**JOHN NAPIER** (1550-1617), the inventor of logarithms, was born at merchiston near Edinburgh in 1550, and was the eighth Napier of Merchiston, The first Napier of Merchiston, "Alexander Napare," acquired the Merchiston estate before the year 1438, from James I. of Scotland. He was provost of Edinburgh in 1437, and was otherwise distinguished. His provost of Edinburgh in 1455, 1457, and 1469; he was knighted and held various important court offices under successive monarchs; at the time of his death in 1473 he was master of the household to James III. His son, John Napier of Rusky, the third of Merchiston, belonged to the royal household in the lifetime of his father. He also was provost of Edinburgh at various times, and it is a remarkable instance of the esteem in which the lairds of Merchiston were held that three of them in immediate lineal succession repeatedly filled so important an office during perhaps the most memorable period in the history of the city. He married a great granddaughter of Duncan, eighth earl of Levenax (or Lennox), and besides this relationship by marriage the Napiers claimed a lineal male cadency from the ancient family of Levenax. His eldest son, Archibald Napier of Edinbillie, the fourth of Merchiston, belonged to the household of James IV. He fought at Flodden and escaped with his life, but his eldest son Alexander (fifth of Merchiston) was killed. Alexander’s eldest son (Alexander, sixth of Merchiston) was born in 1513, and fell at the battle of Pinkie in 1547. his eldest son was Archibald, seventh of Merchiston, and the father of John Napier, the subject of this article.

In 1549 Archibald Napier, at the early age of about fifteen, married Janet, daughter of Francis Bothwell, and in the following year John Napier was born. In the criminal court of Scotland, the earl of Argyll, hereditary justice-general of the kingdom, sometimes presided in person, but more frequently he delegated his functions; and it appears that in 1561 Archibald Napier was appointed one of the justice-deputes. In the register of the court, extending over 1563 and 1564, the justice-deputes named are "Archibald Naper of Merchistoune, Alexander Bannatyne, burtyne, burgess of Edinburgh, James Stirling of Keir, and Mr Thomas Craig.’ About 1565 he was knighted at the same time as James Stirling, his colleague,whose daughter John Napier subsequently married. In 1582 Sir Archibald was appointed master of the mint in Scotland, with the sole charge of superintending the mines and minerals within the realm, and this office he held till his death in 1608. His first wife died in 1563, and in 1572 he married a cousin, Elizabeth Mowbray, by whom he had three sons, the eldest of whom was named Alexander.1

As stated above, John Napier was born in 1550, the year in which the Reformation in Scotland may be said to have commenced. In 1563, the year in which his mother died he matriculated at St Salvator’s College, ST Andrews. He early became a Protestant champion, and the one solitary anecdote of his youth that is known to exist occurs in his address "to be Godly and Christian reader" prefixed to his Plaine Discovery. He writes: -

"In my tender yeares, and barneage in Sanct-Ardrois at the Schooles, having, on the one parte, contracted a loving familiaritie with a certainne Gentleman, &c. a Papist; And on the other part, being attentive to the sermons of that worthie man of God, Maister Christopher Goodman, teaching upon the Apocalyps, I was so moved in admiration, against the blindness of Papists, that could not most evidently see their seven hilled citie Rome, painted out there so lively by Saint John, as the mother of all spiritual whoredome, that not onely bursted I out in continual reasoning against my said familiar, but also from thenceforth, I determined with my selfe (by the assistance of Gods spirit) to employ my studie and diligence to search out the remanent mysteries of that holy Book: as to this hourse (praised be the Lorde) I have bin doing at al such times as conveniently I might have occasion."

The names of nearly all Napier’s classfellows can be traced as becoming determinantes in 1566 and masters of arts in 1568; but his won name does not appear in the lists. The necessary inference is that his stay at the university was short, and that only the groundwork of his education was laid there. Although there is no direct evidence of the fact, there can be no doubt that he left St Andrews to complete his education in the Continent, and that he probably studied at the university of Paris, and visited Italy and Germany. He did not, however, as has been supposed, spend the best years his manhood abroad, for he was certainly at home in 1571, when, being just of age, the preliminaries of his marriage were arranged at Merchiston; and for many years after that event he took an earnest interests in the affairs of the church, the most engrossing element in the politics of the time.

In 1572 John Napier married Elizabeth, daughter of Sir James Stirling of Kier, mentioned above. This marriage took place almost immediately after his father’s second marriage. About the end of the year 1579 John Napier’s wife died, leaving him one son, Archibald, the first Lord Napier, and one daughter, Jane. A few years afterwards he married again, the name of his second wife being Agnes Chisholme, and by her he had ten children, five sons and five daughters.

On the 17th of October 1593 a convention of delegates was held at Edinburgh at which a committee was appointed to follow the king and lay before him in a personal interview certain instruction relating to the punishment of the rebellious Popish earls and the safety of the church. This committee consisted of six members, two barons, two ministers, a nd two burgesses – the two barons selected being John Napier of Merchiston and james Maxwell of Calderwood. The delegates found the king at Jedburgh, and the mission, which was a dangerous one, was successfully accomplished. Shortly afterwards another convention was held at Edinburgh, and it was resolved that the delegates sent to Jedburgh should again meet the king at Linlithgow and repeat their former instructions. This was done accordingly, the number of members of the committee being, however, doubled. These interviews took place in October 1593, and on the 29th of the following January Napier wrote to the king the letter which forms the dedication of the Plaine Discovery.

The full title of this first work of Napier’s is given below.1 It was written in English instead of Latin in order that "hereby the simple of this Iland may be instructed"; and the author states that he "was constrained of compassion, leaving the Latine, to haste out in English this present worke." He apologizes also for the language and his own mode of expression in the following sentences: --

"Whatsoever therefore through hast, is here rudely and is base language set downe, I doubt not to be pardoned thereof by all good men, who, considering the necessitie of this time, will esteem it more meete to make hast to prevent the rising againe of Anti-hristian darknes within this Iland, then to prolong the time in painting of language"; and "I garunt indeede, and am sure, that in the style of wordes and utterance of language, we shall greatlie differ, for therein I do judge my selfe inferior to all men: so that scarcely in these high matters could I with long deliberation finde wordes to expenses my minde"2

It is not to be supposed that Napier’s Plaine Discovery was in any respect due to a visionary cloud passing over his mind, or to any temporary infatuation; on the contrary, it is a serious and laborious work, to which he had devoted years of care and thoughts, and which is closely connected with the history of the times. In one sense it may be said to stand to theological literature in Scotland in something of the same position as that occupied by the Canon Mirificus with respect to the scientific literature, for it is the first published original work relating to theological interpretation, and is quite without a predecessor in its own field. In judging of the book, the circumstance of the time and the state of the country have to be taken into account. Napier lived in the very midst of fiercely contending religious factions, and his home was situated in a district which was the scene of constant wars an disturbances; there was but little theological teaching of any kind, and the work related to what were then the leading political and religious questions of the day.

