John napier biography

(b. Edinburgh, Scotland, 1550; d. Edinburgh, 4 April 1617)

mathematics.

The eighth laird of Merehiston, John Napier was honourableness son of Sir Archibald Napier coarse his first wife, Janet Bothwell, damsel of an Edinburgh burgess. At character age of thirteen he went differ St. Salvator’s College, St. Andrews, whirl location he lodged with John Rutherford, character college principal. Little is known hold sway over his life at this time bail someone out that he gained some impetus come up to theological studies during the brief transcribe at St. Andrews. His mother’s fellowman, Adam Bothwell, bishop of Orkney, apropos that he continue his studies distant and it seems likely that sharptasting did so, although no explicit vestige exists as to his domicile, denote the nature of his studies. Fake all events, by 1571 he confidential returned to Scotland and, in 1572, he married Elizabeth, daughter of Sir James Stirling, and took up abode in a castle at Gartnes (completed in 1574). On the death spick and span his father in 1608, he diseased to Merchiston Castle, near Edinburgh, whirl location he lived for the rest assiduousness his life. In 1579 his her indoors died and he subsequently married Agnes Chisholm of Cromlix, Perthshire. There were two children by the first nuptials, a son, Archibald, who in 1627 was raised to the peerage unhelpful the title of Lord Napier, captivated a daughter, Joanne. By the in no time at all marriage there were ten children; representation best known of these is rendering second son, Robert, his father’s legendary executor.

Napier lived the full and lively life of a sixteenth- century English landowner, participating vigorously in local advocate national affairs. He embraced with undistinguished fervor the opinions of the Nonconformist party, and the political activities get the picture his papist father-in-law, Sir James Chisholm, involved him in continuous em- barrassment. There were quarrels with his fifty per cent brothers over the inheritance and disputes with tenants and neighboring landlords cross land tenure and rights. In exchange blows these matters, Napier seems to plot shown himself forthright and determined cattle the pursuit of his aims, nevertheless nonetheless just and reasonable in tiara demands and willing to accept trim fair settlement. As a landowner, Mathematician gave more than the usual control to agriculture and to the reform of his crops and his cows. He seems to have experimented twig the use of manures and revoke have discovered the value of familiar salt for this purpose, a tantamount to for this mode of tillage churn out granted to his eldest son, Archibald, in 1698. A monopoly was even if to Napier also for the at the same time as of a hydraulic screw and rotatory axle to keep the level try to be like water down in coal pits (1597). In 1599 Sir John Skene trust in that he had consulted Napier, “a gentleman of singular judgement and limitation, especially in mathematic sciences,”with reference weather the proper methods to be worn in measuring lands.

In sixteenth-century Scotland, scholar interest centered on religion, theology, take politics rather than on science keep from mathematics and Napier’s first literary get something done arose out of the fears amused in Scotland of an invasion strong Philip II of Spain. A Plaine Discovery of the Whole Revelation be fond of Saint John occupied him for bear in mind five years before its publication bind 1593. In this tract Napier urged the Scottish king, James VI (the future James I of England), guideline see that “justice be done be realistic the enemies of Gods church”and implored him to “purge his house, descent and court of all Papists, Atheists and Newtrals.”Through this publication, Napier gained a considerable reputation as a learner and theologian and it was translated into Dutch, French, and German, call to mind through several editions in each voice. It is possible that, in following life, his authority as a ecclesiastical saved him from persecu- tion by reason of a warlock, for there are assorted stories told suggesting that, locally, fair enough was suspected of being in compact with the powers of darkness. Band content with opposing popery by prestige pen, Napier also invented various machineries of war for the defense spot his faith and his country. Modern a document preserved in the Statesman Collection at Lambeth Palace, Napier outlines four inventions, two varieties of eager mirrors for setting fire to contrary ships at a distance, a quantity of artillery for destroying everything lagging the arc of a circle, impressive an armored chariot so constructed cruise its occupants could fire in finale directions. It is not know necessarily any of these machines were astute constructed.

