Ancient Greek mathematics

"If the early Greeks were cognizant of Babylonian algebra, they made no attempt to develop or even to use it, and thereby they stand convicted of the supreme stupidity in the history of mathematics. ...The ancient Babylonians had a rare capacity for numerical calculation; the majority of Greeks were either mystical or obtuse in their first approach to number. What the Greeks lacked in number, the Babylonians lacked in logic and geometry, and where the Babylonians fell short, the Greeks excelled. Only in the modern mind of the seventeenth and succeeding centuries were number and form first clearly perceived as different aspects of one mathematics."

"Had the early Greek mind been sympathetic to the algebra and arithmetic of the Babylonians, it would have found plenty to exercise its logical acumen, and might easily have produced a masterpiece of the deductive reasoning it worshipped logically sounder than Euclid's greatly overrated Elements. The hypotheses of elementary algebra are fewer and simpler than those of synthetic geometry. ...they could have developed it with any degree of logical rigor they desired. Had they done so, Apollonius would have been Descartes, and Archimedes, Newton. As it was, the very perfection... of Greek geometry retarded progress for centuries."

"The Greeks ordinarily are regarded as the founders of mathematics in the strict sense... for they emphasized the value of abstract generalizations... and the deductive elaboration of these. ...this early intellectual revolution occurred at about the time of a distinct geographical shift in the centers of civilization. The focal points previously had river valleys, such as the Nile, or of the Tigris and Euphrates; but by the middle of the eighth century B.C. these ancient potamic civilizations were confronted with a vigorous young thalassic civilization established about the Mediterranean Sea."

"The Greek search for essences had led the Pythagoreans to picture the universe as a multitude of mathematical points completely subject to the laws of number—a sort of arithmetic geometry... The rival Eleatic philosophy of Parmenides upheld the essential "oneness" of the universe and the impossibility of analyzing it in terms of the "many." Zeno of Elea sought dialectically to defend his master's doctrine by demolishing the Pythagorean association of multiplicity with number and magnitude. ...The paradoxes, as one sees now, involve such notions as infinite sequence, limit, and continuity, concepts for which Zeno nor any of the ancients gave precise definition. ...their influence was profound. The Greeks banned from their mathematics any thought of an arithmetic continuum or of an algebraic variable, ideas which might have led to analytic geometry; and they refused to place any confidence in infinite processes, the methods which would have led to calculus. Whereas the Pythagoreans had envisioned a union of arithmetic and geometry, Greek mathematicians after Zeno saw only the mutual incompatibility of the two fields."

"That the discovery of incommensurability of lines made a strong impression on Greek thought is indicated by the story of Hippasus... It is demonstrated more reliably by the prominence given to the theory of irrationals by Plato and his school [e.g., Eudoxus of Cnidus]. ...the Greeks were led by Zeno and Hippasus to abandon the pursuit of a full arithmetization of geometry... there was no such thing as algebraic analysis. Geometry was the domain of continuous magnitude, arithmetic was concerned with the discrete set of integers; and the two fields were irreconcilable."

"In mathematics... the Greek attitude differed sharply from that of the earlier potamic cultures. The contrast was clear in... Thales and Pythagoras, and it continues to show... in Athens during the Heroic Age. ...while Anaxagoras was in prison he occupied himself with an attempt to square the circle... the first mention of a problem that was to fascinate mathematicians for more than 2000 years. ...Here we see a type of mathematics that is quite unlike that of the Egyptians and Babylonians. It is not the practical application of a science of number... but a theoretical question involving a... distinction between accuracy in approximation and exactitude in thought. ...no more the concern of the technologist than those he raised... concerning the ultimate structure of matter."

"In the Greek world mathematics was more closely related to philosophy than to practical affairs, and this kinship has persisted to the present day."

"These three problems—the , the duplication of the cube, and the trisection of the angle—have since been known as the "three famous (or classical) problems" of antiquity. More than 2200 years later it was proved that all three... were unsolvable by means of straightedge and compass alone. ...the better part of Greek mathematics, and much of later mathematical thought, was suggested by efforts to achieve the impossible—or failing this, to modify the rules."

"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."

