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IMPOSSIBLE QUANTITIES.

more up the natural slope of a hill, than a steam engine of a thousand horse' power could do upon wheels and rails up the

same.

The power which tends to stop the motion of all machines upon the earth's surface is, then, a power which acts constantly and uniformly, never pausing an instant, nor abating a jot; and therefore, in order to get the better of this gravitation, we must have a counteracting power as continually new as itself; and we are not acquainted with any such power, or any kind of matter in which such a power could reside. It is not difficult to calculate (upon mathematical principles), that if we could give any piece of matter a motion round the earth at the rate of about five miles in a second, or one thousand eight hundred miles in an hour, and keep up the motion at this rate, we should overcome the gravitation of that piece of matter. This is what may be regarded as the possible case of the perpetual motion; in this case, the piece of matter must move round the earth, and in no other direction, and it must move unconnected with anything else; and, taking all these circumstances into the account, it will be admitted that the accomplishment is hopeless, and would be useless if it were not.

In the case of a fixed machine,—and the more complicated that the machine is, it is the less likely to succeed,—the impossible element, in the most simple view we can take of it, is this:-to find a piece of matter which, of itself, shall be alternately greater and less than itself, and which shall also remain equal to itself all the time; and if this is not an impossibility, it is not easy to see where impossibility is to be found.

The knowledge of impossible or absurd quantities, and the method of readily discovering them, are often of great use to us, not only in preventing us from wasting our time in attempting to do that which cannot in the nature of things be done, but in

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enabling us to prove or demonstrate truth in cases where that cannot be done directly; for it is easy to see, that if an impossibility or an absurdity would be the necessary consequence of anything else than one particular state of things, then that particular state is the true one. This method of proof is, of course, not so simple as the direct method, but it is often not less convincing; and we shall see afterwards that, in many cases, it is the only species of proof which we can obtain.

The use of Mathematics, as a general exercise for the mind, and a general guide to the art of thinking correctly, may be in part seen from what has been stated in this section; and the more direct and immediate uses of the different parts can be better explained when we notice those parts themselves; therefore we shall close this section with the names and very short definitions of the principal branches into which mathematical science is divided. Of these, in the very simplest view of the matter, there are three :

First, ALGEBRA, or the science of quantity in its most general sense, applying equally to every quantity, whatever may be its nature, and whether possible or impossible; and also to all relations of one quantity to another; and being, on this account, the proper foundation of the whole.

Secondly, GEOMETRY, or the science of extended quantity or magnitude; that is, quantity considered as existing in and occupying space. Geometry is thus a particular branch of that general science which Algebra comprises; and though, so far as Geometry extends, both it and Algebra may be applied to the very same quantities, yet geometrical quantities are always such, that we can imagine them to exist and be visible, which is not the case with all quantities to which Algebra applies. It very often happens, however, that the very same mode of reasoning applies to quantities which have a geometrical form

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DIVISIONS OF MATHEMATICS.

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