“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.”

Achimedes (1920)

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Thomas Little Heath46

British civil servant and academic 1861–1940

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“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

“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)

“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

“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)

“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 (1860–1944) American mathematician

History of Mathematics (1923) Vol.1

“There are here two assumptions about 'evaluations ', which I will call assumption (1) and assumption (2). Assumption (1) is that some individual may, without logical error, base his beliefs about matters of value entirely on premises which no one else would recognise as giving any evidence at all. Assumption (2) is that, given the kind of statement which other people regard as evidence for an evaluative conclusion, he may refuse to draw the conclusion because this does not count as evidence for him.”

Philippa Foot (1920–2010) British philosopher

"Moral Beliefs"

“Fermat died with the belief that he had found a long-sought-for law of prime numbers in the formula 2^{2^n} + 1 = a prime, but he admitted that he was unable to prove it rigorously. The law is not true, as was pointed out by Euler in the example 2^{2^5} + 1 = 4,294,967,297 = 6,700,417 times 641. The American lightning calculator Zerah Colburn, when a boy, readily found the factors but was unable to explain the method by which he made his marvellous mental computation.”

Florian Cajori book A History of Mathematics

Source: A History of Mathematics (1893), p. 180; also cited in Moritz, Memorabilia Mathematica; Or, The Philomath's Quotation-book (1914) pp. 156-157. https://books.google.com/books?id=G0wtAAAAYAAJ&pg=PA156

“Placing your stick at the end of the shadow of the pyramid, you made by the sun's rays two triangles, and so proved that the pyramid [height] was to the stick [height] as the shadow of the pyramid to the shadow of the stick.”

Thales (-624–-547 BC) ancient Greek philosopher and mathematician

W. W. Rouse Ball, A Short Account of the History of Mathematics (1893, 1925)

“Good values are 1) reality-based, 2) socially constructive, and 3) immediate and controllable.
Bad values are 1) superstitious, 2) socially destructive, and 3) not immediate and controllable.”

Mark Manson (1984) American writer and blogger

Source: The Subtle Art of Not Giving a F*ck (2016), Chapter 4, “The Value of Suffering” (p. 86)

“Theorem II. Any feel which feeling has not informed me of, is unknown to me. Comments. 1. Words are sensibly intelligent to a man of only such words as he has experienced. 2. The intellectual signification of words discriminated from the sensible signification. 3. Intellectual intimations discriminated from sensible revelations.”

Alexander Bryan Johnson (1786–1867) United States philosopher and banker

Part II. Of the Extent of Sensible Knowledge.
The Physiology of the Senses: Or, How and what We See, Hear, Taste, Feel and Smell (1856)

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Thomas Little Heath46

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