Mathematicians From Greece

"The quadratures of lunes, which were considered to belong to an uncommon class of propositions on account of the close relation (of lunes) to the circle, were first investigated by Hippocrates, and his exposition was thought to be in correct form... He started with, and laid down as the first of the theorems useful for his purpose, the proposition that similar segments of circles have the same ratio to one another as the squares on their bases have... And this he proved by first showing that the squares on the diameters have the same ratio as the circles. For, as the circles are to one another, so also are similar segments of them. For similar segments are those which are the same part of the circles respectively, as for instance a semicircle is similar to a semicircle, and a third part of a circle to a third part... It is for this reason also... that similar segments contain equal angles...'"

"If on the sides of a right-angled triangle ACB semi-circles are described on the same side, the sum of the areas of the two lunes AEC, BDC is equal to that of the triangle ACB. If the right-angled triangle is isosceles, the two lunes are equal, and each of them is half the area of the triangle. Thus the area of the lunula is found."

"Bodily, material things are... continuously involved in continuous flow and change—in imitation of the nature and peculiar quality of that eternal matter and substance which has been from the beginning... The bodiless things, however, of which we conceive in connection with or together with matter, such as qualities, quantities, configurations, largeness, smallness, equality, relations, actualities, dispositions, places, times, all those things... whereby the qualities in each body are comprehended—all these are of themselves immovable and unchangeable, but accidentally they share in and partake of the affections of the body to which they belong. Now it is with such things that 'wisdom' is particularly concerned, but accidentally also with... bodies."

"Since of quantity, one kind is viewed by itself, having no relation to anything else, as 'even,' 'odd,' 'perfect,' and the like, and the other is relative to something else, and is conceived of together with its relationship to another thing, like' double,' , greater,' 'smaller,' 'half,' 'one and one-half times,' 'one and one-third times,' and so forth, it is clear that two scientific methods will lay hold of and deal with the whole investigation of quantity: arithmetic, [with] absolute quantity; and music, [with] relative quantity."

"Things... are some of them continuous...which are properly and peculiarly called 'magnitudes'; others are discontinuous, in a side-by-side arrangement, and, as it were, in heaps, which are called 'multitudes,' a flock, for instance, a people, a heap, a chorus, and the like. Wisdom, then, must be considered to be the knowledge of these two forms. Since, however, all multitude and magnitude are by their own nature of necessity infinite—for multitude starts from a definite root and never ceases increasing; and magnitude, when division beginning with a limited whole is carried on, cannot bring the dividing process to an end... and since sciences are always sciences of limited things, and never of infinites, it is accordingly evident that a science dealing with magnitude... or with multitude... could never be formulated.... A science, however, would arise to deal with something separated from each of them, with quantity, set off from multitude, and size, set off from magnitude."

"Some... agreeing with , believe that the proportion is called harmonic because it attends upon all geometric harmony, and they say that 'geometric harmony' is the cube because it is harmonized in all three dimensions, being the product of a number thrice multiplied together. For in every cube this proportion is mirrored; there are in every cube 12 sides, 8 angles and 6 faces; hence 8, the [harmonic] mean between 6 and 12, is according to harmonic proportion..."

"Everything that is harmoniously constituted is knit together out of opposites..."

"Scientific number, being set over such things as these, should be harmoniously constituted, in accordance with itself; not by any other but by itself."

"Number is limited multitude or a combination of units or a flow of quantity made up of units; and the first division of number is even and odd."

"And once more is this true in the case of music; not only because the absolute is prior to the relative, as 'great' to 'greater' and 'rich' to 'richer' and 'man' to 'father,' but also because the musical harmonies, diatessaron, diapente, and diapason, are named for numbers; similarly all of their harmonic ratios are arithmetical ones, for the diatessaron [] is the ratio of 4 : 3, the diapente [] that of 3 : 2, and the diapason [perfect ] the double ratio [2 : 1]; and the most perfect, the di-diapason [], is the quadruple ratio [4 : 1]."

"The even is that which can be divided into two equal parts without a unit intervening in the middle; and the odd is that which cannot..."

"The ancients, who under the leadership of Pythagoras first made science systematic, defined philosophy as the love of wisdom... [Οἱ παλαιοὶ καὶ πρώτοι μεθοδεύσαντες ἐπιστήμην κατάρξαντος Πυθαγόρου ὡρίζοντο φιλοσοφίαν εἶναι φιλίαν σοφίας...] This 'wisdom' he defined as the knowledge, or science, of the truth in real things, conceiving 'science' to be a steadfast and firm apprehension of the underlying substance. and 'real things' to be those which continue uniformly and the same in the universe and never depart even briefly from their existence; these real things would be things immaterial..."

