Augmentations surpassing, are ratios of greater inequa¬ lity, viz. when the greater is compared to the less, and are multiples, super-particulars, super-partients, multiple su¬ per-particulars, and multiple super-partients. But mul¬ tiple ratio is, as we have elsewhere shown, when a greater quantity contains a less many times ; super-particular ratio is, when the greater contains the less quantity once, and some part of it besides ; and super-partient ratio is, when the greater contains the less quantity once, and certain parts of it likewise. Again, multiple super-parti¬ cular ratio is, when the greater contains the less many times, and some part of it besides ; and multiple super- partient ratio is, when the greater contains the less many times, and also some of its parts. But augmentations surpassed, are ratios of less inequality, viz. when the less is compared with the greater quantity ; as for instance, sub-multiples, sub-super-particulars, and sub-super-par- tients, and those which are composed from these three. Those numbers are called by Plato assimilating and dis¬ similating, which are denominated by arithmeticians simi¬ lar and dissimilar. And similar numbers are those whose sides are proportional, but dissimilar numbers those whose sides are not proportional. Plato also calls those
numbers increasing and decreasing, which arithmeticians
denominate super-perfect and deficient, or more than per¬
fect and imperfect.
Things correspondent and effable are boundaries which correspond in ratio with each other, and can be expressed in numbers either integral or fractional, — such as these four terms or boundaries, 27, 18, 12, 8, which are in sesquialter and sub-sesquialter ratios ; since these mutually correspond in ratio and are effable. For effable quantities are those which can be expressed in whole numbers or fractions ; and, in like manner, ineffable quantities are such as cannot be expressed in either of these, and are called by modern mathematicians surds.
In the fourth place let U3 consider what we are to un¬ derstand by the sesquitertian progeny when conjoined with the pentad, and thrice increased, affording two harmonies . By the sesquitertian progeny, then, Plato means the num¬ ber 95. For this number is composed from the addition of the squares of the numbers 4 and 3, (i. e. 25,) which form the first sesquitertian ratio, and the number 70, which is composed from 40 and 30, and therefore consists of two numbers in a sesquitertian ratio. Hence, as 95 is com¬ posed from 25 and 70, it may with great propriety be called a sesquitertian progeny. This number conjoined with 5, and thrice increased, produces ten thousand and a
million. For 100x100=10,000, and 10,000x100= 1,000,000. But it must here be observed, that these two numbers, as will shortly be seen, appear to be considered by Plato as analogous to two parallelopipedons, the for¬ mer, viz. ten thousand, being formed from lOx 10 x 100, and the latter from 1000 X lOx 100. These two numbers are called by Plato two harmonies, for the following rea¬ son : — Simplicius, in his commentary on Aristotle’s trea¬ tise De Coelo, informs us, that a cube was denominated by the Pythagoreans harmony , because it consists of 12
bounding lines, 8 angles, and 6 sides ; and 12, 8, and 6, are
in harmonic proportion. As a parallelopipedon, therefore, has the same number of sides, angles, and bounding lines, as a cube, the reason is obvious why the numbers 10,000 and 1,000,000, are called by Plato harmonies. Hence also, it is evident why he says, “ that the other of these harmonies (viz. a million,) is of equal length indeed, but more oblong.” For if we call 100 the breadth, and 10 the depth, both of ten thousand and a million, it is evi¬ dent that the latter number, when considered as produced by 1000x10x100, will be analogous to a more oblong parallelopipedon than the former.
Again, when he says, “ that the number 1,000,000 con¬ sists of a hundred numbers from effable diameters of pentads, each being deficient by unity, and from two that are ineffable, and from a hundred cubes of the triad,” his
meaning is as follows. The number 1,000,000 consists of a hundred numbers, (i. e. of a hundred such numbers as 10,000,) each of which is composed from effable dia¬ meters of pentads, Sic. But in order to understand the truth of this assertion, the reader must recollect what has been delivered in chap. 34, of my Theoretic Arithmetic, viz. that there are certain numbers which are called by arithme¬ ticians effable diameters. These, also, are twofold ; for some are the diameters of even, and others of odd squares. And the diameters of effable even squares, when multiplied into themselves, produce square numbers, double of the squares of which they are the diameters, with an excess of unity. Thus, for instance, the number 3, multiplied into itself, produces 9, which is double of the square of the number 4, with an excess of unity ; and therefore 3 will be the diameter of the even square 4. But the diameters of effable odd square numbers, are in power double of the squares of w hich they are the diameters by a deficiency of unity. This being premised, it follows, that the num¬ ber 10,000 will consist of a certain number of heptads ; for 7 is the effable diameter of the square number 25. And, from what follows, it will be found that this number
is 9S9-
But the number 10,000, not only consists of 989 hep-
tads, but Plato also adds, “ from two numbers that are
ineffable,” viz. from two numbers the roots of which can-
not be exactly obtained nor expressed, either in whole
numbers or fractions, such as the roots of the numbers 2
and 3. The numbers 300 and 77 are also of this kind ;
and, as we shall see, appear to be the numbers signified
by Plato. In the last place, he adds, <l and from a hun¬
dred cubes of the triad/’ viz. from the number 2700 ;
for this is equal to a hundred times 27, the cube of 3.
The numbers, therefore, that form 10,000 are as below ;
viz. 989 heptads, two ineffable numbers, 300 and 77/
and a hundred times the cube of 3, i. e. 2,700. And the
whole geometric number is a million.
1 The reader, who may have my Plato and Aristotle in his pos¬ session, is requested to correct, by the above numbers, an error in the derelopement of this geometric number, which is given in the Republic of the former, and the Politics of the latter. This error originated from mis-stating the product of a hundred times t7 to be 270, instead of 2,700.
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