Mathematics Books

"Proclus was obliged to defend his "pagan" religion, and especially the belief in political equity, against the communal mob violence, systematic book burning, and state repression targeted especially against his school by the new ruler-priests. It is in this context that Proclus turns to mathematics, for he regarded mathematics, and reason generally, as a key instrument in persuading those who violently advocated faith and inequity. It is not incidental that most theorems in the book are about equality (later reinterpreted as "congruence"). Read in the manner explained by Proclus, the Elements is a text which refutes point by subtle point all the key elements in the changed Christian doctrine of the 4th c. CE (reason vs faith, immanence vs transcendence, equity vs inequity, learning as reminiscence, hence past lives vs creation in the recent past, reincarnation vs resurrection, eternal truths hence an eternal cosmos vs apocalypse, images as aids to learning vs charges of idolatry)."

"Euclid is a Platonist by persuasion and intimate with this system of philosophy, and so he has set before himself as goal of his system of Elements the constitution of the Platonic figures, as they are called."

"Now in the discovery of lemmas the best aid is a mental aptitude for it. For we may see many who are quick at solutions and yet do not work by method; thus Cratistus in our time was able to obtain the required result from first principles, and those the fewest possible, but it was his natural gift which helped him to the discovery."

"The property of similar triangles, and that contained in Prop. 47, Book I [the ], are most important theorems in Geometry."

"To solve [Book I] Prop. XI algebraically, or to find the point H in AB such that the rectangle contained by the whole line AB and the part HB shall be equal to the square of the other part AH. Let AB contain a linear units, and AH one of the unknown parts contain x units, then the other part HB contains a - x units. And \therefore a(a - x) = x^2, by the problem, or x^2 + ax = a^2, a quadratic equation.Hence [by the ] x = \frac{\pm\, a \sqrt{5} - a}{2}.The former of these values of x determines the point H. So [substituting a = AB] that x = \frac{\sqrt{5}-1}{2} \cdot AB = AH, one part, and a - x = a - AH = \frac{3 - \sqrt{5}}{2} \cdot AB = HB, the other part. It may be observed that the parts AH and HB cannot be numerically expressed by any rational number. Approximation to their true values in terms of AB, may be made to any required degree of accuracy, by extending the extraction of the square root of 5 to any number of decimals."

"There is a traditional story about Newton: as a young student, he began the study of geometry, as was usual in his time, with the reading of the Elements of Euclid. He read the theorems, saw that they were true, and omitted the proofs. He wondered why anybody should take pains to prove things so evident. Many years later, however, he changed his opinion and praised Euclid. The story may be authentic or not..."

"Newton had so remarkable a talent for mathematics that Euclid’s Geometry seemed to him “a trifling book,” and he wondered that any man should have taken the trouble to demonstrate propositions, the truth of which was so obvious to him at the first glance. But, on attempting to read the more abstruse geometry of Descartes, without having mastered the elements of the science, he was baffled, and was glad to come back again to his Euclid."

"The doctrine of Proportion, in the Fifth Book of Euclid's Elements, is obscure, and unintelligible to most readers. It is not taught either in foreign or American colleges, and is now become obsolete. It has therefore been omitted in this edition of Euclid's Elements, and a different method of treating Proportion has been substituted for it. This is the common algebraical method, which is concise, simple, and perspicuous; and is sufficient for all useful purposes in practical mathematics. The method is clear and intelligible to all persons who know the first principles of algebra. The rudiments of algebra ought to be taught before geometry, because algebra may be applied to geometry in certain cases, and facilitates the study of it."

"Abraham now thinks that the aggregate of all his schooling did not amount to one year. He was never in a college or academy as a student, and never inside of a college or academy building till since he had a law license. What he has in the way of education he has picked up. After he was twenty-three and had separated from his father, he studied English grammar—imperfectly of course, but so as to speak and write as well as he now does. He studied and nearly mastered the six books of Euclid since he was a member of Congress. He regrets his want of education, and does what he can to supply the want. In his tenth year he was kicked by a horse, and apparently killed for a time."

"Let us for the present admit, that a new work were written on a plan different from that of Euclid, constructed upon different principles, built upon different data, and exhibiting the leading results of geometrical science of a different order. Let us wave also the great improbability, that even an experienced instructor should execute a work superior to that which has been stamped with the approbation of ages, and consecrated, as it were, by the collected suffrage of the whole civilised globe. Still it may be questioned whether, on the whole, any real advantage would be gained. It is certain that all would not agree in their decision on the merits of such a work. Euclid once superseded, every teacher would esteem his own work the best, and every school would have its own class book. All that rigour and exactitude, which have so long excited the admiration of men of science, would be at an end. These very words would lose all definite meaning. Every school would have a different standard; matter of assumption in one, being matter of demonstration in others; until, at length, Geometry, in the ancient sense of the word, would be altogether frittered away, or be only considered as a particular application of Arithmetic and Algebra."

