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and existence, and to those which have not. Thus, for instance, the globe of the earth, considered as a piece of matter of a certain form and magnitude, is not only a geometrical quantity, but the very name, Geometry, means measuring the earth" (it originally meant what we now call land-measuring); but the attraction of gravitation, by means of which bodies fall to the earth, and are retained on its surface, is not in itself a geometrical quantity, because we cannot say that it has either size or shape, and yet the law according to which it acts is a geometrical law. Thus all geometrical quantities must be such as that we can imagine them to exist in space; but it is not necessary that they should actually fill any portion of that space. Thus, the surface of the table is a geometrical quantity, and so is the length or the breadth of the table; and these quantities are so related, that we can find the extent of the surface if we know the length and the breadth. But none of these quantities occupies any space, for the surface of the table merely separates the table from the air over it, and the length and breadth are mere expressions for how far it extends in two directions across each other.

Thirdly, ARITHMETIC, or the science of quantities expressed in numbers, either exactly or as nearly so as may be possible. This is the practical application of both Algebra and Geometry; and while those sciences express quantities in a general manner, and in such a way as that any conclusion at which we arrive concerning them, is equally applicable to all quantities of the same kind, Arithmetic takes with it the particular values of quantities; and thus arithmetical conclusions have not that general character which belongs to Algebra and Geometry.

Each of those great branches of mathematical science admits of many subdivisions, according to the nature of the quantities, and the relations in which they are viewed; and it may be said,

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generally, that the grand object of Algebra and of Geometry, besides their great use in teaching the art of accurate thinking, is the preparation of all subjects of which the values can be expressed in numbers, in such a manner as that we can apply Arithmetic to them, and thus ascertain their real values in terms of that known standard by which we are accustomed to measure the kind of quantities to which they belong.

In a civilised country, there is nobody so humble or so illiterate as not to have occasion for a little arithmetic; that is, to be able to express the values of a few quantities in terms of some standard, and therefore a little of the practice of Arithmetic forms a necessary part of every body's education, whether it is acquired at school, or picked up by ourselves in the same way as we learn to speak, and whether it is or is not accompanied by the capacity of reading and writing. Such arithmeticians do not, however, understand any of the principles of that science of which they can thus make a little use; neither are they aware of the advantages which they derive from the science, even in their humble way. It is a fact, however, that the inhabitants of countries in which there never has been any science, or any scientific men, find counting, even to a very limited amount, an operation altogether beyond their power. It is generally said, that many tribes of the North American Indians, when they were first known to Europeans, were quite incapable of counting beyond the number three; and yet it is admitted that these tribes were exceedingly shrewd people, and much more dexterous in the use of their senses than the peasantry of civilised countries. Indeed, even if we take those beginnings which are obtained in our own schools, and in consequence of which the possessor is considered qualified for being a countinghouse calculator, we should find it to be exceedingly difficult to arrive, by means of them, at the establishment of any one arith

metical truth, to say nothing of truths of a more general nature; and therefore, in order to understand the principles, we must make another and a more general beginning.

But, in order to do this properly, it is necessary that we should understand the simpler operations of Arithmetic-the way of expressing quantities arithmetically, and of performing on them those few general changes of which Arithmetic admits. This is necessary, for the very same reason that it is necessary to learn the alphabet, the spelling, and the words of a language, before we begin to study the grammar of that language, so as to understand its structure, its power, its beauty, and its deficiencies, and make ourselves master of its spirit and its extent, so as to express what we wish to say or write in the clearest, most forcible, and most impressive manner; and perhaps it is as desirable that we should not attempt to mix up any of the principles with the learning of this first and simplest alphabet of Mathematics, as it is to avoid confounding the infant which is drudging at its Christ-cross row, with lectures about adverbs and pronouns. We shall assume that the least informed reader whose attention is drawn to this volume, is in possession of this arithmetical alphabet, and of a good deal more, and consequently we shall pass very lightly over this part of the subject.

SECTION III.

ARITHMETICAL NOTATION, AND SCALE AND DISTINCTIONS OF NUMBERS.

LITTLE as we are accustomed to think of our common arithmetical notation, and lightly as we esteem the value of that classification of numbers which it represents, it is really, (second

only to the alphabet of language, and second to that only because it is more confined in its application,) among the most wonderful contrivances of human ingenuity. Let us take an instance, and consider in what situation we should have been placed, if we had been without our arithmetical scale of arrangement, and our corresponding method of notation, or expressing numbers by a limited number of characters applicable to that purpose, and that purpose only. From what has been said of the state of the American Indians, it is not at all probable that we could have had any means of arriving, not at the knowledge only, but even at the name of any such number as we are to instance; but, for the sake of the argument, let us suppose the thing possible. Well, the average distance of the sun from the earth, expressed in words according to our scale of numbers, is, ninety-five millions of miles; and the same in the notation of Arithmetic is, 95,000,000. Both of these expressions are very short; but if we had had no system, and so had been obliged to express this distance by a repetition of a separate name for every individual number, from 1 to 95,000,000, these names alone would have filled nearly four hundred volumes of about the same size and style of printing as the present one; and therefore, to have made any use of the number, or even to have formed any guess respecting its nature or amount, would have been wholly out of the question.

It is worthy of remark, as a proof of the value of Mathematics, even in the very alphabet of Arithmetic, that we are enabled to get the better of the difficulty by means of what we may strictly call a geometrical principle. Thus, in the expression

111111111,

each of the characters has a different value; this value is smallest in the character nearest our right hand, and it increases at a

regular series of ten times in all the others.

The second from the right is ten times the first, the third ten times the second, the fourth ten times the third, and so of the others, whatever may be their number; for as there is nothing to limit it except space to write the characters on, we may make it as extended as we please. This character, while it preserves the same form, and always means one of something, may thus have an endless variety of meanings, according to the characters that are on the right of it; and it is evident that it is the number of those characters only, and not their particular values considered in themselves, upon which this second element of the value of the character depends. Thus, if we take a character which in itself stands for no number, and thus marks place in the expression, but not value, we shall be enabled to express any one of the above characters singly, with exactly the same value as it has in the combination. What character we use for this purpose is of no consequence, provided it be one to which we never attach value as a number; and thus 0 has been used, probably because it is a character which can be very easily written. By means of 0, we can express the above nine repetitions of the character 1, by the following nine expressions, all of which taken together would have exactly the same value as the expression from which they were all derived. Thus :

100000000, 10000000, 1000000, 100000, 10000, 1000, 100, 10, 1.

This is what we may term the system or series of the scale of numbers, and the root, or ratio; that is, the relation of any one term, or place, to the one next it, is ten. But we must understand that it is not the number ten counted in individual ones, as we count when we wish to know "how many;" it is ten times greater, if the place is next on the left hand, and ten times less if it is next on the right.