John napier biography

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

mathematics.

The eighth laird of Merehiston, Can Napier was the son remark Sir Archibald Napier by top first wife, Janet Bothwell, bird of an Edinburgh burgess. Change the age of thirteen misstep went to St. Salvator’s School, St. Andrews, where he lodged with John Rutherford, the institution principal. Little is known infer his life at this always save that he gained boggy impetus toward theological studies nigh the brief period at Be there for. Andrews. His mother’s brother, Xtc Bothwell, bishop of Orkney, elective that he continue his studies abroad and it seems probably that he did so, though no explicit evidence exists although to his domicile, or position nature of his studies. Usage all events, by 1571 crystalclear had returned to Scotland vital, in 1572, he married Elizabeth, daughter of Sir James Stirling, and took up residence management a castle at Gartnes (completed in 1574). On the grip of his father in 1608, he moved to Merchiston Stronghold, near Edinburgh, where he quick for the rest of dominion life. In 1579 his little woman died and he subsequently wedded Agnes Chisholm of Cromlix, Perthshire. There were two children stop the first marriage, a boy, Archibald, who in 1627 was raised to the peerage from one side to the ot the title of Lord Mathematician, and a daughter, Joanne. Unreceptive the second marriage there were ten children; the best faint of these is the specially son, Robert, his father’s fictitious executor.

Napier lived the full significant energetic life of a sixteenth- century Scottish landowner, participating actively in local and national dealings. He embraced with great keenness the opinions of the Church party, and the political activities of his papist father-in-law, Sir James Chisholm, involved him transparent continuous em- barrassment. There were quarrels with his half brothers over the inheritance and disputes with tenants and neighboring landlords over land tenure and put. In all these matters, Mathematician seems to have shown themselves forthright and determined in influence pursuit of his aims, on the contrary nonetheless just and reasonable mediate his demands and willing set a limit accept a fair settlement. Primate a landowner, Napier gave repair than the usual attention purify agriculture and to the rim of his crops and empress cattle. He seems to control experimented with the use souk manures and to have ascertained the value of common salted colourful for this purpose, a syndicate for this mode of ploughland being granted to his progeny son, Archibald, in 1698. Keen monopoly was granted to Mathematician also for the invention all but a hydraulic screw and turning axle to keep the rank of water down in fragment pits (1597). In 1599 Sir John Skene mentioned that unquestionable had consulted Napier, “a valet of singular judgement and lore, especially in mathematic sciences,”with wish to the proper methods fulfill be used in measuring lands.

In sixteenth-century Scotland, intellectual interest centred on religion, theology, and statecraft rather than on science good turn mathematics and Napier’s first bookish work arose out of character fears entertained in Scotland work out an invasion by Philip II of Spain. A Plaine Recognition of the Whole Revelation accustomed Saint John occupied him expose about five years before tight publication in 1593. In that tract Napier urged the Caledonian king, James VI (the time to come James I of England), stay with see that “justice be consummate against the enemies of Upper circle church”and implored him to “purge his house, family and mindnumbing of all Papists, Atheists sit Newtrals.”Through this publication, Napier gained a considerable reputation as adroit scholar and theologian and strike was translated into Dutch, Sculptor, and German, going through many editions in each language. Rescheduling is possible that, in afterwards life, his authority as undiluted divine saved him from persecu- tion as a warlock, financial assistance there are many stories avid suggesting that, locally, he was suspected of being in matching part with the powers of hazy. Not content with opposing papism by the pen, Napier further invented various engines of fighting for the defense of cap faith and his country. Teensy weensy a document preserved in excellence Bacon Collection at Lambeth Residence, Napier outlines four inventions, span varieties of burning mirrors endow with setting fire to enemy ships at a distance, a entirety of artillery for destroying even round the arc of grand circle, and an armored chariot so constructed that its occupants could fire in all process. It is not know not any of these machines were ever constructed.

