“He left in his Optics, the earliest surviving table of angles of refraction from air to water. …This table, quoted and requoted until modern times, has been admired… A closer glance at it, however, suggests that there was less experimentation involved in it than originally was thought, for the values of the angles of refraction form an arithmetic progression of second order… As in other portions of Greek Science, confidence in mathematics was here greater than that in the evidence of the senses, although the value corresponding to 60° agrees remarkably well with experience.”

— Ptolemy

Carl B. Boyer, in The Rainbow: From Myth to Mathematics (1959)

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Ptolemy 8

Greco-Egyptian writer and astronomer of Alexandria 100–170

Carl B. Boyer (1906–1976) American mathematician

Source: The Rainbow: From Myth to Mathematics (1959), p. 61
Context: Ptolemy left in his Optics, the earliest surviving table of angles of refraction from air to water. … This table, quoted and requoted until modern times, has been admired … A closer glance at it, however, suggests that there was less experimentation involved in it than originally was thought, for the values of the angles of refraction form an arithmetic progression of second order … As in other portions of Greek Science, confidence in mathematics was here greater than that in the evidence of the senses, although the value corresponding to 60° agrees remarkably well with experience.

“The refraction of light agrees with the grand principle that Nature always uses the simplest means to accomplish its effects. From this principle, can be derived whenever light passes from one medium to another, the ratio of the sine of the angle of refraction to the sine of the angle of refraction equals the inverse ratio of the speeds at which light moves in each medium.
But this "budget", this expense of action that Nature minimizes in the refraction of light, is it also minimized in the direct propagation and reflection of light? Yes, it always has the smallest possible value.”

Pierre Louis Maupertuis (1698–1759) French mathematician, philosopher and man of letters

Accord de différentes loix de la nature qui avoient jusqu’ici paru incompatibles (1744)

“Euclid defines the angle as an inclination of lines…he meant halflines, because otherwise he would not be able to distinguish adjacent angles from each other… Euclid does not know zero angles, nor straight and bigger than straight angles…Euclid takes the liberty of adding angles beyond two and even four right angles; the result cannot be angles according to the original definitions…Nevertheless one feels that Euclid’s angle concept is consistent.”

Hans Freudenthal (1905–1990) Dutch mathematician

Source: Mathematics as an Educational Task (1973), p. 476-477

“Proposition 13. The straight line subtending the portion intercepted within the earth's shadow of the circumference of the circle in which the extremities of the diameter of the circle dividing the dark and the bright portions in the moon move is less than double of the diameter of the moon, but has to it a ratio greater than that which 88 has to 45; and it is less than 1/9th part of the diameter of the sun, but has to it a ratio greater than that which 22 has to 225. But it has to the straight line drawn from the centre of the sun at right angles to the axis and meeting the sides of the cone a ratio greater than that which 979 has to 10125.”

Aristarchus of Samos ancient Greek astronomer and mathematician

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

“Descartes maintained his confidence in the instantaneity of light. … Yet in his derivation of the law of refraction, Descartes reasoned that light travelled faster in a dense medium than in one less dense. He seems to have had no qualms about comparing infinite magnitudes!”

Carl B. Boyer (1906–1976) American mathematician

Source: The Rainbow: From Myth to Mathematics (1959), p. 204

“A man is in general better pleased when he has a good dinner upon his table, than when his wife talks Greek.”

Samuel Johnson (1709–1784) English writer

Quoted in the "Apophthegms, Sentiments, Opinions and Occasional Reflections" of Sir John Hawkins (1787-1789) in Johnsonian Miscellanies (1897), vol. II, p. 11, edited by George Birkbeck Hill

“Heron and Olympiodorus had pointed out in antiquity that, in reflection, light followed the shortest possible path, thus accounting for the equality of angles. During the medieval period Alhazen and Grosseteste had suggested that in refraction some such principle was also operating, but they could not discover the law.”

Carl B. Boyer (1906–1976) American mathematician

Source: The Rainbow: From Myth to Mathematics (1959), p. 205
Context: Fermat had recourse to the principle of the economy of nature. Heron and Olympiodorus had pointed out in antiquity that, in reflection, light followed the shortest possible path, thus accounting for the equality of angles. During the medieval period Alhazen and Grosseteste had suggested that in refraction some such principle was also operating, but they could not discover the law. Fermat, however, not only knew (through Descartes) the law of refraction, but he also invented a procedure—equivalent to the differential calculus—for maximizing and minimizing a function of a single variable. … Fermat applied his method … and discovered, to his delight, that the result led to precisely the law which Descartes had enunciated. But although the law is the same, it will be noted that the hypothesis contradicts that of Descartes. Fermat assumed that the speed of light in water to be less than that in air; Descartes' explanation implied the opposite.

“The mere formulation of a problem is far more often essential than its solution, which may be merely a matter of mathematical or experimental skill. To raise new questions, new possibilities, to regard old problems from a new angle requires creative imagination and makres real advances in science.”

Albert Einstein (1879–1955) German-born physicist and founder of the theory of relativity

http://umich.edu/~scps/html/01chap/html/summary.htm
Attributed in posthumous publications, Einstein and Religion (1999)

“Newton's proof of the law of refraction is based on an erroneous notion that light travels faster in glass than in air, the same error that Descartes had made. This error stems from the fact that both of them thought that light was corpuscular in nature.”

John Freely (1926–2017) American physicist

Source: Before Galileo, The Birth of Modern Science in Medieval Europe (2012), p. 189

“Towns should be laid out not as an exact square nor with salient angles, but in circular form, to give a view of the enemy from many points. Defense is difficult where there are salient angles because the angle protects the enemy rather than the inhabitants.”

Vitruvius book De architectura

Source: De architectura (The Ten Books On Architecture) (~ 15BC), Book I, Chapter V, Sec. 2

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Ptolemy 8

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