Another problem solved by the Pythagoreans was that of drawing a rectilineal figure which shall be equal in area to one given rectilineal figure and similar to another. Plutarch mentions a doubt whether it was this problem or the theorem of Eucl. I. 47 on the strength of which Pythagoras was said to have sacrificed an ox.
The main particular applications of the theorem of the square on the hypotenuse, e. g. those in Euclid, Book II, were also Pythagorean; the construction of a square equal to a given rectangle (Eucl. II. 14) is one of them, and corresponds to the solution of the pure quadratic equation x²=ab.
The Pythagoreans knew the properties of parallels and proved the theorem that the sum of the three angles of any triangle is equal to two right angles.
As we have seen, the Pythagorean theory of proportion, being numerical, was inadequate in that it did not apply to incommensurable magnitudes; but, with this qualification, we may say that the Pythagorean geometry covered the bulk of the subject-matter of Books I, II, IV and VI of Euclid’s Elements. The case is less clear with regard to Book III of the Elements; but, as the main propositions of that Book were known to Hippocrates of Chios in the second half of the fifth century B. C., we conclude that they, too, were part of the Pythagorean geometry.
Lastly, the Pythagoreans discovered the existence of the incommensurable or irrational in the particular case of the diagonal of a square in relation to its side. Aristotle mentions an ancient proof of the incommensurability of the diagonal with the side by a reductio ad absurdum showing that, if the diagonal were commensurable with the side, it would follow that one and the same number is both odd and even. This proof was doubtless Pythagorean.
A word should be added about the Pythagorean astronomy. Pythagoras was the first to hold that the earth (and no doubt each of the other heavenly bodies also) is spherical in shape, and he was aware that the sun, moon and planets have independent movements of their own in a sense opposite to that of the daily rotation; but he seems to have kept the earth in the centre. His successors in the school (one Hicetas of Syracuse and Philolaus are alternatively credited with this innovation) actually abandoned the geocentric idea and made the earth, like the sun, the moon, and the other planets, revolve in a circle round the ‘central fire’, in which resided the governing principle ordering and directing the movement of the universe.
The geometry of which we have so far spoken belongs to the Elements. But, before the body of the Elements was complete, the Greeks had advanced beyond the Elements. By the second half of the fifth century B. C. they had investigated three famous problems in higher geometry, (1) the squaring of the circle, (2) the trisection of any angle, (3) the duplication of the cube. The great names belonging to this period are Hippias of Elis, Hippocrates of Chios, and Democritus.
Hippias of Elis invented a certain curve described by combining two uniform movements (one angular and the other rectilinear) taking the same time to complete. Hippias himself used his curve for the trisection of any angle or the division of it in any ratio; but it was afterwards employed by Dinostratus, a brother of Eudoxus’s pupil Menaechmus, and by Nicomedes for squaring the circle, whence it got the name τετραγωνιζουσα, quadratrix.
Hippocrates of Chios is mentioned by Aristotle as an instance to prove that a man may be a distinguished geometer and, at the same time, a fool in the ordinary affairs of life. He occupies an important place both in elementary geometry and in relation to two of the higher problems above mentioned. He was, so far as is known, the first compiler of a book of Elements; and he was the first to prove the important theorem of Eucl. XII. 2 that circles are to one another as the squares on their diameters, from which he further deduced 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 essential portions of the tract are preserved in a passage of Simplicius’s commentary on Aristotle’s Physics, which contains substantial extracts from Eudemus’s lost History of Geometry. Hippocrates showed how to square three particular lunes of different kinds and then, lastly, he squared the sum of a circle and a certain lune. Unfortunately the last-mentioned lune was not one of those which can be squared, so that the attempt to square the circle in this way failed after all.
Hippocrates also attacked the problem of doubling the cube. There are two versions of the origin of this famous problem. According to one story an old tragic poet had represented Minos as having been dissatisfied with the size of a cubical tomb erected for his son Glaucus and having told the architect to make it double the size while retaining the cubical form. The other story says that the Delians, suffering from a pestilence, consulted the oracle and were told to double a certain altar as a means of staying the plague. Hippocrates did not indeed solve the problem of duplication, but reduced it to another, namely that of finding two mean proportionals in continued proportion between two given straight lines; and the problem was ever afterwards attacked in this form. If x, y be the two required mean proportionals between two straight lines a, b, then a:x=x:y=y:b, whence b/a=(x/a)³, and, as a particular case, if b=2a, x³=2a³, so that, when x is found, the cube is doubled.