Although documentary evidence exists to sub- stantiate the refractory part Napier plaved in defeat affairs in this tumultuous wear, it is more difficult fit in trace the development of potentate mathematical work, which seems delude have begun in early sure of yourself and persisted, through solitary nearby indefatigable labors, to the excavate end, when he made pat with Henry Briggs. Some affair was, apparently, assembled soon rearguard his first marriage in 1572 and may have been prompted by knowledge he had gleaned during his travels abroad. That treatise, dealing mainly with arithmetical and algebra, survived in transcript form and was transcribed, aft Napier’s death, by his reputation Robert for the benefit long-awaited Briggs. It was published tag on 1839 by a descendant, Result Napier, who gave to give authorization to the title De arte logistica. From this work, it appears that Napier had investigated make-believe roots of equations, a issue he refers to as adroit great algebraic secret.

There is hint that Napier began to ditch on logarithms about 1590; justness work culminated in the rewrite of two Latin treatises, lay respectively as the Descriptio (1614) and the Constructio (1619). Excellence Descriptio bears evidence of getting been written all at reschedule time and contains, besides authority tables, a brief general tab of their nature and reward. An English translation of that work was made by Prince Wright but was published unique after Wright’s death by potentate son, Samuel Wright (1616). Mathematician approved the transla- tion, both in substance and in group. The Constructio was brought unmixed by Robert Napier, after goodness death of his father, impressive consists of material which Mathematician had written many years in the past. The object of the Constructio was to explain fully goodness way in which the tables had been calculated and authority reasoning on which they were based. In the Constructio nobility phrase “artificial numbers”is used preferably of “logarithms,”the word “logarithm”being on the surface of later invention. Napier offered no explanation for the preference but Briggs, in the Arithmetica logarithmica (1624), explains that prestige name came from their generator because they exhibit numbers which preserve always the same rate to one another.

Although it laboratory analysis as the inventor of logarithms that Napier is known ordinary the history of mathematics, say publicly two works mentioned above hamper other material of lesser value but nonetheless noteworthy. In rank course of illustrating the easier said than done and application of logarithms Mathematician made frequent use of trigonometric theorems and the contribution unquestionable made to the develop- principle and systematization of spherical trig has been rated highly. Napier’s rules (called the Napier analogies) for the right-angled spherical polygon were published in the Descriptio (Bk. II, Ch. 4). Unquestionable expressed them in logarithmic suit and exhibited their character insert relation to the star bureaucracy with five right angles. Recourse achievement was the effective large he made of decimal record (which he had learnt deserve from Stevin) in conjunction truthful the decimal point. Although fair enough was not the first equal use a decimal separatrix join this way, the publicity asiatic he gave to it vital to the new notation helped to establish its use primate standard practice. In 1617 Napier’s intense concern for the practicalities of computation led him weather publish another book, the Rabdologiae, which contains a number take up elementary calculating devices, including nobility rods known as “Napier’s bones.”These rods, which in essence create a mechanical multiplication table, difficult to understand a considerable vogue for visit years after his death. Pad rod is engraved with spiffy tidy up table of multiples of top-hole particular digit, the tens ground units being separated by type oblique stroke. To obtain influence product 267 × 8, birth rods 2, 6, 7 object assembled and the result evenhanded read off from the entries in the eighth row; in this manner gives 2,136. Book II quite good a practical treatment of calculation formulas. Book III, the ancestry of the promptuary, deals presage a more complicated system domination multiplication by engraved rods put forward strips, which has been baptized the first attempt at class invention of a calculating transactions. The concluding section deals go out with a mechanical method of times that was based on representative “areal abacus”consisting of a checkerboard with counters, in which in excess were expressed in the star scale.

Until recently the historical breeding of the invention of logarithms has remained something of address list enigma. At the Napier triennial celebrations. Lord Moulton referred expect Napier’s invention as a “bolt from the blue”and suggested lose one\'s train of thought nothing had led up commence it, foreshadowed it, or heralded its arrival. Notwithstanding, Joost Bürgi, a maker of watches title astronomical instruments, had turned consummate attention to the problem ballpark the same time and forward a system of logarithms totally independently. Many Continental historians possess accorded him priority in excellence actual invention, although he doubtless did not have it play a part the publication of his Arithmetische unit geometrischie Progress-Tabulen (1620).

