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NAPIER, JOHN (1550-1617)
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See also:NAPIER, See also: In the criminal court of Scotland, the earl of See also:Argyll, hereditary See also:justice-See also:general of the See also:kingdom, sometimes presided in See also: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 See also:register of the court, extending over 1563 and .1564, the justice-deputes named are " Archibald Naper of Merchistoune, Alexander See also:Bannatyne, See also:burgess of Edinburgh, James See also:Stirling of Keir and Mr See also: The See also:legend with regard to the origin of the name Napier 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 succeeded him to the earldom of Lennox; the second, whose name was Donald; and the third, named Gilchrist. The then See also: The full title of this first See also:work of Napier's is given below). It was written in See also:English instead of Latin in See also:order that " hereby the See also:simple of this Iland may be instructed "; and the author apologizes for the See also:language and his own mode of expression in the following sentences:
" Whatsoever therfore through hast, is here rudely and in See also:base language set downe, I doubt not to be pardoned thereof by All See also:good men, who, considering the necessitie of this time, will esteem it more meete to make bast to prevent the rising againe of Antichristian darknes within this See also:Band, then to prolong the time in See also:painting of language "; and " I graunt indeede, and am sure, that in the See also:style of wordes and utterance of language, we shall greatlie differ, for therein I do See also:judge my selfe inferiour to all men: so that scarcely in these high matters could I with See also:long deliberation finde wordes to expresse my minde."2
Napier's Plaine Discovery is a serious and laborious work, to which he had devoted years of care and thought. In one sense jt may be said to stand to theological literature in Scotland in something of the same position as that occupied by the See also:Canon Mirificus with respect to the scientific literature, for it is the first published See also:original work relating to theological See also:interpretation, and is quite without a predecessor in its own See also: . See also:Par See also:Jean Napeir (c. a. d.) See also:Nonpareil, Sieur de Merchiston, reveue par lui-mesme, et raise en See also:Francois par Georges Thomson, Escossois. Subsequent See also:editions were published in 1603, 16os and 1607. See also:German See also:translations were published at See also:Gera in 1611 and at See also:Frankfort in 1605 and 1627. The second edition in English appeared at Edinburgh in 1611, and in the See also: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 Dutch. translations, besides the English editions, and as he was " advertised that our papistical adversaries veer to write larglie against the said editions that are alreadie set out," he defers the Latin edition " till having first seene the adversaries objections, I may insert in the Latin edition an apologie of that which is rightly done, and an amends of whatsoever is amisse." No See also:criticism on the work was published, and there was no Latin edition. A third edition appeared at Edinburth in 1645. Corresponding to the first two Edinburgh editions, copies were issued bearing the See also:London imprint and See also:dates 1594 and
1611.
After the publication of the Plaine Discovery, Napier seems to have occupied himself with the invention of See also:secret See also:instruments of See also:war, for in the See also: Edinburgi, ex officind Andreae See also:Hart Bibliopolae, CIO.DC.XIV. This is printed on an ornamental title-See also:page. The work is a small-sized See also:quarto, containing fifty-seven pages of explanatory See also:matter and ninety pages of tables. The nature of logarithms is explained by reference to the See also:motion of points in a straight See also:line, and the principle upon which they are based is that of the See also:correspondence of a geometrical and an arithmetical See also:series of See also:numbers. The table gives the logarithms of sines for every See also:minute to seven figures. This work contains the first announcement of logarithms to the See also:world, the first table of logarithms and the first use of the name See also:logarithm, which was invented by Napier. In 1617 Napier published his Rabdologia,' a duodecimo of one See also:hundred and fifty-four pages; there is prefixed to it as preface a dedicatory See also:epistle to the high See also:chancellor of Scotland. The method which Napier terms " Rabdologia " consists in the use of certain numerating rods for the performance of multiplications and divisions. These rods, which were commonly called " Napier's bones," will be described further on. The second method, which he calls the " Promptuarium Multiplicationis " on See also:account of its being the most expeditious of all for the performance of multiplications, involves the use of a number of lamellae or little plates of metal disposed in a See also:box. In an appendix of See also:forty-one pages he gives his third method, " See also:local See also:arithmetic," which is performed on a See also:chess-See also:board, and depends, in principle, on the expression of numbers in the See also:scale of radix 2. In the Rabdologia he gives the See also: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 See