"The Elements of Euclid is as small a part of mathematics as the Iliad is of literature; or as the sculpture of Phidias is of the world’s total art."

"It is certain that from its completeness, uniformity and faultlessness, from its arrangement and progressive character, and from the universal adoption of the completest and best line of argument, Euclid’s “Elements” stand pre-eminently at the head of all human productions. In no science, in no department of knowledge, has anything appeared like this work: for upward of 2000 years it has commanded the admiration of mankind, and that period has suggested little toward its improvement."

"As to the need of improvement there can be no question whilst the reign of Euclid continues. My own idea of a useful course is to begin with arithmetic, and then not Euclid but algebra. Next, not Euclid, but practical geometry, solid as well as plane; not demonstration, but to make acquaintance. Then not Euclid, but elementary vectors, conjoined with algebra, and applied to geometry. Addition first; then the scalar product. Elementary calculus should go on simultaneously, and come into the vector algebraic geometry after a bit. Euclid might be an extra course for learned men, like Homer. But Euclid for children is barbarous."

"All our Greek texts of the Elements up to a century ago…purport in their titles to be either ‘from the edition of Theon’…or ‘from the lectures of Theon... [Greek commentaries] commonly speak of the writer of the Elements instead of using his name."

"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."

"The “elements” of the Great Alexandrian remain for all time the first, and one may venture to assert, the only perfect model of logical exactness of principles, and of rigorous development of theorems. If one would see how a science can be constructed and developed to its minutest details from a very small number of intuitively perceived axioms, postulates, and plain definitions, by means of rigorous, one would almost say chaste, syllogism, which nowhere makes use of surreptitious or foreign aids, if one would see how a science may thus be constructed one must turn to the elements of Euclid."

"Professor Klein then speaks of "that artistic finish that we admire in Euclid's Elements," and mentions Allman's important historical work. I heartily concur in this estimate of Euclid, and desire to contrast it with the error of Charles S. Peirce, in the Nation, where he speaks of "Euclid's proof (Elements Bk. I., props. 16 and 17)" as "really quite fallacious, because it uses no premises not as true in the case of spherics." Our bright American seems to have forgotten Euclid's Postulate 6 (Axiom 12 in Gregory, Axiom 9 in Heiberg), "Two straight lines cannot enclose a space;" that is, two straights having crossed never recur."

"Euclid himself is an elusive figure. We know nothing of him from contemporary references … Euclid’s Elements, which is believed to have been compiled around 300 BC, gives us an edited and anonymous treatment of some of the mathematics of Euclid’s predecessors, in a text that has passed through the hands of an unknown number of further scribes and editors before arriving at the versions that we now possess. ...Our evidence concerning pre-Euclidean mathematics is so indirect and fragmentary that the greatest part of the stories that are now told about the period must be invention."

"If we consider him [Euclid] as meaning to be what his commentators have taken him to be, a model of the most unscrupulous formal rigour, we can deny that he has altogether succeeded, though we admit that he made the nearest approach."

"The thirteen books of Euclid must have been a tremendous advance, probably even greater than that contained in the “Principia” of Newton,"

"The sacred writings excepted, no Greek has been so much read and so variously translated as Euclid."

"No one has ever given so easy and natural a chain of geometrical consequences [as Euclid]. There is a never-erring truth in the results."

"In England the geometry studied is that of Euclid, and I hope it never will be any other; for this reason, that so much has been written on Euclid, and all the difficulties of geometry have so uniformly been considered with reference to the form in which they appear in Euclid, that the study of that author is a better key to a great quantity of useful reading than any other."

"I want now to glance for a moment at the development of the theoretical method, and while doing so especially to observe the relation of pure theory to the totality of the data of experience. Here is the eternal antithesis of the two inseparable constituents of human knowledge, Experience and Reason, within the sphere of physics. We honour ancient Greece as the cradle of western science. She for the first time created the intellectual miracle of a logical system, the assertions of which followed one from another with such rigor that not one of the demonstrated propositions admitted of the slightest doubt—Euclid's geometry]]. This marvellous accomplishment of reason gave to the human spirit the confidence it needed for its future achievements. ...But yet the time was not ripe for a science that could comprehend reality, was not ripe until a second elementary truth had been realized, which only became the common property of philosophers after Kepler and Galileo. Pure logical thinking can give us no knowledge whatsoever of the world of experience; all knowledge about reality begins with experience and terminates in it."

