In some passages of that remarkable work, the Ninth Bridgewater Treatise (pp. 113–115), Babbage has pointed out that if we had power to follow and detect the minutest effects of any disturbance, each particle of existing matter would furnish 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 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. Of 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 might be invented. Common trigonometry consists of a great series of useful formulæ, all of which arise out of the relation of the sine and cosine expressed in one equation, sin 2x + cos 2x = 1. But this is not the only trigonometry which may exist; mathematicians also recognise hyperbolic trigonometry, of which the fundamental equation is cos 2x - sin 2x = 1. De Morgan has pointed out that the symbols of ordinary algebra form but three of an interminable series of conceivable systems.17 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 immense, and to us inconceivable, advances will be made by the human intellect, in the absence of any catastrophe to the species or the globe. 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. How 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 commonplaces of our school books. Step by step we can trace upwards the acquirement of new mental powers. What could be more wonderful than Napier’s discovery of logarithms, a new mode of calculation which has multiplied perhaps a hundredfold the working powers of every computer, and has rendered easy calculations which were before impracticable? Since the time of Newton and Leibnitz 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, a Sylvester, or a Clifford may lead to some new development of new mathematical power, at the simplicity 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.”