this, that to every one in the first group there corresponds one and only one in the second group. When it is not possible to compare the two groups together directly, we can compare them indirectly by means of a third group. For it is clear from the definition that two groups will contain the same number of objects when each contains the same number as a third group. Those objects which in the former groups correspond with the same object in the latter group will also correspond with one another. It is therefore possible to adopt a standard set of things once for all and to use this set constantly for the purpose of comparison. The groups we form from the standard set are called specially numbers, and when we are asked the number of objects in any other group it is meant that we are to give that one of these special groups which contains the same number of objects.

We must next see what are the fundamental relations between different numbers. The first of these relations arises from Addition. If we join two groups of objects to form a third group the number of objects in the whole group is said to be the sum of the numbers of objects in each of the partial groups, and the formation of this sum is called addition. It is clear that the sum of two numbers is independent of the special objects to which the numbers apply. In other words, if we replace each of the partial groups by another group containing the same number of different objects then the whole number of the new objects will be equal to the whole number of the former. For those objects which corresponded to one another in the partial groups may still be considered

to correspond when these groups are joined together. The other fundamental relations between numbers arise from Subtraction, Multiplication, and Division. The definitions of these operations will be given hereafter. The relations arising from them are also independent of special objects, and hence an abstract science of arithmetic dealing with these relations is possible.

Arithmetical theorems may be of two kinds, special or general. They may be either the results of the addition, subtraction, &c. of particular numbers, such as the theorem that 3 and 5 are 8, or they may be general truths relating to numbers, such as the theorem that the product of two numbers is unaltered when the multiplier and multiplicand are interchanged. The resulting division of arithmetic into two parts was made long ago by the Greeks. The former part they called Logistics and reserved the name 66 Arithmetic" for the latter part. We shall begin with the study of the former part, since the more special questions are easier than the more general.

We must first of all shew how to find names for numbers. All that is essential for this purpose is a series of words such as one, two, three, four, or visible signs such as 1, 2, 3, 4 following one another in a definite order. When we wish to give a name to the number of objects in a group we repeat the words one, two, three (or write the signs 1, 2, 3) in their order, saying one and only one for each of the objects in the group. The last word we say will be the required name. This is the ordinary process of counting, and we are really comparing the number of objects with the number of words in the series one, two,

three. We take these words in fact as our standard set of things. The names for numbers are not all arbitrary. They are formed on a systematic plan, and the systematic formation of numerical words is called Numeration. This will be explained in the first section.

Any combination of numbers by means of Addition, Subtraction, Multiplication or Division might be taken as a name for a new number, and we find as a matter of fact that all these operations have been employed in this way. Thus in Latin two from twenty, in French four twenties, in English half a dozen are names for numbers. It is purely the result of a convention that we use a hundred and five and do not use eight and six as a numerical word. But when once a standard form has been adopted we must reduce all other modes of expression to that form. The method by which this is done is shewn in turn for Addition, Subtraction, Multiplication and Division, in the following sections.

We then come to Arithmetic in the Greek sense of the word, and consider some general theorems relative to whole numbers.

We shall then consider numbers in their application to continuous quantities, and shall examine in succession the two cases of quantities commensurable and incommensurable with the unit of measure. We must explain the notation employed for this class of numbers, and the application of the four fundamental operations.

In the last chapter we shall again return to the properties of whole numbers.

CHAPTER I.

NUMERATION.

We have seen already that to provide names for numbers all that is necessary is a series of words or symbols following one another in a definite order. If however these words or symbols were all arbitrary it would soon become impossible to remember them. The problem then is to express all numbers on a systematic plan by means of a few numbers to which arbitrary names are assigned. But we can only express one number in terms of another by means of the operations of Addition, Subtraction, Multiplication and Division. We must consider then which of these four is the most suitable for the systematic formation of numbers. Subtraction and Division may be put aside at once, since by their means we could only express lower numbers in terms of higher, and this would be open to two objections: Ist, since the smaller numbers are the best known to us we should be expressing the more familiar in terms of the less familiar; 2nd, we could not express higher numbers than the highest to which an arbitrary name had been assigned.

Neither could we express all our numbers by multiplication alone, for then we should want an indefinite

series of arbitrary names for those numbers which are not the product of any other numbers, such for example as twenty-three, seventy-one, &c.

These are called prime numbers and will be spoken of later.

It is clear then that we must use addition as a mode of formation of numbers. In fact every number may be considered the sum of smaller numbers and these as the sums of others smaller still, till we arrive at numbers as small as we can imagine.

So that if we give names to the first few numbershow few does not matter—all others could be expressed by means of addition alone. The names obtained in this way would often be very long, but they can be shortened by the use of multiplication, for the numbers added together may be made for the most part equal, and the addition of equal numbers is multiplication. For example, supposing we wished to express twenty-three in a language which had no arbitrary words beyond six, we could say either Ist, one and two and three and four and five and six and two; or 2nd, four and four and four and four and four and three; or 3rd, six and six and six and five; and the latter two modes could be shortened into five fours and three in the one case and into three sixes and five in the other. Again, the names could be shortened by giving arbitrary signs to some high numbers, just as in weighing a heavy body the weights required will be fewer if some of them are great.

We see then that addition alone is essential for naming numbers, but that multiplication may also be employed.