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 Arithmetic consists of three books, entitled-(1) De Computationibns Quantitatum omnibus Logistici-e speciebus communium ; (2) De Logistics Arithmetica; (3) De Logistics Geometries. 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 Merehistonii Baronis consists of two books :-(1) "De nominata Algebraz parte ; (2) De positiva sive cossica Algebrie 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 Rabdologia ; for they contain no allusion to the principle of logarithms, even where we should expect to find such a reference, and the one solitary sentence where the Rabdologia is mentioned (" sive omnium facillime per ossa Rhabdologime nostra3") was no doubt added afterwards. It is worth while to notice that this reference occurs in a chapter "De Multiplicationis et Partitionis compcndiis 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 dcfcetim for positive and negative, defining them as meaning greater or less than nothing (" Abundantes sunt quantitates majores nihilo : defectivie stint quantitates minores nihilo"). The same definitions occur also in the Canon Minfieus (1614), p. 5:-" Logarithmos sinuum, qui semper majores nihilo sunt, abundantes vocamus, et hoc signo +, aut nullo prenotamus. Logarithmos autem minores nihilo defectivos vocamus, prmcnotantes eis hoc signum - ." Napier may thus have been the first to use the expression "quantity less than nothing." Ile uses "radieatum" 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 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 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 digit's) 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 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 bet performed each time the rods are used. The Rabdologia 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. Ursinns published his Rhabdologia Neperiana at 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 welcomed by all classes and regarded as a real aid to calculation.

Napier also describes in the Rabdologia two other larger rods to facilitate the extraction of square and cube roots. In the llabdologia the rods are called virgula," but in the passage quoted above from the manuscript on arithmetic they are referred to as " bones ".(ossa).

Besides the logarithms and the calculating rods or bones, Napier's name is attached to certain rules and formulae in spherical trigonometry. "Napier's rules of circular parts," which include the complete system of formula for the solution of right-angled triangles, may be enunciated as follows. Leaving the right angle out of consideration, the sides including the right angle, the complement of the hypotenuse, and the complements of the other angles are called the circular parts of the triangle. Thus there are five circular parts, a, b, 90° - c, 90°--13, and these are supposed to be arranged in this order (i.e., the order in which they occur in the triangle) round a circle. Selecting any part and calling it the middle part, the two parts next, it arc called the adjacent parts, and the remaining two parts the opposite parts. The rules then are - sine of the middle part product of tangents of adjacent parts - product of cosines of opposite parts.

These rules were published in the Canon Miri*us (1614), and Napier has there given a figure, and indicated a method, by means of which they may be proved directly. The rules arc curious and interesting, but of very doubtful utility, as the formulae arc best remembered by the practical calculator in their unconnected form.

"Napier's analogies" are the four formula - tau 1(i +B)--cos i(a - b)eos(a+ t)cotiC , sin i(a - b) tan 1(A B) - 4,(a + cot 4C ; tan .i(a+b) - cosi(A - cos + B) C sin /(A B) tan Li (a - b) - sin mA tan I e .

They were first published after his death in the Constructio among the formula in spherical trigonometry, which were the results of his latest work. Robert Napier says that these results would have been reduced to order and demonstrated consecutively but for his father's death. Only one of the four analogies is actually given by Napier, the other three being added by Briggs in the remarks which are appended to Napier's results. The work left by Napier is, however, rough and unfinished, and it is uncertain whether he knew of the other formulae or not. They are, however, so simply deducible from the results he has given that all the four analogies may be properly called by his name. An analysis of the formula contained in the Descriptio and Construetio is given by Delambre in vol. i. of his JIisloire de )'Astronomic modern,,.

To Napier seems to be duo the first use of the decimal point in arithmetic. Decimal fractions were first introduced by Stevinus in his tract Lee Disme, published in 1535, hut he used cumbrous exponents (numbers enclosed in circles) to distinguish the different denominations, primes, seconds, thirds, Sze. Thus, for example, he would have written 123-456 as 12300 4 0 5 0 6 C.3). In the Babdologia Napier gives an " Admonitio pro Decimali Arithmetica," in which he commends the fractions of Stevinus and gives an example of their use, the division of 861094 by 432. The quotient is written 1993,273 in the work, and 1993,2'7"3'" in the text. This single instance of the use of the decimal point in the midst of an arithmetical process, if it stood alone, would not suffice to establish a claim for its introduction, as the real introducer of the decimal point is the person who first saw that a point or line as separator was all that was required to distinguish between the integers and fractions, and used it as a permanent notation and not merely iu the course of performing an arithmetical operation. The decimal point is, however, used systematically in the Constructio (1619), there being perhaps two hundred decimal points altogether in the book.