"Euclid, Archimedes, and Apollonius brought geometry to as high a state of perfection as it perhaps could be brought without first introducing some more general and more powerful method than the old method of exhaustion. A briefer symbolism, a Cartesian geometry, an infinitesimal calculus, were needed. The Greek mind was not adapted to the invention of general methods. Instead of a climb to still loftier heights we observe, therefore, on the part of later Greek geometers, a descent during which they paused here and there to look around for details which had been passed by in the hasty ascent."

"Thus, it is again a conclusion to be assumed in advance, only waiting for a confirmation, that Greek mathematics had brought its traces into those Indian works."

"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"

"The inspiration of Fermat's discussion of the conic sections, and that is practically the whole of his analytic geometry, comes direct from Apollonius. The same had been true of Pappus, fourteen centuries before. His point of departure is the famous four-line problem... This question seems to have stumped both Euclid and Aristaeus, and to have been first solved by Apollonius. In Apollonius's own work we find what is rather the converse of this problem. Almost the first piece of geometrical writing which Fermat did was to prove the three-line case."

"It is important to remember that the ancient Greeks did not have an abstract system of number symbols, and used the letters of the alphabet as number symbols. They also commonly manipulated pebbles to learn arithmetic and used small stones on calculating boards. In this case, number patterns were their common experience of arithmetic. From this use of pebbles, we have inherited the word 'calculation,' from the Latin calculus, which means 'pebble.'"

"How is it that the nation which gave us geometry and carried this science so far, did not create even a rudimentary algebra?"

"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 the 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 Greeks... were well aware of geometric magnitudes that we call "irrational," but simply did not think of them as numbers."

"Logistic is the theory which deals with numerable objects and not with numbers; it does not, indeed, consider number in the proper sense of the term, but assumes 1 to be unity, and anything which can be numbered to be number (thus in place of the triad, it employs 3; in place of the decad, 10), and discusses with these the theorems of arithmetic. ... It treats, then, on the one hand, that which Archimedes called 'The Cattle Problem,' and on the other hand 'melite' and 'phialite' numbers, the one discussing vials (measures, containters) and the other flocks; and when dealing with other kinds of problems it has regard to the number of sensible bodies and makes its pronouncements as though it were for absolute objects. ... It has for material all numerable objects, and as subdivisions the so-called Greek and Egyptian methods for multiplication and division, as well as the summation and decomposition of fractions, whereby it investigates the secrets lurking in the subject-matter of the problems by means of the procedure that employs triangles and polygons. ... It has for its aim that which is useful in the relations of life in business, although it seems to pronounce upon sensible objects as if they were absolute."

"It is well known that the commentary of Proclus on Eucl. Book I is one of the two main sources of information as to the history of Greek geometry which we possess, the other being the Collection of Pappus."

"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."

"There is here, as in all great Greek mathematical masterpieces, no hint as to the kind of analysis by which the results were first arrived at; for it is clear that they were not discovered by the steps which led up to them in the finished treatise. If the geometrical treatises had stood alone, Archimedes might seem, as Wallis said, "as it were of set purpose to have covered up the traces of his investigations, as if he has grudged posterity the secret of his method of inquiry, while he wished to extort from them assent to his results.""

"Nicomachus... mentions the customary Pythagorean divisions of quantum and the science that deals with each. Quantum is either discrete or continuous. Discrete quantum in itself considered, is the subject of Arithmetic; if in relation, the subject of Music. Continuous quantum, if immovable, is the subject of Geometry; if movable, of Spheric (Astronomy). These four sciences formed the of the Pythagoreans. With the (which Nicomachus does not mention) of Grammar, Logic, and Rhetoric, they composed the seven liberal arts taught in the schools of the Roman Empire."

"Arithmetic is fundamentally associated by modern readers, particularly by scientists and mathematicians, with the art of computation. For the ancient Greeks after Pythagoras, however, arithmetic was primarily a philosophical study, having no necessary connection with practical affairs. Indeed the Greeks gave a separate name to the arithmetic of business, λογιστική [accounting or practical logistic]... In general the philosophers and mathematicians of Greece undoubtedly considered it beneath their dignity to treat of this branch, which probably formed a part of the elementary instruction of children."

"And do you not know also that although they make use of the visible forms and reason about them, they are thinking not of these, but of the ideals which they resemble; not of the figures which they draw, but of the absolute square and the absolute diameter, and so on --the forms which they draw or make, and which have shadows and reflections in water of their own, are converted by them into images, but they are really seeking to behold the things themselves, which can only be seen with the eye of the mind?"

