Ancient Greek mathematics

Ancient Greek mathematics was developed from the 7th century BC to the 4th century AD by Greek speaking peoples along the shores of the Eastern Mediterranean. The period following Alexander the Great is sometimes referred to as Hellenistic mathematics. The word "mathematics" itself derives from the ancient Greek μάθημα (mathema), meaning "subject of instruction". The use of generalized mathematical theories and proofs is the key difference between Greek mathematics and those of preceding civilizations.

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CONTENT
A-B - C-D - E-G - H-K - L-S - T-Z
A Short History of Greek Mathematics (1884)
A Short Account of the History of Mathematics (1888)
Famous Geometrical Theorems and Problems: With Their History (1900-1901)
A History of Greek Mathematics (1921) Vol. 1. From Thales to Euclid
Science Awakening (1954)
See also-External links

Quotes

A-B

• 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.
  • Eric Temple Bell, The Development of Mathematics (1940)

• 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.
  • Eric Temple Bell, The Development of Mathematics (1940)

Portion of The School of Athens
by Raphael (c. 1510)

• 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.
  • Carl B. Boyer, History of Analytic Geometry (1956)

• 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.
  • Carl B. Boyer, History of Analytic Geometry (1956)

• 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.
  • Carl B. Boyer, History of Analytic Geometry (1956)

• 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.
  • Carl B. Boyer, A History of Mathematics (1968)

• In the Greek world mathematics was more closely related to philosophy than to practical affairs, and this kinship has persisted to the present day.
  • Carl B. Boyer, A History of Mathematics (1968)

• These three problems—the squaring of the circle, 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.
  • Carl B. Boyer, A History of Mathematics (1968)

C-D

• 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.
  • Florian Cajori, A History of Mathematics (1893)

• 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.
  • Florian Cajori, A History of Mathematics (1893)

• 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
  • Julian Lowell Coolidge, A History of Geometric Methods (1940)

• 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.
  • Julian Lowell Coolidge, A History of Geometric Methods (1940)

• 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.'
  • Keith Critchlow Forward, The Theology of Arithmetic by Iamblichus (1988) Tr. Robin Waterfield, p. 11.

E-G

• 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).
  • C. H. Edwards, Jr., The Historical Development of the Calculus (1979)

• The Greeks... were well aware of geometric magnitudes that we call "irrational," but simply did not think of them as numbers.
  • C. H. Edwards, Jr., The Historical Development of the Calculus (1979)

• 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.
  • Based on a scholium by Geminus (fl. 1st century BC), or possibly through Anatolius, on Plato's Charmides as quoted by Louis Charles Karpinski, Nicomachus of Gerasa: Introduction to Arithmetic (1926) Tr. Martin Luther D'Ooge, Part 1, Ch. 1. pp. 3-4. Also see Proclus, In Primum Euclidis Elementorum Librum Commentarii or A Commentary on the First Book of Euclid's "Elements" (ca. 460 AD), pp. 38, 1-42 (ed. Friedlein).

H-K

• 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.
  • Thomas Little Heath, The Thirteen Books of Euclid's Elements, "Proclus and His Sources" (1908) Vol. 1. p. 29.

• 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.
  • Thomas Little Heath, Achimedes (1920)

• 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."
  • Thomas Little Heath, A Manual of Greek Mathematics (1931)

• 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 quadrivium of the Pythagoreans. With the trivium (which Nicomachus does not mention) of Grammar, Logic, and Rhetoric, they composed the seven liberal arts taught in the schools of the Roman Empire.
  • George Johnson, The Arithmetical Philosophy of Nicomachus of Gersa (1916) p.5.

• 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.
  • Louis Charles Karpinski, Nicomachus of Gerasa: Introduction to Arithmetic (1926) p. 3.

L-S

• 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?
  • Plato, Republic (c. 380 BC) Tr. Benjamin Jowett (1871)

• The ancient Geometry had no symbols, nor any notation beyond ordinary language and the specific terms of the science.
  • Robert Potts, Preface, Euclid's Elements of Geometry: Chiefly from the Text of Dr. Simson, with Explanatory Notes (1845) p. iii.

• 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 Theon of Smyrna. 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.
  • Charles Rollin, The Ancient History of the Egyptians, Carthaginians, Assyrians, Babylonians, Medes and Persians, Grecians and Macedonians (1754) as quoted in Vol. 2, p. 598, 1842 edition; originally published in French, Histoire Ancienne... in 12 Volumes (1730-1738).

• 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."
  • David Eugene Smith, History of Mathematics (1925) Vol. 1.

• 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.
  • Socrates in Plato's Charmides (ca. 370 BC) source

• 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 ${\sqrt {2}}$, 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.
  • John Stillwell, Mathematics and Its History (1989)

• 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.
  • Dirk Jan Struik, A Concise History of Mathematics (1948)

• 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.
  • Dirk Jan Struik, A Concise History of Mathematics (1948) with a reference to S. Luria, "Die Infinitesimaltheorie der antiken Atomisten" Quellen und Studien B 2 (1932) pp.106-185.