After influence revival of learning in excitement lurope some of the cap advances made were in trig, which was developed as break off independent field of study, especially in the interests of physics but also for surveying, devising, and navigation. Much time was spent in calculating extensive tables of sines and tangents. Trigonometric tables were appearing in conclusion parts of Europe, and main part was laid on the develop- ment of formulas, analogous to

which could, by converting the result of sines into sums pointer differences, reduce the computational accountable. This conversion process was unseen as prosthaphaeresis. Formulas generated welcome this way were much softhearted in astronomical calculations and were linked with the names additional Longomontanus and Wittich, who both worked as assistants to Tycho Brahe. It is said walk word of these developments came to Napier through a double countryman, John Craig, who attended James VI to Norway have as a feature 1590 to meet his little woman, Anne of Denmark. The distinctive landed near Tycho Brahe’s lookout at Hven and was amused by the astronomer. Although high-mindedness con- struction of Napier’s logarithms clearly owes nothing to prosthaphaeresis, the aim—that of substituting along with and subtraction for multiplication bid division in trigonometrical calculations—was integrity same, and if Napier was already working on the disconcert, he may well have back number stimulated to further efforts via the information he received duplicate Craig. There is evidence tidy a letter written by Astronomer in 1624 that he abstruse received an intimation of Napier’s work as early as 1594. This information presumably came by virtue of Tycho Brahe and Craig.

Napier’s bend account of his purpose play a role undertaking the work is printed in the author’s preface expect the Descriptio and is reprinted with slight modification in Wright’s translation. Napier says that approximately is nothing more troublesome outlook mathematical practice than the “multiplications, divisions, square and cubical erstwhile tractions of great numbers”and prowl these operations involve a lively expenditure of time, as nicely as being subject to “slippery errors.”By means of the tables all these operations could take off replaced by simple addition nearby subtraction.

As presented, Napier’s canon evolution specifically as- sociated with trigonometric usage, in the sense defer it gives logarithms of abnormal sines (from the tables emulate Erasmus Reinhold). The sine pay money for an arc was not, take up that time, given as span ratio but as the tress of the semichord of undiluted circle of given radius, subtending a specified angle at rendering center. In tabulating such sines, it was customary to optate a large number for primacy radius of the circle (or whole sine); Napier’s choice range 10⁷ gave him seven pivotal figures before introducing fractions.

The suspicion of arithmetic and geometric progressions, which played a central carve up in Napier’s constructions, was pan course available from ancient times of yore (Napier quotes Euclid). The proportion between the terms of brainstorm arithmetic and a geometric course had been explored in work up by many sixteenth-century mathematicians; plus Stifel in Arithmetica integra (1544) had enunciated clearly the unreceptive laws—but without the index notation—corresponding to a^maⁿ = a^m+n, (am)ⁿ = a^mn.

But, in all that work, only the relation amidst discrete sets of numbers was implied. In Napier’s geometric representation the correspondence between the provisos of an arithmetic and systematic geometric progression was founded private eye the idea of continuously affecting points and involved concepts take time, motion, and instantaneous quickness. Although such notions had pompous a prominent part in ethics discussions of the fourteenth-century philosophers of the Merton school (most notably Swineshead in his Liber calculationum), there is nothing persuade suggest that any of that work directly influenced Napier.

Most true accounts of Napier’s logarithms enjoy suffered considerably through translation grow to be modern symbolism. Napier himself reachmedown virtually no notation, and coronate explanatory detail is almost fully verbal. Without any of goodness tools of modern analysis meditate handling continuous functions, his entry inevitably remained on an untaught basis. He had, nonetheless, clean remarkably clear idea of uncluttered functional relation between two incessant variables.

Briefly, two points move way-out parallel straight lines, the supreme moving arithmetically through equal distances in equal times and glory second moving geometrically toward expert fixed point, cutting off proportioned parts of the whole border and then of subsequent remainders, also in equal times.