After the publication of the Plaine Discovery, Napier seems to have occupied himself with the invention of secret instruments of war, for in the Bacon collection at Lambeth Palace there is a document, dated June 7, 1596, and signed by Napier, giving a list of this inventions for the defence of the country against the anticipated invasion by Philip of Spain. The document is entitled "Secrett Inventionis, proffitabill and necessary in theis dayes for defence of this Iland, and withstanding of strangers, enemies of God’s truth and religion,"3 and the inventions consist of (1) a mirror for burning the enemies’ ships at any distance, (2) a piece of artillery destroying everything round an arc of a circle, and (3) a round metal chariot, so constructed that its occupants could move it rapidly and easily, while firing out through small holes in it. It has been asserted (by Sir Thomas Urquhart) that the piece of artillery was actually tried upon a plain in Scotland with complete success, a number of sheep and cattle being destroyed.

In 1614 appeared the work which in the history of British science can be placed as second only to Newton’s Principia. The full title is as follows: -- Mirifici Logarithmorum Canonis description, Ejusque usus, in utraque Trigonometria; ut, etiam in omni Logistica Mathematica, Amplissimi, Facillimi & expeditissimi explication. Authore ac Inventore Ioanne Nepero, Barone Merchistonii, &c., Scoto. Edinburgi, ex official Andreae hart Biblioplae, CI_. DC. XVI. This is printed on an ornamental title-page. The work is a small-sized quarto, containing fifty-seven pages of explanatory matter and ninety pages of tables.

The nature of logarithms is explained by reference to the motion of points in a straights line, and the principle upon which the are based is that of the correspondence of a geometrical and an arithmetical series of numbers. The table gives the logarithms of sines for every minute to seven figures; it is arranged semi-quadrantally, so that the differentiae, which are the differences of the two logarithms in the same line, are the logarithms of the tangents. Napier’s logarithms are not the logarithms now termed Napieria, that is to say, logarithms to the base e where e =2·7182818…; but they are closely related to this system =, the connexion being expressed by the equation –

Log Nap. N=10,000,000 loge (10,000,000) – 10,000,000 loge n;

Or log Nap. N=107 loge (107/n).

A translation of the Canon Mirificus into English was made by Edward Wright, and published after his death by his son Samuel Wright, at London, in 1618, under the title A Description of the admirable Table of Logarithms. Edward Wright, who was a fellow of Caius College, Cambridge, occupies a conspicuous place in the history of navigation. In 1599 he published Centaine errors in Navigation detected and corrected, and he was the author of other works; to him also is chiefly due the invention of the method known as Mercator’s sailing. He at once saw the value of logarithms as an aid to navigation, and lost no time in preparing a translation, which he submitted to Napier himself. The preface to Wright’s edition consists of a translation of the preface to the Canon Mirificus, together with the addition of the following sentences written by Napier himself: --"But now some of our Countrymen in this Island well affected to these studies, and the more publique good, procured a most learned Mathematician to more to translate the same into our vulgar English tongue, who after he had finished it, sent the Coppy of it to me, to bee seene and considered on by myselfe. I having most willingly and gladly done the same, finde it to bee most exact and precisely conformable to my minde and the originall. Therefore it may please you who are inclined to these studies, to receive it from me and the Translator, with as much good will a we recommend it unto you."

There is a short "preface to the reader" by Briggs, and a description of a triangular diagram invented by Wright for finding the proportional parts. The table is printed to one figure less than in the Canon Mirificus. Edward Wright died in 1615, and his son in the preface states that his father "gave much commendation of this work (and often in my hearing) as of every great use for mariners"; and with respect to the translation he says that "shortly after he had it returned our of Scotland, it pleased God to call him away afore he could publish it."

In 1617 Napier published his Rabdologia,1 a duodecimo of one hundred and fifty-four pages; there is prefixed to it as preface a dedicatory epistle to the high chancellor of Scotland. The method which Napier terms "Rabdologia" consists in the use of certain numerating rods for the performance of multiplications and division. These rods, which were commonly called "Napier’s bones," will be described further on. The second method, which he calls the "Promptuarium Multiplications" on account of its being the most expeditions of all for the performance of multiplications, involves the use of a number of lamellae or little plates of metal disposed in a box. In an appendix of forty-one pages he gives his third method, "local arithmetic," which is performed on a chess-board, and depends, in principle, on the expression of numbers in the scale of radix 2. In the Rabdologia he gives the chronological order of his inventions. He speaks of the canon of logarithms as "a me longo tempore elaboratum." The other three methods he devised for the sake of those who would prefer to work with natural numbers; and he mentions that the promptuary was his latest invention. In the preface to the appendix containing the local arithmetic he states that, while devoting all his leisure to the invention o these abbreviations of calculation, and to examining by what methods the toil of calculation might be removed, in addition to the logarithms, rabdologia, and promptuary, he had ht upon a certain tabular arithmetic, whereby the more troublesome operations of c ommon arithmetic are performed on an abacus or chess-board, and which may be regarded as an amusement rather than a labour, for, by means of it, addition, subtraction, multiplication, division, and even the extraction of roots are accomplished simply by the motion of counters. He adds that he had appended it to the Rabdologia, in addition to the promptuary, because he did not wish to bury it in silence, not to publish so small a matter by itself. With respect to the calculating rods, Napier mentions in the dedication that they had already found so much favour as to be almost in common use, and even to have been carried to foreign countries; and that he has been advised to publish his little work relating to their mechanism and use, lest they should be put forth in some one else’s name.

