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that which regards the number of times, or parts of times, the one quantity is contained in the other, the latter regarding only the difference between the two quantities. We have already stated that the property of four quantities arranged in geometrical proportion is, that the product of the second and third, divided by the first, gives the fourth. But when four quantities are in arithmetical proportion, the sum of the second and third, diminished by the subtraction of the first, gives the fourth. Thus, in the geometrical proportion, 1 is to 2 as 2 is to 4, if 2 be multiplied by 2 it gives 4; which, divided by 1, still remains 4: while in the arithmetical proportion 1 is to 2 as 2 is to 3, if 2 be added to 2 it gives 4; from which, if 1 be subtracted, there remains the fourth term 3. It is plain, therefore, that, especially where large numbers are concerned, operations by arithmetical must be much more easily performed than operations by geometrical proportion; for in the one case you have only to add and subtract, while in the other you have to go through the greatly more laborious process of multiplication and division.
Now it occurred to Napier, reflecting upon this important distinction, that a method of abbreviating the calculation of a geometrical proportion might perhaps be found, by substituting, upon certain fixed principles, for its known terms, others in arithmetical proportion, and then finding, in the quantity which should result from the addition and subtraction of these last, an indication of that which would have resulted from the multiplication and division of the original figures. It had been remarked before this, by more than one writer, that if the series of numbers 1, 2, 4, 8, &c., that proceed in geometrical progression, that is, by a continuation of geometrical ratios, were placed under, or alongside of, the series 0, 1, 2, 3, &c., which are in arithmetical progression, the addition of any two terms of the latter se. ries would give a sum, which would stand opposite
to a number in the former series indicating the product of the two terms in that series, which corresponded in place to the two in the arithmetical series first taken. Thus, in the two lines
1, 2, 4, 8, 16, 32, 64, 128, 256,
0, 1, 2, 3, 4, 5, 6, 7, 8, the first of which consists of numbers in geometrical, and the second of nunibers in arithmetical progression, if any two terms, such as 2 and 4, be taken from the latter, their sum 6, in the same line will stand opposite to 64 in the other, which is the product of 4 multiplied by 16, the two terms of the geometrical series which stand opposite to the 2 and 4 of the arithmetical. It is also true, and follows directly from this, that if any three terms, as, for instance, 2, 4, 6, be taken in the arithmetical series, the sum of the second and third, diminished by the subtraction of the first, which makes 8, will stand opposite to a number (256) in the geometrical series which is equal to the product of 16 and 64 (the opposites of 4 and 6), divided by 4 (the opposite of 2).
Here, then, is, to a certain extent, exactly such an arrangement or table as Napier wanted. Having any geometrical proportion to calculate, the known terms of which were to be found in the first line or its continuation, he could substitute for them at once, by reference to such a table, the terms of an arithmetical proportion which, wrought in the usual simple manner, would give him a result that would point out or indicate the unknown term of the geometrical proportion. But, unfortunately, there were many numbers which did not occur in the upper line at all, as it here appears. Thus, there were not to be found in it either 3, or 5, or 6, or 7, or 9, or 10, or any other numbers, indeed, except the few that happen to result from the multiplication of any of its terms by 2. Between 128
and 256, for example, there were 127 numbers want ing, and between 256 and the next term (512) there would be 255 not to be found.
We cannot here attempt to explain the methods by which Napier's ingenuity succeeded in filling up these chasms, but must refer the reader, for full information upon this subject, to the professedly scientific works which treat of the history and conconstruction of logarithms.* Suffice it to say, that he devised a mode by which he could calculate the proper number to be placed in the table over against any number whatever, whether integral or fractional. The new numerical expressions thus found he called Logarithms, a term of Greek etymology which signifies the ratios of numbers.
Napier's discovery was very soon known over Europe, and was everywhere hailed with admiration by men of science. The great Kepler, in particular, honoured the author by the highest commendation, and dedicated to him his Ephemerides for 1617. This illustrious astronomer, also, some years afterward, rendered a most important service to the new calculus, by first demonstrating its principle on purely geometrical considerations. Napier's own demonstration, it is to be observed, though exceedingly ingenious, had failed to satisfy many of the mathematicians of that age, in consequence of its proceeding upon the supposition of the movement of a point along a line, a view analogous, as has been remarked, to that which Newton afterward adopted in the exposition of his doctrine of fluxions, but one of which no trace is to be found in the methods of the ancient geometers.
Napier did not expound the process by which he constructed his logarithms in his first publication. This appeared only in a second work, published at Edinburgh in 1619, after the death of the author, by his third son, Robert. In this work also the logarithmic tables appeared in the improved form in which, however, they had previously been published at London, by Mr. Briggs, in 1617. They have since then been printed in numberless editions in every country of Europe. Nay, in the year 1721, a magnificent edition of them, in their most complete form, issued from the imperial press of Pekin, in China, in three volumes, folio, in the Chinese language and character. As for the invention itself, its usefulness and value have grown with the progress of science ; and, in addition to serving still as the grand instrument for the abridgment of calculation in almost every department in which figures are employed, it is now found to be applicable to several important cases which could not be managed at all without its assistance. Some of the greatest names in the history of science, we may also remark, since Napier's time, have occupied themselves with the subject of the theory and construction of logarithms; and the labours of Newton, James Gregory, Halley, and Eüler, have especially contributed to simplify and improve the methods for their investigation.
* The most complete history of logarithms is to be found in the Preface to Hutton's Mathematical Tables.
Napier, however, did not live long to enjoy the reputation of his discovery, having died at Merchiston on the 3d of April, 1617, in the sixty-eighth year of his age. He was twice married, and had iwelve children, of whom Archibald, the eldest, was raised to the peerage, by the title of Lord Napier, in 1627. There are said to be still in the posses. sion of the family some productions of their distinguished ancestor on scientific subjects, which have not been printed, especially a treatise, in English, on Arithmetic and Algebra, and another on Algebra, in Latin.
The lise which we have thus sketched may be considered as affording us an eminent example of the manner in which the many advantages enjoyed by the wealthy may be turned to account in the pursuit of learning and philosophy. A good education, access to all the best means of improvement, uninterrupted leisure, comparative freedom from the ordinary anxieties of life, the means of engaging in inquiries and experiments, the expense of which cannot be afforded by the generality of students ; the possession of all these things, to the mind that knows how to profit by them, is indeed invaluable. We have seen what they produced in Napier's case. In dedicating his time and his fortune to pursuits so much nobler than those that have usually occupied persons of his station, this illustrious individual had his ample reward. We can scarcely doubt that he led a happier life in his studious retirement, in the midst of his books and his experiments, than if he had given himself either to the ordinary pleasures of the world, or to the hazards and vexations of political ambition. The more useful and more honourable path he certainly chose. By his great and fortunate discovery he made the science of all succeeding times his debtor, and constituted himself the benefactor of every generation of posterity. And then for fame, which our very nature has made dear to us, that, too, this philosopher found in his closet of meditation. Even in his own day, his renown was spread abroad over Europe, and he was greeted with the publicly expressed admiration of some of the most distinguished of his contemporaries; and the time that has since elapsed has only served to throw an increasing light around his name, which is now sure to retain its distinction, so long as the sciences which he loved shall continue to be cultivated among men.