The names of nearly all Napier's classfellows can be traced as becoming determinantes in 1566 and masters of arts in 1568; but his own name does not appear in the lists.
The full title of this first work of Napier's is given below.'
2 Napier's Plaine Discovery is a serious and laborious work, to which he had devoted years of care and thought.
These rods, which were commonly called "Napier's bones," will be described further on.
Il mourut l'an 1616, et fut enterre hors la Porte Occidentale d'Edinbourg, dans l'Eglise de Sainct Cudbert.'" There can be no doubt that Napier's devotion to mathematics was not due to old age and the gout, and that he died in 1617 and not in 1616; still these sentences were written within eighteen years of Napier's death, and their author seems to have had some special sources of information.
It has been stated that Napier's mathematical pursuits led him to dissipate his means.
Briggs was greatly excited by Napier's invention and visited him at Merchiston in 1615, staying with him a whole month; he repeated his visit in 1616 and, as he states, "would have been glad to make him a third visit if it had pleased God to spare him so long."
(See Logarithm.) Napier's Descriptio of 1614 contains no explanation of the manner in which he had calculated his table.
Macdonald at Edinburgh in 1889, and that there is appended to this edition a complete catalogue of all Napier's writings, and their various editions and translations, English and foreign, all the works being carefully collated, and references being added to the various public libraries in which they are to be found.
Napier's priority in the publication of the logarithms is unquestioned and only one other contemporary mathematician seems to have conceived the idea on which they depend.
In that article it is mentioned that a Scotsman in 1594 in a letter to Tycho Brahe held out some hope of logarithms; it is likely that the person referred to is John Craig, son of Thomas Craig, who has been mentioned as one of the colleagues of John Napier's father as justice-depute.
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.
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.
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.
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.
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.
The principle of "Napier's bones" may be easily explained by imagir:ing ten rectangular slips of cardboard, each divided into nine squares.
The figures as written down are 12510 6255 14595 1534560 Napier's rods or bones consist of ten oblong pieces of wood or other material with square ends.
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.
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.
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.
"Napier's analogies" are the four formulae - tan (A +B) cos = cos l(a - b) cotzC, tan2(A - B) - sin(a - b)cot2C; a (a+b) sine (a+b) tan a(a +b) = cos(A - B) tan1c, tan2(a - b) = s in(A - B) tan2c.
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.
He has been sometimes erroneously called "Peer of Merchiston," and in the 1645 edition of the Flamm 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 workattached to W.
Napier's three mathematical works are reprinted by N.
The Boers, however, strongly resented the contention of the British that they could not shake off British nationality though beyond the bounds of any recognized British possession, nor were they prepared to see their only port garrisoned by British troops, and they rejected Napier's overtures.
Among the English histories of Florence, Napier's Florentine History (6 vols., London,1846-1847) and A.Trollope's History of the Commonwealth of Florence (4 vols., London, 1865) are not without value although out of date.
The first announcement of the invention was made in Napier's Mirifici Logarithmorum Canonis Descriptio..
Napier's logarithms are not the logarithms now termed Napierian or hyperbolic, that is to say, logarithms to the base e where e= 2.7182818 ...; the relation between N (a sine) and L its logarithm, as defined in the Canonis Descriptio, being N=10 7 e L/Ip7, so that (ignoring the factors re, the effect of which is to render sines and logarithms integral to 7 figures), the base is C".
Napier's logarithms decrease as the sines increase.
If 1 denotes the logarithm to base e (that is, the so-called "Napierian " or hyperbolic logarithm) and L denotes, as above, " Napier's " logarithm, the connexion between 1 and L is expressed by L = r o 7 loge 10 7 - 10 7 / or e t = I 07e-L/Ia7 Napier's work (which will henceforth in this article be referred to as the Descriptio) immediately on its appearance in 1614 attracted the attention of perhaps the two most eminent English mathematicians then living - Edward Wright and Henry Briggs.
Inl,1616 Briggs again visited Napier and showed him the work he had accomplished, and, he says, he would gladly have paid him a third visit in 1617 had Napier's life been spared.
It was published, probably privately, in 1617, after Napier's death,' and there is no author's name, place or date.
The first calculation or publication of Briggian or common logarithms of trigonometrical functions was made in 1620 by Edmund Gunter, who was Briggs's colleague as professor of 1 It was certainly published after Napier's death, as Briggs mentions his " librum posthumum."
It now remains to refer in more detail to the invention itself and to examine the claims of Napier and Briggs to the capital improvement involved in the change from Napier's original logarithms to logarithms to the base ro.
Rejecting therefore, those which he had prepared already, Briggs began, at Napier's advice, to consider seriously the question of the calculation of new tables.
There seems, however, no ground whatever for supposing that Briggs meant to express anything beyond his hope that the reason for the alteration would be explained in the posthumous work; and in his own account, written seven years after Napier's death and five years after the appearance of the work itself, he shows no injured feeling whatever, but even goes out of his way to explain that he abandoned his own proposed alteration in favour of Napier's, and, rejecting the tables he had already constructed, began to consider the calculation of new ones.
When the Descriptio was published Briggs was fiftyseven years of age, and the remaining seventeen years of his life were devoted with steady enthusiasm to extend the utility of Napier's great invention.
As regards priority of publication, Napier has the advantage by six years, and even fully accepting Bramer's statement, there are grounds for believing that Napier's work dates from a still earlier period.
560; de Morgan's article on " Tables " in the English Cyclopaedia; Mark Napier's Memoirs of John Napier of Merchiston (1834), p. 39 2, and Cantor's Geschichte der Mathematik, ii.
Dr Craig was John Craig, the third son of Thomas Craig, who was one of the colleagues of Sir Archibald Napier, John Napier's father, in the office of justice-depute.
We may infer therefore that as early as 1594 Napier had communicated to some one, probably John Craig, his hope of being able to effect a simplification in the processes of arithmetic. Everything tends to show that the invention of logarithms 2 See Mark Napier's Memoirs of John Napier of Merchiston (1834), p. 362.
An account has now been given of Napier's invention and its publication, the transition to decimal logarithms, the calculation of the tables by Briggs, Vlacq and Gunter, as well as of the claims of Byrgius and the method of prosthaphaeresis.
To Napier's Descriptio in order to describe its reception on the continent, and to mention the other logarithmic tables which were published while Briggs was occupied with his calculations.
He says that, although Napier's book had been published five years, he first saw it at Prague two years before; he was then unable to read it, but last year he had met with a little work by Benjamin Ursinus 4 containing the substance of the method, and he at once recognized the importance of what had been effected.
This letter was written two years after Napier's death (of which Kepler was unaware), and in the same year as that in which the Constructio was published.
In the same year (1620) Napier's Descriptio (1614) and Constructio (1619) were reprinted by Bartholomew Vincent at Lyons and issued together.5 Napier calculated no logarithms of numbers, and, as already stated, the logarithms invented by him were not to base e.