John Napier died on April 4, 1617, the same year as that in which the Rabdologia was published, so that his death must have taken place very soon after its appearance. His will, which is extant, was signed on the fourth day before his death. No particulars are known of his last illness, but it seems likely that death came upon him rather suddenly at least. In both the Canon Mirificus and the Rabdologia, however, he makes reference to his ill-health. In the dedication of the former he refers to himself as "mihi jam morbis penè confecto," and in the "Admonitio" att he end he speaks of his "infirma valetudo"; while in the latter he says he has been obliged to leave the calculation o the new canon of logarithms to others "ob infirmam corporis nostril valetudinem."

It is usually stated that John Napier was buried in St Giles’schurch, Edinburgh, and there can be no doubt that some of the family of Napier were buried there in the `6th or 17th century, but the late Professor Wallace in a paper read before the Society of Antiquaries of Scotland in 1832. and quoted by Mr Mark Napier on pp. 425-427 of his Memoris of John Napier of Merchiston, gives evidence for believing that he was buried in St Cuthbert’s church. Professor Wallace’s words are –

"My authority for this belief is unquestionable. It is a Treatise on trigonometry, by a Scotsman, James Hume of Godscroft, Berwickshire, a place still in possession of the family of Hume. The work in question, which is rare, was printed at Paris, and has the date 1636 on the title-page, but the royal privilege which secured it to the author is dated in October 1635, and it may have been written several years earlier. In his treatise page (page 116) Hume says, speaking of logarithms, "L’inuenteur estoit un Seigneur de grande condition, et duquel la posterité est aujourd’huy en possession de grandes dignitéz dans le royaume, qui estant sur l’age, et grandement trauaillé des gouttes ne pouvait faire autre chose que de s’adonner aux sciences, et principlament aux mathematiques et à la logistique, à quoy il se plaisoit infiniment, et auec estrange peine, a construict ses Tables des Logarymes, imprimees à Edinbourg en l’an 1614… Il mourut l’an 1616, et fut enterré hors la Porte Occidentale d’Edinbourg, dans l’Eglise de Saint Cudbert."

There can be no doubt that Napier’s devotion to mathematics was not due to old 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. Additional probability is given to Hume’s assertion by the fact that Merchiston is situated in St Cuthbert’s parish. It is nowhere else recorded that napier suffered from the gout.

The canon Mirific us contains only an explanation of the sue of the logarithms without any account of the manner in which the canon was constructed. In an "Admonitio" on the seventh page he states that, although in that place the mode of construction should be explained, he proceeds at once to the use of the logarithms, "ut praelibatis prius use, et rei utilitate, caetera aut magis placeant posthac edenda, aut minus saltem displiceant silentio sepulta." He awaits therefore the judgement and censure of the learned "priusquam caetera in lucem temerè prolata lividorum detrectationi exponantur"; and in an "Admonitio" on the last page of the book he states that he will publish the mode of construction of the canon "si huius inventi usum eruditis gratum fore intellexero."

Napier, as we have seen, died in 1617, immediately after the appearance of the Rabdologia, and before he had published the promised account of the method of construction of the canon. This work was, however, issued by Robert Napier, his second son by his second marriage, in 1619.1 The Constructio consists of a preface of two pages, followed b y sixty-seven pages of text. In the preface Robert Napier says that he has been assured form undoubted authority that the new invention is much though of by the ablest mathematicians, and that nothing would delight them more than the publication of the mode of construction of the canon. He therefore issues the work to satisfy their desires, although, he states, it is manifest that it would have seen the light in a far more perfect state if his father could have put the finishing touches to it; and he mentions that, in the opinion of the best judges, his father possesses, among other most excellent gifts, in the highest degree the power of explaining the most difficult matters by a certain and easy method in the fewest possible words.

It is important to notice that in the Constructio logarithms are called artificial numbers; and Robert Napier states that the work was composed several years (aliquot annos) before Napier had invented the name logarithm. The Constructio therefore may have been written a good many years previous to the publication of the Descriptio in 1614.2 The Canon Mirificus, on its appearance in 1614, at once attracted the attention of perhaps the two most eminent English mathematicians then living – Edward Wright and Henry Briggs. The former, as we have seen, translated the work into English, but died in 1615 before he could publish his translation. The latter was concerned with Napier in the change of the logarithms from those originally invented to decimal or common logarithms, and it is to him that the original calculation of the logarithmic tables now in use is mainly due (see BRIGGS). He died on January 26, 1631, aged about seventy-four years, so that at the time of the publication of the canon Mirificus he was about fifty-seven years of age. In a letter to Archbishop Ussher, dated Gresham House, March 10, 1615, Briggs wrote, "napper, lord of MArkinston, hath set my head and hands a work his new and admirable logarithms. I hope to see him this summer, if it please God, for I never saw book which pleased me better, or made me more wonder.3 I purpose to discourse with him concerning eclipse, for what is there which we may not hope for at his hands"; and he also states "that he was wholly taken up and employed about the noble invention of logarithms, lately discovered." In the summer of 1615 he went to Merchiston and stayed with Napier a whole month; he repeated his visit in 1616, and, he states, he ‘would have been glad to make him a third visit, if it had pleased God to spare him so long." William Lilly, the astrologer, in his life and Times, 1721, gives the following account of the meeting between Napier and Briggs on the occasion of the first visit: --

"I will acquaint you with one memorable story, related unto me By Mr John Marr, an excellent mathematician and geometrician, whom I conceive you remember: he was servant to King James and Charles I. at first, when the Lord Napier, or Marchiston, first made publick his logarithms, Mr Briggs, then reader of the Astronomy lecture at Gresham College in London, was so surprised with admiration of them that he could have no quietness in himself, until he had seen that noble person the Lord Marchiston whose went into Scotland before Mr Briggs, purposely to be there when these two so learned persons should meet. Mr Briggs appoints a certain day when to meet at Edinburgh: but failing thereof, the Lord Napier was doubtful he would not come. It happened one day as John Marrand the Lord Napier were speaking of Mr Briggs; ‘Ah, John,’ saith Marchiston, ‘Mr Briggs will not come: at the very instant one knocks at the gate; John Marr hasted down and it proved Mr Briggs to his great contentment. He brings Mr Briggs up into my Lord’s chamber, where almost one quarter of an hour was spent, each beholding the other almost with admiration, before one word was spoke. At last Mr Briggs began – ‘My Lord, I have undertaken this long journey purposely to see your person, and to know by what engine of wit or ingenuity you came first to think of this most excellent help unto Astronomy, viz, the Logarithms; but my Lord, being by you found out, I wonder nobody else fund it out before, when now known it is so easy.’ He was nobly entertained by the Lord Napier, and every summer after that, during the Lord’s being alive, this venerable man, Mr Briggs went purposely into Scotland to visit him."

