8 or proportion, between one thing and another, is therefore the

most valuable of all operations; the only means which we have
of understanding it fully is the study of mathematics, and did
that study lead to nothing else, it would be worthy of all the
attention we can bestow upon it.

But we have not only to examine and understand quantities and the relations of quantities which can or which do exist; we have to consider those that have no existence, and even those of which the existence is utterly impossible. Every plan or scheme which we form is a quantity which does not exist until we have put it in execution; and, even with every desire and every effort on our part to carry those plans into execution, they very frequently fail because they involve impossible quantities of which we were not aware when we formed them. In the common business of life, where we have not all the elements under our controul-within ourselves as it were, but must be controlled by other people and by the general events of the world, which never give us full warning of their coming, we cannot in the nature of things avoid all these impossible elements; but still it is of the utmost advantage to be "in the way" of doing it, and there is nothing which puts us so much in this way as a mathematical habit that of estimating the value of every circumstance and every probability in terms of some known standard.

Even when we ourselves have or should have perfect controul over all the elements which enter into our scheme, there is often some lurking impossible quantity which insinuates itself into the chain, and mars the purpose of the whole; and we may be sure that when any scheme fails, without negligence on our part or prevention from any external cause, there has been an impossible element in that scheme at its formation. Of the vast number of inventions and projects which are every day brought before

1

X 24

FAILURE OF SCHEMES.

the public, not as mere bubbles or impostures, but with perfect honesty and zeal on the part of the projectors, we speak with most charitable liberality when we say that not one in the hundred proves to be of any use, and nine out of every ten are altogether impracticable. The reason clearly is, that neither the projectors, nor those by whom they are encouraged, are able to see the impossible elements which their schemes involve; that they look at the possible and promising ones only; and thus a large quantity of well-meant labour and ingenuity is constantly wasted.

It may be worth while to mention one case by way of illustration; but before we do this, we may mention that every scheme, process in reasoning, and other plan or project, whatever it may be, is like a machine, no stronger than its weakest part, or like a chain, of which if one link is broken the whole is broken. This consideration is as universal as it is important, and it is the want of attention to this which causes so many sad failures after long and arduous labour with every prospect of success. If the study of mathematics (we speak of the mode of mathematical reasoning, and not of any one branch or application of the science) had no other value than that of enabling us to detect the one cause of failure amid the thousand prospects and promises of success, the time and labour bestowed upon the study would be amply repaid; but this, though a great and perhaps the greatest advantage, is an indirect one, and accompanies the others without our pursuing it as one of our specific purposes, at least if we go to the general principles of the science, and do not confine ourselves to the mere details and mechanical operations.

Now for our case in illustration:----Perhaps we cannot select a better one than that of the "Perpetual Motion," that is, a selfmoving machine which shall not involve any cause of stoppage save the wearing out of the materials of which it is composed. We believe that the fonder votaries of this visionary project do not take even the wearing out of the materials into the account; but it is necessary to do this; and even this necessity, when analysed, involves the necessity of the machine stopping before the parts are worn out.

It may be useful to those who are not acquainted with the method of analysis, or of separating the parts of compound subjects and estimating their values singly, and not taking them simply in the mass, when the favourable ones are sure to hide the unfavourable, to point out by how little either of thought or of trouble we arrive at the truth of this case. Every thing on or near the surface of the earth gravitates towards the earth's centre, and this gravitation, which depends on the mass or quantity of matter of the earth, is a power acting constantly and uniformly, so that the tendency of it is to bring every piece and combination of pieces of matter to a state of rest. Thus if an animal walks or a wheel rolls along, this gravitation is continually attempting to stop it, in the proportion of the weight of the animal or the wheel. It is this gravitation which causes animals to make hoof-prints and wheels to make ruts in soft ground; and though we make the surfaces ever so hard or so smooth, though we could get rid of the mere friction or rubbing as not arising from weight, yet the weight is not thereby lessened, that is, the gravitation toward the earth is not one jot the less. Thus in the case of a rail-road, the same power can pull far more upon a level than where the surface is rough or soft, or both; and if there is even a very small declivity, the weight alone will bring down the load without any pull, and bring it down the faster the smoother that both the rails and the wheels are. But when we come to an ascent, even a very trifling one, the disadvantage is just as great, and one horse would pull

26

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 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