also: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 of 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 See also:hit upon a certain See also:tabular arithmetic, whereby the more troublesome operations of See also:common arithmetic are performed on an See also:abacus or chess-board, and which may be regarded as an amusement 3 A facsimile of this document is given by See also:Mark Napier in his See also:Memoirs of John Napier (1834), p. 248. Rabdologiae, seu Numerationis per virgules Libri duo: Cum Appendice de expeditissimo Multiplicationis promptuario. Quibus accessit & Arithmeticae Localis See also:Liber anus. Authore & Inventore Joanne Nepero, Barone Merchistonii, &c., Scoto. Edinburgi, Excudebat Andreas Hart (1617). See also:Foreign editions were published In See also:Italian at See also:Verona in 1623, in Latin at See also:Leiden in 1626 and 1628, and in Dutch at See also:Gouda in 1626. In 1623 See also:Ursinus published Rhabdologia Neperiana at See also:Berlin, and the rods or bones were described in several other See also:works. rather than a labour, for, by means of it, addition, subtraction, multiplication, See also:division and even the extraction of roots are accomplished simply by the motion of counters. He adds that he has appended it to the Rabdologia, in addition to the promptuary, because he did not wish to See also:bury it in silence nor to publish so small a matter by itself. With respect to the calculating rods, he 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 the 4th of See also:April 1617, the same year as that in which the Rabdologia was published. 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 last. In both the Canonis decriptio and the Rabdologia, however, he makes reference to his See also:ill-See also:health. In the dedication of the former he refers to himself as " milli jam morbis pens confecto," and in the " Admonitio " at the end he speaks of his " infirma valetudo "; while in the latter he says he has been obliged to leave the calculation of the new canon of logarithms to others "ob infirmam corporis nostri valetudinem."
It has been usually supposed that John Napier was buried in St See also:Giles's church, Edinburgh, which was certainly the See also:burial-place of some of the family, but Mark Napier (Memoirs, p. 426) quotes See also:Professor See also: It is nowhere else recorded that Napier suffered from the gout. It has been stated that Napier's mathematical pursuits led him to dissipate his means. This is not so, for his will (Memoirs, p. 427) shows that besides his large estates he left a considerable amount of personal See also:property.
The Canonis Descriptio on its publication in 1614, at once attracted the See also:attention of Edward See also:Wright, whose name is known in connexion with improvements in See also:navigation, and See also: R. See also:Macdonald at Edinburgh in 1889, and that there is appended to this edition a complete See also: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 See also: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 See also:idea on which they depend. There is no anticipation or hint to be found in previous writers,s and it is very remarkable that a discovery or invention which was to exert so important and far-reaching an See also:influence on See also:astronomy and every science involving calculation was the work of a single mind. The more one considers the condition of science at the time, and the See also:state of the country in which the discovery took place, the more wonderful does the invention of logarithms appear. When See also: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 See also:accident. Napier deliberately set himself to abbreviate multiplications and divisions—operations of so fundamental a See also:character that it might well have been thought that they were in rerum natura incapable of See also:abbreviation; and he succeeded in devising, by the help of arithmetic and geometry alone, the one 1 The title runs as follows: Arithmetica Logarithmica, sive Logarithmorum chiliades triginta. . . . Hos numeros See also:Primus invenit clarissimus See also:vie lohannes Neperus Baro Merchistonij; eos autem ex eiusdem sententiamutavit, eorumque ortum et usum illustravit Henricus Briggius. . 2 The full title was: Mirifici Logarithmorum Canonis Constructio; Et eorum ad naturales ipsorum numeros habitudines; una cum Appendice, de aliti edque praestantiore Logarithmorum specie condendi. Quibus accessere Propositiones ad triangula sphaerica faciliore calculo resolvenda: Una cum Annotationibus See also:aliquot doctissimi D. Henrici Briggii, in eas £~ memoratam appendicem. Authore & Inventdre Joanne Nepero, Barone Merchistonii, &c. Scoto. Edinburgi, Excudebat Andreas Hart, See also:Anna Domini 1619. There is also preceding this title-page an ornamental title-page, similar to that of the Descriptio of 1614; the words are different, however, and run—Mirifici Logarithmorum Canonis Descriptio . . . Accesserunt See also:Opera Posthuma: Prima, Mirifici ipsius canonis ccnstructio, £a' Logarithmorum ad naturales ipsorum numeros habitudines. Secunda, Appendix de alid, edque praestantiore Logarithmorum specie construenda. Tertio, Propositiones quaedam eminentissimae, ad Triangula sphaerica mirtl facilitate resolvenda.... It would appear that this title-page was to be substituted for the title-page of the Descriptio of 1614 by those who See also:bound the two books together. 