OF MERCHISTON.] NAPIER 185 Napier also describes in the Eabdologia two other larger rods to facilitate the extraction of square and cube roots. In the Rabdo- lof/ia the rods are called "virgulfe," 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 formula in spherical trigono metry. "Napier s rules of circular parts," which include the com plete system of formula for the solution of right-angled triangles, may be enunciated as follows. Leaving the right angle out of con sideration, 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, j ; 90 A, 90 c, 90 B, 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 are 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 Mirificus (1614), and Napier has there given a figure, and indicated a method, by means of which they may be proved directly. The rules are curious and interesting, but of very doubtful utility, as the formula! are best remembered by the practical calculator in their unconnected form. " Napier s analogies" are the four formulae tanJ(A + tani(A-B)=^ tantfo- cotiC; cosi(A-B). , T tan 4 c, ,. sini(A-B), . tan i (a - b} = T . , { tan I c . sm|(A+B) They were first published after his death in the Constructio among the formulae 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 formulas contained in the Descriptio and Constructio is given by Delambre in vol. i. of his Histoire de V Astronomic moderne. To Napier seams to be due the first use of the decimal point in arithmetic. Decimal fractions were first introduced by Stevinus in his tract La Dismc, published in 1585, but he used cumbrous exponents (numbers enclosed in circles) to distinguish the different denominations, primes, seconds, thirds, &c. Thus, for example, he would have written 123 456 as 123@ 4 (T) 5 (T) 6 (V). In the Rab- dologia Napier gives an "Admonitio pro Decimal! 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 arith metical 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 in the course of performing an arithmetical operation. The decimal poinl 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 Constructio in the words: "In numeris periodo sic in se distinctis, quicqnid pos periodum notatur fractio est, cujus denominator est unitas cum to cyphris post se, quot sunt figime post periodum. Ut 10000000 "0- valet idem, quod 10000000^. Item 25 803, idem quod 25-^7 Item 9999998-0005021, idem valet quod 9999998^^1^, & sic d. creteris." On p. 8, 10 502 is multiplied by 3 216, and the resul found to be 33774432; and on pp. 23 and 24 occur decimals no attached to integers, viz., 4999712 and 0004950. These example show that Napier was in possession of all the conventions and attri butes that enable the decimal point to complete so symmetrical!} our system of notation, viz., (1) he saw that a point or separatrix wa quite enough to separate integers from decimals, and that no sign to indicate primes, seconds, &c., were required ; (2) he used cipher after the decimal point and preceding the first significant figure and (3) he had no objection to a decimal standing by itself withou any integer. Napier thus had complete command over decima 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 iar under the decimals ; this notation first appears without any xplanation in his " Lucubrationes " appended to the Constructio. 3riggs used the notation all his life, but in writing it, as appears rom manuscripts of his, he added also a small vertical line just ugh enough to fix distinctly which two figures it was intended to eparate : thus he might have written 63J0957379. The vertical line vas printed by Oughtred and some of Briggs s successors. It was a ong time before decimal arithmetic came into general use, and all hrough the 17th century exponential marks were in common use. "here seems but little doubt that Napier was the first to make use of i decimal separator, and it is curious that the separator which he ised, 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 Tohn Napier. The office, Mr Mark Napier states, is repeatedly nentioned in the family charters as appertaining to the "pultrc andis " near the village of Dene in the shire of Linlithgow. The luties were to be performed by the possessor or his deputy ; and the dug was entitled to demand the yearly homage of a present of joultry from the feudal holder. The pultrelands and the office vere s old by John Napier in 1610 for 1700 marks. It has been erroneously asserted that Napier dissipated his means ; there is no
- ruth in this statement. With the sole exception of the pultre-
auds 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 chil dren, 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 Mer chiston." 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 some times erroneously called "Peer of Merchiston," 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 super sunt, 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 Merchiston 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 supcrsunt. 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 Lynford, Buckingham shire. 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 astro logy, 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 1787 An Account of the Life, Writings and Inventions of John Napier ofMer- expla: arc vei all the v scanty, but, such as they are, they form the source frcm which nearly notices of Napier which have appeared since have been drawn. The work XVII. 24