With respect to the change of the logarithms to decimal logarithms, the concluding paragraph of the "Admonitio" which appears on the last page of the canon of 1614 is "Verum si huis inventi usum eruditis gratum fore intelexero, dabo fortasse brevi (Deo aspirante) rationem ac methodum aut hunc canonem ememdandi, aut ememdatiorem de novo condendi, ut ita plurum Logistarum diligentia, limatior tandem et accuratior, quam unius opera fieri potuit, in lucem prodeat. Nihil in ortu perfectum." In some copies, however, this "Admonitio" is absent. In Wright’s translation of 1616 Napier has added the sentence – "But because the addition and subtraction of these former numbers may seeme somewhat painfull, I intend (if it shall please God) in a second Edition, to set out such Logarithmes as shall make those numbers above written to fall upon decimal numbers, such as 100,000,000, 200,000,000, 300,000,000, &c., which are easie to be added or abated to or from any other number" (p. 19); and in the dedication to the Rabdologia (1617) he wrote "Quorum quidem Logarithmorum speciem aliam multò praestantiorem nunc etiam invenimus, & creandi methodum, unà cum eorum usu (si Deus longiorem vitae & valetudinis usuram oncsserit) evulgare statuimus; ipsam autem novi canonis supputationem, ob infirmam corporis nostril valetudinem, viris in hoc studii genere versatis relinquimus: imprimis verò doctrissimo viro D. Henrico Briggio Londini publico Geometriae Professori, et amico mihi longé charissimo."

Briggs published in 1617, after Napier’s death, his Logarithmorum Chilias Prima, containing the decimal logarithms of the first thousand numbers to 14 places of decimals. This is the first table of common (or Briggian) logarithms calculated or published. In 1624 he published his Arithmetica Logarithmica, containing the logarithms of the first 20,000 numbers and of the numbers from 90,000 to 100,000 to 14 places of decimals. In the short preface to the Chilias (1617) Briggs states that the reason why his logarithms are different from those introduced by Napier "sperandum, ejus librum posthumu, abunde nobis propediem satisfacturum." The "liber posthumus" was the Constructio (1619), in the preface to which Robert Napier states that he has added an appendix relating to another and more excellent species of logarithms referred to by the inventor in the Rabdologia, and in which the logarithm of unity is 0. He also mentions that he has published some remarks upon the propositions in spherical trigonometry and upon the new species of logarithms by Henry Briggs, "qui novi hujus Canonis supputandi laborem gravissimum, pro singulari amicitiâ quae illi cum Patre meo L.M. intercessit, animo libentissimo in se suscepit; creandi methodo, et usuum explanatione Inventori relictis. Nunc autem ipso ex hâc vita evocato, totius negotii onus doctissimi Briggii humeris incumbere, et Sparta baec ornanda illi sorte quadam obtigisee videtur." In the address prefixed to the Arithmetica Logarithmica (1625) Briggs bids the reader not to be surprised that these logarithms are different from those published in the Canon Mirificus:-

"Ego enim, cum meis auditoribus Londini, publice in Collegio Greshamensi horum doctrinam explicarem; animadverti multo futurum commodius, si Logarithmus sinus totius servaretur ) (ut in Canone mirifico) Logarithmus autem parties decimae ejusdem sinus totius, nempe sinus 5 graduum, 44. 21, s. esset 10000000000. atque ea de re scripsi statim ad ipsum authorem, et quamprimum per anni tempus et vacationem a publico docendi munere licuit, profectus sum Edinbugum; ubi humanissime ab so acceptus haesi per integram mensem. Cum autem inter nos de horum mutatione sermo haberetur; ille se idem dudum sensisse, et cupivisse dicebat: verumtanem istos, quos jam paraverat edendos curasse, donec alios, si per negotia et valetudinem licerat, magis commodes confedisset. Istam autem mutationem ita faciendam censebat, ut 0 esset Logarithmus unitatis, et 10000000000 sinus totius: quod ego longe commodissimum ease non potui non agnoscere. Coepi igitur, ejus hortatu, rejectis illis anteà paraveram, de horum calculo serio cogitare; et sequenti aestate iterum profectus Edinburgum, horum quos hic exhibeo praecipuos illt ostendi, idem etiam tertia aestate libentissime facturus, si Deus illum nobis tamdiu superstitem esse voluisset."

There is also a reference to the change of the logarithms on the title-page of the work.1

These extracts contain all the original statements made by Napier, Robert Napier, and Briggs which have reference to the origin of decimal logarithms. It will be seen that they are all in perfect agreement. Briggs pointed out in his lectures at Gresham College that it would be more convenient that 0 should stand for the logarithm of the whole sine as in the Canon Mirificus, but that the logarithm of the tenth part of the whole sine (that is to say, of the sine of 5° 44' 21") should be 10,000,000,000. He wrote also to Napier at once; and as soon as he could he went to Edinburgh to visit him, where, as he was most hospitably received by him, he remained for a whole month. When they conversed about the change of system, Napier said that he had felt and desired the same thing, but that he had published the tables which he had already prepared, so that they might be used until he could construct others more convenient. But he considered that the change ought to be so made that 0 should be the logarithm of unity and 10,000,000,000 that if the whole sine which Briggs could not but admit was by far the most convenient of all. Rejecting therefore those which he had prepared already, Briggs began, at Napier’s advice, he consider seriously the question of the calculation of new tables. In the following summer he went to Edinburgh and showed Napier the principal portion of the logarithms which he published in 1624. There probably included the logarithms of the first chiliad which he published in 1617.

