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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 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 9×9. 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 Rabdologia attracted more general attention than the logarithms, and as has been mentioned, there were several editions on the Continent. 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 Rabdologia the rods are called “virgulae,” 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 formulae 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°—,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
⁠sine of the middle part = product of cosines of opposite parts.

These rules were published in the Canonis Descriptio (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 formulae are best remembered by the practical calculator in their unconnected form.

“Napier’s analogies” are the four formulae—

$\tan {\tfrac {1}{2}}\left({\mbox{A}}+{\mbox{B}}\right)={\frac {\cos {\tfrac {1}{2}}\left(a-b\right)}{\cos {\tfrac {1}{2}}\left(a+b\right)}}\cot {\tfrac {1}{2}}{\mbox{C}}$,⁠	$\tan {\tfrac {1}{2}}\left({\mbox{A}}-{\mbox{B}}\right)={\frac {\sin {\tfrac {1}{2}}\left(a-b\right)}{\sin {\tfrac {1}{2}}\left(a+b\right)}}\cot {\tfrac {1}{2}}{\mbox{C}}$;
$\tan {\tfrac {1}{2}}\left(a+b\right)={\frac {\cos {\tfrac {1}{2}}\left({\mbox{A}}-{\mbox{B}}\right)}{\cos {\tfrac {1}{2}}\left({\mbox{A}}+{\mbox{B}}\right)}}\tan {\tfrac {1}{2}}c$,	$\tan {\tfrac {1}{2}}\left(a-b\right)={\frac {\cos {\tfrac {1}{2}}\left({\mbox{A}}-{\mbox{B}}\right)}{\cos {\tfrac {1}{2}}\left({\mbox{A}}+{\mbox{B}}\right)}}\tan {\tfrac {1}{2}}c$.

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 formulae contained in the Descriptio and Constructio is given by Delambre in vol. i. of his Histoire de l’Astronomie moderne.

To Napier seems to be due the first use of the decimal point in arithmetic. Decimal fractions were first introduced by Stevinus in his tract La Disme, 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 123⓪4①5②6③. In the Rabdologia 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 in 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 Constructio in the words: “In numeris periodo sic in se distinctis, quicquid post periodum notatur fractio est, cujus denominator est unitas cum tot cyphris post se, quot sunt figurae post periodum. Ut 10000000·04 valet idem, quod 100000004/100. Item 25·803, idem quod 25803/1000 Item 9999998·0005021, idem valet quod 99999985021/10000000, & sic de caeteris.” 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, &c., 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 the use 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 seems to have 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, 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. With the exception of the pultrelands all the estates he inherited descended to his posterity.

With regard to the spelling of the name, 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 Plaine 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 Merchiston,” and in the 1645 edition of the Plaine Discovery he is so styled (see Mark Napier’s Memoirs, pp. 9 and 173, and Libri qui supersunt, p. xciv.).

The bibliography of Napier’s work attached to W. R. Macdonald’s translation of the Canonis Constructio (1889) is complete and valuable. Napier’s three mathematical works are reprinted by N. L. W. A. Gravelaar in Verhandelingen der Kon. Akad. van Wet te Amsterdam, 1. sectie, deel 6 (1899).  (J. W. L. G.) 

NAPIER, SIR WILLIAM FRANCIS PATRICK (1785–1860), British soldier and military historian, third son of Colonel George Napier (1751–1804), and brother of Sir Charles James Napier (see above), was born at Celbridge, near Dublin, on the 17th of December 1785. He became an ensign in the Royal Irish Artillery in 1800, but at once exchanged into the 62nd, and was put on half-pay in 1802. He was afterwards made a cornet in the Blues by the influence of his uncle the duke of Richmond, and for the first time did actual military duty in this regiment, but he soon fell in with Sir John Moore’s suggestion that he should exchange into the 52nd, which was about to be trained in the famous camp of Shorncliffe. Through Sir John Moore he soon obtained a company in the 43rd, joined that regiment at Shorncliffe and became a great favourite with Moore. He served in Denmark, and was present at the engagement of Kioge, and his regiment being shortly afterwards sent to Spain, he bore himself nobly through the retreat to Corunna, the hardships of which permanently impaired his health. In 1809 he became