(John), baron of Merchiston in Scotland, inventor of the logarithms, was the eldest son of Sir Archibald Napier of Merchiston, and born in the year 1550. Having given early discoveries of great natural parts, his father was careful to have them cultivated by a liberal education. After going through the ordinary courses of philosophy at the university of St Andrew's, he made the tour of France, Italy, and Germany. Upon his return to his native country, his literature and other fine accomplishments soon rendered him conspicuous, and might have raised him to the highest offices in the state; but declining all civil employments, and the battle of the court, he retired from the world to pursue literary researches, in which he made an uncommon progress, so as to have favoured mankind with sundry useful discoveries. He applied himself chiefly to the study of mathematics; but at the same time did not neglect that of the Holy Scriptures. In both these he hath discovered the most extensive knowledge and profound penetration. His essay upon the book of the Apocalypse, indicates the most acute investigation, and an uncommon strength of judgment; though time hath discovered, that his calculations concerning particular events had proceeded upon fallacious data. This work has been printed abroad in several languages; particularly in French at Rochelle in the year 1693, 8vo, announced in the title as revised by himself. Nothing, says Lord Buchan, could be more agreeable to the Rochellers or to the Huguenots of France at this time, than the author's annunciation of the pope as antichrist, which in this book he has endavoured to set forth with much zeal and erudition.—But what has principally rendered his name famous, was his great and fortunate discovery of logarithms in trigonometry, by which the ease and expedition in calculation have so wonderfully assisted the science of astronomy and the arts of practical geometry and navigation. That he had begun about the year 1593 the train of enquiry which led him to that great achievement in arithmetic, appears from a letter to Crugerus from Kepler in the year 1624; wherein, mentioning the Canon Mirificus, he writes thus: "Nilhautem supra Neperianam rationem esse puto: et si Scotus quidem literis ad Tychonem, anno 1594, scriptis jam spem fecit Canonis illius mirifici;" which allusion agrees with the idle story mentioned by Wood in his Athene Oxoni, and explains it in a way perfectly consonant to the rights of Napier as the inventor.

When Napier had communicated to Mr Henry Briggs, mathematical professor in Gresham college, his wonderful canon for the logarithms, that learned professor set himself to apply the rules in his Imitatio Neperiana; and in a letter to archbishop Usher in the year 1615, he writes thus: "Napier, baron of Merchiston, hath set my head and hands at work with his new and admirable logarithms. I hope to see him this summer, if it please God; for I never saw a book which pleased me better, and made me more wonder." The following passage from the life of Lilly the astrologer is quoted by Lord Buchan as giving a picture. resque view of the meeting betwixt Briggs and the inventor of the logarithms at Merchiston near Edinburgh. "I will acquaint you (says Lilly) 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 I. and Charles I. When Merchiston first published his logarithms, Mr Briggs, then reader of the astronomy lectures at Gresham college in London, was so much surprised with admiration of them, that he could have no quietness in himself until he had seen that noble person whose only invention they were: he acquaints John Marr therewith, who went into Scotland before Mr Briggs, purposely to be there when these two learned persons should meet. Mr Briggs appoints a certain day when to meet at Edinburgh; but failing thereof, Merchiston was fearful he would not come. It happened one day as John Marr and the baron Napier were speaking of Mr Briggs; 'Ah, John (said Merchiston), Mr Briggs will not now come.' At the very instant one knocks at the gate; John Marr hasted down, and it proved to be Mr Briggs to his great contentment. He brings Mr Briggs up to the Baron's chamber, where almost one quarter of an hour was spent, each beholding other with admiration before one word was spoken. At last Mr Briggs began: 'Sir, 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, Sir, being by you found out, I wonder nobody else found it out before, when now being known it appears so easy.' He was nobly entertained by baron Napier; and every summer after that, during the laird's being alive, this venerable man Mr Briggs went purposely to Scotland to visit him."

There is a passage in the life of Tycho Brahe by Gassendi, which may mislead an attentive reader to suppose that Napier's method had been explored by Herwart at Hoenburg: It is in Gassendi's Observations on a Letter from Tycho to Herwart of the last day of August 1599. "Dixit Hervartus nihil morari se solvendi cujusquam trianguli difficulatem; folere se enim multiplicationum, ac divisionum vice additiones solum, subtractiones 93 ufuppare (quod ut fieri posset, docuit postmodum fuo Logarithmorum Canone Nepereus.)" But Herwart here alludes to his work afterwards published in the year 1610, which solves triangles by prosthapheresis; a mode totally different from that of the logarithms.

Kepler dedicated his Ephemerides to Napier, which were published in the year 1617; and it appears from many passages in his letter about this time, that he held Napier to be the greatest man of his age in the particular department to which he applied his abilities. "And indeed (says our noble biographer), if we consider that Napier's discovery was not like those of Kepler or of Newton, connected with any analogies or coincidences which might have led him to it, but the fruit of unassisted reason and science, we shall be vindicated in placing him in one of the highest niches in the temple of fame. Kepler had made many unsuccessful attempts to discover his canon for the periodic motions of the planets, and hit upon it at last, as he himself candidly owns on the 15th of May 1618; and Newton applied the palpable tendency of heavy bodies to the earth to the system of the universe in general; but Napier sought out his admirable rules by a slow scientific progress, arising from the gradual evolution of truth."