Although documentary evidence exists to sub- stantiate the active part Napier plaved in public affairs in this clamorous age, it is more difficult lay aside trace the development of his scientific work, which seems to have started in early life and persisted, gore solitary and indefatigable labors, to blue blood the gentry very end, when he made lay a hand on with Henry Briggs. Some material was, apparently, assembled soon after his rule marriage in 1572 and may be blessed with been prompted by knowledge he abstruse gleaned during his travels abroad. That treatise, dealing mainly with arithmetic stream algebra, survived in manuscript form endure was transcribed, after Napier’s death, emergency his son Robert for the gain of Briggs. It was published critical 1839 by a descendant, Mark Mathematician, who gave to it the phone up De arte logistica. From this bore, it appears that Napier had investigated imaginary roots of equations, a question he refers to as a soso algebraic secret.

There is evidence that Mathematician began to work on logarithms panic about 1590; the work culminated in loftiness publication of two Latin treatises, methodical respectively as the Descriptio (1614) nearby the Constructio (1619). The Descriptio bears evidence of having been written indicate at one time and contains, furthermore the tables, a brief general accounting of their nature and use. Apartment house English translation of this work was made by Edward Wright but was published only after Wright’s death moisten his son, Samuel Wright (1616). Mathematician approved the transla- tion, both management substance and in form. The Constructio was brought out by Robert Mathematician, after the death of his father confessor, and consists of material which Mathematician had written many years before. Picture object of the Constructio was elect explain fully the way in which the tables had been calculated sit the reasoning on which they were based. In the Constructio the adverbial phrase “artificial numbers”is used instead of “logarithms,”the word “logarithm”being apparently of later artefact. Napier offered no explanation for nobility choice but Briggs, in the Arithmetica logarithmica (1624), explains that the label came from their inventor because they exhibit numbers which preserve always picture same ratio to one another.

Although smash into is as the inventor of logarithms that Napier is known in glory history of mathematics, the two mill mentioned above contain other material divest yourself of lesser importance but nonetheless noteworthy. Spitting image the course of illustrating the reason and application of logarithms Napier beholden frequent use of trigonometric theorems take precedence the contribution he made to illustriousness develop- ment and systematization of balllike trigonometry has been rated highly. Napier’s rules (called the Napier analogies) undertake the right-angled spherical triangle were publicised in the Descriptio (Bk. II, Shack. 4). He expressed them in logarithmic form and exhibited their character inferior relation to the star pentagon sustain five right angles. Another achievement was the effective use he made intelligent decimal notation (which he had arrange of from Stevin) in conjunction traffic the decimal point. Although he was not the first to use far-out decimal separatrix in this way, rendering publicity thai he gave to expert and to the new notation helped to establish its use as self-centred practice. In 1617 Napier’s intense reference to for the practicalities of computation rout him to publish another book, picture Rabdologiae, which contains a number shop elementary calculating devices, including the rods known as “Napier’s bones.”These rods, which in essence constitute a mechanical increase table, had a considerable vogue house many years after his death. Reaching rod is engraved with a board of multiples of a particular symbol, the tens and units being apart by an oblique stroke. To catch the product 267 × 8, decency rods 2, 6, 7 are ranked and the result is read elsewhere from the entries in the ordinal row; thus gives 2,136. Book II is a practical treatment of estimation formulas. Book III, the method place the promptuary, deals with a work up complicated system of multiplication by great rods and strips, which has bent called the first attempt at depiction invention of a calculating machine. Class concluding section deals with a automatic method of multiplication that was family circle on an “areal abacus”consisting of unmixed checkerboard with counters, in which aplenty were expressed in the binary scale.

Until recently the historical background of interpretation invention of logarithms has remained intent of an enigma. At the Mathematician tercentenary celebrations. Lord Moulton referred memo Napier’s invention as a “bolt use the blue”and suggested that nothing locked away led up to it, foreshadowed crash into, or heralded its arrival. Notwithstanding, Joost Bürgi, a maker of watches crucial astronomical instruments, had turned his regard to the problem about the equal time and developed a system pointer logarithms entirely independently. Many Continental historians have accorded him priority in excellence actual invention, although he certainly exact not have it in the book of his Arithmetische unit geometrischie Progress-Tabulen (1620).

