(20.) To do this, however, as we shall presently see, requires in many cases a degree of knowledge of mathematics and geometry altogether unattainable by the generality of mankind, who have not the leisure, even if they all had the capacity, to enter into such enquiries, some of which are indeed of that degree of difficulty that they can be only successfully prosecuted by persons who devote to them their whole attention, and make them the serious business of their lives. But there is scarcely any person of good ordinary understanding, however little exercised in abstract enquiries, who may not be readily made to comprehend at least the general train of reasoning by which any of the great truths of physics are deduced, and the essential bearings and connections of the several parts of natural philosophy. There are whole branches too and very extensive and important ones, to which mathematical reasoning has never been at all applied; such as chemistry, geology, and natural history in general, and many others, in which it plays a very subordinate part, and of which the essential principles, and the grounds of application to useful purposes, may be perfectly well understood by a student who possesses no more mathematical knowledge than the rules of arithmetic; so that no one need be deterred from the acquisition of knowledge, or even from active original research in such subjects, by a want of mathematical information. Even in those branches which, like astronomy, optics, and dynamics, are almost exclusively under the dominion of mathematics, and in which no effectual progress can be made without some acquaintance with geometry, the principal results may be perfectly understood without it. To one incapable of following out the intricacies of mathematical demonstration, the conviction afforded by verified predictions must stand in the place of that purer and more satisfactory reliance which a verification of every step in the process of reasoning can alone afford, since every one will acknowledge the validity of pretensions which he is in the daily habit of seeing brought to the test of practice.
(21.) Among the verifications of this practical kind which abound in every department of physics, there are none more imposing than the precise prediction of the greater phenomena of astronomy; none, certainly, which carry a broader conviction home to every mind from their notoriety and unequivocal character. The prediction of eclipses has accordingly from the earliest ages excited the admiration of mankind, and been one grand instrument by which their allegiance (so to speak) to natural science, and their respect for its professors, has been maintained; and though strangely abused in unenlightened ages by the supernatural pretensions of astrologers, the credence given even to their absurdities shows the force of this kind of evidence on men’s minds. The predictions of astronomers are, however, now far too familiar to endanger the just equipoise of our judgment, since even the return of comets, true to their paths and exact to the hour of their appointment, has ceased to amaze, though it must ever delight all who have souls capable of being penetrated by such beautiful instances of accordance between theory and facts. But the age of mere wonder in such things is past, and men prefer being guided and enlightened, to being astonished and dazzled. Eclipses, comets, and the like, afford but rare and transient displays of the powers of calculation, and of the certainty of the principles on which it is grounded. A page of “lunar distances” from the Nautical Almanack is worth all the eclipses that have ever happened for inspiring this necessary confidence in the conclusions of science. That a man, by merely measuring the moon’s apparent distance from a star with a little portable instrument held in his hand, and applied to his eye, even with so unstable a footing as the deck of a ship, shall say positively, within five miles, where he is, on a boundless ocean, cannot but appear to persons ignorant of physical astronomy an approach to the miraculous. Yet, the alternatives of life and death, wealth and ruin, are daily and hourly staked with perfect confidence on these marvellous computations, which might almost seem to have been devised on purpose to show how closely the extremes of speculative refinement and practical utility can be brought to approximate. We have before us an anecdote communicated to us by a naval officer[8], distinguished for the extent and variety of his attainments, which shows how impressive such results may become in practice. He sailed from San Blas on the west coast of Mexico, and after a voyage of 8000 miles, occupying 89 days, arrived off Rio de Janeiro, having, in this interval, passed through the Pacific Ocean, rounded Cape Horn, and crossed the South Atlantic, without making any land, or even seeing a single sail, with the exception of an American whaler off Cape Horn. Arrived within a week’s sail of Rio, he set seriously about determining, by lunar observations, the precise line of the ship’s course and its situation in it at a determinate moment, and having ascertained this within from five to ten miles, ran the rest of the way by those more ready and compendious methods, known to navigators, which can be safely employed for short trips between one known point and another, but which cannot be trusted in long voyages, where the moon is the only sure guide. The rest of the tale we are enabled by his kindness to state in his own words:—“We steered towards Rio de Janeiro for some days after taking the lunars above described, and having arrived within fifteen or twenty miles of the coast, I hove to at four in the morning till the day should break, and then bore up; for although it was very hazy, we could see before us a couple of miles or so. About eight o’clock it became so foggy that I did not like to stand in farther, and was just bringing the ship to the wind again before sending the people to breakfast, when it suddenly cleared off, and I had the satisfaction of seeing the great Sugar Loaf Rock, which stands on one side of the harbour’s mouth, so nearly right ahead that we had not to alter our course above a point in order to hit the entrance of Rio. This was the first land we had seen for three months, after crossing so many seas and being set backwards and forwards by innumerable currents and foul winds.” The effect on all on board might well be conceived to have been electric; and it is needless to remark how essentially the authority of a commanding officer over his crew may be strengthened by the occurrence of such incidents, indicative of a degree of knowledge and consequent power beyond their reach.
