The Arithmetical Philosophy of Nicomachus of Gerasa

Press of the New Era Printing Company, 1916 - Arithmetic - 49 pages

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Page 31 - The doubles, then, will produce 1 sesquialters, the first one, the second two, the third three, the fourth four, the fifth five, the sixth six, and neither more nor less, but by every necessity when the superparticulars that are generated attain the proper number, that is, when their number agrees with the multiples that have generated them, at that point by a divine device, as it were, there is found the number which terminates them all because it naturally is not divisible...
Page 8 - An EVEN NUMBER is that which can be divided into two equal whole numbers.
Page 33 - Now that which is indivisible in quantity is called a unit if it is not divisible in any dimension and is without position, a point if it is not divisible in any dimension and has position, a line if it is divisible in one dimension, a plane if in two, a body if divisible in quantity in all — ie in three — dimensions. And, reversing the order, that which is divisible in two dimensions...
Page 49 - It is evident that the terms of a proportion may undergo any change which will not destroy the equality of the ratios ; or which will leave the product of the means equal to the product of the extremes.
Page 19 - But it happens that, just as the beautiful and the excellent are rare and easily counted but the ugly and the bad are prolific, so also excessive and defective numbers are found to be very many and in disorder, their discovery being unsystematic. But the perfect are both easily counted and drawn up in a fitting order: for only one is found in the units, 6; and only one in the tens, 28; and a third in the depth of the hundreds, 496; as a fourth the one, on the border of the thousands, that is short...
Page 33 - There is a subcontrary mean, which we call 'harmonic', when they are such that the part of the third by which the middle term...
Page 19 - ... and easily counted but the ugly and the bad are prolific, so also excessive and defective numbers are found to be very many and in disorder, their discovery being unsystematic. But the perfect are both easily counted and drawn up in a fitting order: for only one is found in the units, 6; and only one in the tens, 28; and a third in the depth of the hundreds, 496; as a fourth the one, on the border of the thousands, that is short of the ten thousand, 8128. It is their uniform attribute to end...
Page 35 - ... fourth, 5; of the fifth, 6, and so on in general with all that follow. [3] This number also is produced if the natural series is extended in a line, increasing by 1, and no longer the successive numbers are added to the numbers in order, as was shown before, but rather all those in alternate places, that is, the odd numbers. For the first, 1, is potentially the first square; the second, 1 plus 3, is the first in actuality; the third, 1 plus 3 plus 5, is the second in actuality; the fourth, 1...
Page 42 - V, p, </> to the effect that, if three numbers be proportional, the product of the extremes is equal to the square of the mean, and conversely. It does not appear in P in the first hand, B has it in the margin only, and Campanus omits it, remarking that Euclid does not give the proposition about three proportionals as he does in vi.
Page 49 - II. 26. 2. 2 Cf. II. 24. 6. shown to differ from the one next beneath it by the amount whereby this latter differs from the least term, such an array becomes an arithmetic proportion and the sum of the extremes is twice the mean. But if the third term from the greatest exceeds and is exceeded by the same fraction of the extremes, it is harmonic and the product of the mean by the sum of the extremes is double the product of the extremes. [3] Let this be an example of this proportion, 6, 8, 9, 12....