VIII HAVING proved that Euclid was the "father of all such as handle the rule and compasses," we have to inquire what has been the influence which these have exerted on the generations which have intervened between Adelard and ourselves, between the twelfth and the twentieth centuries. In this, it must be admitted, we are faced by two questions, with only one of which we have any means of dealing. We know, in some sort, what the generations have been under the influence of their mathematicians; but we know nothing, and can know nothing, of what the same generations would have been under the influence of the same men, the Newtons and the Leibnitzes, the Laplaces and Lagranges, had these men not been mathematicians at all. They would doubtless have exerted a potent influence, but it would have been an influence utterly different from that which they did actually exert. The history of the world might have been as interesting as it has been; but it must have been wholly different. To tell all that mathematics has done for the world during the eight centuries, would be to recount the history of the world and of mathematics—a task impossible of achievement, though we had a hundred lives to spend in its accomplishment, and a hundred volumes in which to record it. To achieve it well would need in addition that our ability were multiplied a hundredfold. It is but a minute portion of this stupendous task that we can undertake. Any question as to how much of what modern mathematics is, and of what it has done, was known to Euclid, does not concern us. We are quite willing to admit that he knew comparatively very little. We claim not for him that he knew much, but that he handed down to his successors the means of knowing vastly more than he ever knew. He was not the giant to see afar; rather he was the block on which others stood, so that the sphere of their view might be extended far beyond his. Some may deem it a very humble merit that we claim for him. We do not think so; and, what is of far more moment, Lord Bacon would not have thought so. What concerns us is not the amount of the extension of mathematical science, but the manner in which that extension is due to adherence to Euclid's methods. The progress of mathematical science, as probably of all other sciences, has been partly slow and continuous, and partly rapid and brilliant; now by slow and quiet walking,—the snail's pace, if you like,—and now by a notable bound. That the progress, so far as it has been of the former kind, has been in the line of the Euclidean methods, will not be disputed; to use his own expression, it needs no demonstration. But we must examine the other case at some length. When we speak in this connection of leaps and bounds, we do not refer at all to the demonstration of some theorem which has not been demonstrated before. A solution of the problem of the duplication of the cube, or the trisection of an angle, or the rectification and quadrature of the circle, or, in another department of mathematics, the reduction of the irreducible case of cubic equations, would not constitute such a leap as we contemplate. We have not in view the advancement of results, however important that may be; but the discovery of new methods, the invention of new instruments which are fitted to increase the power of the geometer, very much as the invention of gunpowder long ago, and the multitudinous inventions of explosives in our own day, have increased the destructive potency of our troops and our warships. The greatest of these inventions are: (a) that of logarithms by Napier (or Neper) (1550-1617); (b) that of co-ordinate geometry by Descartes (1596–1650); (c) that of the higher calculus, simultaneously by Newton (1642-1727) and Leibnitz (1646-1716). We place among these in the meantime (d) the innovations of Legendre (1752-1833), although he never regarded them otherwise than as improvements on Euclid. We shall have, in the sequel, to consider whether they were such or the contrary. To these, in their order, with reference to their bearing on the position of Euclid, we have now to direct our attention. (a) Logarithms are not now regarded as belonging at all to geometry. They are essentially an arithmetical instrument, an instrument of arithmetical calculation, and no one now would think of computing them otherwise than algebraically. But it was otherwise in the days of old. Euclid himself regarded arithmetic as a branch of geometry; and, as we have seen, the books of the Elements, VII.-X. inclusive, constitute an arithmetical manual, a great portion of whose contents does not go beyond the standard of our ordinary schoolbooks. Things were materially different in the days of Napier; for it cannot be said that algebra was then non-existent, or that arithmetic had no being save as a department of geometrical science. Still algebra had only outlived the embryonic, and had entered on the infantile state of being. We do not quite precisely know what methods Napier adopted in his computation. But it would appear that they were more or less dependent on geometrical relations, and were founded on certain properties of one of the conic sections. Hence his logarithms are still frequently distinguished from other systems by the designation of hyperbolical. The relation to the hyperbola is not really distinctive of the Napierian logarithms, but appertains to logarithms generally; and therefore we suppose that the designation was applied to his system while as yet it was the only system. It may be well to explain to the nonmathematical reader what logarithms substantially are. This can be done very simply If we take two series of numbers, one consisting of the numbers in their order, and the other of the numbers formed by the continuous multiplication of unity by any number, say 2, we shall form a table of logarithms, thus 10, 11, 12, 13, 14. Log. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, Now, suppose we wish to multiply one by another of the numbers contained in the line marked Num. in this perfectly real, though, of course, very brief table, we have to add the logarithms corresponding to these numbers, and below their sum in the line of Log. will be their product in the line of Num. Thus, to find the product of 16 × 64, we add the corresponding loga rithms 4+6=10, and the number corresponding to the logarithm 10 is 1024, which is the product required. This table could be very easily continued to any length, but it would not really be of any use, because the line of numbers contains only comparatively few of the actual numbers, and the table gives no logarithm for 3, 5, 6, 7, 15, etc. Now Napier's work was to compute a table in which a logarithm should be given for every number between 1 and 100,000. Any arithmetician will see that as our little table gives the product of two numbers by addition of their logarithms, so it will give the quotient of two numbers by subtracting the logarithm of the divisor from that of the dividend; that the logarithm of the square root, cube root, etc., of a number will be,, etc., of the logarithm of the number. Thus, to find the square root of 16384, we halve its logarithm 14, and the number corresponding to the logarithm 7 is 128, the square root required. We have stated that Napier computed his logarithms in quite a different way from this. He did not make the use of any number that we have made of the number 2 in constructing our little table. But it was found that his results would have been got if he had made that use of a certain number, greater than 2 and less than 3.1 After the publication of his work he perceived that a much more convenient table would be produced by the use of 10 as the "base" of the system. The same thought occurred to Henry Briggs, of Oxford, who at once perceived the marvellous potency and the glaring defect of Napier's table. He paid Napier a visit at Merchiston,2 and at his earnest request agreed to make the transformation, as Napier 1 The number 2.718281828459. . . . 2 A suburb of Edinburgh. |