. The first was his loss of domination in the world, to the advancing power of Russia and the West. The second was the undermining of his authority in his own country, through an invasion of foreign ideas and laws and ways of life and sometimes even foreign rulers or settlers, and the enfranchisement of native non-Muslim elements. The third—the last straw—was the challenge to his mastery in his own house, from emancipated women and rebellious children. It was too much to endure, and the outbreak of rage against these alien, infidel, and incomprehensible forces that had subverted his dominance, disrupted his society, and finally violated the sanctuary of his home was inevitable. It was also natural that this rage should be directed primarily against the millennial enemy and should draw its strength from ancient beliefs and loyalties."

"The overwhelming majority of classical theologians, jurists, and traditionalists [i.e., specialists in the hadith] . . . understood the obligation of jihad in a military sense."

"The whole question of blackness was discussed in a special essay by Jahiz of Basra (ca. 776-869), one of the greatest prose writers in classical Arabic literature and said by some of his biographers to be of partly African descent. Entitled "The Boast of the Blacks against the Whites,"" the essay purports to be a defense of the dark-skinned peoples-and especially of the Zanj, the blacks of East Africa-against their detractors, refuting the accusations commonly brought against them and setting forth their qualities and achievements, with a wealth of poetic illustration... To those who ask, "How is it that we have never seen a Zanji who had the intelligence even of a woman or of a child?" the answer, says Jahiz, is that the only Zanj they knew were slaves of low origin and from outlying and backward areas. If they judged by their experience of Indian slaves, would they have any notion of Indian science, philosophy, and art? Obviously not-and the same is true of the black lands. Jahiz also defends the equality of blacks as marriage partners and notes the paradox that discrimination against them first arose after the advent of Islam: At is part of your ignorance," he makes the blacks say, "that in the time of heathendom [i.e., in pre-Islamic Arabia] you regarded us as good enough to marry your women, yet when the justice of Islam came, you considered this wrong.""

"Of all these offenses the one that is most widely, frequently, and vehemently denounced is undoubtedly imperialism—sometimes just Western, sometimes Eastern (that is, Soviet) and Western alike. But the way this term is used in the literature of Islamic fundamentalists often suggests that it may not carry quite the same meaning for them as for its Western critics. In many of these writings the term "imperialist" is given a distinctly religious significance, being used in association, and sometimes interchangeably, with "missionary," and denoting a form of attack that includes the Crusades as well as the modern colonial empires. One also sometimes gets the impression that the offense of imperialism is not—as for Western critics—the domination by one people over another but rather the allocation of roles in this relationship. What is truly evil and unacceptable is the domination of infidels over true believers. For true believers to rule misbelievers is proper and natural, since this provides for the maintenance of the holy law, and gives the misbelievers both the opportunity and the incentive to embrace the true faith. But for misbelievers to rule over true believers is blasphemous and unnatural, since it leads to the corruption of religion and morality in society, and to the flouting or even the abrogation of God's law. This may help us to understand the current troubles in such diverse places as Ethiopian Eritrea, Indian Kashmir, Chinese Sinkiang, and Yugoslav Kossovo, in all of which Muslim populations are ruled by non-Muslim governments. It may also explain why spokesmen for the new Muslim minorities in Western Europe demand for Islam a degree of legal protection which those countries no longer give to Christianity and have never given to Judaism. Nor, of course, did the governments of the countries of origin of these Muslim spokesmen ever accord such protection to religions other than their own. In their perception, there is no contradiction in these attitudes. The true faith, based on God's final revelation, must be protected from insult and abuse; other faiths, being either false or incomplete, have no right to any such protection."

"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)."

"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 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."

"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."

"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."

"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..."

"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."

"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..."

"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."

"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."

"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."

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

"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."

"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."

"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."

"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)."

"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."

"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."

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

"The method of exhaustion was not discovered all at once; we find traces of gropings after such a method before it was actually evolved. It was perhaps Antiphon. the sophist, of Athens, a contemporary of Socrates, who took the first step. He inscribed a square (or, according to another account, a triangle) in a circle, then bisected the arcs subtended by the sides, and so inscribed a polygon of double the number of sides; he then repeated the process, and maintained that, by continuing it, we should at last arrive at a polygon with sides so small as to make the polygon coincident with the circle. Thought this was formally incorrect, it nevertheless contained the germ of the method of exhaustion."

"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."

"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."

"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 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."

"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."

"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."

"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."

"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."

"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 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}."