"The ancient Geometry had no symbols, nor any notation beyond ordinary language and the specific terms of the science."

"Arithmetic... teaches all the various operations of numbers and demonstrates their properties. ...The Greeks are said have received it from the Phoenicians. The ancients, who have treated arithmetic most exactness, are Euclid, Nicomachus of Alexandria, and . It was difficult either for the Greeks or the Romans to succeed much in arithmetic, as both used only letters of the alphabet for numbers, the multiplication of which, in great calculations, necessarily occasioned abundance of trouble. The Arabic ciphers... are infinitely more commodious, and have contributed very much to the improvement of arithmetic."

"More than any other of his predecessors Plato appreciated the scientific possibilities of geometry... By his teaching he laid the foundation of the science, insisting upon accurate definitions, and logical proof. His opposition to the materialists, who saw in geometry only what was immediately useful to the artisan and the mechanic, is made clear by Plutarch in his Life of Marcellus... "Plato's indignation at it and his invections against it as the mere corruption and annihilation of the one good geometry, which was thus shamefully turning its back upon the unembodied objects of pure intelligence.""

"Each of these sciences has a subject which is different from the science. I can show you that the art of computation has to do with odd and even numbers in their numerical relations to themselves and to each other. ...And the odd and even numbers are not the same with the art of computation? ..The art of weighing, again, has to do with lighter and heavier; but the art of weighing is one thing, and the heavy and the light another. ...what is that which is not wisdom, and of which wisdom is the science? ...wisdom is the only science which is the science of itself as well as of the other sciences. ...But the science of science... will also be the science of the absence of science."

"The theory of proportions is credited to Eudoxus... and is expounded in Book V of Euclid's Elements. The purpose of the theory is to enable lengths (and other geometric quantities) to be treated as precisely as numbers, while only admitting the use of rational numbers. ...To simplify ...let us call lengths rational if they are rational multiples of a fixed length. Eudoxus' idea was to say that a length \lambda is determined by those rational lengths less than it and those greater than it. ...he says \lambda_1 = \lambda_2...if any rational length < \lambda_1 is also < \lambda_2, and vice versa [any rational length > \lambda_2 is also > \lambda_1]. Likewise \lambda_1 < \lambda_2 if there is a rational length > \lambda_1 but < \lambda_2 [between \lambda_1 and \lambda_2]. This definition uses the rationals to give an infinitely sharp notion of length while avoiding any overt use of infinity. ... The theory of proportions was so successful that it delayed the development of a theory of real numbers for 2000 years. This was ironic, because the theory of proportion can be used to define irrational numbers just as well as lengths. It was understandable though, because the common irrational lengths... arise from constructions that are intuitively clear and finite from the geometric point of view. Any arithmetic approach to the \sqrt2, whether by sequences, decimals, or continued fractions, is infinite and therefore less intuitive. Until the nineteenth century this seemed a good reason... Then the problems of geometry came to a head, and mathematicians began to fear geometric intuition as much as they had previously feared infinity."

"The towns which arose along the coast of Asia Minor and on the Greek mainland were no longer administration centers of an irrigation society. They were trading towns in which the old-time feudal landlords had to fight a losing battle with an independent, politically conscious merchant class. ...The merchant trader had never enjoyed so much independence, but he knew that this independence was the result of a constant and bitter struggle. The static outlook of the Orient could never be his. He lived in a period of geographical discovery comparable only to those of sixteenth-century Western Europe; he recognized no absolute monarch or power supposedly vested in a static deity. ...he could enjoy a certain amount of leisure, the result of wealth and of slave labor. He could philosophize...The absence of any well-established religion led many... into mysticism, but also stimulated its opposite, the growth of rationalism and the scientific outlook."

"The "exhaustion method"... was the Platonic school's answer to Zeno. It avoided the pitfalls of the infinitesimals by simply discarding them... It had the disadvantage that the result... must be known in advance ...a letter from Archimedes to Eratosthenes... described a nonrigorous but fertile way of finding results ...known as the "Method." It has been suggested... that it represented a school of mathematical reasoning competing with Eudoxus... In Democritus' school, according to the theory of Luria, the notion of a "geometrical atom" was introduced. ...The advantage of the "atom method" over the "exhaustion method" was that it facilitated the finding of new results. Antiquity had thus the choice between a rigorous but relatively sterile, and a loosely-founded but far more fertile method. ...in practically all classical texts the first [the exhaustion] method was used. This... may be connected with the fact that mathematics had become a hobby of the leisure class which was based on slavery, indifferent to invention, and interested in contemplation. It may also be a reflection of the victory of Platonic idealism over Democritian materialism in the realm of mathematical philosophy."

