16,

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 series 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, 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 numbers 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

* Namely, by H. Grammateus, in his Commercial Arithmetic, published in German, at Vienna, in 1518; and more clearly by M. Stifels, in his Arithmetica Integra, printed at Nuremberg in 1544. See Montucla, Histoire des Mathematiques, ii. 19. Even Archimedes was acquainted with these relations.

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 wanting, 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 construction 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. The table, however, which he published, in the first instance, in his Mirifici Logarithmorum Canonis

See especially Montucla, Histoire des Mathematiques, ii. 16, &c.; Delambre, Histoire de l'Astronomie Moderne, i. 491, &c.; and, where the most complete history of logarithms is to be found, the Preface to Hutton's Mathematical Tables, London, 1785; which is reprinted in the first volume of Baron Maseres's Scriptores Logarithmici.

Descriptio, which appeared at Edinburgh in 1614, contained only the logarithms of the sines of angles for every degree and minute in the quadrant, which shews that he chiefly contemplated, by his invention, facilitating the calculations of trigonometry. These logarithms differed also from those that are now in use, in consequence of Napier having chosen, originally, a different geometrical series from that which has since been adopted. He afterwards fixed upon the progression, 1, 10, 100, 1000, &c., or that which results from continued multiplication by 10, and which is the same according to which the present tables are constructed. This improvement, which possesses many advantages, had suggested itself about the same time to the learned Henry Briggs, then Professor of Geometry in Gresham College,-one of the persons who had the merit of first appreciating the value of Napier's invention, and who certainly did more than any other to spread the knowledge of it, and also to contribute to its perfection. Lilly, the astrologer, gives us, in his Memoirs, a curious account of the intercourse between Briggs and Napier, to which the publication of the logarithmic calculus led. “I will acquaint you," he writes, " with one memorable story, related unto me by John Marr, an excellent mathematician and geometrician, whom I conceive you remember. He was servant to King James and Charles the First. At first, when the Lord Napier, or Marchiston, made public his logarithms, Mr. Briggs, then reader of the Astronomy Lectures at Gresham College, in London, was so surprised with admiration of them, that he could have no quietness in himself until he had seen that noble person, the Lord Marchiston, whose only invention they were; he acquaints John Marr herewith, who went into Scotland before Mr. Briggs,

purposely to be there when these two so learned persons should meet. Mr. Briggs appoints a certain day when to meet at Edinburgh; but failing thereof, the Lord Napier was doubtful he would not come. It happened one day, as John Marr and the Lord Napier were speaking of Mr. Briggs; Ah, John,' said Marchiston, Mr. Briggs will not now come.' At the very instant one knocks at the gate; John Marr hasted down, and it proved Mr. Briggs, to his great contentment. He brings Mr. Briggs up into my lord's chamber, where almost one quarter of an hour was spent, each beholding other, almost with admiration, before one word was spoke. At last Mr. Briggs began; My lord, I have undertaken this long journey purposely to see your person, and to know by what engine of wit or ingenuity you came first to think of this most excellent help into astronomy, viz. the logarithms; but, my lord, being by you found out, I wonder nobody else found it out before, when now known it is so easy.' He was nobly entertained by the Lord Napier; and every summer after that, during the lord's being alive, this venerable man, Mr. Briggs, went purposely into Scotland to visit him."

Napier's discovery was very soon known over Europe, and was every where 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 afterwards, 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 afterwards 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.

Napier, however, did not live long to enjoy the reputation of his discovery, having died at Merchis