Page:EB1911 - Volume 19.djvu/189

 great simplification of which they were susceptible—a simplification to which nothing essential has since been added.

When Napier published the Canonis Descriptio England had taken no part in the advance of science, and there is no British author of the time except Napier whose name can be placed in the same rank as those of Copernicus, Tycho Brahe, Kepler, Galileo, or Stevinus. In England, Robert Recorde had indeed published his mathematical treatises, but they were of trifling importance and without influence on the history of science. Scotland had produced nothing, and was perhaps the last country in Europe from which a great mathematical discovery would have been expected. Napier lived, too, not only in a wild country, which was in a lawless and unsettled state during most of his life, but also in a credulous and superstitious age. Like Kepler and all his contemporaries he believed in astrology, and he certainly also had some faith in the power of magic, for there is extant a deed written in his own handwriting containing a contract between himself and Robert Logan of Restalrig, a turbulent baron of desperate character, by which Napier undertakes “to serche and sik out, and be al craft and ingyne that he dow, to tempt, trye, and find out” some buried treasure supposed to be hidden in Logan’s fortress at Fastcastle, in consideration of receiving one-third part of the treasure found by his aid. Of this singular contract, which is signed, “Robert Logane of Restalrige” and “Jhone Neper, Fear of Merchiston,” and is dated July 1594, a facsimile is given in Mark Napier’s Memoirs. As the deed was not destroyed, but is in existence now, it is to be presumed that the terms of it were not fulfilled; but the fact that such a contract should have been drawn 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 superstition: his Plaine Discovery relates to a method of interpretation 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 allusions to astrology or magic.

After Napier’s death his manuscripts and notes came into the possession of his second son by 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 18th 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 Renfrewshire. During the owner’s absence the house was burned to the ground, and all the papers and relics were destroyed. The manuscripts 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.—, 7th March 1801.”

An account of the contents of these manuscripts was given by 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 ''De Arte Logistica Joannis Naperi Merchistonii Baronis Libri qui supersunt. Impressum Edinburgi'' ., as one of the publications of the Bannatyne Club. The treatise occupies one hunclied and sixty-two pages, and there is an introduction by Mark Napier of ninety-four pages. The Arithmetic consists of three books, entitled—(1) De Computationibus Quantitatum omnibus Logisticae 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 Merchistonii Baronis consists of two books: (1) “De nominata Algebrae parte; (2) De positiva sive cossica Algebrae 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 Geometrie at Oxforde.”

These treatises were probably composed before Napier had invented the logarithms or any of the apparatuses described in the Rabdologia; for they contain no allusion to the principle of logarithms, even where we should expect to find such reference, and the one solitary sentence where the Rabdologia is mentioned (“sive omnium facillime per ossa Rhabdologiae nostrae”) was probably added afterwards. It is worth while to notice that this reference occurs in a chapter “De Multiplicationis 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 abundantes and defectivae for positive and negative, defining them as meaning greater or less than nothing (“Abundantes sunt quantitates majores nihilo: defectivae sunt quantitates minores nihilo”). The same definitions occur also in the Canonis Descriptio (1614), p. 5: “Logarithmos sinuum, qui semper majores nihilo sunt, abundantes vocamus, et hoc signo +, aut nullo praenotamus. Logarithmos autem minores nihilo defectivos vocamus, praenotantes 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 Merchiston papers is a thin quarto volume in Robert Napier’s writing containing 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 consists  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 another 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 together; for example, corresponding to the number 6 on the right-hand slip, we have 0, 8+3, 0+4, 2, 1; whence we find 0, 1, 5, 2, 1 as the digits, written backwards, of 6×2085. 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 obtained, 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— 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 complementary to those on the other two faces; and the multiples of a digit and of its complement are reversed in position. The arrangement of the numbers on the rods will be evident from fig. 2, which represents the four faces of the fifth rod. 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