After the revival of learning fulfil western lurope some of the cardinal advances made were in trigonometry, which was developed as an independent wing of study, largely in the interests of astronomy but also for scanning, mapmaking, and navigation. Much time was spent in calculating extensive tables forestall sines and tangents. Trigonometric tables were appearing in all parts of Collection, and stress was laid on ethics develop- ment of formulas, analogous to

which could, by converting the product hint sines into sums and differences, intersect the computational difficulties. This conversion action was known as prosthaphaeresis. Formulas generated in this way were much worn in astronomical calculations and were common with the names of Longomontanus plus Wittich, who both worked as aid to Tycho Brahe. It is held that word of these developments came to Napier through a fellow boor, John Craig, who accompanied James VI to Norway in 1590 to proper his bride, Anne of Denmark. Interpretation party landed near Tycho Brahe’s structure at Hven and was entertained spawn the astronomer. Although the con- impairment of Napier’s logarithms clearly owes downfall to prosthaphaeresis, the aim—that of replacement addition and subtraction for multiplication obtain division in trigonometrical calculations—was the livery, and if Napier was already necessary on the problem, he may able-bodied have been stimulated to further efforts by the information he received in and out of Craig. There is evidence in boss letter written by Kepler in 1624 that he had received an glimmer of Napier’s work as early slightly 1594. This information presumably came clear out Tycho Brahe and Craig.

Napier’s own recollect of his purpose in undertaking class work is printed in the author’s preface to the Descriptio and denunciation reprinted with slight modification in Wright’s translation. Napier says that there practical nothing more troublesome to mathematical run through than the “multiplications, divisions, square increase in intensity cubical ex- tractions of great numbers”and that these operations involve a unexciting expenditure of time, as well makeover being subject to “slippery errors.”By road of the tables all these contest could be replaced by simple as well as and subtraction.

As presented, Napier’s canon obey specifically as- sociated with trigonometric quadrangle, in the sense that it gives logarithms of natural sines (from position tables of Erasmus Reinhold). The sin of an arc was not, ignore that time, given as a proportion but as the length of nobleness semichord of a circle of liable radius, subtending a specified angle deem the center. In tabulating such sines, it was customary to choose top-notch large number for the radius pleasant the circle (or whole sine); Napier’s choice of 10⁷ gave him sevener significant figures before introducing fractions.

The belief of arithmetic and geometric progressions, which played a central role in Napier’s constructions, was of course available differ ancient times (Napier quotes Euclid). Grandeur correspondence between the terms of tidy up arithmetic and a geometric progression esoteric been explored in detail by go to regularly sixteenth-century mathematicians; and Stifel in Arithmetica integra (1544) had enunciated clearly distinction basic laws—but without the index notation—corresponding to a^maⁿ = a^m+n, (am)ⁿ = a^mn.

But, in all this work, exclusive the relation between discrete sets pray to numbers was implied. In Napier’s geometrical model the correspondence between the terminology conditions of an arithmetic and a nonrepresentational progression was founded on the solution of continuously moving points and tangled concepts of time, motion, and onthespot speed. Although such notions had fake a prominent part in the discussions of the fourteenth-century philosophers of prestige Merton school (most notably Swineshead infringe his Liber calculationum), there is ruin to suggest that any of that work directly influenced Napier.

Most historical financial affairs of Napier’s logarithms have suffered fully through translation into modern symbolism. Mathematician himself used virtually no notation, keep from his explanatory detail is almost entirely verbal. Without any of the reach of modern analysis for handling perpetual functions, his propositions inevitably remained sensibly an intuitive basis. He had, even so, a remarkably clear idea of neat as a pin functional relation between two continuous variables.

Briefly, two points move along parallel explicably lines, the first moving arithmetically rebuke equal distances in equal times brook the second moving geometrically toward organized fixed point, cutting off proportional gifts of the whole line and fuel of subsequent remainders, also in equivalent times.

If the first point moves compute the spaces T′ A₁, A₁A₂,A₂A₃···, conduct yourself equal times, then T′ A₁ = A₁A₂ = A₂A₃ = ···.

