MATHEMATICS. 311 



the golden number, to determine the new and full moons, on which 

 moveable festivals depend. 



The Pythagoreans had paid much attention to arithmetic, i. e., to Arithmetic, 

 the properties of numbers ; indeed, they attributed to numbers a mys- 

 tical meaning, which is not very intelligible. The Platonists also 

 pursued this subject, and invented arrangements of numbers into 

 various classes : thus they were called odd or even, perfect or imper- 

 fect, polygonal, which included triangular, square and pentagonal, 

 pyramidal, &c. Besides these speculations, which are not of very 

 material consequence, the practical art of performing arithmetical 

 operations had been carried to a considerable extent, as we shall see 

 hereafter. 



It has already been mentioned, that, from the time of Pythagoras, Music. 

 music had become a mathematical science. Though there seems to 

 be some error in the account of the inferences which that philosopher 

 drew from the notes struck by the hammers on a blacksmith's anvil, 

 the general fact is probably true, that he made the discovery that two 

 musical strings which gave the most perceptible concord to the ear, 

 exhibited also remarkable mathematical relations to each other in their 

 lengths and tensions. This curious fact, connected with the great 

 importance which the Greeks attached to music, soon led to a variety 

 of speculations, derived from these mathematical proportions, which 

 were assumed to be perfectly exact. This accuracy, however, though 

 a proper subject for theory, is not the foundation of practical music : 

 and though a mathematical exactness in concords is susceptible of 

 being appreciated by the ear, it is rejected by the practice of modern 

 music. Indeed, the unalterable properties of numbers, thus curiously 

 connected with one of the most exquisite gratifications of the senses, 

 make it impossible to preserve the perfect exactness of chords in every 

 part of the musical scale. Therefore, though the ancients reasoned 

 upon their concords as perfect, it is probable that in practice they used 

 them imperfect. The latitude which this allows gives rise to the 

 different expression of the different keys, as they are called, which 

 probably correspond, at least to a considerable extent, with the various 

 modes, Phrygian, Dorian, &c., of the ancient music. It w'as also pro- 

 bably this capacity of the ear to adapt itself to concords slightly 

 imperfect, which caused the separation into two sects of the ancient 

 theorisers on this subject : of one of which sects Pythagoras was the 

 founder, as Aristoxenus, a cotemporary of Aristotle, was of the other. 

 While the former made the simplicity of the arithmetical relations 

 regulate, as it were, the musical ones ; the latter appealed more to 

 experiment, and placed the tones at equal intervals in the scale; 

 perhaps making, much in the same manner as is done in modern keyed 

 instruments, their defects compensate each other. It seems requisite, 

 in speaking of ancient mathematics, to say something of this difficult, 

 and perhaps uninteresting portion of the science, as it was by them 

 considered a most important branch ; and many of their greatest 