The decimal point is defined on p. 6 of the Construetio in the words : - " In numeris periodo sic in se distinctis, quicqnid post periodum notatur fractio est, cujus denominator eat cum tot cyphris post se, quot aunt figurre post periodum. Ut 10000000.04 valet idem, quod 10000000AT. Item 25.803, idem quod 25A947. Item 9999998.0005021, idem valet quod 9999998i-14m,, & sic do cateris." On p. 8, 10.502 is multiplied by 3.216, and the result found to be 33.774432; and on pp. 23 and 24 occur decimals not attached to integers, viz., .4999712 and .0004950. These examples show that Napier was in possession of all the conventions and attributes that enable the decimal point to complete so symmetrically our system of notation, viz., (1) he saw that a point or separatrix was quite enough to separate integers from decimals, and that no signs to indicate primes, seconds, Sze., were required ; (2) he used ciphers after the decimal point and preceding the first significant figure ; and (3) he had no objection to a decimal standing by itself without any integer. Napier thus had complete command over decimal fractions, and understood perfectly the nature of the decimal point.

Briggs also used decimals, but in a form not quite so convenient as Napier. Thus he prints 63.0957379 as 630957379, viz., he prints a bar under the decimals ; this notation first appears without any explanation in his " Lucubrationes" appended to the Constructio. Briggs used the notation all his life, but in writing it, as appears from manuscripts of his, he added also a small vertical line just high enough to fix distinctly which two figures it was intended to separate: thus he might have written 63;0957379. The vertical line was printed by Oughtred and some of Briggs's successors. It was a long time before decimal arithmetic came into general use, and all through the 17th century exponential marks were in common use. There seems but little doubt that Napier was the first to make use of a decimal separator, and it is curious that the separator which he used, the point, should be that which has been ultimately adopted, and after a long period of partial disuse.

The hereditary office of king's poulterer (Pultrie Regis) was for many generations in the family of Merchiston, and descended to John Napier. The office, Mr Mark :Napier states, is repeatedly mentioned in the family charters as appertaining to the " pultre landis " near the village of Dene in the shire of Linlithgow. The duties were to be performed by the possessor or his deputy ; and the king was entitled to demand the yearly homage of a present of poultry from the feudal holder. The pultrelands and the office were sold by John Napier in 1610 for 1700 marks. It has been erroneously asserted that Napier dissipated his means ; there is no truth in this statement. With the sole exception of the pultrelands all the estates he inherited descended undiminished to his posterity.

With regard to the spelling of the name, Mr Mark Napier states that among the family papers there exist a great many documents signed by John Napier. His usual signature was "Jhone Neper," but in a letter written in 1608, and in all deeds signed after that date, he wrote "Jhone Nepair." His letter to the king prefixed to the Plains Discovery is signed "John Napeir." His own children, who sign deeds along with him, use every mode except Napier, the form now adopted by the family, and which is comparatively modern. In Latin he always wrote his name " Neperus." The form " Neper " is the oldest, as John, third Napier of Merchiston, so spelt it in the 15th century.

Napier frequently signed his name "Jhone Neper, Fear of Merchiston." He was "Fear of Merchiston" because, more majorum, he had been invested with the fee of his paternal barony during the lifetime of his father, who retained the liferent. He has been sometimes erroneously called " Peer of Merehiston," and in the 1645 edition of the Plaine Discovery he is so styled, probably by a misprint (see Mr Mark Napier's Memoirs, pp. 9 and 173, and Libri qui supersunt, p. xciv).

Napier's home at Merchiston is thus described by Sir Walter Scott in his Provincial Antiquities of Scotland : - " This fortalice is situated upon the ascent, and nearly upon the summit of the eminence called the Borough-moor-head, within a mile and a half of the city walls. In form it is a square tower of the 14th or 15th century, with a projection on one side. The top is battlemented, and within the battlements, by a fashion more common in Scotland than in England, arises a small building with a steep roof, like a stone cottage erected on the top of the tower. . . The celebrated John Napier of Merehiston was born in this weather-beaten tower ; and a small room in the summit is pointed out as the study in which he secluded himself while engaged in the mathematical researches which led to his great discovery. The battlements of Merchiston tower command an extensive view of great interest and beauty." There is a view of Merchiston tower in Mr Mark Napier's Memoirs of John Napier, and in the Libri qui supersunt.