Unfortunately Hutton in his history of logarithms, which was prefixed to the early edition of his Mathematical Tables, and was also published as one of his Mathematical Tracts, has charged Napier with want of candour in not telling the world of Briggs’s share in the change of system, and he expresses the suspicion that "Napier was desirous that the world ascribe to him alone the merit of this very useful improvement of the logarithms." According to Hutton’s view, the words "it is to be hoped that his posthumous work"…. Which occur in the preface to the Chilias, were a modest hint that the share Briggs had had in changing the logarithms should be mentioned, and that, as no attention was paid to it, he himself gave the account which appears in the Arithmetic a of 1624. 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. The facts, as stated by Napier and Briggs, are in complete accordance, and the friendship existing assisted Robert Napier in the editing of the "posthumous work," the Constructio, and in the account he gives of the alteration of the logarithms in the Arithmetica of 1624 he seems to have been more anxious that justice should be done to Napier than to himself; while on the other hand Napier received Briggs most hospitality and refer to him as "amico mihi longè charissimo."

Hutton’s unfair suggestions are all the more to be regretted as they occur in a history which is the result of a good deal of investigation, and which has been referred to as an authority by many English and foreign writers. He seems to have felt a strong prejudice against Napier for some reason, and all his statements with regard to the origin of logarithms and Napier’s connexion with the are untrustworthy. While speaking of the change of the logarithms, it should be noticed that the "Admonitio" on the last page of the Canon Mirificus, containing the reference to the new logarithms, does not occur in all the copies. It is printed on the back of the last page of the table itself, and so cannot have been torn out from the copies that are without it. As there could have been no reason for omitting it after it had once appeared, we may assume that the copies which do not have it are those which were first issued. It is probable therefore that Brigg’s copy contained no reference to the change, and it is even possible that the "Admonitio" may have been added after Briggs had communicated with Napier. As special attention had not been drawn to the fact that some copies have the "Admonito" and some have not, different writers have assumed that Briggs did or did not known of the promise contained in the "Admonitio" according as it was present or absent in the copies they had themselves referred to, and this has given rise to some confusion. It ought also to be borne in mind that had Napier lived to publish the Constructio himself, he would probably have referred to Briggs in much warmer terms than those used by Robert Napier who doubtless regarded it as due to his father’s memory to simply state the facts as he knew them. The character of Briggs is very amiable and perfect; he states with modesty and simplicity his own share in the improvement: and with complete loyalty to his friend, and with great earnestness, he devoted the rest of his life to extend the utility of Napier’s splendid invention.

Napier’s original canon is a table of logarithms of sines, and it was clearly Briggs’s original intention to calculate logarithms of sines also; it does not appear from the account he gives who it was who first suggested the tabulation of the logarithms of numbers instead of sines.

Kepler received the invention of logarithms with great enthusiasm. His first mention of them occurs in a letter to Schikhart dated March 11, 1618, in which he writes –" Extitit Scotus Barom cuius nomen mihi excidit, qui praeclari quid praestiti, necessitate omni multiplicationum et divisionum in meras additions et subtractions commutate, nec sinibus utitur: at tamen opus est ipsi tangentium canone: et varietas crebritas difficultasque additionum subtractionumque alicubi laborem multiplicandi et dividendi superat." This erroneous estimate was formed when he had seen the Canon Mirificus but had not read it; and his opinion was very different when he became acquainted with the nature of logarithms. The dedication of his Ephemeris for 1620 consists of a letter to Napier dated July 28, 1619, and he there congratulates him warmly on his invention and on the benefit he has conferred upon astronomy generally and also upon his own Rudolphine tables. 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 wotk by Benjamin Ursinus1 containing the substance of the method, and he at once recognized the importance of what bad been effected. He then explains how he verified the canon, and so found that there were no essential errors in it, although there were a few inaccurancies near the beginning o the quadrant, and he proceeds, "Haec te obiter scire volui, ut quibus tu methods incesseris, quas non dubito et plurimas et ingeniosissimas tibi in promptu esse, eas publici juris fieri, mihi saltem (puto et caeteris) scires fore gratissimum; eoque percepto, tua promissa folio 57, in debitum cedisidde intelligeres." This letter was written two years after Napier’s death, of which Kepler was ignorant, and in the same year as that in which the Constructio was published. In 1624 Kepler published a table of Napierian logarithms, with certain modifications and additions.

Ina letter from Kepler to Petrus Cugerus there occurs the remarkable sentence – "Nihil autem supra Neperianam rationem esse puto: etsi quidem Scotus quidam literis ad Tychonem A. GREEK scriptis jam spem fecit Canonis illius Mirifici." It is here distinctly stated that some Scotsman in the yeat 1594, in a letter so Tycho Brahe, gave him some hope of the logarithms; and as Kepler joins Tycho after his expulsion from the island of Huen, and hand been so closely associated with him in his work, he would be likely to be correct in any assertion of this kind. In connexion with Kepler’s statement the following story, told by Anthony Wood in the Athenae Oxonienses, should be noticed: --

"it must be now known, that one Dr. Craig, a Scotchman… coming out of Denmark into his own country, called upon Joh. Neper, Baron of Merchenston, near Edinburgh, and told him, among other discourses, of a new invention in Denmark (by Longomontanus, as ‘tis said), to save the tedious multiplication and division in astronomical calculations. Neper being solicitous to know farther of him concerning this matter, he could give no other account of it than that it was by proportional numbers. Which hint Neper taking, he desired him at his return to call upon him again. Craig, after some weeks had passed, did so, and Neper then showed him a rude draught of what he called Canon mirabilis Logarithmorum. Which draught, with some alternations, he printing in 1614, it came forthwith into the hands of our author Briggs, and into those of Will. Oughtred, from whom the relation of this matter came."

