“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

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

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“Any segment of a right-angled conoid (i. e., a paraboloid of revolution) cut off by a plane at right angles to the axis is 1½ times the cone which has the same base and the same axis as the segment”

Archimedes book The Method of Mechanical Theorems

Proprosition 4.
The Method of Mechanical Theorems

“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)

“I thought fit to… explain in detail in the same book the peculiarity of a certain method, by which it will be possible… to investigate some of the problems in mathematics by means of mechanics. This procedure is… no less useful even for the proof of the theorems themselves; for certain things first became clear to me by a mechanical method, although they had to be demonstrated by geometry afterwards… But it is of course easier, when we have previously acquired, by the method, some knowledge of the questions, to supply the proof than it is to find it without any previous knowledge.”

Archimedes book The Method of Mechanical Theorems

The Method of Mechanical Theorems

“In geometry the following theorems are attributed to him [Thales]—and their character shows how the Greeks had to begin at the very beginning of the theory—(1) that a circle is bisected by any diameter (Eucl. I., Def. 17), (2) that the angles at the base of an isosceles triangle are equal (Eucl. I., 5), (3) that, if two straight lines cut one another, the vertically opposite angles are equal (Eucl. I., 15), (4) that, if two triangles have two angles and one side respectively equal, the triangles are equal in all respects (Eucl. I., 26). He is said (5) to have been the first to inscribe a right-angled triangle in a circle: which must mean that he was the first to discover that the angle in a semicircle is a right angle. He also solved two problems in practical geometry: (1) he showed how to measure the distance from the land of a ship at sea (for this he is said to have used the proposition numbered (4) above), and (2) he measured the heights of pyramids by means of the shadow thrown on the ground”

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

this implies the use of similar triangles in the way that the Egyptians had used them in the construction of pyramids
Achimedes (1920)

“I am persuaded that it [The Method of Mechanical Theorems] will be of no little service to mathematics; for I apprehend that some, either of my contemporaries or of my successors, will, by means of the method when once established, be able to discover other theorems in addition, which have not yet occurred to me.”

Archimedes book The Method of Mechanical Theorems

The Method of Mechanical Theorems

“Proposition 1. Two equal spheres are comprehended by one and the same cylinder, and two unequal spheres by one and the same cone which has its vertex in the direction of the lesser sphere; and the straight line drawn through the centres of the spheres is at right angles to each of the circles in which the surface of the cylinder, or of the cone, touches the spheres.”

Aristarchus of Samos ancient Greek astronomer and mathematician

p, 125
On the Sizes and Distances of the Sun and the Moon (c. 250 BC)

“If two right lines cut one another, they will form the angles at the vertex equal….
This… is what the the present theorem evinces, that when two right lines mutually cut each other, the vertical angles are equal. And it was first invented according to Eudemus by Thales…”

Proclus (412–485) Greek philosopher

Proposition XV. Thereom VIII.
The Philosophical and Mathematical Commentaries of Proclus on the First Book of Euclid's Elements Vol. 2 (1789)

“The ideal of strictly scientific method in mathematics which I have tried to realise here, and which perhaps might be named after Euclid I should like to describe in the following way… The novelty of this book does not lie in the content of the theorems but in the development of the proofs and the foundations on which they are based… With this book I accomplish an object which I had in view in my Begriffsschrift of 1879 and which I announced in my Grundlagen der Arithmetik. I am here trying to prove the opinion on the concept of number that I expressed in the book last mentioned.”

Gottlob Frege (1848–1925) mathematician, logician, philosopher

Vol. 1. pp. 137-140, as cited in: Ralph H. Johnson (2012), Manifest Rationality: A Pragmatic Theory of Argument, p. 87
Grundgesetze der Arithmetik, 1893 and 1903

“I confess that Fermat's Theorem as an isolated proposition has very little interest for me, because I could easily lay down a multitude of such propositions, which one could neither prove nor dispose of.”

Carl Friedrich Gauss (1777–1855) German mathematician and physical scientist

A reply to Olbers' 1816 attempt to entice him to work on Fermat's Theorem. As quoted in The World of Mathematics (1956) Edited by J. R. Newman

“A theorem is a proposition which is a strict logical consequence of certain definitions and other propositions”

Anatol Rapoport (1911–2007) Russian-born American mathematical psychologist

Anatol Rapoport. " Various meanings of “theory”." http://www.acsu.buffalo.edu/~fczagare/PSC%20504/Rapoport%20(1958).pdf American Political Science Review 52.04 (1958): 972-988. <br class="br">1950s

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