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is, makes it ten times greater in number, but not any greater in its whole value; so that every 0 thus added makes each individual figure of the number only one-tenth part of what it was without the 0, or we have ten times as many as we had before; but each 1 of that number is ten times less than it was before, and thus the entire value is not in the least altered. Two Os increase the number one hundred times, and make every 1 of which it is composed only one-hundredth part of what it was; and generally whatever number of O's we annex to the right of a number, without altering the whole value of that number, we divide each individual 1 which the number contains as often by 10 as there are O's.

By the application of this principle any number whatever may be expressed in terms of any lower place in the scale, whether integral or decimal, by annexing as many ciphers to the right of it as shall bring it down to the required place; and therefore, when we have a divisor and dividend which contain different numbers of decimal places, we have only to add to that which has fewest as many O's as shall make its number equal to that in the other, and then divide the one by the other in exactly the same manner as if they were both wholly integers and the right hand figures of them units. Of course, whatever figures of the quotient are obtained from the divisor and dividend so prepared, must be integers, but if more O's be annexed either to the dividend or to the remainders, there must be a decimal place in the quotient answering to each of them.

The principle which this involves is the most important one in the whole science of arithmetic; because it enables us to separate the absolute values even of numbers from the numerical values of them, and thus to consider their relations generally; and, it is the application of those general relations which is our

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chief element in the finding of unknown quantities by means of known ones; for, if we have a quantity of the same kind with that which we seek, and know the relation which this quantity bears to that which we seek, we are in a condition for finding the value of the unknown quantity, in terms of the known one. This part of the subject is called the doctrine of PROPORTION, that is, the doctrine of the relation which the magnitude, or value, of one quantity taken as a whole bears to that of another quantity taken as a whole. In order to have this relation, the quantities must be such that we can at once say that they are equal, or that one of them must be greater than another; and if they are expressed in numbers, those numbers must be expressed in terms of the same place in the scale of numbers; for if the arithmetical expression were not of this description, the numbers would express a different ratio from the quantities. A very simple instance will serve to illustrate this: there is a certain ratio or proportion between 7 sovereigns and 8 guineas; but it will be at once seen that this is not the ratio of the numbers 7 and 8, because 1 in the one of them is not equal to 1 in the other of them, for a sovereign is equal to 20 shillings and a guinea to 21 shillings; and therefore, before we can find two simple numbers which will express the ratio, we must turn both into their values in shillings, which is evidently done by multiplying the sovereigns by 20 and the guineas by 21, which numbers are 140 and 168. But if we look at the numbers which we multiply, we find there is 7 on the one side, and 21, or three 7's, on the other; and again, that there is 20 or five 4's on the one side, and 8 or two 4's on the other. We may therefore throw the 7 and the 4 out of both sides, and there remain 1 and 5 answering to the 7 sovereigns, and 2 and 3 answering to the 8 guineas; and if we take the pro

ducts answering to each, we have 7 sovereigns to 8 guineas, in the ratio of 5 to 6, which is really a more simple ratio than that of the numbers.

Before proceeding to the explanation of this doctrine in such a manner as to have clear notions of its usefulness, and due expertness in the application of it, it becomes necessary to introduce principles more general than can be introduced by means of the common arithmetical figures; because we have seen that the numbers in which quantities are presented to us do not necessarily express the ratios of those quantities of which they are the arithmetical names; and also because, as we have already partially seen, from the impossibility of dividing some numbers exactly by some others as we shall afterwards see more at length—all ratios are not expressible in terms of 1 in any place whatever of the arithmetical scale. We shall therefore only mention farther, that as the product in multiplication, when divided by any one of the two factors, must necessarily give the other factor as quotient, it follows that the quotient of a number accurately expressible by the scale, divided by one not accurately expressible, may be a number accurately expressible. We shall now proceed to the consideration of quantities generally, and without any reference to whether they can or cannot be expressed arithmetically by numbers.

SECTION V.

GENERAL OR ALGEBRAICAL EXPRESSION OF QUANTITIES.

Ir is a general rule in nature, in art, and in science, that everything which is peculiarly well adapted for the accomplishment of some one particular purpose, is, for that very

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reason, the worse adapted for every other purpose. This is remarkably the case with the scale and notation of numbers in our Arithmetic. It is difficult to imagine any means by which individual operations in numbers could be performed with so much ease and certainty; but the great facility and precision with which we are able to manage each particular case, become obstacles in our way when we attempt to investigate general principles. The whole expression-all the characters which are before our eyes, apply to the particular case only, and each number tells us nothing but its relation to the scale. There is no trace of the operation in the result, and very often there is not a vestige of the original or given numbers, by which the result has been obtained. Thus, for instance, 49 is the product of 7 by 7; and the product of no other two integer numbers, except of 49 and 1, which is not, properly speaking, a product at all. But there is nothing in 49, as it appears arithmetically, to let us know that it is the product of 7 by 7, or of any two numbers whatever; and any one who had not learned by rote the products of all numbers up to 9 times 9, or 81, would be just as likely to suppose that 47 were the product of two numbers; but we find by actual trial that 47 is not the product of any two numbers, each greater than the number 1. Even in the very simplest instances which we can take, we find no trace whatever either of the given numbers, or of the operation, in the result. 5 is the sum of 2 and 3; but there is no appearance in it, either of its being connected with 2 and 3, or of its being a sum at all. It is one simple and original mark, and leads us to think only of a number of times 1-of a short expression for the same number of dots ( . . . . . ).

An arithmetical expression does not, thus, give any account of itself, and an arithmetical operation has no story to tell. However long and complicated, however short and plain, however

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tedious from want of the knowledge of principles, or simple from the possession of this knowledge, such an operation may be, it offers no instruction to the ignorant; for though some of the numbers may appear as data, or as being originally given, and others as results; yet all that is done in order to obtain the results from the data is concealed; and, in order to understand the operation, the student must bring to it the very same degree of knowledge which was required for the original performing of it. A book might be filled with such operations, but it would be a book containing no knowledge; and as we cannot separate the principles from the particular numbers contained in the example, the only knowledge that we can obtain is a set of empirical rules, the truth of which we have no means of bringing to the test.

It is for these reasons, that the arithmetic which we usually learn at school is not only wholly useless to us as an instrument of knowledge, but acts as a barrier in our way when we attempt to understand anything of the other branches of mathematics. It is as if we were to sow flowers, and expect them to spring up, and grow, and produce a crop of plants. The principles of vegetable life are in the flowers, but they are not developed: the development is in the seed, and that seed we cannot obtain, if we separate the flower from the parent plant. Just so, when we are conversant with numbers only in the way of the common arithmetical operations, we are without the principles of knowledge in such a state as that one part may be fruitful of other knowledge.

There is this farther disadvantage in the application of empirical rules of which we do not comprehend the meaning and in the performing of operations the reason of which is a mystery to us, that we never enter heartily upon such subjects, and never at all, if we can avoid them. Now, as this repulsive

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