Longomontanus was Tycho’s assistant, and this story, though obviously untrue in its facts, is of important, as it connects Dr. Craig with Napier and Longomontanus. In the early part of this article Thomas Craig was mentioned as one of the colleagues of Sir Archibald Napier, John Napier’s father, in the office of justice-depute.He is well known as the author of a celebrated legal work De Napier a friendship sprang up which may have been due to their common taste for mathematics. There are extant three letters from Dr John Craig to Rycho Brahe, which show that he was on the most friendly terms with him. In the first letter, of which the date is not given, Craig says that Sir William Stewart has safely delivered to him, "about the beginning of last winter," the book which he sent him. Now Mr Mark Napier found in the library of the university of Edinburgh a mathematical work bearing a sentence in Latin of which the translation is "To Doctor John Craig of Edinburgh, in Scotland, a most illustrious man, highly gifted with various and excellent learning, professor of medicine, and exceedingly skilled in the mathematics, Tycho Brahe hath sent this gift, and with his own hand Tycho Brahe hath sent this gift, and with his own hand written this at Uraniburg, 2d November 1588." As Sir William Stewart was sent to Denmark to arrange the preliminaries of King Jame’s marriage, and retuned to Edinburgh on November 15, 1588, there can be little doubt that this was the volume referred to by Craig. It appears from Craig’s letter, to which we may therefore assign the date 1589, that, five years before, he had made an attempt to reach Uraniburg, but had been baffled by the storm and rocks of Norway, and that ever since then he had been longing to visit Tycho. Now John Craig was physician to the king, and in 1590 James Vi. spent some days at Uraniburg before returning to Scotland from his matrimonial expedition. It seems not unlikely therefore that Craig may have accompanied the king in his visit to Uraniburg. In any case it is certain that Craig was a friend and correspondent of Tycho’s, and there can be but little doubt that he was the "Scotus quidam."

It is therefore clear that as early as 1594 Napier must gave communicated to Craig his hope of being able to effect the simplification of the processes of arithmetic. Everything tends to show that the invention of logarithms was the result of many years of labour and thoughts, undertaken with this special object, and it thus appears that Napier had seen some prospect of success nearly twenty years before the publication of the canon Mirificus. It is very evident that no mere hint with regard to the use of proportional numbers could have been of any service to Napier, but it is possible that the news brought by Craig of the difficulties placed in the progress of astronomy by the labour of the calculations may have stimulated him to persevere in his efforts.

The "new invention in Denmark" ton which Anthony Wood refers as having given the hint to Napier was probably the method of calculation called prosthaphaeresis (often written in Greek letter GREEK), which had its origin in the solution fo spherical triangles.2 The method consists in the use of the formula

Sin a sin b = _ {a-b) – cos (a+b)}.

by means of which the multiplication of two sines is

reduced to the addition or subtraction of two tabular results taken form a table of sines; and, as such products occur in the solution of spherical triangles, the method affords the solution of spherical triangles in certain cases by addition and subtraction only. It seems to be due to Wittich of Breslau, who was assistant for a short time to Tycho Brahe; and it was used by them in their calculations in 1582. Wittich in 1584 made known at Cassel the calculation of one case by this prosthaphaeresis; and Justus Byrgius proved it in such a manner that from his proof the extension to the solution of all triangles could be deduced. Clavius generalized the method in his treatise De astrolabio (1593), lib. i. lemma liii. The lemma commences as follows: --

"Quaestiones omnes, quae per sinus, tangents, atque secants absolve solent, per solam prosthaphaeresim, id est, per solam additionem, subtractionem, sine laboriosa numerorum multiplicatione divisioneque expedire.

"Edidit ante tres quatuorve annos Nicolaus Raymarus Ursus Dithmarsus libellum quondam, in quo praeter alia proponit inventum sane acutum, et ingeniosum, quo per solam prosthaphaeresim pleraque triangular sphaerica solvit. Sed quoniam id solum putat fieri posse, quando sinus in regula proportionum obtinet, verum etiam in tangentibus, secantibus , sinibus versis et aliis numeris, et sive sinus totus sit in principio regulae proportionum, sive in medio, sive denique nullo modo interveniat: quae res nova omnino est, ac jucunditatis et voluptatis plena."

The work of Raymarus Ursus, referred to by Clavius, is his fundamentum Astronomicum (1588). Longomontanus, in his Astronomia Danica (1622), gives an account of the method stating that it is not to be found in the writings of the Arabs or Regiomontanus. As Longomontanus is mentioned in Anthony Wood’s anecdote, and as Wittich as well as Longomontanus were assistant’s of Tycho, there seems little room for doubt that Wittich’s prosthaphaeresis is the method referred to by Wood.

In 1610 Herwart ab Hohenburh published at Munich a multiplication table extending to 1000 x 1000., a huge folio colume of more than a thousand pages; and some writers, misled by the title,1 have supposed that it contained logarithms, It appears form a correspondence between Kepler and Herwart,2 which took place at the end of 1608, that Herwart used his table when in manuscript for the performance of multiplications in general, and that the occurrence of the word prosthaphaeresis on the title is due to Kepler, who pointed out that by means of the table spherical triangles could be solved more easily than by Wittich’s prosthaphaeresis.

It is evident that Wittich’s prosthaphaeresis could not be a good method of practically effecting multiplications unless the quantities to be multiplied were sines, on account of the labour of the interpolations. It satisfies the condition, however, equally with logarithms, of enabling multiplication to be performed by the aid of a table of single entry; and, analytically considered, it is not so different in principle form the logarithmic method. In fact, if we put xy=_ (x + y), X being a function of x only and Y a function of y only, we can show that we must have X=Aeqx, y=Beqy; and if we put xy= _ (X+Y) – _(X-Y), the solutions are _(X+Y)= - _ cos (X+Y). The former solution gives a method known as that as that of quartersquares; the latter gives the method of prosthaphaeresis.

An account of the logarithmic table of Justus Byrgius is given in the article LOGARITHMS.

The more one considers the condition of science at the time, and the state of the country in which the discovery took place, the more wonderful does the invention of logarithms appear. When algebra had advanced to the point where exponents were introduced, nothing would be more natural than that their utility as a means of performing multiplications and divisions should be remarked; but it is one of the surprises in the history of science that logarithms were invented as an arithmetical improvement years before their connexion with exponents was known. It is to be noticed also that the invention was not the result of any happy accident. Napier deliberately set himself to abbreviate multiplications and divisions, -- operations of so fundamental a character that it might well have been thought that they were in rerum natura incapable of abbreivaitio; and he succeeded in devising, by the help of arithmetic and geometry alone, the one great simplification of which they were susceptible, -- a simplification to which the following two hundred and seventy years have added nothing.

