We have, I conceive, an instance of the error as well as of the truth, to which such views may lead, in the speculations of Plato concerning Harmony, contained in that part of his writings (the seventh Book of the Republic), in which these views are especially urged. He there, by way of illustrating the superiority of philosophical truth over such exactness as the senses can attest, speaks slightingly of those who take immense pains in measuring musical notes and intervals by the ear, as the astronomers measure the heavenly motions by the eye. “They screw their pegs and pinch their strings, and dispute whether two notes are the same or not.” Now, in truth, the ear is the final and supreme judge whether two notes are the same or not. But there is a case in which notes which are nominally the same, are different really and to the ear; and it is probably to disputes on this subject, which we know did prevail among the Greek musicians, that Plato here refers. We may ascend from a note A1 to a note C3 by two octaves and a third. We may also ascend from the same note A1 to C3 by fifths four times repeated. But the two notes C3 thus arrived at are not the same: they differ by a small interval, which the Greeks called a [501] Comma, of which the notes are in the ratio of 80 to 81. That the ear really detects this defect of the musical coincidence of the two notes under the proper conditions, is a proof of the coincidence of our musical perceptions with the mathematical relations of the notes; and is therefore an experimental confirmation of the mathematical principles of harmony. But it seems to be represented by Plato, that to look out for such confirmation of mathematical principles, implies a disposition to lean on the senses, which he regards as very unphilosophical.
Hero of Alexandria.
The other branches of mathematical science which I have spoken of in the History as cultivated by the Greeks, namely Mechanics and Hydrostatics, are not treated expressly by Plato; though we know from Aristotle and others that some of the propositions of those sciences were known about his time. Machines moved not only by weights and springs, but by water and air, were constructed at an early period. Ctesibius, who lived probably about b. c. 250, under the Ptolemies, is said to have invented a clepsydra or water-clock, and an hydraulic organ; and to have been the first to discover the elastic power of air, and to apply it as a moving power. Of his pupil Hero, the name is to this day familiar, through the little pneumatic instrument called Hero’s Fountain. He also described pumps and hydraulic machines of various kinds; and an instrument which has been spoken of by some modern writers as a steam-engine, but which was merely a toy made to whirl round by the steam emitted from holes in its arms. Concerning mechanism, besides descriptions of Automatons, Hero composed two works: the one entitled Mechanics, or Mechanical Introductions; the other Barulcos, the Weight-lifter. In these works the elementary contrivances by which weights may be lifted or drawn were spoken of as the Five Mechanical Powers, the same enumeration of such machines as prevails to this day; namely, the Lever, the Wheel and Axle, the Pulley, the Wedge, and the Screw. In his Mechanics, it appears that Hero reduced all these machines to one single machine, namely to the lever. In the Barulcos, Hero proposed and solved the problem which it was the glory of Archimedes to have solved: To move any object (however large) by any power (however small). This, as may easily be conceived by any one acquainted with the elements of Mechanics, is done by means of a combination of the mechanical powers, and especially by means of a train of toothed-wheels and axles. [502]
The remaining writings of Hero of Alexandria have been the subject of a special, careful, and learned examination by M. Th. H. Martin (Paris, 1854), in which the works of this writer, Hero the Ancient, as he is sometimes called, are distinguished from those of another writer of the same name of later date.
Hero of Alexandria wrote also, as it appears, a treatise on Pneumatics, in which he described machines, either useful or amusing, moved by the force of air and vapor.
He also wrote a work called Catoptrics, which contained proofs of properties of the rays of reflected light.
And a treatise On the Dioptra; which subject however must be carefully distinguished from the subject entitled Dioptrics by the moderns. This latter subject treats of the properties of refracted light; a subject on which the ancients had little exact knowledge till a later period; as I have [shown] in the History. The Dioptra, as understood by Hero, was an instrument for taking angles so as to measure the position and hence to determine the distance of inaccessible objects; as is done by the Theodolite in our times.
M. Martin is of opinion that Hero of Alexandria lived at a later period than is generally supposed; namely, after b. c. 81.