"The Greeks... were well aware of geometric magnitudes that we call "irrational," but simply did not think of them as numbers. ...a direct result of their unyielding insistence on logical rigor. ...Book X ...devotes 115 propositions... to a comprehensive classification of irrational magnitudes of the forms a \pm \sqrt{b}, \sqrt{a} \pm \sqrt{b}, \sqrt{a \pm \sqrt{b}}, and \sqrt{\sqrt{a} \pm \sqrt{b}}, where a and b are commensurable lengths."

"The discovery of incommensurability meant that there existed geometric magnitudes that could not be measured by numbers! ...It followed that geometric magnitudes could not be manipulated without hesitation in algebraic computations just as though they were numbers. ...The Greek answer ...is presented in the geometric algebra of Books II and VI... The product of two lengths a and b is not a third length, but rather the area of a rectangle with sides a and b. ... The principle Greek technique for the geometric solution of algebraic equations was based on the "application of areas." For example, given a segment... of length a, the construction in Proposition I.44 (Proposition 44 of Book I) ...a rectangle with base AB of length a and area equal to that of a given square of edge b... provides a solution of the equation ax = b^2. This corresponds to geometric division [x = \frac{b^2}{a}], and we say that the given area b^2 has been applied to the given segment AB. Proposition VI.28... shows how to apply a given area b^2 to a given segment of length a, but "deficient" by a square. ...This construction provides a geometric solution of the quadratic equation ax - x^2 = b^2."

"The existence of incommensurable geometric magnitudes... necessitated a thorough reexamination and recasting of the foundations of mathematics, a task that occupied much of the fourth century B.C. During this period Greek algebra and geometry assumed the highly organized and rigorously deductive form that is set forth in the 13 books of the Elements that Euclid wrote about in 300 B.C. This systematic exposition of the Greek mathematical accomplishments of the preceding three centuries is the earliest major Greek mathematical text that is now available...(due perhaps to the extent to which the Elements subsumed previous expositions)."

"The discovery of incommensurable quantities threw an awful wrench in the machinery of geometry... The difficulty was finally overcome by Eudoxus' theory of proportion. But there was an indirect scare... In Euclid the theory of proportion and similar figures is postponed until the last possible moment, quite contrary to our present practice. Meanwhile, theorems which we prove by proportion were handled by the method of Application of Areas... The credit for discovering this seems to belong to the Pythagoreans:'According to the familiars of Eudemus, the inventions respecting the application, excess, and defect of spaces, is ancient, and belongs to the Pythagorean use.'What Proclus means...is... An area is applied to a given line segment, if we construct thereon a parallelogram of the given area and having a given angle. If the side of the parallelogram include not only the segment, but a prolongation, that part which is built on the extension is called the excess. On the other hand, if we use but a part of the segment, the parallelogram of the same height built on the unused part is called the defect. Let us see how the Greeks actually used the method. The deux ex machina was a simple figure called the 'gnomon'... Let us say that two plane figures are equivalent if they can be divide into the same number of figures... congruent, in pairs. They shall be called equivalent by completion if, by adding equivalent figures to them, the results are equivalent. ...we come to the actual use of the gnomon... in Euclid, II. 5... This is the identity\alpha \beta + (\frac{\alpha - \beta}{2})^2 = (\frac{\alpha + \beta}{2})^2Suppose... we wish to find the fourth proportional to... \alpha, \beta, \gamma. Euclid would reword this... apply to the line \alpha an area equal to that included by the lines \beta and \gamma. ...We construct a rectangle with non-parallel sides \beta and \gamma, extend the \beta side by the length \alpha, and complete the gnomon."

"Out of the pictures which are all that we can really see, we imagine a world of solid things; and... this world is constructed so as to fullfil a certain code of rules, some called axioms, and some called definitions, and some called postulates, and some assumed in the course of demonstration, but all laid down in one form or another in Euclid’s Elements of Geometry. ...This book has been for nearly twenty-two centuries the encouragement and guide of that scientific thought which is one... with the progress of man from a worse to a better state. The encouragement; for it contained a body of knowledge that was really known and could be relied on, and that moreover was growing in extent and application. For even at the time this book was written—shortly after the foundation of the Alexandrian Museum—Mathematic was no longer the merely ideal science of the Platonic school, but had started on her career of conquest over the whole world of Phenomena. The guide; for the aim of every scientific student of every subject was to bring his knowledge of that subject into a form as perfect as that which geometry had attained. Far up on the great mountain of Truth, which all the sciences hope to scale, the foremost of that sacred sisterhood was seen, beckoning for the rest to follow her. And hence she was called, in the dialect of the Pythagoreans, “the purifier of the reasonable soul.” Being thus in itself at once the inspiration and the aspiration of scientific thought, this book of Euclid has had a history as chequered as that of human progress itself."