"More evidently still astronomy attains through arithmetic the investigations that pertain to it, not alone because it is later than geometry in origin—for motion naturally comes after rest—nor because the motions of the stars have a perfectly melodious harmony, but also because risings, settings, progressions, retrogressions, increases, and all sorts of phases are governed by numerical cycles and quantities. So then we have rightly undertaken first the systematic treatment of this, as the science naturally prior, more honorable, and more venerable, and, as it were, mother and nurse of the rest; and here we will take our start for the sake of clearness."

"Every number is at once half the sum of the two on either side of itself..."

"Two other sciences in the same way will accurately treat of 'size': geometry, the part that abides and is at rest, [and] astronomy, that which moves and revolves."

"To quote the words of Timaeus, in Plato, "What is that which always is, and has no birth, and what is that which is always becoming but never is? The one is apprehended by the mental processes, with reasoning, and is ever the same; the other can be guessed at by opinion in company with unreasoning sense, a thing which becomes and passes away, but never really is." Therefore, if we crave for the goal which is worthy and fitting for man, namely happiness of life—and this is accomplished by philosophy alone and nothing else, and philosophy means... for us desire for wisdom, and wisdom the science of the truth of things... it is reasonable and most necessary to distinguish and systematize the accidental qualities of things."

"Without the aid of these, then, it is not possible to deal accurately with the forms of being nor to discover the truth in things, knowledge of which is wisdom, and evidently not even to philosophize properly, for "just as painting contributes to the menial arts toward correctness of theory, so in truth lines, numbers, harmonic intervals, and the revolutions of circles bear aid to the learning of the doctrines of wisdom," says the Pythagorean Androcydes. Likewise Archytas of Tarentum, at the beginning of... On Harmony, says... in about these words: "It seems to me that they do well to study mathematics, and it is not at all strange that they have correct knowledge about each thing, what it is. For if they knew rightly the nature of the whole, they were also likely to see well what is the nature of the parts. About geometry, indeed, and arithmetic and astronomy, they have handed down to us a clear understanding, and not least also about music. For these seem to be sister sciences; for they deal with sister subjects, the first two forms of being.""

"Plato, too, at the end of the thirteenth book of the Laws, to which some give the title The Philosopher... adds: "Every diagram, system of numbers, every scheme of harmony, and every law of the movement of the stars, ought to appear one to him who studies rightly; and what we say will properly appear if one studies all things looking to one principle, for there will be seen to be one bond for all these things, and if anyone attempts philosophy in any other way he must call on Fortune to assist him. For there is never a path without these... The one who has attained all these things in the way I describe, him I for my part call wisest, and this I maintain through thick and thin." For it is clear that these studies are like ladders and bridges that carry our minds from things apprehended by sense and opinion to those comprehended by the mind and understanding, and from those material, physical things, our foster-brethren known to us from childhood, to the things with which we are unacquainted, foreign to our senses, but in their immateriality and eternity more akin to our souls, and above all to the reason which is in our souls."

"In Plato's Republic, when the interlocutor of Socrates appears to bring certain plausible reasons to bear upon the mathematical sciences, to show that they are useful to human life, arithmetic for reckoning, distributions, contributions, exchanges, and partnerships, geometry for sieges, the founding of cities and sanctuaries, and the partition of land, music for festivals, entertainment, and the worship of the gods, and the doctrine of the spheres, or astronomy, for farming, navigation and other undertakings, revealing beforehand the proper procedure and suitable season, Socrates, reproaching him says: "You amuse me, because you seem to fear that these are useless studies that I recommend; but that is very difficult, nay, impossible. For the eye of the soul, blinded and buried by other pursuits, is rekindled and aroused again by these and these alone, and it is better that this be saved than thousands of bodily eyes, for by it alone is the truth of the universe beheld.""

"In geometry we begin with the point, which is indimensional. This is the beginning of the first dimensional form, the line, and by movement the point generates the line. Now Nicomachus had a similar idea of the nature of multitude and number; they form a series, as it were a moving stream, which proceeds out of unity, the monad. Just as the point is not part of the line (for it is indimensional, and the line is defined as that which has one dimension), but is potentially a line, so the monad is not a part of multitude nor of number, though it is the beginning of both, and potentially both. The monad is unity, absence of multitude, potentiality; out of it the dyad first separates itself and 'goes forward' and then in succession follow the other numbers."

