“How many theorems in geometry which have seemed at first impracticable are in time successfully worked out!”

On Spirals (225 B.C.)

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Archimedes20

Greek mathematician, physicist, engineer, inventor, and ast… -287–-212 BC

John Rawls book A Theory of Justice

Source: A Theory of Justice (1971; 1975; 1999), Chapter III, Section 21, pg. 126

“How many things there are in this world which do not seem to be, and how many which seem to be and are not.”

Alessandro Pepoli (1757–1796) Italian writer

In questo mondo, quante cose sonc e non sembrano! e quante poi sembrano e non sono!
La Scomessa, Act I., Sc. III. — (Il Marchese.). Translation reported in Harbottle's Dictionary of quotations French and Italian (1904), p. 325.

“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.

“Geometry has two great treasures: one is the Theorem of Phythagoras, the other the division of a line in extreme and mean ratio. The first we can compare to a mass of gold; the other we may call a precious jewel.”

Johannes Kepler book Mysterium Cosmographicum

As quoted by Karl Fink, Geschichte der Elementar-Mathematik (1890) translated as A Brief History of Mathematics https://books.google.com/books?id=3hkPAAAAIAAJ (1900, 1903) by Wooster Woodruff Beman, David Eugene Smith. Also see Carl Benjamin Boyer, A History of Mathematics (1968). <br class="br">Mysterium Cosmographicum (1596)

“If we wished to relate the space of the [Cubist] painters to geometry, we should have to refer it to the non-Euclidean mathematicians; we should have to study, at some length, certain of Riemann's theorems.”

Jean Metzinger (1883–1956) French painter

Du Cubisme (1912)

Albert Gleizes (1881–1953) French painter

1910s, Du Cubisme (1912)

“We find in the history of ideas mutations which do not seem to correspond to any obvious need, and at first sight appear as mere playful whimsies — such as Apollonius' work on conic sections, or the non-Euclidean geometries, whose practical value became apparent only later.”

Arthur Koestler (1905–1983) Hungarian-British author and journalist

as quoted by Michael Grossman in the The First Nonlinear System of Differential and Integral Calculus (1979).
The Sleepwalkers: A History of Man's Changing Vision of the Universe (1959)

“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.”

Proclus (412–485) Greek philosopher

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

“First then I will set out the very first theorem which became known to me by means of mechanics, namely that
Any segment of a section of a right angled cone (i. e., a parabola) is four-thirds of the triangle which has the same base and equal height,
and after this I will give each of the other theorems investigated by the same method. Then at the end of the book I will give the geometrical [proofs of the propositions]…”

Archimedes book The Method of Mechanical Theorems

The Method of Mechanical Theorems

“They seem to have noticed her for the first time”

Jenny Han (1980) American writer

“How many theorems in geometry which have seemed at first impracticable are in time successfully worked out!”

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