"It is the purpose of this paper to show what is historically wrong with the traditional way the history of ancient Greek mathematics has been written and to call to the new generation of historians of Greek mathematics to rewrite the history on a new and historically sane basis."

"One of the central concepts for the understanding of ancient Greek mathematics has customarily been, at least since the time of and , the concept of 'geometric algebra'. What it amounts to is that Greek mathematics, especially after the discovery of the 'irrational'... is algebra dressed up, primarily for the sake of rigor, in geometrical garb. The reasoning... the line of attack... the solutions... etc. all are essentially algebraic... attired in geometrical accouterments. We... look for the algebraic 'subtext'... of any geometrical proof... always to transcribe... any proposition in[to] the symbolic language of modern algebra... [making] the logical structure of the proof clear and convincing, without thereby losing anything, not only in generality but also in any possible sui generis features of the ancient way of doing things. ...[i.e., that] there is nothing unique and (ontologically) idiosyncratic concerning the way... ancient Greek mathematicians went about their proofs, which might be lost... I cannot find any historically gratifying basis for this generally accepted view... those who have been writing the history of mathematics... have typically been mathematicians... largely unable to relinquish and discard their laboriously acquired mathematical competence when dealing with periods in history during which such competence is historically irrelevant and... anachronistic. Such... stems from the unstated assumption that mathematics is a scientia universalis, an algebra of thought containing universal ways of inference, everlasting structures, and timeless, ideal patterns of investigation which can be identified throughout the history of civilized man and which are completely independent of the form in which they happen to appear at a particular junction of time."

"Mathematics as a science commenced when first someone, probably a Greek, proved propositions about any things or about some things, without specification of definite particular things. These propositions were first enunciated by the Greeks for geometry; and, accordingly, geometry was the great Greek mathematical science. After the rise of geometry centuries passed away before algebra made a really effective start, despite some faint anticipations by the later Greek mathematicians."

"The Greeks would have said... we know a much better way of taking a square root. ...the ancient Greeks thought entirely geometrically, not arithmetically. And they would... do the following. If you want to solve x^2 = N, you should first... think of whether N is bigger than or equal to one. Suppose that case 1) N < 1. ...Draw a [horizontal] line segment of length one and then [within and from the end of that segment]... make a segment of size N. And then with the center of the [length one] segment you draw a circle so this is a [unit length] diameter. And you... [draw a vertical line from the end of the N segment inside the circle] up here [to intersect the circle] and then... look at this quantity x... this [top angle of the largest triangle circumscribed by the circle] is a right angle by Thales theorem, so we have some similar triangles. So [side x, side 1 from the large circumscribed triangle] \frac{x}{1} = \frac{N}{x} [side N, side x from the small left triangle] by similar \triangle's . And so x^2 = N. So Geometrically finding a square root is... a relatively simple... rule or construction, but arithmetically much more difficult. What happens if N is bigger than one? Well then you just interchange the roles of the N and the one. Case 2) N \ge 1. So you start by having a diameter of size N and then you make [a line segment of length] 1 here [from the end of the segment of length N to within that segment] and then otherwise do exactly the same thing [as in the above, case 1]. ...x will be square root, x^2 = N, by the same argument."

"The history of Alexandrian mathematics begins with the Elements of Euclid and closes with the Algebra of Diophantus, both of which are founded on the discoveries of several preceding centuries."

"The extraordinary ability of Diophantus appears rather in... the ingenuity with which he reduces every problem to an equation which he is competent to solve."

"The most common and characteristic of Diophantus' methods is his use of tentative assumptions which is applied in nearly every problem of the later books. It consists in assigning to the unknown a preliminary value which satisfies one or two only of the necessary conditions, in order that, from its failure to satisfy the remaining conditions, the operator may perceive what exactly is required..."

"With Diophantus the history of Greek arithmetic comes to an end. No original work, that we know of, was done afterwards."

