Α TREATISE ON ARITHMETIC IN THEORY AND PRACTICE. INTRODUCTORY EXPLANATIONS. NOTE. This chapter of Introductory Explanations is meant rather for the consideration of teachers, and for affording suggestions towards verbal explanations from them to their pupils, and for affording information to advanced learners, than as a passage to be continuously studied by young beginners. ARITHMETIC is the science of numbers and of quantities expressed and considered numerically. It has to do with the question how many; and it has besides to do with the question how much, when the answer is to be by a number or combination of numbers. It may now be asked, What is number, or what are numbers? To this question there is no short reply possible. Notions of numbers are arrived at in many ways; and, indeed, by every intelligent mind considerable progress is made in their acquisition even in early childhood. By consideration of a few examples, such as will now be briefly sketched out, the learner may acquire clear fundamental notions respecting the nature of numbers, or may get his existing knowledge extended or confirmed. If we put a pebble into an empty basket, we say there is one pebble in it. If we put another in, we say there are now two in it, or we say that the number of the pebbles in the basket is two. If we put again another in, we say there are three in it, or we say that the number of the pebbles is three. If, at the same time, a farmer puts a sheep into a field where there was none before, we say there is one sheep in the field. If he puts another in, we say there are two in the field; and if he puts again another in, we say the number of the sheep in the field is three. Now, the group of sheep in the field is very unlike to the group of pebbles in the basket; but, in one respect, the groups can be perceived to be perfectly alike, and that quality or character of likeness is expressed by saying that the The name arithmetic is derived from the Greek word arithmos, number. 37 two groups are alike in number, or are of the same number, or that the number of the sheep is the same as the number of the pebbles. Again, if yet another pebble were put into the basket, and another sheep were put into the field, there would be a new number of pebbles and a new number of sheep, and the two groups so arrived at would be again perceived to be perfectly alike in one character, while quite unlike in every other respect. This character in which sameness can be perceived in the two new groups is still called number, and the new number now arrived at, different from the number which was before attained, and was named three, is called the number four. We can after this go on indefinitely, increasing the number of the pebbles, or of the sheep, or of both, by continually adding one more to those already put together; and we can give a new name to each new number arrived at, which is more by one than the previous number. It is scientifically proper, and it is essential for practical convenience (although sometimes misfitting awkwardly with the arbitrary structure of the English and of many other languages), to call one a number, as well as two, three, four, &c.; and thus to maintain freedom to say, when only one pebble is in the basket, that the number of pebbles in it is one; or freedom to say, that if pebbles are from time to time put in and taken out, so that the number in the basket is to be considered as varying, we may have the number of the pebbles in it, for instance, three, then four, then two, and afterwards one. This is to the same effect as to speak of a group of things being changed in its number till we have to regard the group as being reduced to only a single one of the things which might be brought together into it. Sometimes also even zero, or nought, is called a number; but this application of the term "number" can scarcely be supported, unless occasionally for convenience, and with some correction mentally introduced; since 0, nought, or zero, expresses the absence of any one or more of the things contemplated as counted or expressed. As a matter of convenience, however, it is allowable to make such statements as that the numbers of births in a town on five successive days were 3, 1, 2, 0, and 4; and the extension of the signification of the word "number" in this way is scarcely, if at all, liable to introduce any perplexity or any inaccurate modes of thought. The explanations now given are sufficient to show fully what is the character of groups of objects which can properly be called their number, and to specify the different characters which can properly be called different numbers, and which are distinguishable by different names, as one, two, three, four, five, six, seven, eleven, twenty-four, &c. Besides numbers, properly so called, there are very important numerical expressions, which may be called fractional numerical expressions, such as one half, two thirds, four and three fifths, seven and two tenths and three hundredths-or, as they may also be written,,, 4, 7-23-to which the designation number is frequently applied. The application of the word "number" to such fractional numerical expressions, although sanctioned by very frequent usage, involves, unfortunately, ambiguity in language, and tends to introduce perplexity in thought. It is indeed essentially inconsistent with the employment of the word "number" in the simple sense which has been already explained, and which is certainly sanctioned by universal usage as the primary or fundamental meaning of the word. Fractional numerical expressions have no direct applicability in counting the number of sheep driven into a field, or of rough lumps of broken stones which might be put into a basket, and which, though different in their sizes, and in their forms, and in their material-some being perhaps of basalt, some of flint, and some of slaty rock-might still be perfectly well used as counters to indicate the number of sheep put into the field. A living sheep can only be counted properly as one: it cannot be regarded as made up of three third parts of a sheep, each alike with the other two. Also we cannot put 4 rough fragments of stone into a basket; if we put four in, and then, for the half, break another stone into two parts, and put one of those parts into the basket, we shall find that we have really put five pieces of stone into the basket. But otherwise and indirectly fractional numerical expressions may be used for expressing numbers of individual objects by the device of using groups of those objects and fractional parts of a group, as when we speak of 4 dozens of eggs, or 53 millions of persons; but in each case here we state directly, not the number of individuals, but the number of groups, and we superadd a fraction of a group; and by custom we often slip into speaking of the number and the fraction jointly as being a fractional number. Again, though, as we have seen already, we cannot put four and a half rough fragments of stone into a basket, yet we can perfectly well put four pounds of butter and half a pound of butter, or what for brevity is called four and a half pounds of butter, into a basket. We here meet with a very important distinction in passing from the counting of individual objects, such as fragments of stones, to the numerical statement of quantity of any kind of indefinitely divisible thing, such, for instance, as quantity of butter. We are passing from the one province of arithmetic which has to do with the question how many, and entering on the other province which has to do with the question how much. It is only in giving numerical expression to the muchness or quantity of things that fractional numerical expressions unavoidably arise; they have no applicability in expressing directly the numerousness or the number of individual objects, though indirectly they can be used for making known the number of individual objects by taking the objects in groups, and then stating, not the number of the objects themselves, but the number of groups and the fractional part of another group, as when we say 5 millions of persons and of another million, or briefly 52 millions of persons. Fractional numerical expressions are referred to at this place merely to notice them in connection with the meaning of the word "number," and to make early mention that they are distinct from numbers properly so called, and to state that though they are often spoken of as numbers, yet this arises through imperfections of the language which has been handed down to us from remote times, imperfections which the people at any period living in the world have |