"The Elements has been considered as offering models of scrupulously rigorous demonstrations. It is certainly true that in point of rigour it compares favourably with its modern rivals; but when examined in the light of strict mathematical logic, it has been pronounced by C.S. Peirce to be "riddled with fallacies." The results are correct only because the writer's experience keeps him on his guard."

"Comparatively few of the propositions and proofs in the Elements are his [Euclid's] own discoveries. In fact, the proof of the "Theorem of Pythagoras" is the only one directly ascribed to him."

"It is a remarkable fact in the history of geometry, that the Elements of Euclid, written two thousand years ago, are still regarded by many as the best introduction to the mathematical sciences."

"The most influential mathematics textbook of ancient times is easily named, for the Elements of Euclid has set the pattern in elementary geometry ever since. The most effective textbook of the medieval age is less easily designated; but a good case can be made out for the Al-jabr of Al-Khwarizmi, from which algebra arose and took its name. Is it possible to indicate a modern textbook of comparable influence and prestige? Some would mention the Géométrie of Descartes or the Principia of Newton or the Disquisitiones of Gauss; but in pedagogical significance these classics fell short of a work by Euler titled Introductio in analysin infinitorum."

"We have no reason to think that any special attention was paid to geometry by the Phoenicians, or other neighbours of the Egyptians. A small piece of evidence which tends to show that the Jews had not paid much attention to it is to be found in the mistake made in their sacred books, where it is stated that the circumference of a circle is three times its diameter: the Babylonians also reckoned that was equal to 3."

"The following comprise the chief propositions that can now with reasonable probability be attributed to him [Thales]...(i) The angles at the base of an isosceles triangle are equal (Euc. I, 5). Proclus seems to imply that this was proved by taking another exactly equal isosceles triangle, turning it over, and then superposing it on the first—a sort of experimental demonstration. (ii) If two straight lines cut one another, the vertically opposite angles are equal (Euc. I, 15). Thales may have regarded this as obvious, for Proclus adds that Euclid was the first to give a strict proof of it. (iii) A triangle is determined if its base and base angles be given (cf. Euc. I, 26). Apparently this was applied to find the distance of a ship at sea—the base being a tower, and the base angles being obtained by observation. (iv) The sides of equiangular triangles are proportionals (Euc. VI, 4, or perhaps rather Euc. VI, 2). This is said to have been used by Thales when in Egypt to find the height of a pyramid. "...the pyramid [height] was to the stick [height] as the shadow of the pyramid to the shadow of the stick." …we are told that the king Amasis, who was present, was astonished at this application of abstract science. (v) A circle is bisected by any diameter. This may have been enunciated by Thales, but it must have been recognised as an obvious fact from the earliest times. (vi) The angle subtended by a diameter of a circle at any point in the circumference is a right angle (Euc. III, 31). This appears to have been regarded as the most remarkable of the geometrical achievements of Thales... It has been conjectured that he may have come to this conclusion by noting that the diagonals of a rectangle are equal and bisect one another, and that therefore a rectangle can be inscribed in a circle. If so, and if he went on to apply proposition (i), he would have discovered that the sum of the angles of a right-angled triangle is equal to two right angles, a fact with which it is believed that he was acquainted. It has been remarked that the shape of the tiles used in paving floors may have suggested these results."

"We cannot form any exact idea as to how Thales presented his geometrical teaching. We infer, however, from Proclus that it consisted of a number of isolated propositions which were not arranged in a logical sequence, but that the proofs were deductive, so that the theorems were not a mere statement of an induction from a large number of special instances, as probably was the case with the Egyptian geometricians. The deductive character which he thus gave to the science is his chief claim to distinction."

"Thales... must have had considerable reputation as a man of affairs and as a good engineer, since he was employed to construct an embankment so as to divert the river Halys in such a way as to permit of the construction of a ford."

"Our knowledge of the mathematical attainments of those who preceded the Greeks is very limited; but... the early Greeks learned the use of the abacus for practical calculations, symbols for recording the results, and as much mathematics as is contained or implied in the Rhind papyrus. It is probable that this sums up their indebtedness..."

