66 "AND XXII. ND now," I asked, "will Ellen tell the old Pine the distinction between the odd and the even?" "Numbers," she answered, "have a character of their own. And so, as Ellen thinks, has everything, even the most insignificant. Thus, always, a certain space is occupied by a thing whilst it exists, and it doesn't make any difference whether the thing is at work or at play, awake or asleep. Possibly a thing might be of such character that the whole story of its existence could be summed up in this one fact, that it occupied space. But it is not a bit so with numbers, for they are awfully funny things, look some like rabbits, and act some like them, dodging around and jumping over each other. The nines are so constituted that they may be used to prove a sum in addition. Thus, take "Add the figures in each line of numbers, and find how many nines are contained in their sums. Reject these nines. and set down the remainders, 4, 7, 2, in line with the numbers. Then take the total, 5404, and see how many nines are contained in the sum of its figures. Reject as before and set down the excess 4. If this excess is equal to the excess of nines in the sum of the figures 4, 7, 2, the addition is correct. Thus the sum of the right hand column is 13, the excess of which above 9 is 4. And the excess of nines in the figures in the sum total, 5404, is also 4.* * "And this is always true. Awfully funny things, the 9's are aren't they? and the 3's copy after them. "A number is even or odd, as it can or cannot be divided by two. But practically this is entirely a question of the amount of the quantity considered, whether the unit of measure will go in it just one time, or one and something more, until *This method of proof depends upon a property of the number 9, which, except the number 3, belongs to no other digit whatever; namely, "that any number divided by 9, will leave the same remainder as the sum of its figures or digits divided by 9;" which may be demonstrated in this manner. 'Demonstration.-Let there be any number proposed, as 4658. This, separated into its several parts, becomes 4000+600+50+8. But 4000=4X1000=4X(999+1) =4X999+4. In like manner 600=6X99+6; and 50=5×9+5. Therefore the given number 4658=4X999+4+6X99+6+5×9+5+8=4X999+6X99+5×9+4+6 +5+8; and 4658÷9=(4X999+6X99+5×9+4+6+5+8)÷9. But 4X999+6X99 +5X9 is evidently divisible by 9, without a remainder; therefore if the given number 4658 be divided by 9, it will leave the same remainder as ( 4+6+5+8) divided by 9. And the same, it is evident, will hold for any other number whatever. In like manner, the same property may be shown to belong to the number 3; but the preference is usually given to the number 9, on account of its being more convenient in practice. 'Now, from the demonstration above given, the reason of the rule itself is evident; for the excess of 9's in two or more numbers taken separately, and the excess of 9's taken also out of the sum of the former excesses, it is plain that this last excess must be equal to the excess of 9's contained in the total sum of all these numbers; all the parts taken together being equal to the whole.-This rule was first given by Dr. Wallis in his Arithmetic, published in the year 1657.'-Hutton's Mathematics, p. 7. the something more becomes enough for two times. Passing the number 2, there must be space enough for it to go twice again, before the next even number is reached. This belongs to the nature of numbers. It is very simple. There could hardly be anything simpler, but from it by the increase or diminution of distance the antithesis of the odd and even is generated. And the bottom significance of the distinction is, first, that two things cannot exist in the same place at the same time, for if there is just room for one in a space there isn't room for two; and second, that a thing cannot both be and not be, for if it is a fact that one will completely fill a space it cannot be a fact that two will. And this is the significance of being, one significance, at least. The old Pine has admitted. to Ellen that he expected there would be some significance to being." 'Certainly," I said; "that is self-evident." "Ellen is afraid that it is not very self-evident to scientists; for they are constantly acting as though there was no significance in things, but that they are able both to be and not be. And they are constantly forgetting, apparently never conceived that things are as they are, and do what they do, because it is their nature. It's what they were made for. Ellen thinks that things behave precisely as it was intended they should; nor can she see how it would be possible to have a creation if they didn't. Suppose they were made for one thing, and should go to work doing another; -where would creation be? There couldn't be any, and the old Pine and Ellen would never have been upon this mountain, or anywhere else. For it is most evident that if we are to have this great, big creation, every thing must fit into its place and perform its function. And therefore things are made so that they not only will, but must. And so throughout the universe we see things acting out their nature, and the result is that every part of the universe is maintained, and we have a universe. Ellen sees that things are wonderfully well made, every one of them. The maple trees spread out their gnarled branches and interweave them with each other; and the pines grow up straight, interfering with their neighbors the least possible; and the birches droop their branches as though they were held down by the sunshine; and every little flower fills its peculiar niche, reigning complete within the radius of its beauty. And the old Pine and Ellen behave themselves according to their natures. The old Pine stays up here on this mountain, and Ellen walks all about and has an awfully good time. And thus, whichever way we turn, we find that there's a place for everything, and that everything is in its place, and not only in its place but performing the function of its existence. Possibly some few things, or quite a good many, could be spared; but the old Pine must see that nature's first and greatest law is order, and that order is attained through this principle of everything having its own character which in spite of all difficulties it is determined to assert. Numbers, then, in these regards, perform the functions they were made to perform, and at all times, like everything else, act out their natures. "And now Ellen will instruct the old Pine in the properties of numbers. A factor of a number is any number which is exactly contained in that number. Thus, 2 and 4 are factors of 8. "A prime number is one which has no other factor except |