The last literary exertion of this eminent person was the publication of his Rabdologia and Promptuary in the year 1617, which he dedicated to the Chancellor Seton; and soon after died at Merchiston on the 3rd of April O.S. of the same year, in the 68th year of his age and 23d of his happy invention.—The particular titles of his published works are: 1. A plain discovery of the Revelation of St John. 2. Mirifici ipsius canonis construction et logarithmorum, ad naturales illorum numeros habitudines. 3. Appendix de alia atque praestantiori logarithmorum specie confinienda, in qua sciuntur unitas logarithmus eft. 4. Rhabdologie, seu numerationis per virgulas, libri duo. 5. Propositiones quaedam eminentissime, ad triangula sphærica mira facultate refolventa. To which may be added, 6. His Letter to Anthony Bacon (the original of which is in the archbishop's library at Lambeth), intitled, "Secret inventions, profitable and necessary in these days for the defence of this island, and notwithstanding strangers enemies to God's truth and religion?" which the earl of Buchan has caused to be printed in the Appendix to his Account of Napier's Writings. This letter is dated June 2. 1596, about which time it appears the author had set himself to explore his logarithmic canon.

This eminent person was twice married. By his first wife, who was a daughter of Sir James Stirling of Keir, he had only one son named Archibald, who succeeded to the estate. By his second wife, a daughter of Sir James Chisholm of Cromlix, he had a numerous issue.—Archibald Napier, the only son of the first marriage, was a person of fine parts and learning. Having more a turn to public business than his father had, he was raised to be a privy counsellor by James VI. under whose reign he also held the offices of treasurer-depute, justice-clerk, and senator of the college of justice. By Charles I. he was raised to the peerage by the title of Lord Napier.

Napier's Rods, or Bones, an instrument invented by Baron Napier, whereby the multiplication and division of large numbers is much facilitated.

As to the Construction of Napier's Rods: Suppose the common table of multiplication to be made upon a plate of metal, ivory, or pasteboard, and then conceive the several columns (standing downwards from the digits on the head) to be cut asunder; and these are what we call Napier's rods for multiplication. But then there must be a good number of each; for as many times as any figure is in the multiplicand, so many rods of that species (i.e. with that figure on the top of it) must we have; though six rods of each species will be sufficient for any example in common affairs: there must be also as many rods of o's.

But before we explain the way of using these rods, there is another thing to be known, viz. that the figures on every rod are written in an order different from that in the table. Thus the little square space or division in which the several products of every column are written, is divided into two parts by a line across from the upper angle on the right to the lower on the left; and if the product is a digit, it is set in Napier's lower division; if it has two places, the first is set in the lower, and the second in the upper division; but the spaces on the top are not divided; also there is a rod of digits, not divided, which is called the index rod, and of this we need but one single rod. See the figure of all the different rods, and the index, separate from one another, in Plate CCCXLIV.

Multiplication by Napier's Rods. First lay down the index-rod; then on the right of it set a rod, whose top is the figure in the highest place of the multiplicand: next to this again, set the rod whose top is the next figure of the multiplicand; and so on in order to the first figure. Then is your multiplicand tabulated for all the nine digits; for in the same line of squares standing against every figure of the index rod, you have the product of that figure; and therefore you have no more to do but to transfer the products and sum them. But in taking out these products from the rods, the order in which the figures stand obliges you to a very easy and small addition: thus, begin to take out the figure in the lower part, or units place, of the square of the first rod on the right; add the figure in the upper part of this rod to that in the lower part of the next, and so on; which may be done as fast as you can look on them. To make this practice as clear as possible, take the following example:

Example: To multiply 4768 by 385. Having set the rods together for the number 4768 (ibid. n° 2.) against 5 in the index, I find this number, by adding according to the rule,

\[ \begin{array}{c} \text{Against 5, this number} & 23840 \\ \text{Against 8, this number} & 38144 \\ \text{Against 3, this number} & 14324 \\ \end{array} \]

Total product 183,680

To make the use of the rods yet more regular and easy, they are kept in a flat square box, whose breadth is that of ten rods, and the length that of one rod, as thick as to hold six (or as many as you please) the capacity of the box being divided into ten cells, for the different species of rods. When the rods are put up in the box (each species in its own cell distinguished by the first figure of the rod set before it on the face of the box near the top), as much of every rod stands without the box as shows the first figure of that rod: also upon one of the flat sides without and near the edge, upon the left hand, the index rod is fixed; and along the foot there is a small ledge; so that the rods when applied are laid upon this side, and supported by the ledge, which makes the practice very easy; but in case the multiplicand should have more than nine places, that upper face of the box may be made broader. Some make the roads with four different faces, and figures on each for different purposes.

Division by Napier's Rods. First tabulate your divisor; then you have it multiplied by all the digits, out of which you may choose such convenient divisors as will be next less to the figures in the dividend, and write the index answering in the quotient, and so continually till the work is done. Thus 2179788, divided by 6123, gives in the quotient 356.

Having tabulated the divisor 6123, you see that 6123 cannot be had in 2179; therefore take five places, and on the rods find a number that is equal or next less to 21797, which is 18369; that is, 3 times the divisor: wherefore set 3 in the quotient, and subtract 18369 from the figures above, and there will remain 34288; to which add 8, the next figure of the dividend, and seek again on the rods for it, or the next less, which you will find to be five times; therefore set 5 in the quotient, and subtract 30615 from 34288, and there will remain 3673; to which add 8, the last figure in the dividend, and finding it to be just 6 times the divisor, set 6 in the quotient.

\[ \begin{array}{c} 6123 \times 2179788 = 356 \\ 18369 \\ 34288 \\ 30615 \\ 3673 \\ 30738 \\ \end{array} \]