“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.”

— Proclus

Source: The Philosophical and Mathematical Commentaries of Proclus on the First Book of Euclid's Elements Vol. 1 (1788), Ch. IV.

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Greek philosopher 412–485

Proclus (412–485) Greek philosopher

Source: The Philosophical and Mathematical Commentaries of Proclus on the First Book of Euclid's Elements Vol. 1 (1788), Ch. IV.

“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.”

Proclus (412–485) Greek philosopher

Source: The Philosophical and Mathematical Commentaries of Proclus on the First Book of Euclid's Elements Vol. 1 (1788), Ch. IV.

“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.”

Proclus (412–485) Greek philosopher

Source: The Philosophical and Mathematical Commentaries of Proclus on the First Book of Euclid's Elements Vol. 1 (1788), Ch. IV. On the Origin of Geometry, and its Inventors.

“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.”

Proclus (412–485) Greek philosopher

Source: The Philosophical and Mathematical Commentaries of Proclus on the First Book of Euclid's Elements Vol. 1 (1788), Ch. IV.

“A multitude of whelps came forth from the lair of this barbaric lioness, in three cyuls, as they call them, that is, in their ships of war.”
Tum erumpens grex catulorum de cubili laeanae barbarae, tribus, ut lingua eius exprimitur, cyulis, nostra longis navibus.

Gildas De Excidio et Conquestu Britanniae

Section 23.
De Excidio Britanniae (On the Ruin of Britain)

“Menæchmus, a pupil of Eudoxus, and a contemporary of Plato, found the two mean proportionals by means of conic sections, in two ways, (α) by the intersection of two parabolas, the equations of which in Cartesian co-ordinates would be x2=ay, y2=bx, and (β) by the intersection of a parabola and a rectangular hyperbola, the corresponding equations being x2=ay, and xy=ab respectively. It would appear that it was in the effort to solve this problem that Menæchmus discovered the conic sections, which are called, in an epigram by Eratosthenes, "the triads of Menæchmus."”

Thomas Little Heath (1861–1940) British civil servant and academic

Achimedes (1920)

“Not much younger than these (sc. Hermotimus of Colophon and Philippus of Mende) is Euclid, who put together the Elements, collecting many of Eudoxus' theorems, perfecting many of Theaetetus', and also bringing to irrefragable demonstration the things which were only somewhat loosely proved by his predecessors. This man lived in the time of the first Ptolemy. For Archimedes, who came immediately after the first (Ptolemy), makes mention of Euclid: and, further, they say that Ptolemy once asked him if there was in geometry any shorter way than that of the elements, and he answered that there was no royal road to geometry. He is then younger than pupils of Plato but older than Eratosthenes and Archimedes; for the latter were contemporary with one another, as Eratosthenes somewhere says.”

Proclus (412–485) Greek philosopher

As quoted by Sir Thomas Little Heath, The Thirteen Books of Euclid's Elements (1908) Vol.1 https://books.google.com/books?id=UhgPAAAAIAAJ Introduction and Books I, II p.1, citing Proclus ed. Friedlein, p. 68, 6-20.

“In mathematics there is a certain way of seeking the truth, a way which Plato is said first to have discovered and which was called "analysis" by Theon and was defined by him as "taking the thing sought as granted and proceeding by means of what follows to a truth which is uncontested"; so, on the other hand, "synthesis" is "taking the thing that is granted and proceeding by means of what follows to the conclusion and comprehension of the thing sought." And although the ancients set forth a twofold analysis, the zetetic and the poristic, to which Theon's definition particularly refers, it is nevertheless fitting that there be established also a third kind, which may be called rhetic or exegetic, so that there is a zetetic art by which is found the equation or proportion between the magnitude that is being sought and those that are given, a poristic art by which from the equation or proportion the truth of the theorem set up is investigated, and an exegetic art by which from the equation set up or the proportion, there is produced the magnitude itself which is being sought. And thus, the whole threefold analytic art, claiming for itself this office, may be defined as the science of right finding in mathematics…. the zetetic art does not employ its logic on numbers—which was the tediousness of the ancient analysts—but uses its logic through a logistic which in a new way has to do with species [of number]…”

François Viète (1540–1603) French mathematician

Source: In artem analyticem Isagoge (1591), Ch. 1 as quoted by Jacob Klein, Greek Mathematical Thought and the Origin of Algebra (1934-1936) Appendix.

“There are three relations [between thee and other things]: the one to the body which surrounds thee; the second to the divine cause from which all things come to all; and the third to those who live with thee.”

Marcus Aurelius book Meditations

Source: Meditations (c. AD 121–180), Book VIII, 27

“This Being, whether properly to be called "That which is above comprehension," or the "Type of things existing," or "The One," (inasmuch as Unity appears to be the most ancient of all things), or "The Good," as Plato regularly designates Him, This, then, is the Single Principle of all things, and which serves to the universe as a model of indescribable beauty, perfection, unity, and power.”

Julian (emperor) (331–363) Roman Emperor, philosopher and writer

Upon the Sovereign Sun (362)
Context: That divine and all-beauteous World, which from the highest vault of Heaven down to the lowest Earth is held together by the immutable providence of God, and which has existed from all eternity, without creation, and shall be eternal for all time to come, and which is not regulated by anything, except approximately by the Fifth Body (of which the principle is the solar light) placed, as it were, on the second step below the world of intelligence; and finally by the means of the "Sovereign of all things, around whom all things stand." This Being, whether properly to be called "That which is above comprehension," or the "Type of things existing," or "The One," (inasmuch as Unity appears to be the most ancient of all things), or "The Good," as Plato regularly designates Him, This, then, is the Single Principle of all things, and which serves to the universe as a model of indescribable beauty, perfection, unity, and power. And after the pattern of the primary substance that dwells within the Principle, He hath sent forth out of Himself, and like in all things unto Himself, the Sun, a mighty god, made up of equal parts of intelligible and creative causes. And this is the sense of the divine Plato, where he writes, "You may say (replied I) that I mean the offspring of the Good, whom the Good has produced, similar to itself; in order that, what the Good is in the region of intelligence, and as regards things only appreciable by the mind, its offspring should be the same in the region that is visible, and in the things that are appreciable by the sight." For this reason I believe that the light of the Sun bears the same relation to things visible as Truth does to things intelligible. But this Whole, inasmuch as it emanates from the Model and "Idea" of the primal and supreme Good, and exists from all eternity around his immutable being, has received sovereignty also over the gods appreciable by the intellect alone, and communicates to them the same good things, (because they belong to the world of intelligence), as are poured down from the Supreme Good upon the other objects of Intelligence. For to these latter, the Supreme Good is the source, as I believe, of beauty, perfection, existence, and union; holding them together and illuminating them by its own virtue which is the "Idea" of the Good.

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