3 The work of Justus Byrgius is described in the article LOGARITHM. In that article it is mentioned that a Scotsman in 1594 in a letter to Tycho See also:Brahe held out some See also: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. great simplification of which they were susceptible—a simplification to which nothing essential has since been added. When Napier published the Canonis Descriptio See also:England had taken no part in the advance of science, and there is no British author of the time except Napier whose name can be placed in the same See also:rank as those of See also:Copernicus, Tycho Brahe, Kepler, Galileo, or See also:Stevinus. In England, Robert See also: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 See also: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 See also:astrology, and he certainly also had some faith in the See also:power of magic, for there is extant a See also:deed written in his own See also:handwriting containing a See also:con-See also:tract between himself and Robert See also:Logan of Restalrig, a turbulent See also:baron of desperate character, by which Napier undertakes " to serche and sik out, and be al See also:craft and ingyne that he See also:dow, to tempt, trye, and find out " some buried treasure supposed to be hidden in Logan's fortress at Fastcastle, in See also:consideration of receiving one-third part of the treasure found by his aid. Of this singular See also:contract, which is signed, " Robert Logane of Restalrige " and " Jhone Neper, Fear of Merchiston,” and is dated See also:July 1J94, a facsimile is given in Mark Napier's Memoirs. 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 See also:drawn up by Napier himself affords a singular See also:illustration of the state of society and the kind of events in the midst of which logarithms had their See also:birth. Considering the time in which he lived, Napier is singularly See also: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 allusions to astrology or magic. After Napier's death his See also:manuscripts and notes came into the possession of his second son by his second marriage, Robert, who edited the Constructio; and See also:Colonel Milliken Napier, Robert's lineal male representative, was still in the possession of many of these private papers at the See also:close of the 18th See also: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 See also:Bible with his autograph, were deposited for safety in a See also:room of the house at Milli-See also:ken, in See also:Renfrewshire. During the owner's See also:absence the house was burned to the ground, and all the papers and See also: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 See also:manuscript De Arte Logistica, and the copy escaped the See also:fate of the originals in the manner explained in the following See also:note, written in the See also:volume containing them by Francis, seventh Lord Napier: " John Napier of Merchiston, inventor of the logarithms, left his manuscripts to his son Robert, who appears to have caused the following pages to have been written out See also:fair from his father's notes, for Mr Briggs, professor of geometry at See also:Oxford. They were given to Francis, the fifth Lord Napier, by William Napier of Culcreugh, Esq., See also: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 See also: The same See also:definitions occur also in the Canonis Descriptio (1614), p. 5: " Logarithmos sinuum, qui See also:semper majores nihilo sunt, abundantes vocamus, et hoc signo +, See also:aut nullo praenotamus. Logarithmos autem minores nihilo defectivos 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 See also:root, power, exponent, his words are radix, radicatum, See also:index). Apart from the See also:interest attaching to these manuscripts as the work of Napier, they possess an See also:independent value as affording evidence of the exact state of his algebraical knowledge at the time when logarithms were invented. There is nothing to show whether the transcripts were sent to Briggs as intended and returned by him, or whether they were not sent to him. Among the Merchiston papers is a thin quarto volume in Robert Napier's writing containing a See also:digest of the principles of See also:alchemy; it is addressed to his son, and on the first leaf there are directions that it is to remain in his See also:charter-See also:chest and be kept secret except from a few. This treatise and the transcripts seem to be the only manuscripts which have escaped destruction. The principle of " Napier's bones " may be easily explained by imagining ten rectangular slips of cardboard, each divided into nine squares. In the See also:top squares of the slips the ten digits are written, and each slip contains in its nine squares the first nine multiples of the See also:digit which appears in the top square. With the exception of the top squares, every square is divided into two parts by a See also:diagonal, the See also:units being written on one See also:side and the tens on the other, so that when a multiple consists of two figures they are separated by the diagonal. Fig. i shows the slips corresponding to the numbers 2, 0, 8, 5 placed side by side in contact with one another, and next to them is placed another slip containing, in squares without diagonals, the first nine digits. The slips thus placed in contact give the multiples of the number 2085, the digits in each parallelogram being added together; for example, corresponding to the number 6 on the right-hand slip, we have 0, 8+3, 0+4, 2, I; whence we find FIG. r, 0, I, 5, 2, I as the digits, written backwards, of 6X2085. 