This observation, or one very similar to it, was made by Archimedes, and was employed by that great geometer to convey an idea of a number too vast to be correctly • expressed by the arithmetical notation of the Greeks. Thus far, however, there was no difficulty, and the discovery might certainly have been made by men much inferior either to Napier or Archimedes. What remained to be done, what Archimedes did not attempt, and what Napier completely performed, involved two great difficulties. It is plain, that the resource of the geometrical progression was sufficient, when the given numbers were terms of that progression ; but if they were not, it did not seem that any advantage could be derived from it. Napier, however, perceived, and it was by no means obvious, that all numbers whatsoever might be inserted in the progression, and have their places assigned in it. Af ter conceiving the possibility of this, the next difficulty was, to discover the principle, and to execute the arithmetical process, by which, these places were to be ascertained. It is in these two points that the peculiar merit of his invention consists ; and at a period when the nature of series, and when every other resource of which he could avail himself were so little known, his success argues a depth and originality of thought which, I am persuaded, have rarely been surpassed.
The way in which he satisfied himself that all numbers might be intercalated be tween the terms of the given progression, and by which he found the places they must oc cupy, was founded on a most ingenious supposition,—that of two points describing two dif ferent lines, the one with a constant velocity, and the other with a velocity always increas ing in the ratio of the space the point had already gone over : the first of these would gene rate magnitudes in arithmetical, and the second magnitudes in geometrical progression. It is plain, that all numbers whatsoever would find their places among the magnitudes so generat ed; and, indeed, this view of the subject is as simple and profound as any which, after two hun dred years, has yet presented itself to mathematicians. The mode of deducing the results has been simplified; but it can hardly be said that the principle has been more clearly developed.
I need not observe, that the numbers which indicate the places of the terms of the geo metrical progression are called by Napier the logarithms of those terms. ' Various systems of logarithms, it is evident, may be constructed according to the geo metrical progression assumed ; and of these, that which was first contrived by Napier, though the simplest, and the foundation of the rest, was not so convenient for the purposes of calculation, as one which soon afterwards occurred, both to himself and his friend Briggs, by whom the actual calculation was performed. The new system of logarithms was an im provement, practically considered ; but in as far as it was connected with the principle of the invention, it is only of secondary consideration. The original tables had been also some what embarrassed by too close a connection between them and trigonometry. The new tables were free from this inconvenience.
It is probable, however, that the greatest inventor in science was never able to do more than to accelerate the progress of discovery, and to anticipate what time, " the author of authors," would have gradually brought to light. Though logarithms had not been in
vented by Napier, they would have been discovered in the progress of the algebraic analysis, when the arithmetic of powers and exponents, both integral and fractional, came to be fully understood. The idea of considering all numbers, as powers of one given number, would then have readily occurred, and the doctrine of series would have greatly facilitated the calculations which it was necessary to undertake. Napier had none of these advantages, and they were all supplied by the resources of his own mind. Indeed, as there never was , any invention for which the state of knowledge had less prepared the way, there never was any where more merit fell to the share of the inventor.
His good fortune, also, not less than his great sagacity, maybe remarked. Had the invention of logarithms been delayed to the end of the seventeenth century, it would have come about without effort, and would not have conferred on the author the high celebrity which Napier so justly derives from it. In another respect he has also been fortunate. Many inventions have been eclipsed or obscured by new discoveries ; or they have been so altered by subsequent improvements; that their original forM can hardly be recognised, and, in some instances, has been entirely forgotten. This has almost always happened to the discoveries made at an early period in the progress of science, and before their princi ples were fully unfolded. It has been quite otherwise with the invention of logarithms, which came out of the hands of the author so perfect, that it has never received but one material improvement, that which it, derived, as has just been said, from the ingenuity of his friend in ponjunction with his own. Subsequent improvements in science, instead of - offering any thing that could supplant this invention, have only enlarged the circle to which its utility extended. Logarithms have been applied to numberless purposes, which were not thought of at the time of first construction. Even the sagacity of their author did not see the immense fertility of the principle he had discovered ; he calculated his tables merely to facilitate arithmetical, and chiefly trigonometrical computation, and little imagin ed that he was at the same time constructing a scale whereon to measure the density of the strata of the atmosphere, and the heights of mountains ; that he was actually computing the areas and the lengths of innumerable curves, and was preparing for a calculus which was yet to be discovered, many of the most refined and most valuable of its resources. Of Na pier, therefore, if of any man, it may safely be pronounced, that his name will never be eclip sed by any one more conspicuous, or his invention superseded by any thing more valuable.
As a geometrician, Napier has left behind him a noble monument in the two tri gonometrical theorems, which are known by his name, and which appear first to have been communicated in writing to Cavalleri, who has mentioned them with great eulogy. ', They are theorems not a little difficult, and of much use, as being particularly adapted to logarithmic calculation. They were published in the Canon Miriftcus Logarithmorum, at Edinburgh, in 1614. a