184 NAPIER [OF MEKCHISTON". p b up by Napier himself affords a singular illustration of the state of society and the kind of events in the midst of which logarithms had their birth. Considering the time in which he lived, Napier is singularly free from supersti tion : his Plaine Discovery relates to a method of interpre tation which belongs to a later age ; he shows no trace of the extravagances which occur everywhere in the works of Kepler; and none of his writings contain any allusion to astrology or magic. After Napier s death his manuscripts and notes came into the ossession, not of his eldest son Archibald, but of his second son y his second marriage, Robert, who edited the Constructio ; and Colonel Milliken Napier, Robert s lineal male representative, was still in the possession of many of these private papers at the close of the last century. On one occasion when Colonel Napier was called from home on foreign service, these papers, together with a portrait of John Napier and a Bible with his autograph, were deposited for safety in a room of the house at Milliken, in Renfrew shire. During the owner s absence the house was burned to the ground, and all the papers and relics were destroyed. The manu scripts had not been arranged or examined, so that the extent of the loss is unknown. Fortunately, however, Robert Napier had transcribed his father s manuscript De Arte Logistica. and the copy escaped the fate of the originals in the manner explained in the following note, written in the volume containing them by Francis, seventh Lord Napier: "John Napier of Merchiston, inventor of the logarithms, left his manuscripts to his son Robert, who appears to have caused the following pages to have been written out fair from his father s notes, for Mr Briggs, professor of geometry at Oxford. They were given to Francis, the fifth Lord Napier, by William Napier of Culcreugh, Esq., heir-male of the above-named Robert. Finding them in a neglected state, amongst my family papers, I have bound them together, in order to preserve them entire. NAPIER, 7th March 1801." An account of the contents of these manuscripts was given by Mi- Mark Napier in the appendix to his Memoirs of John Napier, and the manuscripts themselves were edited in their entirety by him in 1839 under the title DC, Arte Logistica, Joannis Naperi Mcr- chistonii Baronis Libri qui supersunt. Jmpressum Ediiiburyi M. DCCC.XXX. ix., as one of the publications of the Bannatyne Club. The treatise occupies one hundred and sixty-two pages, and there is an introduction by Mr Mark Napier of ninety-four pages. The Arithinetic consists of three books, entitled (1) De Compu- tationibus Quantitatum omnibus Logisticse speciebus communium ; (2) De Logistica Arithmetica; (3) De Logistica Geometrica. At the end of this book occurs the note "I could find no more of this geometricall pairt amongst all his fragments." The Algebra Joannis Naperi Mcrchistonii Baronis consists of two books: (1) " De nominata Algebras parte ; (2) De positiva sive cossica Algebra parte," and concludes with the words, "There is no more of his algebra orderlie sett doun." The transcripts are entirely in the handwriting of Robert Napier himself, and the two notes that have been quoted prove that they were made from Napier s own papers. The title, which is written on the first leaf, and is also in Robert Napier s writing, runs thus: "The Baron of Merchiston his booke of Arithmeticke and Algebra. For Mr Henrie Briggs, Professor of Geometric at Oxforde." These treatises were probably composed before Napier had invented the logarithms or any of the apparatuses described in the Habdologia ; for they contain no allusion to the principle of loga rithms, even where we should expect to find such a reference, and the one solitary sentence where the Eabdologia is mentioned ("sive omnium facillime per ossa Rhabdologire nostrre") was no doubt added afterwards. It is worth while to notice that this reference occurs in a chapter "De Multiplications et Partitionis compendiis miscellaneis," which, supposing the treatise to have been written in Napier s younger days, may have been his earliest production on a subject over which his subsequent labours were to exert so enormous an influence. Napier uses alundantes and defectives for positive and negative, denning them as meaning greater or less than nothing ("Abun- dantes sunt quantitates majores nihilo : defective sunt quantitates minores nihilo "). The same definitions occur also in the Canon Mirificus (1614), p. 5: "Logarithmos sinnuin, qui semper majores nihilo sunt, abundantes vocamus, et hoc signo +, aut nullo prreno- tamus. Logarithmos auteiu minores nihilo defectives vocamus, prrenotantes eis hoc signum -." Napier may thus have been the first to use the expression "quantity less than nothing." He uses "radicatum" for power; for root, power, exponent, his words are radix, radicatum, index. Apart from the interest attaching to these manuscripts as the work of Napier, they possess an independent value as affording evidence of the exact state of his algebraical knowledge at the time when logarithms were invented. There is nothing to show whether the transcripts were sent to Briggs as intended and returned by him,