"Nicomachus planned his Introduction so as to explain the mathematical principles involved in the difficult Platonic passages concerning the world-soul in the Timaeus and the marriage-number in the Republic."

"Nicomachus gives three definitions of number. The first, "determinate multitude," is by genus and difference, and is ascribed by Iamblichus to Eudoxus. The second, "a group (i.e., organized complex) of units," [Iamblichus] ascribes to Thales and the Egyptians. The third stream of quantity composed of units Philoponus explains as another attempt to distinguish the particular kind of quantum treated in Arithmetic. The Unit was conceived either mystically as an Idea whose "essence" passes in some way into concrete individuals and even into the Ideas to organize them, or spatially and temporally as the boundary of individuals. The former conception gave rise to fantastic speculations on the cosmic meaning of number, examples of which Nicomachus has... given us; the latter gave rise to the s..."

"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."

"The philosophical arithmetic of the Greeks, αριθμητική, of which the arithmetic of Nicomachus is a specimen, corresponds in a measure to our number theory; the subject was designed for mature students as a preparation for the study of philosophy, and was not at all intended for children. Arithmetica is... the study of that which is implied in number."

"In mathematical studies he is among the first to attempt a systematic treatment of Arithmetic distinct from Algebra."

"His native place was Gerasa probably the modern Jerash about 56 miles northeast of Jerusalem. Two treatises bear the name of Nicomachus, the Introductionis arithmeticæ libri duo and the Enchiridion harmonikon. ...A third treatise called Theologoumena arithmetica is anonymous but is probably the work of Nicomachus."

"Nicomachus is preparing for the answer, mathematical knowledge, and so he says, the knowledge which distinguishes accurately "the accidents of the Existents." Accident may refer either to a contingent or to a necessary attribute... Here it is the latter. To distinguish such accidents seems at first glance an incredibly difficult task. But Nicomachus, like his descendants, simplifies it by reducing the accidents to two: Magnitude and Multitude."

"The Introduction to Arithmetic of Nicomachus is but a restatement of facts which were common property... long before him, and that, except for the few unimportant propositions the discovery of which our author... claims for himself, the book is largely unoriginal. This naturally leads to the inference that the Introduction must be closely connected with other mathematical treatises, which served as the fountains... Because so little remains of this literature, [our inference] is difficult to demonstrate..."

""Arithmetic Introduction" is the most complete exposition extant of Pythagorean arithmetic. It deals in great part with the same subjects as the arithmetical books of Euclid's "Elements," but where Euclid represents numbers by straight lines, Nicomachus uses arithmetical notation with ordinary language when undetermined numbers are expressed. His treatment of s and s was of influence on medieval arithmetic, especially through Boetius."

"The sixth century [BC] was the time, and Greece the place, for human beings to reject once and for all the pernicious number mysticism of the East. Instead, Pythagoras and his followers eagerly accepted it as the celestial revelation of a higher mathematical harmony. Adding vast masses of sheer numerological nonsense of their own to an already enormous bulk, they transmitted this ancient superstition to the golden age of Greek thought, which passed it on to the first century A.D. to the decadent arithmologist Nicomachus. He, enriching his already opulent legacy with a wealth of original rubbish, left it to be sifted by the Roman Boethius, the dim mathematical light of the Middle Ages, thereby darkening the mind of Christian Europe with the venerated nonsense, and encouraging the gemaria of the Talmudists to flourish like a weed."

"Boetius wrote mathematical texts which were considered authoritative in the Western world for more than a thousand years. ...His "institutio arithmetica," a superficial translation of Nicomachus, did provide some Pythagorean number theory which was absorbed in medieval instruction as part of the ancient and : arithmetic, geometry, astronomy, and music."

"If geometry exists, arithmetic must also needs be implied... But on the contrary 3, 4, and the rest might be 5 without the figures existing to which they give names. Hence arithmetic abolishes geometry along with itself, but is not abolished by it, and while it is implied by geometry, it does not itself imply geometry."