"Our acquaintance with the early attainments of the Chinese... serves to illustrate the fact that a nation may possess considerable skill in the applied arts while they are ignorant of the sciences on which those arts are founded."

"Their [the ancient Chinese] arithmetic was decimal in notation, but their knowledge seems to have been confined to the art of calculation by means of the swan-pan."

"The only geometrical theorem with which we can be certain that the ancient Chinese were acquainted is that in certain cases (namely, when the ratio of the sides is 3 : 4 : 5, or 1 : 1 : √2) the area of the square described on the hypotenuse of a right-angled triangle is equal to the sum of the areas of the squares described on the sides. It is barely possible that a few geometrical theorems which can be demonstrated in the quasi-experimental way of superposition were also known to them."

"The idea that the Chinese had made considerable progress in theoretical mathematics seems to have been due to a misapprehension of the Jesuit missionaries who went to China in the sixteenth century. ...they failed to distinguish between the original science of the Chinese and the views which they found prevalent on their arrival|the latter being founded on the work and teaching of Arab or Hindoo missionaries who had come to China in the course of the thirteenth century or later, and while there introduced a knowledge of spherical trigonometry."

"At a very early period the Chinese were acquainted with several geometrical or rather architectural implements, such as the rule, square, compasses, and level; with a few mechanical machines, such as the wheel and axle; that they knew of the characteristic property of the magnetic needle; and were aware that astronomical events occurred in cycles. But the careful investigations of L. A. Sédillot have shown that the Chinese made no serious attempt to classify or extend the few rules of arithmetic or geometry with which they were acquainted, or to explain the causes of the phenomena which they observed."

"We know nothing of the applied mathematics (if there were any) of the Egyptians or Phoenicians. The astronomical attainments of the Egyptians and Chaldaeans were no doubt considerable, though they were chiefly the results of observation: the Phoenicians are said to have confined themselves to studying what was required for navigation."

"Ahmes's book gives us much that idea of Egyptian mathematics which we should have gathered from statements about it by various Greek and Latin authors, who lived centuries later. Previous to its translation it was commonly thought that these statements exaggerated the acquirements of the Egyptians, and its discovery must increase the weight to be attached to the testimony of these authorities."

"All the specimens of Egyptian geometry which we possess deal only with particular numerical problems and not with general theorems; and even if a result be stated as universally true, it was probably proved to be so only by a wide induction. ...Greek geometry was from its commencement deductive. There are reasons for thinking that Egyptian geometry and arithmetic made little or no progress subsequent to the date of Ahmes's work; and though for nearly two hundred years after the time of Thales Egypt was recognised by the Greeks as an important school of mathematics, it would seem that, almost from the foundation of the Ionian school, the Greeks outstripped their former teachers."

"Ahmes gives some problems on pyramids. ...Ahmes was attempting to find, by means of data obtained from the measurement of the external dimensions of a building, the ratio of certain other dimensions which could not be directly measured: his process is equivalent to determining the trigonometrical ratios of certain angles. The data and the results given agree closely with the dimensions of some of the existing pyramids. Perhaps all Ahmes's geometrical results were intended only as approximations correct enough for practical purposes."

"Ahmes then goes on to find the area of a circular field … and gives the result as (d - 1/9d)2, where d is the diameter of the circle: this is equivalent to taking 3.1604 as the value of π, the actual value being very approximately 3.1416."

"That some geometrical results were known at a date anterior to Ahmes's work seems clear if we admit... that, centuries before it was written, the following method of obtaining a right angle was used in laying out the ground-plan of certain buildings. The Egyptians were very particular about the exact orientation of their temples; and they had therefore to obtain with accuracy a north and south line, as also an east and west line. By observing the points on the horizon where a star rose and set, and taking a plane midway between them, they could obtain a north and south line. To get an east and west line, which had to be drawn at right angles to this, certain professional "rope-fasteners" were employed. These men used a rope... divided by knots or marks... in the ratio 3 : 4 : 5. ...A similar method is constantly used at the present time by practical engineers for measuring a right angle. ...But though these are interesting facts in the history of the Egyptian arts we must not press them too far as showing that geometry was then studied as a science. Our real knowledge of the nature of Egyptian geometry depends mainly on the Rhind papyrus."

"He [Ahmes] concludes the work with some arithmetico-algebraical questions, two of which deal with arithmetical progressions and seem to indicate that he knew how to sum such series."