Sir George Airy has also recorded a curious case, in which he accidentally fell by trial on a new geometrical property of the sphere. Many of the most important and now trivial propositions in geometry, were probably thus discovered by the ancient Greek geometers; and we have pretty clear evidence of this in the Commentaries of Proclus. But discovery in such cases means nothing more than suggestion, and it is always by pure deduction that the general law is really established. As Proclus puts it, we must pass from sense to consideration, Given, for instance, the series of figures in the accompanying diagram, a little examination and measurement will show that the curv ed lines approximate to semicircles, and the rectilineal figures to rightangled triangles. These figures may seem to suggest to the mind the general law that angles in scribed in semicircles are right angles; but no number of instances, and no possible accuracy of measurement would really establish the truth of that general law. Availing ourselves of the suggestion furnished by the figures, we can only investigate deductively the consequences which flow from the definition of a circle, until we discover among them the property of containing right angles. Many persons, after much labour, have thought that they had discovered a method of trisecting angles by plane geometrical construction, because a certain complex arrangement of lines and circles had appeared to trisect an angle in every case tried by them, and they inferred, by a ¶ 'Philosophical Transactions,' [1866] vol. 146, p. 334. supposed act of induction, that it would succeed in all other cases. Professor de Morgan has recorded a proposed mode of trisecting the angle which could not be discriminated by the senses from a true general solution, except when it was applied to very obtuse angles. In all such cases, it has always turned out either that the angle was not trisected at all, or that only certain particular angles could be thus trisected. They were misled by some apparent or special coincidence, and only deductive proof could establish the truth and generality of the result. In this case, deductive proof shows that the problem, as attempted, is impossible, and that angles generally cannot be trisected by common geometrical methods. Geometrical Reasoning. This view of the matter is strongly supported by the further consideration of geometrical reasoning. No skill and care could ever enable us to verify absolutely any one geometrical proposition. Rousseau, in his Emile,t tells us that we should teach a child geometry by causing him to measure and compare figures by superposition. While a child was yet incapable of general reasoning, this would doubtless be an instructive exercise; but it never could teach geometry, nor prove the truth of any one proposition. All our figures are rude approximations, and they may happen to seem unequal when they should be equal, equal when they should be unequal. Moreover, fres may from chance be equal in case after case, and ere may be no general reason why they should be The results of deductive geometrical reasoning are Budget of Paradoxes,' p. 257. 12mo. Amsterdam, 1762, vol. i. p. 401. absolutely certain, and are either exactly true or capable of being carried to any required degree of approximation. In a perfect triangle, the angles must be equal to one halfrevolution precisely; even an infinitesimal divergence would be impossible; and I believe with equal confidence, that however many are the angles of a figure, provided there are no re-entrant angles, the sum of the angles will be precisely and absolutely equal to twice as many rightangles as the figure has sides, less by four right-angles. In such cases, the deductive proof is absolute and complete; empirical verification can at the most guard against accidental oversights. There is a second class of geometrical truths which can only be proved by approximation; but, as the mind sees no reason why that approximation should not always go on, we arrive at complete conviction. We thus learn that the surface of a sphere is equal exactly to two-thirds of the whole surface of the circumscribing cylinder, or to four times the area of the generating circle. The area of a parabola is exactly two-thirds of that of the circumscribing parallelogram. The area of the cycloid is exactly three times that of the generating circle. These are truths that we could never ascertain, nor even verify by observation ; for any finite amount of difference, vastly less than what the senses can discern, would falsify them. There are again geometrical relations which we cannot assign exactly, but can carry to any desirable degree of approximation. Thus, the ratio of the circumference to the diameter of a circle is that of 314159265358979323846.... to I, and the approximation may be carried to any extent by the expenditure of sufficient labour, as many as 607 places of figures having been calculated." Some years since, I amused myself by trying how near I could get to this ratio, by the careful use of compasses, and I did not come u English Cyclopædia,' art. Tables. nearer than 1 part in 540. We might imagine measurements so accurately executed as to give us eight or ten places correctly. But the power of the hands and senses must soon stop, whereas the mental powers of deductive reasoning can proceed to an unlimited degree of approximation. Geometrical truths, then, are incapable of verification; and, if so, they cannot even be learnt by observation. How can I have learnt by observation a proposition of which I cannot even prove the truth by observation, when I am in possession of it? All that observation or empirical trial can do is to suggest propositions, of which the truth may afterwards be proved deductively. By drawing a number of right-angled triangles on paper, with squares upon their sides, and cutting out and weighing these squares very accurately, I might have reason to suspect the existence of the relation of equality proved in Euclid's 47th Proposition; but no process of weighing or measuring could ever prove it, nor could it ever assure me that the like degree of approximation would exist in untried cases. Much has been said about the peculiar certainty of mathematical reasoning, but it is only certainty of deductive reasoning, and equal certainty attaches to all correct logical deduction. If a triangle be right-angled, the square on the hypothenuse will undoubtedly equal the sum of the two squares on the other sides; but I can never be sure that a triangle is right-angled: so I can be certain that nitric acid will not dissolve gold, provided I know that the substances employed really correspond to those on which I tried the experiment previously. Here is like certainty of inference, and like doubt as to the facts. Discrimination of Certainty and Probability in the Inductive Process. We can never recur too often to the truth that our knowledge of the laws and future events of the external world is only probable. The mind itself is quite capable of possessing certain knowledge, and it is well to discriminate carefully between what we can and cannot know with certainty. In the first place, whatever feeling is actually present to the mind is certainly known to that mind. If I see blue sky, I may be quite sure that I do experience the sensation of blueness. Whatever I do feel, I do feel beyond all doubt. We are indeed very likely to confuse what we really feel with what we are inclined to associate with it, or infer inductively from it; but the whole of our consciousness, as far as it is the result of pure intuition and free from inference, is certain knowledge beyond all doubt. In the second place, we may have certainty of inference ; the first axiom of Euclid, the fundamental laws of thought, and the rule of substitution (p. 11), are certainly true; and if my senses could inform me that A was indistinguishable in colour from B, and B from C, then I should be equally certain that A was indistinguishable from C. In short, whatever truth there is in the premises, I can certainly embody in their correct logical result. But practically the certainty generally assumes a hypothetical character. I never can be quite sure that two colours are exactly alike, that two magnitudes are exactly equal, or that two bodies whatsoever are identical even in their apparent qualities. Almost all our judgments involve quantitative relations, and, as will be shown in succeeding chapters, we can never attain exactness and certainty where continuous quantity enters. Judgments concerning |