“Wallis was in sympathy with Greek mathematics and astronomy, editing parts of the works of Archimedes, Eutocius, Ptolemy, and Aristarchus; but at the same time he recognized the fact that the analytic method was to replace the synthetic, as when he defined a conic as a curve of the second degree instead of as a section of a cone, and treated it by the aid of coordinates.”

— David Eugene Smith

History of Mathematics (1923) Vol.1

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David Eugene Smith 33

American mathematician 1860–1944

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“Eudoxes… not only based the method [of exhaustion] on rigorous demonstration… but he actually applied the method to find the volumes (1) of any pyramid, (2) of the cone, proving (1) that any pyramid is one third part of the prism which has the same base and equal height, and (2) that any cone is one third part of the cylinder which has the same base and equal height. Archimedes, however, tells us the remarkable fact that these two theorems were first discovered by Democritus, though he was not able to prove them (which no doubt means, not that he gave no sort of proof, but that he was not able to establish the propositions by the rigorous methods of Eudoxes. Archimedes adds that we must give no small share of the credit for these theorems to Democritus… another testimony to the marvellous powers, in mathematics as well as in other subjects, of the great man who, in the words of Aristotle, "seems to have thought of everything"…. Democritus wrote on irrationals; he is also said to have discussed the question of two parallel sections of a cone (which were evidently supposed to be indefinitely close together), asking whether we are to regard them as equal or unequal… Democritus was already close on the track of infinitesimals.”

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

Achimedes (1920)

“Of the contemporaries of Newton one of the most prominent was John Wallis. …Wallis was a voluminous writer, and not only are his writings erudite, but they show a genius in mathematics… He was one of the first to recognize the significance of the generalization of exponents to include negative and fractional as well as positive and integral numbers. He recognized also the importance of Cavalieri's method of indivisibles, and employed it in the quadrature of such curves as y=xn, y=x1/n, and y=x0 + x1 + x2 +… He failed in his attempts at the approximate quadrature of the circle by means of series because he was not in possession of the general form of the binomial theorem. He reached the result, however, by another method. He also obtained the equivalent of ds = \! dx \sqrt{1+(\frac{dy}{dx})^2} for the length of an element of a curve, thus connecting the problem of rectification with that of quadrature.”

David Eugene Smith (1860–1944) American mathematician

History of Mathematics (1923) Vol.1

“He moved to Egypt and supported himself by teaching and by copying Arabic translations of Greek mathematical classics such as Euclid’s Elements and Ptolemy’s Almagest.”

Alhazen (965–1038) Arab physicist, mathematician and astronomer

Abdelhamid I. Sabra, in “Ibn al-Haytham Brief life of an Arab mathematician: died circa 1040 (September-October 2003)”

“There is perhaps no question that occupies, comparatively, a larger space in the history of Greek geometry than the problem of the Doubling of the Cube. The tradition concerning its origin is given in a letter from Eratosthenes of Cyrene to King Ptolemy Euergetes quoted by Eutocius…
"Eratosthenes to King Ptolemy greeting.
"There is a story that one of the old tragedians represented Minos as wishing to erect a tomb for Glaucus and as saying, when he heard that it was a hundred feet every way,Too small thy plan to bound a royal tomb.
Let it be double; yet of its fair form
Fail not, but haste to double every side.But he was clearly in error; for when the aides are doubled, the area becomes four times as great, and the solid content eight times as great. Geometers also continued to investigate the question in what manner one might double a given solid while it remained in the same form.”

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

Apollonius of Perga (1896)

“The absolute scholar is in fact a rather uncanny being. He is instinct with Nietzsche's finding that to be interested in something, to be totally interested in it, is a libidinal thrust more powerful than love or hatred, more tenacious than faith or friendship — not infrequently, indeed, more compelling than personal life itself. Archimedes does not flee from his killers, he does not even turn his head to acknowledge their rush into his garden when he is immersed in the algebra of conic sections.”

George Steiner (1929–2020) American writer

"The Cleric of Treason".
George Steiner: A Reader (1984)

“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.”

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Book I, Chapter III, p.184
Nicomachus of Gerasa: Introduction to Arithmetic (1926)

“It is true that in recent years a number of attractive histories of mathematics have been published in England and America, but these have only dealt with Greek mathematics as part of the larger subject, and in consequence the writers have been precluded… from presenting the work of the Greeks in suflicient detail. The same remark applies to the German histories of mathematics, even to the great work of Moritz Cantor…”

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

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

“Joseph Ratzinger has stood still because as a Bavarian Catholic in the Hellenistic tradition, interpreted in Roman terms, he wanted to stand still. To this degree he represented and represents a different basic model of theology and church, as different from mine as in astronomy Ptolemy's geocentric picture of the world is different from Copernicus' heliocentric picture.”

Hans Küng (1928) Swiss Catholic priest, theologian and author

Disputed Truth: Memoirs Volume 2 (2008), p. 329

“The mathematical activity of Ancient Greece reached its peak during the glorious era of Euclid, Eratosthenes, Archimedes and Apollonius, a time when Greek letters, art and philosophy were already on the decline. …it was not Greece proper but its outposts in Asia Minor, in Lower Italy, in Africa that had contributed most to the development of mathematics.”

Tobias Dantzig (1884–1956) American mathematician

The Bequest of the Greeks (1955)

“If we wish to express our ideas in terms of the concepts synthetic and analytic, we would have to point out that these concepts are applicable only to sentences that can be either true of false, and not to definitions. The mathematical axioms are therefore neither synthetic nor analytic, but definitions. …Hence the question of whether axioms are a priori becomes pointless since they are arbitrary.”

Hans Reichenbach (1891–1953) American philosopher

The Philosophy of Space and Time (1928, tr. 1957)

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