(22.) But even such results as these, striking as they are, yet fall short of the force with which conviction is urged upon us when, through the medium of reasoning too abstract for common apprehension, we arrive at conclusions which outrun experience, and describe beforehand what will happen under new combinations, or even correct imperfect experiments, and lead us to a knowledge of facts contrary to received analogies drawn from an experience wrongly interpreted or overhastily generalised. To give an example:—every body knows that objects viewed through a transparent medium, such as water or glass, appear distorted or displaced. Thus, a stick in water appears bent, and an object seen through a prism or wedge of glass seems to be thrown aside from its true place. This effect is owing to what is called the refraction of light; and a simple rule discovered by Willebrod Snell enables any one to say exactly how much the stick will be bent, and how far, and in what direction, the apparent situation of an object seen through the glass will deviate from the real one. If a shilling be laid at the bottom of a basin of water and viewed obliquely, it will appear to be raised by the water; if instead of water spirits of wine be used it will appear more raised; if oil, still more:—but in none of these cases will it appear to be thrown aside to the right or left of its true place, however the eye be situated. The plane, in which are contained the eye, the object, and the point in the surface of the liquid at which the object is seen, is an upright or vertical plane; and this is one of the principal characters in the ordinary refraction of light, viz. that the ray by which we see an object through a refracting surface, although it undergoes a bending, and is, as it were, broken at the surface, yet, in pursuing its course to the eye, does not quit a plane perpendicular to the refracting surface. But there are again other substances, such as rock-crystal, and especially Iceland spar, which possess the singular property of doubling the image or appearance of an object seen through them in certain directions; so that instead of seeing one object we see two, side by side, when such a crystal or spar is interposed between the object and the eye; and if a ray or small sunbeam be thrown upon a surface of either of these substances, it will be split into two, making an angle with each other, and each pursuing its own separate course,—this is called double refraction. Now, of these images or doubly refracted rays, one always follows the same rule as if the substance were glass or water: its deviation can be correctly calculated by Snell’s law above mentioned, and it does not quit the plane perpendicular to the refracting surface. The other ray, on the contrary, (which is therefore said to have undergone extraordinary refraction) does quit that plane, and the amount of its deviation from its former course requires for its determination a much more complicated rule, which cannot be understood or even stated without a pretty intimate knowledge of geometry. Now, rock-crystal and Iceland spar differ from glass in a very remarkable circumstance. They affect naturally certain regular figures, not being found in shapeless lumps, but in determinate geometrical forms; and they are susceptible of being cleft or split much easier in certain directions than in others—they have a grain which glass has not. When other substances having this peculiarity (and which are called crystallized substances) were examined, they were all, or by far the greater part, found to possess this singular property of double refraction; and it was very natural to conclude, therefore, that the same thing took place in all of them, viz. that of the two rays, into which any beam of light falling on the surface of such a substance was split, or of the two images of an object seen through it, one only was turned aside out of its plane and extraordinarily refracted, while the other followed the ordinary rule. Accordingly this was supposed to be the case; and not only so, but from some trials and measurements purposely made by a philosopher of great eminence, it was considered to be a fact sufficiently established by experiment.
(23.) Perhaps we might have remained long under this impression, for the measurements are delicate, and the subject very difficult. But it has lately been demonstrated by an eminent French philosopher and mathematician, M. Fresnel, that, granting certain principles or postulates, all the phenomena of double refraction, including perhaps the greatest variety of facts that have ever yet been arranged under one general head, may be satisfactorily explained and deduced from them by strict mathematical calculation; and that, when applied to the cases first mentioned, these principles give a satisfactory account of the want of the extraordinary image; that when applied to such cases as those of rock-crystal or Iceland spar, they also give a correct account of both the images, and agree in their conclusions with the rules before ascertained for them: but so far from coinciding with that part of the previous statement, which would make these conclusions extend to all crystallised substances, M. Fresnel’s principles lead to a conclusion quite opposite, and point to a fact which had never been observed, viz. that in by far the greater number of crystallized substances which possess the property of double refraction, neither of the images follows the ordinary law, but both undergo a deviation from their original plane. Now this had never been observed to be the case in any previous trial, and all opinion was against it. But when put to the test of experiment in a great variety of new and ingenious methods, it was found to be fully verified; and to complete the evidence, the substances on whose imperfect examination the first erroneous conclusion was founded, having been lately subjected to a fresh and more scrupulous examination, the result has shown the insufficiency of the former measurements, and proved in perfect accordance with the newly discovered laws. Now it will be observed in this case, first, that, so far from the principles assumed by M. Fresnel being at all obvious, they are extremely remote from ordinary observation; and, secondly, that the chain of reasoning by which they are brought to the test is one of such length and complexity, and the purely mathematical difficulty of their application so great, that no mere good common sense, no general tact or ordinary practical reasoning, would afford the slightest chance of threading their mazes. Cases like this are the triumph of theories. They show at once how large a part pure reason has to perform in our examination of nature, and how implicit our reliance ought to be on that powerful and methodical system of rules and processes which constitute the modern mathematical analysis, in all the more difficult applications of exact calculation to her phenomena.
(24.) To take an instance more within ordinary apprehension. An eminent living geometer had proved by calculations, founded on strict optical principles, that in the centre of the shadow of a small circular plate of metal, exposed in a dark room to a beam of light emanating from a very small brilliant point, there ought to be no darkness,—in fact, no shadow at that place; but, on the contrary, a degree of illumination precisely as bright as if the metal plate were away. Strange and even impossible as this conclusion may seem, it has been put to the trial, and found perfectly correct.[9]
(25.) We shall now proceed to consider more particularly, and in detail,—
I. The nature and objects immediate and collateral of physical science, as regarded in itself, and in its application to the practical purposes of life, and its influence on the well-being and progress of society.
II. The principles on which it relies for its successful prosecution, and the rules by which a systematic examination of nature should be conducted, with examples illustrative of their influence.
III. The subdivision of physical science into distinct branches, and their mutual relations.