"The oldest definition of Analysis as opposed to Synthesis is that appended to Euclid XIII. 5. It was possibly framed by Eudoxus. It states that "Analysis is the obtaining of the thing sought by assuming it and so reasoning up to an admitted truth: synthesis is the obtaining of the thing sought by reasoning up to the inference and proof of it." In other words, the synthetic proof proceeds by shewing that certain admitted truths involve the proposed new truth: the analytic proof proceeds by shewing that the proposed new truth involves certain admitted truths."

"The history of the Athenian school begins with the teaching of Hippocrates about 420 B.C.; the school was established on a permanent basis by the labours of Plato and Eudoxus; and, together with the neighboring school of Cyzicus, continued to extend on the lines laid down by these three geometricians until the foundation (about 300 B.C.) of the university at Alexandria drew thither most of the talent of Greece."

"Eudoxus... is also reckoned as the founder of the school at Cyzicus. The connection [with the school] of Athens was very close, and it is impossible to disentangle their histories. It is said that Hippocrates, Plato, and Theaetetus belonged to the Athenian school; while Eudoxus, , and Aristaeus belonged to that of Cyzicus. There was always constant intercourse between the two schools, the earliest members of both had been under the influence either of Archytas or of his pupil ..."

"The geometricians of these schools... were especially interested in three problems: namely (i), the duplication of the cube... (ii) the trisection of an angle; and (iii) the squaring of a circle... Now the first two... (considered analytically) require the solution of a quadratic equation; and, since a construction by means of circles (whose equations are of the form x^2 + y^2 + ax + by + c = 0 and straight lines (whose equations are of the form \alpha x + \beta y + \gamma = 0) cannot be equivalent to the solution of a cubic equation, the problems are insoluble if in our constructions we restrict ourselves to the use of circles and straight lines, that is, to Euclidean geometry. If the use of s be permitted, both of these questions can be solved in many ways. The third problem is equivalent to finding a rectangle whose sides are equal respectively to the radius and to the semiperimeter of the circle. These lines have long been known to be incommensurable, but it is only recently that it has been shown by Lindemann that their ratio cannot be the root of a rational algebraical equation. Hence the problem also is insoluble by Euclidean geometry. The Athenians and Cyzicians were thus destined to fail in all three problems, but the attempts to solve them led to the discovery of many new theorems and processes."

"The sum of the three angles of every plane triangle is equal to two right angles. The mathematical truth enunciated in the above theorem is not new It has been known for more than two thousand years. ...Our reason for believing that Thales was not ignorant of the theorem under consideration is found in the beautiful demonstration by which he proved that every angle in a semicircle is a right angle. This appears to have been regarded as the most remarkable of the geometrical achievements of Thales, and it is stated that on inscribing a right angled triangle in a circle he sacrificed an ox to the immortal gods."

"Before giving the proof by which Thales probably established the truth... it will be well to consider the geometrical capital which this Grecian mathematician had at his command. ...i. The angles at the base of an are equal. ...ii. If two straight lines cut one another the vertically opposite angles are equal."

"vi. The angle in a semicircle is a right angle. It is believed that Thales proved this proposition in the following manner: Let ABCH be a circle of which the diameter is BC, and the centre E. ...Draw AE and produce BA to F. Because BE is equal to EA [both being radii of the circle], the angle EAB is equal to EBA; also, because AE is equal to EC, the angle EAC is equal to ECA [being angles at the base of an isosceles triangle]; wherefore, the whole angle BAC is equal to the two angles ABC, ACB. But FAC, the exterior angle of the triangle ABC, is also equal to the two angles ABC, ACB [since the sum of the three angles of the triangle is equal to two right angles, i.e., a straight line]; therefore the angle BAC is equal to the angle FAC, and each of them is therefore a right angle; wherefore the angle BAC in a semicircle is a right angle. Thales's demonstration, if we may call this his, is quite different from the one given in modern text-books; but it is certainly neither less rigid nor less beautiful. The demonstration is the one given in Euclid, but his work, we must remember, is to a large extent compiled from the works of previous writers. It will be seen, however, that this demonstration implies a knowledge of a seventh proposition,—"If one side of a triangle be produced, the exterior angle is equal to the sum of the two interior and opposite angles." Thales must have been familiar with this truth."

"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."

"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."

"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."