CHAPTER I. ABSTRACT NUMBERS. ARITHMETIC has been frequently defined to be the Science of Number, and this it certainly is; but in the general acceptation of the word it includes something more. As a science it has for its subject the construction and development of all systems of expressing numbers, or parts of numbers, the explanation of all the consequences that follow from the adoption of these systems, and the investigation of the mutual relations of the numbers themselves. By means of Calculation, which is usually considered as a part of Arithmetic, these principles are applied to practical purposes. Hence Arithmetic includes both a science and an art. The first part of the science is the construction of systems of expressing numbers. There have been many such systems, which may be divided into two main classes, each containing several varieties. It is not the object of the present work to consider any other of these than that which is generally used in the world now. This system belongs to a class where the principle of 'local value' is recognised in contradistinction to systems such as the Greek and Roman, where it is not found; and the particular variety of this class is determined by the fact of the number ten being chosen as the scale upon which the local value increases or diminishes. For the purposes of expression we have ten figures or digits, namely O signifying nothing, and called nought, a cipher, or zero, and 1, 2, 3, 4, 5, 6, 7, 8, 9, expressing the numbers one, a unit or unity, two, three, four, five, six, seven, eight, nine. B The system will be best explained by showing its application to a particular case. Let there be a large heap of corn, and let it be required to count the number of grains. If ten grains be counted and put aside by themselves, and afterwards ten more, and then ten more, and so on as far as possible; then the whole heap will have been separated into a large number of smaller heaps, each containing ten grains, and probably one containing some number less than ten, say seven, besides. Now the number of these heaps of ten, though large, is yet not nearly so large as the original number of grains, so that by this separation a great reduction has been effected in the number with which we have to do. The same process may evidently be repeated. Ten of the heaps of ten may be counted and put aside, and then ten more, and ten more, and so on until as the result we have a number of heaps each containing ten tens, and one containing some number less than ten, say five, of tens, besides. For ten tens we use the word hundred, and therefore the original heap has thus been resolved into some number of hundreds, five tens, and seven. Taking the hundreds, and continuing the process, we should have some tens of hundreds, or thousands, and a number less than ten, say three, of hundreds, besides. Afterwards, still proceeding in the same manner, suppose that we have a number of tens of thousands, and four thousands besides; then a number of hundreds of thousands and six tens of thousands besides; then a number of thousands of thousands, or millions, and eight hundreds of thousands besides. Lastly, suppose that we cannot proceed further in this way, the number of millions being less than ten, suppose seven. Then the whole number of grains in the heap would be expressed as seven millions, eight hundreds of thousands, six tens of thousands, four thousands, three hundreds, five tens, and seven. Let this be written with figures instead of words, and the names of the different heaps be placed above the corresponding figures. Then the number will be |