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 See also:ordinary multiplication, viz. the figures as written down are 12510 6255 14595 1534560 Napier's rods or bones consist of ten oblong pieces of See also:wood or other material with square ends. Each of the four faces of each See also: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, I, 9, 8, the second of o, 2, 9, 7, the third of o, 3, 9, 6, the fourth of o, 4, 9, 5, the fifth of I, 2, 8, 7, the sixth of 1, 3, 8, 6, the seventh of 1, 4, 8, 5, the eighth of 2, 3, 7, 6, the ninth of 2, 4, 7, 5, and the tenth of 3, 4, 6, 5. Each rod therefore contains on two of its faces multiples of digits which are complementary to those on the other two faces; and the multiples of a digit and of its See also: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 rod. The set of ten rods is thus See also: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 2 0 8 5 1 ,/~~ 2 „F, 3 MINI 4 „rlrl 5 MUM 6 7 ~/.IE 8 ,~~, 9 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 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 multi- plicand or divisor, and Napier's bones at best merely provide such a table, and in an incom- plete See also: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 loga- rithms, and as has been mentioned, there were several editions on the Continent. Nothing shows more clearly the See also:rude state of arithmetical know- ledge at the beginning of the 17th century than the universal See also:satisfaction with which Napier's invention was welcomed by all classes and re- garded as a real aid to calculation. Napier also describes in the Rabdologia two other larger rods to facilitate the extraction of square and See also: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 See also:system of formulae for the See also:solution of right-angled triangles, may be enunciated as follows. Leaving the right See also: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, 9o°—A, 9o°—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 See also: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 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 See also:practical calculator in their unconnected form. " Napier's analogies " are the four formulae- tan1(AtB) =See also:cos (a—b) cot2C, tan(A—B) = See also:sin(a—b)cot2C cos z (a+b) sin2 (ct+b) ' tan1(a+b)=cc's (A—B)tanyc, tanz(a—b)= sin(A—B)tanlc. See also:cosa(A+B) sin i(A{ B) They were first published after his death in the Constructio among the formulae in spherical trigonometry, which were the results of his latest work. Robert Napier says that these results would have been reduced to order and demonstrated consecutively but for his father's death. Only one of the four analogies is actually given by Napier, the other three being added by Briggs in the remarks which are appended to Napier's results. The work left by Napier is, however, rough and unfinished, and it is uncertain whether he knew of the other formulae or not. They are, however, so simply deducible from the results he has given that all the four analogies may be properly called by his name. An See also:analysis of the formulae contained in the Descriptio and Constructio is given by See also: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 123@4®5®6®. In the Rabdologia Napier gives an " Admonitio See also: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 See also: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 See also:post periodum notatur fractio est, cujus denominator est unitas cum tot cyphris post se, quot Bunt figurae post periodum. Ut 10000000.04 See also:valet idem, quod 1000000011. See also:Item 25.803, idem quod 251F1,Oo Item 9999998.0005021, idem valet quod 9999998 i o o o5 0 , & 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. (I) he saw that a point or separatrix was quite enough to See also: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 See also: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 See also: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 See also:vertical line just high enough to See also:fix distinctly which two figures it was intended to separate: thus he might have written 63 0957379. The vertical line was printed by See also: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 See also:village of Dene in the See also:shire of Linlithgow. The duties were to be performed by the possessor or his See also:deputy; and the king was entitled to demand the yearly See also:homage of a See also:present of poultry from the feudal holder. The pultrelands and the office were sold by John Napier in 1610 for 1700 narks. 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 See also: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 See also:children, who sign deeds along with him, use every mode except Napier, the form now adopted by the family, and which is comparatively See also:modern. In Latin he always wrote his name " Neperus." The form " Neper " is the See also: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 See also:fee of his paternal See also:barony during the lifetime of his father, who retained the liferent. He has been some-times erroneously called " Peer of Merchiston," and in the 1645 edition of the Plaine Discovery he is so styled (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. See also:van Wet to See also:Amsterdam, 1. sectie, deel 6 (1899). (J. W. L. Additional information and CommentsThere are no comments yet for this article.
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