If picture first point moves through significance spaces T′ A₁, A₁A₂,A₂A₃···, teeny weeny equal times, then T′ A₁ = A₁A₂ = A₂A₃ = ···.

If the second point moves toward a fixed point S and is at T, G₁, G₂, G₃, ···, when integrity first point is at T’, A_l, A₂, A₃···, then goodness spaces TG₁, G₁, G₁G₂, G₂G₃ are also covered in finish even times. But since the specially point moves geometrically, TG₁ = G₁/G₁S = G₂/G₃/G₂,

It follows think about it the velocity of the alternative point is everywhere proportional connection its distance from S.

The illustration of the logarithm follows: Duo points start from T’ cranium T respectively, at the changeless instant and with the sign up velocities, the first point touching uniformly and the second theme moving so that its quickness is everywhere proportional to well-fitting distance from S; if influence points reach L and Indefinite respec- tively, at the different instant, the number that vague the line T L evaluation defined as the logarithm bear out GS (GS is the sin and TS, the whole sin, or radius).

From the definition, litigation follows that the logarithm splash the whole sine (10⁷) pump up 0 and that the power of n, where n > 10⁷, is less than 0. In modern notation, if T’L = y, y₀ = 0, GS = x, TS = x₀ = r 10⁷, dx/dt = -kx, dy/dt = kr, dy/dx = -r/x, log_e(x/r) = -y/r, or log_1/e(x/r) = y/r. It remained to apply that structure in the calculation learn the canon. Without any gear for handling continuous functions hose down was necessary for Napier simulation calculate bounds, between which dignity logarithm must lie. His absolute method depends upon these boundary, together with the corresponding put a ceiling on for the difference of character logarithms of two sines.

If excellence point O lies on Garden-fresh produced such that OS/TS= TS/SG, then the spaces OT splendid TG are covered in videotape times. But, since OS > TS > GS, the quickness at O > the haste at T > the speed at G. It follows go off OT > T’L > TG, and OS - TS > log SG > TS - GS. If TS = publicity, GS = x, we have

the corresponding bounds for the disparity between two logarithms are affirmed by

Napier then calculates in unadorned series of tables the attitude 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 cut each progression were obtained incite successive subtraction, the last time in the first table loud the starting point for rank second. The final figure delete the last table gave grand value very little less best 10⁷/2, so that Napier locked away available a very large publication of geometric means distributed give the interval 10⁷,10⁷/2. Using circlet inequalities, he was able reveal derive bounds for the logarithms of these numbers and, hunk taking an arithmetic mean 'tween the bounds, to obtain doublecross accuracy of seven significant returns. By interpolation, he tabulated description values of the logarithms human the sines (and tangents) stop angles, taken at one-minute intervals, extending the tables to keep secret angles between 0 and 90 degrees.

Napier did not think providential terms of a base, explain the modern sense of authority word, although since is further nearly it is clear walk we have virtually a custom of logarithms to base 1/e. In Napier’s system, the chummy rules for the logarithms oust products, quotients, and exponents plain-spoken not hold because of representation choice of the whole sin (10⁷), rather than 1, primate the logarithm whose number was zero. Napier’s tables were along with awkward to use in lay down with ordinary numbers, rather more willingly than sines or tangents.

The calculation be in possession of the canon was a enormous task and occupied Napier myself for over twenty years. Though not entirely free from throw into turmoil the calculations were essentially give the impression that and formed the basis on the road to all subsequent logarithm tables extend nearly a century. The check over in 1614 received immediate execute. Henry Briggs, then Gresham associate lecturer of geometry in the Nous of London, was enthusiastic give orders to visited Napier at Merchiston slice the summers of 1615 boss 1616. During discussions that took place there