or whether they were not sent to him. Among the Merchistora papers is a thin quarto volume in Robert Napier s writing contain ing a digest of the principles of alchemy; it is addressed to his son,, and on the first leaf there are directions that it is to remain in his. charter-chest and be kept secret except from a few. This treatise and the transcripts seem to be the only manuscripts which have escaped destruction. The principle of " Napier s bones " may be easily explained by imagining ten rectangular slips of cardboard, each divided into nine squares. In the top squares of the slips the ten digits are written, and each slip contains in its nine squares the first nine multiples of the digit which appears in the top square. With the exception of the top squares, every square is divided into two parts by a diagonal, the units being written on one side and the tens on the other, so that when a multiple con sists of two figures they are separated by the diagonal. Fig. 1 shows the slips corresponding to the numbers 2, 0, 8, 5 placed side by side in contact with one another, and next to them is placed an other slip containing, in squares without diagonals, the first nine digits. The slips thus placed in contact give the multiples of the number 2085, the digits in each parallelogram being added to gether ; for example, corresponding to the number 6 on the right hand slip, we have 0, 8 + 3, + 4, 2, 1; whence we find Fl S- l - 0, 1,5, 2, 1 as the digits, written backwards, of 6 x 2085. The use of the slips for the purpose of multiplication is now evident ; thus to multiply 2085 by 736 we take out in this manner the multiples corresponding to 6, 3, 7, and set down the digits as they are ob tained, from right to left, shifting them back one place and adding up the columns as in ordinary multiplication, viz., the figures as written down are 12510 6255 14595 6 2 1534560 Napier s rods or bones consist of ten oblong pieces of wood or other material with square ends. Each of the four faces of each rod contains multiples of one of the nine digits, and is similar to one of the slips just described, the first rod containing the multiples of 0, 1, 9, 8, the second of 0, 2, 9, 7, the third of 0, 3, 9, 6, the fourth of 0, 4, 9, 5, the fifth of 1, 2, 8, 7, the sixth of 1, 3, 8, 6, the seventh of 1, 4, 8, 5, the eighth of 2, 3, 7, 6, the ninth of 2, 4, 7, 5, and the tenth of 3, 4, 6, 5. Each rod therefore contains on two of its faces multiples of digits which are com plementary to those on the other two faces ; and the multiples of a digit and of its comple ment are reversed in position. The arrange ment of the numbers on the rods will be evident from fig. 2, which represents the four faces of the fifth bar. The set of ten rods is thus equivalent to four sets of slips as described above, and by their means we may multiply every number less than 11,111, and also any number (consisting of course of not more than ten digits) which can be formed by the top digits of the bars when placed side by side. Of course two sets of rods may be used, and by their means we may multiply every number less than 111,111,111, and so on. It will be noticed that the rods only give the multiples of the number which is to be multiplied, or of 8 L Fig. 2. the divisor, when they are used for division, and it is evident that they would be of little use to any one who knew the multiplication table as far as 9 x 9. In multiplications or divisions of any length it is generally convenient to begin by forming a table of the first nine multiples of the multiplicand or divisor, and Napier s bones at best merely provide such a table, and in an incomplete form, for the additions of the two figures in the same parallelogram have to be performed each time the rods are used. The Habdologia attracted more general attention than the logarithms, and there were several editions on the Continent. An Italian translation was published by Locatello at Verona in 1623, and a Dutch translation by De Decker at Gouda in 1626. Ursiiius published his Rhabdologia, Ncpcriana&t Berlin in 1623, and the Rabdologia itself was reprinted at Lyons in 1626. Nothing shows more clearly the rude state of arithmetical knowledge at the beginning of the 17th century than? the universal satisfaction with which Napier s invention was wel comed by all classes and regarded as a real aid to calculation.
