By Sir Thomas Heath
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Extra resources for A History of Greek Mathematics, Vol. 1: From Thales to Euclid
Now it is obvious that, if all the recorded solutions are collected together, it is much easier to see the relations, amounting in some cases to substantial identity, between them, and to get a comprehensive view of the history of the problem. I have therefore dealt with this problem in a separate section of the chapter devoted to ‘Special Problems’, and I have followed the same course with the other famous problems of squaring the circle and trisecting any angle. Similar considerations arise with regard to certain well-defined subjects such as conic sections.
Archytas, in the passage quoted, specifies the four subjects of the Pythagorean quadrivium, geometry, arithmetic, astronomy, and music (for ‘sphaeric’ means astronomy, being the geometry of the sphere considered solely with reference to the problem of accounting for the motions of the heavenly bodies); the same list of subjects is attributed to the Pythagoreans by Nicomachus, Theon of Smyrna, and Proclus, only in a different order, arithmetic, music, geometry, and sphaeric; the idea in this order was that arithmetic and music were both concerned with number (ποσόν), arithmetic with number in itself, music with number in relation to something else, while geometry and sphaeric were both concerned with magnitude (πηλίκον), geometry with magnitude at rest, sphaeric with magnitude in motion.
E. a little downwards on all sides instead of merely in the plane of the horizon, and so might see both the sun and the moon although the latter was in the earth’s shadow. Cleomedes denies this, and prefers to regard the whole story of such cases as a fiction designed merely for the purpose of plaguing astronomers and philosophers; but it is evident that the cases had actually been observed, and that astronomers did not cease to work at the problem until they had found the real explanation, namely that the phenomenon is due to atmospheric refraction, which makes the sun visible to us though it is actually beneath the horizon.
A History of Greek Mathematics, Vol. 1: From Thales to Euclid by Sir Thomas Heath