"A Jew, Nicomachus, of Gerasa, published an Arithmetic, which, or rather a Latin translation of it) remained for a thousand years a standard authority on the subject. Geometrical demonstrations are here abandoned, and the work is a mere classification of the results then known, with numerical illustrations: the evidence of the truth of the propositions enunciated, for I cannot call them proofs, being in general an induction from numerical instances. The object of the book is the study of the properties of numbers, and particularly of their ratios. Nicomachus commences with the usual distinctions between even, odd, prime, and perfect numbers; he next discusses fractions in a somewhat clumsy manner; he then turns to polygonal and to solid numbers; and finally treats of ratio, proportion, and progressions. Arithmetic of this kind is usually termed Boethian, and the work of Boethius on it was a recognised text-book in the middle ages."

"Neo-Pythagoreans, like Nicomachus of Gerasa... and Iamblichus... revel in... number mysticism. ...The purpose of the layman's Introduction to Arithmetic... is to explain in a manner intelligible to every one, the wonderful and divine properties of numbers. In a very entertaining way he tells about s... square, rectangular and s... gnomic numbers and spatial numbers, about ratios and the distinction between multiple and epimoric."

"Iamblichus, when he refers to the Introduction as the... Art of Arithmetic, exactly describes it... The Introduction belongs... among the artes or... concise, practical descriptions and systematic expositions of the principles of various arts and sciences, a type of treatise exceedingly common in ancient times, and one which... made scant claim to originality. Designed for the use of students... they may best be compared to the modern school and college text-book. ...we must consider that the Introduction to Arithmetic differs from the great original treatises of Diophantus and Heron. Because in clearness, conciseness, compendiousness, orderly arrangement and adaptability for scholastic use, it satisfied the demands of seekers after education or general information, it remained the standard work of its class for many centuries."

"All that has by nature, with systematic method, been arranged in the universe, seems both in part and as a whole to have been determined and ordered in accordance with number, by the forethought and the mind of him that created all things; for the pattern was fixed, like a preliminary sketch, by the domination of number preëxistent in the mind of the world-creating God, number conceptual only and immaterial in every way, but at the same time the true and the eternal essence, so that with reference to it, as to an artistic plan, should be created all these things: time, motion, the heavens, the stars, all sorts of revolutions."

"After Pythagoras, Anaxagoras the Clazomenian succeeded, who undertook many things pertaining to geometry. And Oenopides the Chian, was somewhat junior to Anaxagoras, and whom Plato mentions in his Rivals, as one who obtained mathematical glory. To these succeeded Hippocrates, the Chian, who invented the quadrature of the lunula, and Theodorus the Cyrenean, both of them eminent in geometrical knowledge. For the first of these, Hippocrates composed geometrical elements: but Plato, who was posterior to these, caused as well geometry itself, as the other mathematical disciplines, to receive a remarkable addition, on account of the great study he bestowed in their investigation. This he himself manifests, and his books, replete with mathematical discourses, evince: to which we may add, that he every where excites whatever in them is wonderful, and extends to philosophy. But in his time also lived Leodamas the Thasian, Architas the Tarentine, and Theætetus the Athenian; by whom theorems were increased, and advanced to a more skilful constitution. But Neoclides was junior to Leodamas, and his disciple was Leon; who added many things to those thought of by former geometricians. So that Leon also constructed elements more accurate, both on account of their multitude, and on account of the use which they exhibit: and besides this, he discovered a method of determining when a problem, whose investigation is sought for, is possible, and when it is impossible."

"But after these, Pythagoras changed that philosophy, which is conversant about geometry itself, into the form of a liberal doctrine, considering its principles in a more exalted manner; and investigating its theorems immaterially and intellectually; who likewise invented a treatise of such things as cannot be explained in geometry, and discovered the constitution of the mundane figures."

"But Eudoxus the Cnidian, who was somewhat junior to Leon, and the companion of Plato, first of all rendered the multitude of those theorems which are called universals more abundant; and to three proportions added three others; and things relative to a section, which received their commencement from Plato, he diffused into a richer multitude, employing also resolutions in the prosecution of these."

"Let us now explain the origin of geometry, as existing in the present age of the world. For the demoniacal Aristotle observes, that the same opinions often subsist among men, according to certain orderly revolutions of the world: and that sciences did not receive their first constitution in our times, nor in those periods which are known to us from historical tradition, but have appeared and vanished again in other revolutions of the universe; nor is it possible to say how often this has happened in past ages, and will again take place in the future circulations of time. But, because the origin of arts and sciences is to be considered according to the present revolution of the universe, we must affirm, in conformity with the most general tradition, that geometry was first invented by the Egyptians, deriving its origin from the mensuration of their fields: since this, indeed, was necessary to them, on account of the inundation of the Nile washing away the boundaries of land belonging to each. Nor ought It to seem wonderful, that the invention of this as well as of other sciences, should receive its commencement from convenience and opportunity. Since whatever is carried in the circle of generation proceeds from the imperfect to the perfect."