T-Z

• 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.
  • Sabetai Unguru, On the Need to Rewrite the History of Greek Mathematics, Archive for History of Exact Sciences, Vol. 15, No. 1, 30.XII (1975) pp. 67-114, as quoted, with Introduction in Classics in the History of Greek Mathematics (2004) ed. Jean Christianidis, p. 386.

• One of the central concepts for the understanding of ancient Greek mathematics has customarily been, at least since the time of Paul Tannery and Hieronymus Georg Zeuthen, 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.
  • Sabetai Unguru, On the Need to Rewrite the History of Greek Mathematics, Archive for History of Exact Sciences, Vol. 15, No. 1, 30.XII (1975) pp. 67-114, as quoted, with Introduction in Classics in the History of Greek Mathematics (2004) ed. Jean Christianidis, pp. 386-390.

• 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.
  • Alfred North Whitehead, An Introduction to Mathematics (1911)

Ancient Greek square root construction by similar triangles.

• 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) $N\geq 1$. 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.
  • N. J. Wildberger, "A Brief Course on the History of Mathematics," MathHistory6a: Polynomial equations (2011) 28:48-31:30.

A Short History of Greek Mathematics (1884)

James Gow

• 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.
  • Preface

• 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.

A Short Account of the History of Mathematics (1888)

W. W. Rouse Ball, source

• 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, Menaechmus, 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 Theodorus of Cyrene...

• 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 conic sections 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.

Famous Geometrical Theorems and Problems: With Their History (1900-1901)

William Whitehead Rupert, Parts 1-4

• 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 isosceles triangle are equal. ...ii. If two straight lines cut one another the vertically opposite angles are equal.

Thales' Theorem: Rt. angle in a semicircle

• 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.

A History of Greek Mathematics (1921) Vol. 1. From Thales to Euclid

Thomas Little Heath

• 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 Moritz Cantor (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.

• 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.

• Theon of Smyrna was the author of a book purporting to be a manual of mathematical subjects such as a student would require to enable him to understand Plato.

Science Awakening (1954)

Bartel Leendert van der Waerden, Ontwakende wetenschap (1950) translated as Science Awakening (1954) by Arnold Dresden

• Thales and Pythagoras took their start from Babylonian mathematics but gave it a very different... specifically Greek character... in the Pythagorean school and outside, mathematics was brought to... ever higher development and began gradually to satisfy the demands of stricter logic... through the work of Plato's friends Theaetetus and Eudoxus, mathematics was brought to a state of perfection, beauty and exactness, which we admire in the elements of Euclid. ...the mathematical method of proof served as a prototype for Plato's dialectics and for Aristotle's logic.

• Plato is very fond of appealing to mathematics to show that exact reasoning is possible, not about things which are seen and heard, but about ideal objects which exist in thought only. ...
  Plato is apparently convinced that the mathematicians will agree with him. And indeed, when we deal with line segments which one sees and which one measures empirically, the question as to the existence of a common measure has no sense; a hair's breadth will measure integrally every line that is drawn. The question of commensurability makes sense only for line segments as objects of thought. It is therefore clear that Theodorus could not appeal to intuition to prove the incommensurability of the sides of his squares.

• After Apollonius Greek mathematics came to a dead stop. It is true that there were some epigones, such as Diocles and Zenodorus... But apart from trigonometry, nothing great nothing new appeared. The geometry of the conics remained in the form Apollonius gave it, until Descartes. ...The "Method" of Archimedes was lost sight of, and the problem of integration remained where it was, until it was attacked anew in the 17th century... Germs of projective geometry were present, but it remained for Desargues and Pascal to bring these to fruition. ...Higher plane curves were studied only sporadically... Geometric algebra and the theory of proportions were carried over into modern times as inert traditions, of which the inner meaning was no longer understood. The Arabs started algebra anew, from a much more primitive point of view... Greek geometry had run into a blind alley.

• Any one can use our algebraic notation, but only a gifted mathematician can deal with the Greek theory of proportions and with geometric algebra.

• An oral tradition makes it possible to indicate the line segments with the fingers; one can emphasize essentials and point out how the proof was found. All of this disappears in the written formulation... as soon as some external cause brought about an interruption in the oral tradition, and only books remained, it became very difficult to assimilate the work of the great predursors, and next to impossible to pass beyond it.

See also

• Archimedes
• Euclid
• Euclid’s Elements
• Geometry
• James Gow
• Thomas Little Heath
• History of mathematics
• Pythagoras
• Thales

External links

• James Gow, A Short History of Greek Mathematics (1884) @Internet Archive
• Thomas Little Heath, A History of Greek Mathematics, Vol. 1, From Thales to Euclid, Vol. 2 From Aristarchus to Diophantus (1921) @Internet Archive