“Hippocrates… is said to have proved the theorem that circles are to one another as the squares on their diameters, and it is difficult to see how he could have done this except by some form, or anticipation, of the method”

—  Thomas Little Heath

of exhaustion
Achimedes (1920)

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Thomas Little Heath46
British civil servant and academic 1861–1940

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“Hippocrates himself is an example of the concurrent study of the two departments. On the one hand, he was the first of the Greeks who is known to have compiled a book of Elements. This book, we may be sure, contained in particular the most important propositions about the circle included in Euclid, Book III. But a much more important proposition is attributed to Hippocrates; he is said to have been the first to prove that circles are to one another as the squares on their diameters (= Eucl. XII., 2) with the deduction that similar segments of circles are to one another as the squares on their bases. These propositions were used by him in his tract on the squaring of lunes, which was intended to lead up to the squaring of the circle. The latter problem is one which must have exercised practical geometers from time immemorial. Anaxagoras for instance is said to have worked at the problem while in prison.”

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

p, 125
Achimedes (1920)

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“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)

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“No one ever squared the circle with so much genius, or, excepting his principal object, with so much success.”

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Attributed to Montucla in Augustus De Morgan, A Budget of Paradoxes, (London, 1872), p. 96; Cited in: Robert Edouard Moritz. Memorabilia mathematica; or, The philomath's quotation-book, (1914) p. 366
About Gregory St. Vincent, described by De Morgan as "the greatest of circle-squarers, and his investigations led him into many truths: he found the property of the arc of the hyperbola which led to Napier's logarithms being called hyperbolic."

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“How can a man be said to have a country where he has no right to a square inch of soil…”

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“Yes it was 1949. How I came to that. That's like how one gets to know a human being. It so happens that I've always had a preference – as everyone has prejudices and preferences – for the square as a shape in preference to the circle as a shape. And I have known for a long time that a circle always fools me by not telling me whether it's standing still or not. And if a circle circulates you don't see it. The outer curve looks the same whether it moves or does not move. So the square is much more honest and tells me that it is sitting on one line of the four, usually a horizontal one, as a basis. And I have also come to the conclusion that the square is a human invention, which makes it sympathetic to me. Because you don't see it in nature. As we do not see squares in nature, I thought that it is man-made. But I have corrected myself. Because squares exist in salt crystals, our daily salt. We know this because we can see it in the microscope. On the other hand, we believe we see circles in nature. But rarely precise ones. Mature, it seems, is not a mathematician. Probably there are no straight lines either. Particularly not since Einstein says in his theory of relativity that there is no straight line, rod knows whether there are or not, I don't. I still like to believe that the square is a human invention. And that tickles me. So when I have a preference for it then I can only say excuse me.”

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“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.”

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The Method of Mechanical Theorems

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“As a square number is known to be the product of a number multiplied by itself, so every polygonal number, multiplied by one number and added to another, both of which depend upon the number of its angles, produces a square number. I shall prove this, and shall show also how from a given side to find its polygon and conversely. Some auxiliary propositions must first be proved.”

Diophantus Arithmetica

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