One well-known character of the time, Dr Richard Napier, was cousin to John Napier. The eldest son of Alexander, sixth Napier of Merchiston, ; was Archibald, the father of John Napier • his second son, named Alexander, settled at Exeter, and married an English lady by whom he had two sons, the eldest of whom, Robert, was the merchant, mentioned in the note near the beginning of this article as having been created a baronet. The second son was a fellow of Exeter College, Oxford, and became rector of Lynforcl, Buckinghamshire. He was a friend and pupil of Dr Simon Forman, a well-known Rosicrucian adept of the time, and at his death became the possessor of his secret manuscripts. Dr Richard Napier, who was more of a physician than a divine, was a great pretender to astrology, necromancy, and magical cures. There is a portrait of him in the Ashmolean Museum, Oxford (engraved in Mr Mark Napier's Memoirs), which is interesting on account of the similarity of some of the features to those of John Napier. It does not appear that there was ever any friendship or correspondence between John Napier and Richard Napier.

In 17S7 An Account of The Life, Writings, and Inrentions of John 1'apterofIferli chiston was published at Perth by David Stewart, earl of Buchan, and Walter Mint°, LL.D. It Is a quarto work of one hundred and thirty-four pages, only twelve of which relate to the life of Napier, the rest being devoted to a careful explanation of the contents of his works. The particulars given of Napier's life are very scanty, but, such as they arc, they form the source frem which nearly all the notices of Napier which have appeared since have been drawn. The work has also given rise to the impression that there was but little chance of further Information being obtained with respect to Napier's life. In 1831 Mr Mark Napier published his Memoirs of John ropier of Herchiston, his Life, Lineage, and Times, with a history of the Invention of Logarithms, a large quarto volume of five hundred and thirty-four pages. Mr Mark Napier, who had already devoted great attention to the history of Scotland with special reference to the families of Lennox and Napier, had full access to all the charters and papers in the possession of the family, and he spared no pains in examining every document and Investigating every point which seemed likely to throw light upon the life of Napier and the circumstances amidst which his life was passed. The work contains a vast mass of general information relating to Napier and his relatives, and the people with whom he was brought into contact, besides much collateral matter which serves to illustrate the state of the country at the time. The facts relating to Napier's own life are so interwoven with the other contents of the volume, and the work is so large, that in the absence of an index it Is very difficult to extract the comparatively small portion that relates to Napier himself. From this work, which is the sole authority upon the private events of Napier's life, all the facts given in this article with respect to his descent and personal history have been derived. In 1839 Mr Mark Napier completed his labours by editing Napier's unpublished manuscripts, of which he had only been able to give a rdsmnd in the Memoirs, and to this he prefixed an introduction, the greater part of which, however, Is included In the Memoirs. Three different portraits of Napier are known to be In existence ; one was engraved as the frontispiece to the earl of Buchan's Account, and another forms the frontispiece to the Memoirs. There is also an engraving of Napier in Lilly's Life and Times (1822). Foran account of the contents of Napier's mathematical works and their place in the history of science, the reader ii ref et red to Delambre's Histoire de l'Astronomie moderne.

It maybe useful to give, in conclusion, a list of Napier's works with a brief statement of the contents of each. The works published in his lifetime are - (1) The Plaine Discovery (1593), containing an interpretation of the Book of Revelation ; (2) the Canonis Minfici Logarithmorum Descriptio, containing the first announcement of the invention of logarithms and a table of log sines, also the rules of circular parts ; (3) the flabdologia (1617), containing the description of Napier's bones, the promptnary; and the method of local mithmetic, - all three designed for the simplification of multiplications and divisions. The posthumous works are - (1) the Canonis Minfici Logarithmorum Constructio (1619), edited by his son Robert, containing an account of the mode of construction of the canon, and Napler's analogies ; this book is the first in which the decimal point was systematically employed ; (2) the treatise De Arte Logistica, edited by Mr Mark Napier in 1839, containing treatises on arithmetic and algebra, transcribed from Napler's notes by his son Robert. (J. W. L. G.)