Many extraordinary instances have been recorded of extremely high degrees of innate sensitiveness to particular classes of ideas, especially those relating to numbers and sounds. 'The case of Zerah Colburn, the son of an American peasant, is especially remarkable among these, not only for the immediateness and correctness with which he gave the answers to questions resolvable by simple but prolonged computation-such as the product of two numbers, each consisting of two, three, or four figures; the exact number of minutes and seconds in a given period of time; the raising of numbers up to high powers; or the extraction of the square and cube roots-but, still more, for his power of at once answering questions to which no rules known to mathematicians would apply.' 'On being interrogated as to the method by which he obtained these results, the boy constantly declared that he did not know how the answers came into his mind. In the act of multiplying two numbers together, and in the raising of powers, it was evident (alike from the facts just stated and from the motions of his lips) that some operation was going forward in his mind; yet that operation could not (from the readiness with which the answers were furnished) have been at all allied to the usual modes of procedure, of which, indeed, he was utterly ignorant, not being able to perform on paper a simple sum in multiplication or division. But in the extraction of roots and in the discovery of factors of large numbers it did not appear that any operation could take place, since he answered immediately, or in a very few seconds, questions which, according to the ordinary methods, would 'have required very difficult and laborious calculations; and prime numbers cannot be recognised as such by any known rule.' INNATE MATHEMATICAL GENIUS. 41 'The same faculty, improved by cultivation, appears to have been possessed by the illustrious Euler, who had not only a most extraordinary memory for numbers-to the extent, it is said, of being able to recall the first six powers of any number under 100-but also a kind of divining power, by which he perceived, almost at a glance, the factors of which his formulæ were composed; the particular system of factors belonging to the question under consideration; the various artifices by which that system might be simplified and reduced; and the relation of the several factors to the conditions of the hypothesis. This power of divining truths in advance of existing knowledge is the special attribute of those mathematicians who have done most for the development of their science. A notable instance of it is furnished by the celebrated formula devised by Newton for the solution of equations; for although its correctness was proved experimentally by the results of its application in every conceivable variety of case, its rationale seems to have been unknown to Newton himself, and remained a puzzle to succeeding mathematicians until discovered by the persevering labours of Professor Sylvester, who is himself specially distinguished for the possession of this highest form of mathematical genius. That such a power as Zerah Colburn's should exist in a child who had never been taught even the rudiments of arithmetic, seems to point (as Mr. Baily remarks) to the existence of properties of numbers as yet undiscovered, somewhat analogous to those on which the system of logarithms is based. And if, as he grew older, he had become able to make known to others the methods by which his results were obtained, a real advance in knowledge might have been looked for. But it seems to have been the case with him, as with George Bidder and other "calculating boys," that with the general culture of Mozart was as the mind this special power faded away.'' intuitively and highly sensitive to musical ideas as Colburn to arithmetical ones; but in his case the power was largely improved and developed by education and exercise.2 The selection of a profession and general occupation of life is often determined by us in accordance with our particular inherited tendencies; and it is in consequence of inherited fitness for one class of thoughts and actions in preference to others, that each man has largely a right to select his own sphere and kind of useful employment. Every idea is conformable in some respect to its cause, and ideas are usually images or resemblances of the existences, attributes, or relations they represent, but not necessarily so, because we often have false ones. Ideas are frequently not actual resemblances either in kind or degree of the objects, actions, or attributes, &c., they indicate. A round figure indeed produces the idea of roundness, and a large one that of largeness, but a heated body, although it produces a sensation and idea of pain, does not possess pain, nor is the property of an orange which produces an idea of sweetness, sweetness itself; and an idea of the sun also is not equal in brightness to the image of the sun itself, nor is that of a mountain proportioned in size to the mountain. For every important existence, real or imaginary, a representative idea is usually sooner or later discovered. 6 It was held and maintained by Locke, and is now generally admitted, that the sole original source of all our ideas is experience, and that we have no innate ideas,' but only innate tendencies. We each derive our scientific ideas not only from our own personal experience and observation, but also from that of others, by means of 1 Carpenter, Mental Physiology, pp. 232–235. reading, pictorial representation, &c., and what we are Perception of ideas is also essentially automatic, and the only direct volitional power we possess over it is to direct attention to an idea already present, thus increasing its strength and permanence. Of our most common ideas, a few are probably acquired by means of one sense alone, some by the combined action of several, and others by the additional aid of the judgment. Most are acquired by the aid of several senses, and are abstracts of many impressions. Those of light, shade, and colour are first acquired only by means of vision, and some only by the aid of the organ of hearing. The great bulk of our ideas can be formed only by the help of the intellect; and most of our perception is mingled with inference. Fixed vision with one eye, without the aid of comparison, excites only an idea of flatness, and requires a greater aid of the judgment to correctly interpret than that with both eyes. The ideas of form in relief, and solidity, are each acquired by means of the combined sensations of sight and touch, aided by inference. Judgment (in one of its meanings) is a perception of the connection of two things; and when Ideas are of many kinds, and may be divided into true and false, real and fanciful, simple and complex, strong and feeble, distinct and indistinct, complete and incomplete, adequate and inadequate, relative and absolute, disorderly, confused, axiomatic, abstract, ultimate, essential, immediate and mediate, qualitative and quantitative, &c.; and some of these are treated of in the chapter on 'Scientific Terms.' An idea, associated with almost any common name, may differ in kind, and its implicit contents differ in quality. It may either be what is logically termed in extension or intension. Thus we may either think of a steamship in extension as any single vessel propelled by the expansion of steam, or in intension as any one of the individual steamships that we are acquainted with. In the former case our idea includes a large number of objects, and in the latter a large number of marks or attributes; and the completeness of our idea depends upon our clearly conceiving it in both these aspects. As, however, the human mind can only think of a few marks at a time, and cannot realise those with a vividness equal to that excited by the original object, there is a limit to the degree of clearness of every mental conception; the more objects or ideas also we perceive at once, the less we perceive of each. Scientific ideas may be either of real or possible |