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Classical Arithmetic, Lesson 14. Book I, Chapter 14

The following tasks are required to complete this lesson:

    1. Study the Lesson carefully.
    2. Complete the lesson Assessment.

Lesson

Note that this chapter is concerned only with even numbers. Be sure to use the Oxford English Dictionary for definitions of unfamiliar words.

Again, of even numbers a second division is as follows:   Of these, some are superperfect, and others are deficient, according to each habitude of inequality. For all inequality is considered either in greater or less terms.

In the next paragraph, when we read “as it were”, the author speaks of a comparison that will be explained later in the lesson.

But the superperfect numbers are such as by an immoderate plenitude, exceed, as it were, by the numerosity of their parts, the measure of their proper body.  On the contrary, the deficient numbers being as it were, oppressed by poverty, are less than the sum of their parts.

And the superperfect numbers indeed, are such as 12 and 24; for these will be found to be more than the aggregate of their parts. For the half of 12 is 6; the third part is 4; the fourth part is 3; the sixth part is 2 ; and the twelfth part is 1.  And the aggregate of all these parts is 16, which surpasses the multitude of its whole body. Again, of the number 24, the half is 12; the third 8; the fourth 6; the sixth 4; the eighth 3; the twelfth 2; and the twenty-fourth 1; the aggregate of all which is 36.  And it is evident in this instance also, that the sum of the parts is greater than, and overflows as it were, its proper body.  And this number indeed, because the parts surpass the sum of the whole number, is called superabundant.

On the contrary, that number is called deficient, the parts of which are surpassed by the multitude of the whole; and such are the numbers 8 and 14. For the half of 8 is 4; the fourth is 2; and the eighth is 1; the aggregate of all which is 7; a sum less than the whole number. Again, the half of 14 is 7; the seventh is 2 ; and the fourteenth is 1; the aggregate of which is 10; a sum less than the whole term. Such therefore are these numbers, the former of which in consequence of being surpassed by its parts, resembles one born with a multitude of hands in a manner different from the common order of nature, such as the hundred-handed giant Briareus, or one whose body is formed from the junction of three bodies, such as the triple Gerion, or any other production of nature which has been deemed monstrous by the multiplication of its parts. But the latter of these numbers resembles one who is born with a deficiency of some necessary part, as the one-eyed Cyclops, or with the want of some other member.

Between these however, as between things equally immoderate, the number which is called perfect is allotted the temperament of a middle limit, and is in this respect the emulator of virtue; for it is neither extended by a superfluous progression, nor remitted by a contracted diminution; but obtaining the limit of a medium, and being equal to its parts, it is neither overflowing through abundance, nor deficient through poverty.  Of this kind are the numbers 6 and 28. For the half of 6 is 3; the third is 2; and the sixth is 1, which if reduced into one sum, the whole body of the number will be found to be equal to its parts.  Again, the half of 28 is 14; the seventh is 4; the fourth is 7; the fourteenth is 2; and the twenty-eighth is 1; the aggregate of which is 28.

Nor must we omit to observe, that all the multiples of a perfect number are superabundant, but on the contrary all the submultiples are deficient. Thus, for instance, 3 the subduple of 6 is a deficient, but 12 the double of 6 is superabundant.  Thus also 2 which is subtriple of 6 is a deficient, but 18 which is the triple of it is a superabundant number. And the like will take place in other multiples, and submultiples. Hence also it is evident that a perfect number is a geometric medium between the superabundant and the deficient number. Thus in the three numbers 3, 6 and 12, 6 is the geometrical mean between 3 and 12; for as 3 is to 6, so is 6 to 12.  Thus, too, 28 is the geometric mean between 14 and 56, the former of which is a deficient, and the latter a superabundant number.

Perfect numbers therefore, are beautiful images of the virtues which are certain media between excess and defect, and are not summits, as by some of the ancients they were supposed to be.  And evil indeed is opposed to evil, but both are opposed to one good. Good however, is never opposed to good, but to two evils at one and the same time. Thus timidity is opposed to audacity, to both which the want of true courage is common; but both timidity and audacity are opposed to fortitude. Craft also is opposed to fatuity, to both which the want of intellect is common; and both these are opposed to prudence. Thus, too, profusion is opposed to avarice, to both which illiberality is common; and both these are opposed to liberality. And in a similar manner in the other virtues; by ail which it is evident that perfect numbers have a great similitude to the virtues. But they also resemble the virtues on another account; for they are rarely found, as being few, and they are generated in a very constant order. On the contrary, an infinite multitude of superabundant and diminished numbers may be found, nor are they disposed in any orderly series, nor generated from any certain end; and hence they have a great similitude to the vices, which are numerous, inordinate, and indefinite.

Assessment

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