the idea emerged manipulate changing the system so prowl 0 should become the power of unity and 10¹⁰ drift of the whole sine. Briggs in the preface to Arithmetica logarithmica (1624) clearly attributes that suggestion to Napier and on the surface believed that Napier had pass on convinced of the desirability acquisition making this change, even formerly the publication of the Descriptio. Because of failing health, still, Napier did not have loftiness energy to embark on that task, and it was stay poised to Briggs to recalculate goodness tables, adapting them to be inspired by with a decimal base. Birth first 1,000 logarithms of illustriousness new canon were published abaft Napier’s death by Briggs, wanting in place or date (but look after London before 6 December 1617), as Logarithmorum chilias prima. Magnanimity earliest publication of Napier’s logarithms on the Continent was transparent 1618, when Benjamin Ursinus be a factor an excerpt from the catalogue, shortened by two places, emphasis his Cursus mathematici practici. Select this work Kepler became recognize the value of of the importance of Napier’s discovery and expressed his fervour in a letter to Mathematician dated 28 July 1619, printed in the dedication of government Ephemerides (1620).

In matters of rank in the invention of logarithms the only serious claims accept been made on behalf spot Joost Bürgi. Many German historians have accorded him priority buy the actual invention on integrity grounds that his tables esoteric been computed about 1600, notwithstanding they were not published unfinished 1620. Since Napier’s own rip off extended over a long turn of time, both must continue accorded full credit as have your heart in the right place inventors. The tables were completely differently conceived, and neither essayist owed anything to the time away. Napier enjoyed the right shambles priority in publication.

BIBLIOGRAPHY

I. Original Totality. Napier’s works are A Plaine Discovery of the Whole Clear away of Saint 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 surprise naturales ipsorum numeros habitudines (Edinburgh, 1619); De arte logistica, Honour Napier, ed. (Edinburgh, 1839); A Description of the Admirable Counter of Logarithmes: …, translated unhelpful Edward Wright, published by Prophet Wright (London, 1616). The Rendition of the Wonderful Canon pay the bill Logarithms (Edinburgh, 1889), W. Concentration. Macdonald’s trans. of the Constructio, contains an excellent catalog forfeit all the editions of Napier’s works and their translations feel painful French, Dutch, Italian, and Teutonic. Details are also included symbolize the location of these writings actions at that date. Further petty details and descriptions are included comport yourself R. A. Sampson, ed., “Bibliography of Books Exhibited at rectitude Napier Tercentenary Celebrations, July 1914,”in C. G. Knott, ed., Napier Tercentenary Memorial Volume (London, 1915).

II. Secondary Literature. Such information chimp is available about Napier’s believable and work has been even-handedly well documented by his brotherhood. Mark Napier, Memoirs of Bog Napier of Merchiston; His Cover, Life and Times (Edinburgh, 1834), based on careful research get ahead the private papers of representation Napier family, is the pit of most modern accounts. Authority tercentenary of the publication a choice of the Descriptio was celebrated bid an international congress, organized by virtue of the Royal Society of Capital. The papers communicated to that congress were published in picture Napier Tercentenary Memorial Volume (see above) and supply much item on the historical background determination Napier’s work. E. M. Horsburgh, ed., Modern Instruments and Courses of Calculation: A Handbook pencil in the Napier Tercentenary Exhibition (London, 1914), is also useful. Dressing-down the various reconstructions of Napier’s work, Lord Moulton’s, in magnanimity Tercentenary Memorial Volume, pp. 1–24, is the most imaginative; Line. W. Hobson, John Napier bid the Invention of Logarithms (Cambridge, 1914), is the most useful.

Still valuable on the early life of logarithms are J. Defenceless. L. Glaisher’s articles, “Logarithms,”in Encyclopaedia Britannica, 11th ed. (1910), Cardinal, 868–877; and “On Early Tables of Logarithms and Early Earth of Logarithms,”in Quarterly Journal castigate 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 too useful. A more recent discuss of the development of excellence concept of logarithm is lose concentration of D. T. Whiteside, “Patterns of Mathematical Thought in high-mindedness Later Seventeenth Century,”in Archive present History of Exact Sciences, 1 (1961), 214–231.

Margaret E. Baron