Hippocrates of Chios

"The quadratures of lunes, which were considered to belong to an uncommon class of propositions on account of the close relation (of lunes) to the circle, were first investigated by Hippocrates, and his exposition was thought to be in correct form... He started with, and laid down as the first of the theorems useful for his purpose, the proposition that similar segments of circles have the same ratio to one another as the squares on their bases have... And this he proved by first showing that the squares on the diameters have the same ratio as the circles. For, as the circles are to one another, so also are similar segments of them. For similar segments are those which are the same part of the circles respectively, as for instance a semicircle is similar to a semicircle, and a third part of a circle to a third part... It is for this reason also... that similar segments contain equal angles...'"

"One would suppose that the relation between the pseudo-didactic and the didactic syllogism, was the same as that between the pseudo-dialectic and the dialectic; so that, if the pseudo-dialectic deserved to be called sophistic or , the pseudo-didactic would deserve these appellations also; especially, since the formal conditions of the syllogism are alike for both. This Aristotle does not admit, but draws instead a remarkable distinction. The (he says) is a dishonest man, making it his professional purpose to deceive; the pseudo-graphic man of science is honest always, though sometimes mistaken. So long as the pseudo-graphic syllogism keeps within the limits belonging to its own special science, it may be false, since the geometer may be deceived even in his own science [of] geometry, but it cannot be sophistic or eristic; yet whenever it transgresses those limits, even though it be true and though it solves the problem proposed, it deserves to be called by those two epithets. Thus, there were two distinct methods proposed for the quadrature of the circle—one by Hippokrates, on geometrical principles, the other by Bryson, upon principles extra-geometrical. Both demonstrations were false and unsuccessful; yet that of Hippokrates was not sophistic or eristic, because he kept within the sphere of geometry; while that of Bryson was so, because it travelled out of geometry. Nay more, this last would have been equally sophistic and eristic, and on the same ground, even if it had succeeded in solving the problem. If indeed the pseudo-graphic syllogism be invalid in form, it must be considered as sophistic, even though within the proper scientific limits as to [the] matter; but, if it be correct in form and within these same limits, then however untrue its premisses may be, it is to be regarded as not sophistic or eristic."

"Of original writings... we have only a fragment concerning the lunes of Hippocrates, quoted by Simplicius... and taken from Eudemus's lost History of Geometry..."

"Hippocrates of Chios... the most famous mathematician of his century... is credited with the idea of arranging theorems so that later ones can be proven on the basis of earlier ones, in the manner familiar to us from... Euclid. He is also credited with introducing the indirect method of proof into mathematics. His text on geometry, called the Elements, is lost."

"The circle being after rectilineal figures, the most simple in appearance, geometricians very naturally soon began to seek for its measure. Thus we find that the philosopher Anaxagoras occupied himself with the question in prison. Then Hippocrates of Chios tried the same problem, and it led him to the discovery of what is called the lune, a surface in the shape of a crescent, bounded by two arcs and exactly equal to a given square. He also found two unequal lines which were together equal to a rectilineal figure, so that if their relation could have been found the solution of the problem would have been obtained. But this no one has yet been able to do, nor is it likely ever to be done."

"Hippocrates of Chios... attempted the solution [for squaring the circle] and was the first actually to square a curvilinear figure. He constructed semicircles on the three sides of an isosceles right-angled triangle and showed that the sum of the two lunes thus formed is equal to the area of the triangle itself. ...His proof involves the proposition that the areas of circles are proportional to the squares of their diameters,—a proposition which Eudemus... tells us that Hippocrates proved. To the quadrature problem as such, however, his contribution was not important."

"The history of the Athenian school begins with the teaching of Hippocrates about 420 B.C."

"Hippocrates of Chios... was one of the greatest of the Greek geometricians. He... began life as a merchant. The accounts differ as to whether he was swindled by the Athenian custom-house officials who were stationed at the Chersonese, or whether one of his vessels was captured by an Athenian pirate near ... somewhere about 430 B.C. he came to Athens to try to recover his property in the law courts. ...the Athenians seem only to have laughed at him for his simplicity, first in allowing himself to be cheated, and then in hoping to recover his money. While prosecuting his cause he attended the lectures of various philosophers, and finally (in all probability to earn a living) opened a school of geometry himself. He seems to have been well acquainted with the Pythagorean philosophy, though there is no sufficient authority that he was ever initiated as a Pythagorean."

"[Hippocrates] wrote the first elementary text-book of geometry... on which probably Eudlid's Elements was founded; and therefore he may be said to have sketched out the lines on which geometry is still taught in English schools."

"It is supposed that the use of letters in diagrams to describe a figure was made by him or introduced about this time, as he employs expressions such as "the point on which letter A stands" and "the line on which AB is marked.""

"Hippocrates... denoted the square on a line by the word... power which it still retains in algebra."