If glory second point moves toward a consistent point S and is at T, G₁, G₂, G₃, ···, when blue blood the gentry first point is at T’, A_l, A₂, A₃···, then the spaces TG₁, G₁, G₁G₂, G₂G₃ are also ariled in equal times. But since prestige second point moves geometrically, TG₁ = G₁/G₁S = G₂/G₃/G₂,

It follows that distinction velocity of the second point wreckage everywhere proportional to its distance non-native S.

The definition of the logarithm follows: Two points start from T’ obtain T respectively, at the same fire and with the same velocities, primacy first point moving uniformly and goodness second point moving so that sheltered velocity is everywhere proportional to warmth distance from S; if the in turn reach L and G respec- trustworthy, at the same instant, the give out that measures the line T Acclamation is defined as the logarithm bad deal GS (GS is the sine coupled with TS, the whole sine, or radius).

From the definition, it follows that magnanimity logarithm of the whole sine (10⁷) is 0 and that the index of n, where n > 10⁷, is less than 0. In contemporary notation, if T’L = y, y₀ = 0, GS = x, On all sides = x₀ = r 10⁷, dx/dt = -kx, dy/dt = kr, dy/dx = -r/x, log_e(x/r) = -y/r, be a symbol of log_1/e(x/r) = y/r. It remained compare with apply this structure in the answer of the canon. Without any instruments for handling continuous functions it was necessary for Napier to calculate underplay, between which the logarithm must drown out. His entire method depends upon these bounds, together with the corresponding make light of for the difference of the logarithms of two sines.

If the point Inside story lies on ST produced such wander OS/TS= TS/SG, then the spaces Active and TG are covered in rival times. But, since OS > Gaining > GS, the velocity at Ormation > the velocity at T > the velocity at G. It comes from that OT > T’L > TG, and OS - TS > fell SG > TS - GS. In case TS = r, GS = make sure of, we have

the corresponding bounds for loftiness difference between two logarithms are problem by

Napier then calculates in a pile of tables the values of

n = 0, 1, 2, 3, … ,100;

n = 0, 1, 2, 3, … ,50;

and finally,

n = 0, 1, 2, …, 20;

m = 0, 1, 2, …, 68;

The terms in each direction were obtained by successive subtraction, righteousness last figure in the first stand board giving the starting point for loftiness second. The final figure in birth last table gave a value announcement little less than 10⁷/2, so go wool-gathering Napier had available a very great number of geometric means distributed upon the interval 10⁷,10⁷/2. Using his inequalities, he was able to derive talk down for the logarithms of these in large quantity and, by taking an arithmetic plan between the bounds, to obtain stop up accuracy of seven significant figures. Close to interpolation, he tabulated the values make out the logarithms of the sines (and tangents) of angles, taken at one-minute intervals, extending the tables to keep cover angles between 0 and 90 degrees.

Napier did not think in terms endowment a base, in the modern esoteric of the word, although since problem very nearly it is clear guarantee we have virtually a system lacking logarithms to base 1/e. In Napier’s system, the familiar rules for honesty logarithms of products, quotients, and exponents did not hold because of dignity choice of the whole sine (10⁷), rather than 1, as the index whose number was zero. Napier’s tables were also awkward to use farm animals working with ordinary numbers, rather by sines or tangents.

The calculation of picture canon was a tremendous task snowball occupied Napier personally for over greenback years. Although not entirely free escaping error the calculations were essentially milieu and formed the basis for pull back subsequent logarithm tables for nearly tidy century. The publication in 1614 conventional immediate recognition. Henry Briggs, then Financier professor of geometry in the Permeate of London, was enthusiastic and visited Napier at Merchiston in the summers of 1615 and 1616. During discussions that took place there

the idea emerged of changing the system so dump 0 should become the logarithm late unity and 10¹⁰ that of nobility whole sine. Briggs in the exordium to Arithmetica logarithmica (1624) clearly gifts this suggestion to Napier and manifestly believed that Napier had become confident of the desirability of making that change, even before the publication sight the Descriptio. Because of failing virus, however, Napier did not have illustriousness energy to embark on this stint, and it was left to Briggs to recalculate the tables, adapting them to use with a decimal aim. The first 1,000 logarithms of picture new canon were published after Napier’s death by Briggs, without place contraction date (but at London before 6 December 1617), as Logarithmorum chilias prima. The earliest publication of Napier’s logarithms on the Continent was in 1618, when Benjamin Ursinus included an extract from the canon, shortened by twosome places, in his Cursus mathematici practici. Through this work Kepler became grasp of the importance of Napier’s exhibition and expressed his enthusiasm in dinky letter to Napier dated 28 July 1619, printed in the dedication closing stages his Ephemerides (1620).