When Napier published the canon Mirificus England had taken no part in the advanced of science, an 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 extent 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. In the deed Logan also agrees to conduct Napier from Edinburgh to fastcastle and back again, without his being despoiled of his third part or otherwise harmed, when the deed is to be cancelled and destroyed as a discharge in full. "And incaiss the said Jhone sal find na poiss to be thair efit all tryall and utter diligens tane; he referris the satisfactione of his trawell and painis to the discreetione of the said Robert." Of this singular contract, which is signed "Robert Logane of Restalige" and "Jhone Neper, Fear of Merchiston," and is dated July 1594, a facsimile is given in Mr Mark Napier’s Memoirs.3 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 nay illusion to astrology or magic.

After Napier’s death his manuscripts and notes came into the possession, not of his eldest son Archibald, but 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 last 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 original 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 manuscript 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.—NAPIER, 7th March 1801."

An account of the contents of these manuscript was given by Mr 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 M. 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 Computationibus Quantitatum omnibus Logistacae speciebus communium ; (2) De Logistica Arthmetica ; (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 Merchistonni Baronis consists of two books :—(1) "De nominate Algebrae parte ; (2) De positive sive conssica 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 Roberty Napier himself, and two notes that have been quoted prove that they were made from Napier’s own papers. The titles, which is written on the first leaf, and is also in Robert Napier’s writing, runs thus :—"Thus Baron of Merchiston his books of Arthmeticke and Algebra. For Mr Henrie Briggs, Professor of Geometrie at Oxforde."

These treatises were probably composed before Napier had inventedthe logarithms or any of the apparatuses described in the Rabdologia ;for they contain no allusion to the principle of logarithms, even where we should except to find such a reference, and the one solitary sentence where the Rabdologia is mentioned ("sive omnium facillime per ossa Rhabdologiaenostrae") was no doubt added afterwards. It is worth while to notice that this reference occurs in the chapter "De Multiplicationis et Partitionis compendiis miscellanies," 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 Canon Mirificus (1614,) p. 5: —"Logarithmos sinuum, qui simper majores nihilo sunt, abundantes vocamus, et hoc signo +, aut nullo praenotamus. Logarithmos autem minores nihilo defectives 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 manuscript 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 is Robert Napier’s writing containing a digest of the priciples of alchemy; it is addressed to his son and on the first leaf 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 carboard, each divide into nine squares. In the top squares of the slips the ten digitsare written, and each slip contains in its TABLE 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 an other slip containing, in squares without diagonals, the first nine digit. The slips thus placed in contact give the multiples of the numbers 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 6x 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—

12510

6255

4595

_______

1534560

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, 0 6,the fourth of 0, 4, 9, 5, the fifth of 1, 2, 8, 7, the sixth o 1, 3, 8, 6, the seventh of 1, 4, 8, 5, he 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 multiple 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 digits) which can be formed by the top digits of the bars when place 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 9x9. 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 attacted more general attention than the logarithms, and there was 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. Ursinus 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 calculation. TABLE

Napier also describe 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 = 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 formulae are best remembered by the practical calculator in their unconnected form. ‘Napier’s analogies" are the four formulae—

TABLE

They were first published after his death in the Constructio among the foumulae in spherical trigonometry, which were the results of his latest work. Robert Napiers 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 Historire de l’ Astronomic mderne.

To Napier seems to be due the first use of 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 distinguished the different denominations, primes, seconds, thirds, &c. Thus, for example, he would have written 123·456 as 123 TABLE. In the Rabdologia Napier gives an "Admonitio pro Decimali Arithmetica," in which he commends the fractions of Stevinus and gives and 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, cujas denominator est unitas cum tot cyphris post se, quot sunt figurae post periodum. Ut 100000000·04 valet idem, quod 10000000 4/100· Item 25·803, idem quod 25 803/1000· Item 9999998·005021, idem valet quod 9999998 5021/0000000, & 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., ·499712 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 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 notations 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 630957379. The vertical line was printed by Oughtred and some of Brigg’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) for many generations in the family f Merchiston, and descended to John Napier. The office, Mr Mark Napier states, is repeatedly mentioned in the family charters as appertaining to the "pultrelandis 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 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 "Neporus." 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, probably by a misprint (see Mr Mark Napier’s Memoirs, pp. 9 and 173, and Libri qui supersunt, p. xciv).

Napier’ 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 half of the city walls. In form it is a square tower oft eh 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 Napieir 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 this great discovery. The battlements of Merschiston tower command and 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 cousins 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, 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 1787 An Account of the Life, Writings, and Inventions of John Napier of Merchiston was published at Perth by David Stewart, earl of Buchan, and Walter Minto, 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 are, form the source from which nearly all the notices of which have appeared since have been drawn. The work has also given arise t the impression that there was but little chance of further information being obtained with respect to Napier’s life. In 1834 Mr Mark Napier published his Memoirs of John Napier of Merchiston, his life Lineage, and Times, with a History of the Invention of Logarithms, a large quarto volume of five hundred and thirty-fours 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 no pains in examining every documents 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 Napiers’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 this article with respect to his descent and personal history have been derived. In 1839 Mr Napier completed his labours by editing Napier’s unpublished manuscripts, of which he had only been able to give a rèsumè 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 frontishpiece 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 is referred to Delambre’s Histoire de l’ Astronomie moderne.

It may be useful to give, in conclusion, a list of Napier’s work 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 Mirifici 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 Rabdologia (1617), containing the description of Napier’s bones, the promptuary, and the method oflocal arithmetic,--all three designed for the simplication of multiplications and divisions. The posthumous work are—(1) the canonis Mirifici Logarithmorum Constructio (1619), edited by his son Robert, containing an account of the mode of construction of the canon, and Napier’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 Napier’s notes by his son Robert. (J. W. L. G.)