"The mathematician speculates the causes of a certain sensible effect, without considering its actual existence; for the contemplation of universals excludes the knowledge of particulars; and he whose intellectual eye is fixed on that which is general and comprehensive, will think but little of that which is sensible and singular."

"A transition, therefore, is not undeservedly made from sense to consideration, and from this to the nobler energies of intellect. Hence, as the certain knowledge of numbers received its origin among the Phœnicians, on account of merchandise and commerce, so geometry was found out among the Egyptians from the distribution of land. When Thales, therefore, first went into Egypt, he transferred this knowledge from thence into Greece: and he invented many things himself, and communicated to his successors the principles of many. Some of which were, indeed, more universal, but others extended to sensibles."

"Again, Amyclas the Heracleotean, one of Plato's familiars, and Menæchmus, the disciple, indeed, of Eudoxus, but conversant with Plato, and his brother Dinostratus, rendered the whole of geometry as yet more perfect. But Theudius, the Magnian, appears to have excelled, as well in mathematical disciplines, as in the rest of philosophy. For he constructed elements egregiously, and rendered many particulars more universal. Besides, Cyzicinus the Athenian, flourished at the same period, and became illustrious in other mathematical disciplines, but especially in geometry. These, therefore, resorted by turns to the Academy, and employed themselves in proposing common questions."

"This, therefore, is mathematics: she reminds you of the invisible form of the soul; she gives life to her own discoveries; she awakens the mind and purifies the intellect; she brings light to our intrinsic ideas; she abolishes oblivion and ignorance which are ours by birth."

"If we listen to those who like to record antiquities, we shall find them attributing this theorem to Pythagoras and saying that he sacrificed an ox on its discovery. For my part, though I marvel at those who first noted the truth of this theorem, I admire more the author of the Elements for the very lucid proof by which he made it fast."

"But Hermotimus, the Colophonian, rendered more abundant what was formerly published by Eudoxus and Theætetus, and invented a multitude of elements, and wrote concerning some geometrical places. But Philippus the Mendæan, a disciple of Plato, and by him inflamed in the mathematical disciplines, both composed questions, according to the institutions of Plato, and proposed as the object of his enquiry whatever he thought conduced to the Platonic philosophy."

"The Platonic doctrine of Ideas has been, in all ages, the derision of the vulgar, and the admiration of the wife. Indeed, if we consider that ideas are the most sublime objects of speculation, and that their nature is no less bright in itself, than difficult to investigate, this opposition in the conduct of mankind will be natural and necessary; for, from our connection with a material nature, our intellectual eye, previous to the irradiations of science, is as ill adapted to objects the most splendid of all, "as the eyes of bats to the light of day.""

"And thus far historians produce the perfection of this science. But Euclid was not much junior to these, who collected elements, and constructed many of those things which were invented by Eudoxus; and perfected many which were discovered by Theætetus. Besides, he reduced to invincible demonstrations, such things as were exhibited by others with a weaker arm. But he lived in the times of the first Ptolemy: for Archimedes mentions Euclid, in his first book, and also in others. Besides, they relate that Euclid was asked by Ptolomy, whether there was any shorter way to the attainment of geometry than by his elementary institution, and that he answered, there was no other royal path which led to geometry. Euclid, therefore, was junior to the familiars of Plato, but more ancient than Eratosthenes and Archimedes (for these lived at one and the same time, according to the tradition of Eratosthenes) but he was of the Platonic sect, and familiar with its philosophy: and from hence he appointed the constitution of those figures which are called Platonic, as the end of his elementary institutions."

"The regular solids were studied so extensively by the Platonists that they received the name of Platonic figures The statement of Proclus that the whole aim of Euclid in writing the Elements was to arrive at the construction of the regular solids is obviously wrong The fourteenth and fifteenth books treating of solid geometry are apocryphal."

"The term 'axiom' was used by Proclus, but not by Euclid. He speaks, instead, of 'common notions'—common either to all men or to all sciences."