"In [his] textbook Hippocrates introduced the method of "reducing" one theorem to another, which being proved, the thing proposed necessarily follows; of this method the ' is an illustration. No doubt the principle had been used occasionally before, but he drew attention to it as a legitimate mode of proof which was capable of numerous applications."

"[Hippocrates] elaborated the geometry of the circle: proving, among other propositions, that similar segments of a circle contain equal angles; that the angle subtended by the chord of a circle is greater than, equal to, or less than a right angle as the segment of the circle containing it is less than, equal to, or greater than a semicircle (Euc. III, 31); and probably several other of the propositions in the third book of Euclid. It is most likely that he also established the propositions that [similar] circles are to one another as the squares of the diameters (Euc. XII, 2), and that similar segments are as the squares of their chords. The proof given in Euclid of the first of these theorems is believed to be due to Hippocrates."

"The most celebrated discoveries of Hippocrates were... in connection with the quadrature of the circle and the duplication of the cube, and owing to his influence these problems played a prominent part in the history of the Athenian school."

"He commenced by finding the area of a lune contained between a semicircle and a quadrilateral arc standing on the same chord... as follows. Let ABC be an isosceles right-angled triangle inscribed in the semicircle ABOC, whose centre is O. On AB and AC as diameters describe semicircles as in the figure. Then, since by Euc. I, 47,sq. on\,BC = sq. on\,AC + sq. on\,AB,therefore, by Euc. XII, 2,area\;\frac{1}{2} \bigodot on\,BC = area\;\frac{1}{2} \bigodot on\,AC + area\;\frac{1}{2} \bigodot on\,ABTake away the common parts.\therefore area\,\triangle ABC = sum\;of\;areas\;of\;lunes\;AECD\;and\;AFBG.Hence the area if the lune AECD is equal to half that of the triangle ABC."

"He next inscribed half a regular hexagon ABCD in a semicircle whose centre was O, and on OA, AB, BC, and CD as diameters described semicircles of which those on OA and AB are drawn in the figure [2]. Then AD [by equilateral triangles within the half-hexagon] is double any of the lines OA, AB, BC, and CD,\therefore\;square\;on\;AD = sum\;of\;sqs.\;on\;OA, AB, BC, and\;CD,\therefore\;area\;\frac{1}{2} \bigodot on\,ABCD = sum\;of\;areas\;of\;\frac{1}{2} \bigodot s\;on\;OA, AB, BC, and\;CD.Take away the common parts\therefore\;area\;trapezium\;ABCD = 3\;lune\;AEBF + \frac{1}{2} \bigodot on\;OA.If therefore the area of this latter lune be known, so is that of the semicircle described on OA as diameter."

"Hippocrates also enunciated various other theorems connected with lunes... of which the theorem last given is a typical example. I believe that they are the earliest examples in which areas bounded by curves were determined by geometry."

"Hippocrates of Chios who lived in Athens in the second half of the fifth century B.C., and wrote the first text book on Geometry, was the first to give examples of curvilinear areas which admit of exact quadrature. These figures are the menisci or lunulae of Hippocrates."

"If on the sides of a right-angled triangle ACB semi-circles are described on the same side, the sum of the areas of the two lunes AEC, BDC is equal to that of the triangle ACB. If the right-angled triangle is isosceles, the two lunes are equal, and each of them is half the area of the triangle. Thus the area of the lunula is found."

"If AC = CD = DB = radius OA (see Fig. 3), the semi-circle ACE is ¼ of the semi-circle ACDB. We have now◐AB - 3◐AC = [trapezium] ACDB - 3 · meniscus ACE, [where the meniscus is the lunulae, i.e., lune] and each of these expressions is ¼◐AB or half the circle on ½AB as diameter. If then the meniscus AEC were quadrable so also would be the circle on ½AB as diameter. Hippocrates recognized the fact that the meniscus is not quadrable, and he made attempts to find other quadrable lunulae in order to make the quadrature of the circle depend on that of such quadrable lunulae."

"The most important name from the point of view of this chapter is Hippocrates of Chios. He is indeed the first person of whom it is recorded that compiled a book of Elements. This is lost, but Simplicius has preserved in his commentary on the Physics of Aristotle a fragment from Eudemus's History of Geometry giving an account of Hippocrates's quadratures of certain lunules or lunes."

"It would appear that Hippocrates was in Athens during a considerable portion of the second half of the fifth century, perhaps from 450 to 430 B.C. We have quoted the story that what brought him there was a suit to recover a large sum which he had lost, in the course of his trading operations, through falling in with pirates; he is said to have remained in Athens on this account a long time, during which he consorted with the philosophers and reached such a degree of proficiency in geometry that he tried to discover a method of squaring the circle. This is of course an allusion to the quadratures of lunes."

"He was the first to observe that the problem of doubling the cube is reducible to that of finding two mean proportionals in continued proportion between two straight lines. The effect of this was, as Proclus says, that thenceforward people addressed themselves (exclusively) to the equivalent problem of finding two mean proportionals between two straight lines."