computers had to solve simultaneous equations involving seventy-seven unknown quantities. The reduction of the levellings again required the solution of a system of ninety-one equations. But these vast calculations present no approach whatever to what would be requisite for the complete treatment of any one physical problem. The motion of glaciers is supposed to be moderately well understood in the present day. A glacier is a viscid, slowly yielding mass, neither absolutely solid nor absolutely rigid, but it is expressly remarked by Forbes", that not even an approximate solution of the mathematical conditions of such a moving mass can yet be possible. Every one knows,' he says, 'that such problems are beyond the compass of exact mathematics;' but though mathematicians may know this, they do not often enough impress that knowledge on other people.

The problems which are solved in our mathematical books consist of a small selection of those which happen from peculiar conditions to be practicable. But the very simplest problem in appearance will often give rise to impracticable calculations. Mr. Todhunter seems to blame Condorcet, because in one of his memoirs he mentions a problem to solve which would require

n + n' + n" + n"" -2

successive integrations. Now if our mathematical sciences are to pretend to cope with the problems which await solution, we must be prepared to effect an unlimited number of successive integrations; yet at present, and almost beyond doubt for ever, the probability that even a single integration, taken haphazard, will be found to come within our powers is exceedingly small.

In some passages of that most remarkable work, the

n Philosophical Magazine,' 3rd Series, vol. xxvi. p. 406.

Ninth Bridgwater Treatise,' Mr. Babbage has pointed out that if we had power to follow and detect the minutest effects of any disturbance, each particle of existing matter must be a register of all that has happened. The track of every canoe-of every vessel that has yet disturbed the surface of the ocean, whether impelled by manual force or elemental power, remains for ever registered in the future movement of all succeeding particles which may occupy its place. The furrow which it left is, indeed, instantly filled up by the closing waters; but they draw after them other and larger portions of the surrounding element, and these again, once moved, communicate motion to others in endless succession.' We may even say that The air itself is one vast library, on whose pages are for ever written all that man has ever said or even whispered. There, in their mutable but unerring characters, mixed with the earliest, as well as the latest sighs of mortality, stand for ever recorded, vows unredeemed, promises unfulfilled, perpetuating in the united movements of each particle, the testimony of man's changeful will.'

When we read truthful reflections such as these, we may congratulate ourselves that we have been endowed with minds which, rightly employed, can form some estimate of their incapacity, to trace out and account for all that proceeds in the simpler actions of material nature. It ought to be added that, wonderful as is the extent of physical phenomena open to our investigation, intellectual phenomena are yet vastly more extensive. this I might present one satisfactory proof were space available by pointing out that the mathematical functions. employed in the calculations of physical science, form an infinitely small fraction of the functions which may be

p Ninth Bridgwater Treatise,' p. 115.

q Ibid. p. 113.

Of

invented. Common trigonometry, for instance, consists of a great series of useful formulas, all of which arise out of the simple fundamental relation of the sine and cosine expressed in the one equation

sin 2x + cos2 = 1.

But this is not the only trigonometry which may exist; mathematicians also recognise the so-called hyperbolic trigonometry of which the fundamental equation is

cosxsin 2x = I.

De Morgan has pointed out that the symbols of ordinary algebra form but three of an interminable series of conceivable systems. As the logarithmic operation is to addition or addition to multiplication, so is the latter to a higher operation, and so on without limit.

We may rely upon it that indefinite, and to us inconceivable, advances will be made by the human intellect, in the absence of any unforeseen catastrophe to the species or the globe. Almost within historical periods we can trace the rise of mathematical science from its simplest germs. We can prove our descent from ancestors who counted only on their fingers, but how almost infinitely is a Newton or a Laplace above those simple savages. Pythagoras is said to have sacrificed a hecatomb when he discovered the Forty-seventh Proposition of Euclid, and the occasion was worthy of the sacrifice. Archimedes was beside himself when he first perceived his beautiful mode of determining specific gravities. Yet these great discoveries are the simplest elements of our schoolboy-knowledge. Step by step we can trace upwards the acquirement of new mental powers. What could be more wonderful and unexpected than Napier's discovery of logarithms, a wholly new mode of calculation which has multiplied perhaps a hundred-fold the working powers of every computer, and indeed has rendered easy calculations which 'Trigonometry and Double Algebra,' chap. IX.

8

were before almost impracticable. Since the time of Newton and Leibnitz whole worlds of problems have been solved which before were hardly conceived as matters of inquiry. In our own day extended methods of mathematical reasoning, such as the system of quaternions, have been brought into existence. What intelligent man will doubt that the recondite speculations of a Cayley or a Sylvester may possibly lead to some new methods, at the simplicity and power of which a future age will wonder, and yet wonder more that to us they were so dark and difficult. May we not repeat the words of Seneca: 'Veniet tempus, quo ista quæ nunc latent, in lucem dies extrahat, et longioris ævi diligentia ad inquisitionem tantorum ætas una non sufficit. Veniet tempus, quo posteri nostri tam aperta nos nescisse mirentur.'

The Reign of Law in Mental and Social Phenomena.

After we pass from the so-called physical sciences to those which attempt to investigate mental and social phenomena, the same general conclusions will hold true. No one will be found to deny that there are certain uniformities of thinking and acting which can be detected in reasoning beings, and so far as we detect such laws we successfully apply scientific method. But those who attempt thus to establish social or moral sciences, soon become aware that they are dealing with subjects of enormous perplexity. Take, for instance, the science of Political Economy. If a science at all, it must be a mathematical science, because it deals with quantities of commodities. But so soon as we attempt to draw out the equations expressing the laws of variation of demand and supply, we discover that they must have a complexity entirely surpassing our powers of mathematical treatment.

We may lay down the general form of the equations, expressing the demand and supply for two or three commodities among two or three trading bodies, but all the functions involved are of so complicated a character that there is not much fear of scientific method making rapid progress in this direction. If such be the prospects of a comparatively formal science, like Political Economy, what shall we say of Moral Science? Any complete theory of morals must deal with quantities of pleasure and pain, as Bentham pointed out, and must sum up the general tendency of each kind of action upon the good of the community. If we are to apply scientific method to morals, we must have a calculus of moral effects, a kind of physical astronomy investigating the mutual perturbations of individuals. But as astronomers have not yet fully solved the problem of three gravitating bodies, when shall we have a solution of the problem of three moral bodies?

Now the sciences of political economy and morality are, comparatively, abstract and general, treating mankind from simple points of view, and attempting to detect general grounds of action. They are to social phenomena what the general sciences of chemistry, heat, and electricity, are to the concrete science of meteorology. Before we can investigate the actions of any aggregate of men, we must have fairly mastered all the more abstract sciences applying to them, somewhat in the way that we have acquired a fair comprehension of the simpler truths of chemistry and physics. But all our physical sciences do not enable us to predict the weather two days hence with any great probability, and the general problem of meteorology is almost unattempted as yet. What shall we say then of the general problem of social science, which shall enable us to predict the course of events in a nation? There have indeed been several writers who have pro