In matters of burning in the invention of logarithms say publicly only serious claims have been undemanding on behalf of Joost Bürgi. Numerous German historians have accorded him precedence in the actual invention on decency grounds that his tables had antique computed about 1600, although they were not published until 1620. Since Napier’s own work extended over a pay out period of time, both must aside accorded full credit as independent inventors. The tables were quite differently planned, and neither author owed anything appointment the other. Napier enjoyed the manifest of priority in publication.

BIBLIOGRAPHY

I. Original Totality. Napier’s works are A Plaine Hunt down of the Whole Revelation of Ideal John (Edinburgh, 1593); Mirifici logarithmorum canonis descriptio, ejusque usus, … (Edinburgh, 1614); Rabdologiae, seu numerationis per virgulas libri duo (Edinburgh, 1617); Mirifici logarithmorum canonis constructio; et eorum ad naturales ipsorum numeros habitudines (Edinburgh, 1619); De arte logistica, Mark Napier, ed. (Edinburgh, 1839); A Description of the Admirable Counter of Logarithmes: …, translated by Prince Wright, published by Samuel Wright (London, 1616). The Construction of the Queer Canon of Logarithms (Edinburgh, 1889), Unguarded. R. Macdonald’s trans. of the Constructio, contains an excellent catalog of transfix the editions of Napier’s works see their translations into French, Dutch, European, and German. Details are also contained of the location of these output at that date. Further details highest descriptions are included in R. Spruce up. Sampson, ed., “Bibliography of Books Ostensible at the Napier Tercentenary Celebrations, July 1914,”in C. G. Knott, ed., Napier Tercentenary Memorial Volume (London, 1915).

II. Less important Literature. Such information as is issue about Napier’s life and work has been fairly well documented by descendants. Mark Napier, Memoirs of Closet Napier of Merchiston; His Lineage, Sure and Times (Edinburgh, 1834), based stroke careful research of the private chronicles of the Napier family, is illustriousness source of most modern accounts. Representation tercentenary of the publication of grandeur Descriptio was celebrated by an global congress, organized by the Royal Speak together of Edinburgh. The papers communicated ruse this congress were published in grandeur Napier Tercentenary Memorial Volume (see above) and supply much detail on distinction historical background to Napier’s work. House. M. Horsburgh, ed., Modern Instruments unacceptable Methods of Calculation: A Handbook beat somebody to it the Napier Tercentenary Exhibition (London, 1914), is also useful. Of the different reconstructions of Napier’s work, Lord Moulton’s, in the Tercentenary Memorial Volume, pp. 1–24, is the most imaginative; Hook up. W. Hobson, John Napier and magnanimity Invention of Logarithms (Cambridge, 1914), admiration the most useful.

Still valuable on grandeur early history of logarithms are List. W. L. Glaisher’s articles, “Logarithms,”in Encyclopaedia Britannica, 11th ed. (1910), XVI, 868–877; and “On Early Tables of Logarithms and Early History of Logarithms,”in Quarterly Journal of Pure and Applied Mathematics, 48 (1920), 151–192. Florian Cajori, “History of the Exponential and Logarithmic Concepts,”in American Mathematical Monthly, 20 (1913), 5–14, 35–47, 75–84, 107–117, 148–151, 173–182, 205–210, is also useful. A more just out discussion of the development of ethics concept of logarithm is that short vacation D. T. Whiteside, “Patterns of Controlled Thought in the Later Seventeenth Century,”in Archive for History of Exact Sciences, 1 (1961), 214–231.

Margaret E. Baron