**Footnotes**

FOOTNOTES (page 177)

(1) The descent of the first Napier of Merchiston has been traced to "Johan le Naper de Counte de Dunbretan," who was one of those show swore fealty to Edward I. in 1296 and defended the castle of Stirling against him in 1304; but there is no authority for this genealogy. The legend with regard to the origin of the name Napler was given by Sir Alexander Napier, eldest son of John Napier. In 1625, in these words: -- "One of the ancient earls of Lennox in Scotland had issue three sons: the eldest, that succedded him to the earldom of Lennox; the second, whose name was Donald; and the third, named Gilchrist. The then king of Scotland having wars, did convocate his lieges to battle, amongst whom that was commanded was the earl of Lennox, who keeping his eldest son at home, sent his two sons to serve for him with the forces that were under his command. This battle went hard with the Scots; for the enemy pressing furiously upon them forced them to lose ground until it came to flat running away, which being perceived by Donald, he pulled his father’s standard from the bearer thereof, and valiantly encountering the foe, being well followed by the earl of Lennox’s men, he repulsed the enemy and changed the fortune of the day, whereby a great victory was got. After the battle, as the manner is, every one drawing and setting forth his own acts, the king said unto them, ye have all done valiantly, but there is one amongst you who hath Na-Peer [i.e. no equal];and calling Donald into his presence commanded him, in regard to his worthy service, and in augmentation of his honour, to change his name from Lennox to Napier, and gave him the lands of Gosford, and lands in Fife, and made him his own servant, which discourse is confirmed by evidence of mine, wherein we are called Lennox alias Napier." Sir Archibald adds that this is "the origin of our name, as, by tradition form father to son, we have generally and without any doubt received the same." This written statement of the legend was occasioned by the following circumstances. Robert Napier, a cousin of John Napier, had amassed riches abroad as a merchant; he was created a baronet in 1612, and in order to out his genealogy formally on record in the herald’s books, he applied for an authentic certificate to Sir Archibald, afterwards Lord Napier, who resided at Merchiston, as the head of the family; and Sir Archibald in reply wrote out in his own hand the document from which the preceding extract had been made.

FOOTNOTES (page 178)

(1) A plaine discovery of the whole Revelation of Saint Iohn: set downe in two treatises: The one searching and proving the true interpretation thereof: The other applying the same paraphrastically and Historically to the text. Set fourth by John Napeir L. of Marchistoun younger. Wheneunto are annexed certaine Oracles of Sibylla, agreeing with the Revelation and other places of Scripture. Edinburgh, printed by Robert Walde-grave, printer to the King’s Majestie, 1593. Cum privilegio Regali.

(2) The work was translated into French by George Thomson, a naturalized Scotman residing in La Rochelle, and publishing by him at that town in 1602, under the title Ouverture de taus les secrets de l’Apocalypse, &c. Par Jean Napier (c.a.d) Nonpareil, Sieur de Merchiston, reveue par lui-meme, et mise en Francois par Georges Thomson, Escossois. There was a second edition of the translation in 1605, and a third edition in 1607. There was also a German translation published at Frankfort, which reached its third edition in 1627. The second English edition appeared in 1611, and in the preface to it Napier states he intended to have published an edition in Latin soon after the original publication in 1593, but that, as the work had now been made public by the French and German translation, and as he was "advertised that our papistical adversaries wer to write larglie against the said editions that are aldreadie set out," he defers the Latin edition "till having first seene the adversaries objections, I may insert in the Lation edition an apologie of that which is rightly done, and an amends or whatsoever is amisse." No criticism on the work was ever published, and there was no Latin edition. A third edition appeared in 1645.

(3) A facsimile of this document is given by Mr Mark Napier in his Memoirs of John Napier.

FOOTNOTES (page 179)

(1) Rabdologiae, seu Numerationis per virgules Libri duo: Cum Appendice de expeditissimo Multiplicationis Promptuario. Quibus accessit & Arithmeticae Localis Liber unus. Authore & Inventore Ioanne Nepero, Barone Merhistonii, &c., Scoto. Edinburgi, Excudebat Andreas Hart, 1617.

FOOTNOTES (page. 181)

(1) The title runs as follows: -- Arithmetica Logarithmica, sive Logarithmorum chiliads triginta… Hos numeros primus invenit clarissimus vir Iohannes Neperus Baro Merchistonij; eos autem ex eiusdem sentential mutavit, eorumque ortum et usum illustravit Henricus Briggius…

FOOTNOTES (page 182)

(1) The title of this work is – Benjaminis Ursini… Cursus Mathematici Practici volumen Primum continens Illustr. & Generosi Dn. Dn. Johannis Neperi Baronis Merchistonij &c. Scoti. Trigomometriam logarithmicam Usibus discentium accomodatam… Coloniae… GREEK. At the end, Napier’s table is reprinted, but to two figures less. As this work was published in 1619, and Vincent’s reprint of the Descriptio and Constructio not ill 1620, it forms the earliest publication of logarithms on the Continent.

FOOTNOTES (page 182)

(2) A careful examination of the history of the method is given by Scheibel in his Einleitung zur mathematischen Bücherkenntniss, Stück vii. (Breslau, 1775), pp. 13-20; and there is also an account in Kästner’s Geshichte der Mathematik, vol. i. (1796,) pp. 566-569; in Montucla’s Histoire des Mathématiques, vol. i. pp. 583-585 and 617-619 and in Klügel’s Wörterbuch (1808), article "Prosthaphaeresis."

FOOTNOTES (page. 183)

(1) Tabulae arithmeticae GREEK universals, quarum subsidio numerus quilibet, ex multiplicatione producendus, per solam additionem; et quotiens quilibet, e divisone eliciendus, per solam subtractionem, sine taediosâ Multiplicationis, atque Divisionis operatione, etiam ab eo, qui arithmetices non admodum sit qnarus, exacte, celeriter & nullo negotio invenitur.

(2)The correspondence is printed in Frish’s edition of Kepler’s works, vol. iv. pp. 527-530. See also a paper ("On Multiplication by a Table of Single Entry," in the Philosophical Magazine for November 1878.

FOOTNOTES (page. 183)

(3) Of the contract itself Mr Mark Napier writes: "the singularity of his holding conference with one who had just been proclaimed an outlaw, and whose lawless violence is alluded to and provided against by Napier himself, must be accounted for by the rude state of society, and the simplicity of our philosopher’s character. He took care to word the contract itself, however, and there is not an expression which indicates an idea beyond the most legitimate purpose; but, under the shield of his own innocence, he never dreamed of contamination from his company, was fond of the romance of science, and not averse (nothing derogatory in his times) to the prospect of gold." – Memoirs, p. 223.

**The above article was written by:** J. W. L. Glaisher.