John Napier

John Napier (1550–1617)

John Napier [Neper, Nepair] of Merchiston (1550 – 4 April 1617) was a Scottish mathematician, physicist, astronomer and theologian recognized for the discovery of logarithms and the invention of "Napier's bones." He was the 8th Laird of Merchiston.

Quotes

• In my tender years and bairn-age, at schools, having on the one part contracted a loving familiaritie with a certain gentleman a papist, and on the other part being attentive to the sermons of that worthy man of God, Maister Christopher Goodman, teaching upon the Apocalyps, I was moved in admiration against the blindness of papists that could not most evidentlie see their seven hilled Citie of Rome, painted out there so lively by Saint John, as the Mother of all Spiritual Whoredome: that not onlie bursted I oute in continuall reasoning against my said familiar, but also from thence forth I determined with myself by the assistance of God's spirit to employ my study and diligence to search out the remanent mysteries of that holy booke (as to this houre praised be the Lord I have bin doing at all such times as convenientlie I might have occasion) &c.
  • David Stewart Erskine Earl of Buchan, Walter Minto, An Account of the Life, Writings, and Inventions of John Napier, of Merchiston (1787) a reference to his education at the University of St. Andrews

Memoirs of John Napier of Merchiston (1834)

His Lineage, Life, and Times, with a History of the Invention of Logarithms by Mark Napier, Esq.

• Seeing there is nothing, (right well beloved students of mathematics,) that is so troublesome to mathematical practice, nor that doth more molest and hinder calculations, that the multiplications, divisions, square and cubical extractions of great numbers, which besides the tedious expence of time, are for the most part subject to many slippery errors, I began, therefore, to consider in my mind, by what certain and ready art I might remove these hindrances. And having thought upon many things to this purpose, I found at length some excellent brief rules to be treated of perhaps hereafter: But amongst all, none more profitable than this, which together with the hard and tedious multiplications, divisions, and extractions of roots, doth also cast away even the very numbers themselves that are to be multiplied, divided, and resolved into roots, and putteth other numbers in their place which perform as much as they can do, only by addition and substraction, division by two, or division by three. Which secret invention being, (as all other good things are,) so much the better as it shall be the more common, I thought good heretofore, to set forth in Latin for the public use of mathematicians.
  • Canon Mirificus, Englsh edition (1616)

• But now, some of our countrymen in this island, well affected to these studies, and the more public good, procured a most learned mathematician to translate the same into our vulgar English tongue, who after he had finished it, sent a copy of it to me, to be seen and considered on by myself. I having most willingly and gladly done the same, find it to be most exact and precisely conformable to my mind and the original. Therefore it may please you who are inclined to these studies, to receive it from me and the translator, with as much good will as we recommend it unto you.—Fare thee well.
  • Canon Mirificus, Englsh edition (1616)

Quotes about Napier

• The invention of logarithms, without which many of the numerical calculations which have constantly to be made would be practically impossible, was due to Napier of Merchiston. The first public announcement of the discovery was made in his Mirifici Logarithmorum Canonis Descriptio, published in 1614, and of which an English translation was issued in the following year; but he had privately communicated a summary of his results to Tycho Brahe as early as 1594. In the work Napier explains the nature of logarithms by a comparison between corresponding terms of an arithmetical and geometrical progression. He illustrates their use, and gives tables of the logarithms of the sines and tangents of all angles of the first quadrant, for differences of every minute, calculated to seven decimal places. His definition of the logarithm of a quantity n was what we should now express by $10^{7}\log _{e}\!\left({\frac {10^{7}}{n}}\right)$. This work... is the first valuable contribution to the progress of mathematics which was made by any British writer.
  • W. W. Rouse Ball, The History of Mathematics (1912)

• The method by which the logarithms were calculated was explained in the Constructio, a posthumous work issued in 1619: it seems to have been very laborious, and depended either on direct involution and evolution, or on the formation of geometrical means. The method by finding the approximated value of a convergent series was introduced by Newton, Cotes, and Euler. Napier had determined to change the base to one which was a power of 10, but died before he could effect it
  • W. W. Rouse Ball, The History of Mathematics (1912)

• The rapid recognition throughout Europe of the advantages of using logarithms in practical calculations was mainly due to Briggs, who was one of the earliest to recognize the value of Napier's invention. Briggs at once realized that the base to which Napier's logarithms were calculated were inconvenient; he accordingly visited Napier in 1616, and urged the change to a decimal base, which was recognized by Napier as an improvement. On his return Briggs immediately set to work to calculate tables to a decimal base, and in 1617 he brought out a table of logarithms of the numbers from 1 to 1000 calculated to fourteen decimal places.
  • W. W. Rouse Ball, The History of Mathematics (1912)

• Napier, in 1614, ...employed the idea of the fluxion of a quantity to picture by means of lines the relation between logarithms and numbers.
  • Carl Benjamin Boyer, The Concepts of the Calculus, a Critical and Historical Discussion of the Derivative and the Integral (1949)

• Napier, Lord of Merchiston, hath set my head and hands at work with his new and admirable Logarithms. I hope to see him this Summer, if it please God, for I never saw a book [Mirifici Logarithmorum Canonis Descriptio] which pleased me better, and made me more wonder.
  • Henry Briggs, Letter to Archbishop Usher (1615) as quoted by David Stuart & John Minto in "Account of the Life of John Napier of Merchiston," The Edinburgh Magazine, or Literary Miscellany (1787) Vol.6

• Many computing devices have been used since the invention of the abacus. These include Napier's bones, sector compasses, slide rules, calculators, and computers.
  • Lucas N. H. Bunt, Phillip S. Jones, Jack D. Bedient, The Historical Roots of Elementary Mathematics (1976)

• Let our judgment not be too harsh. The period under consideration is too near the Middle Ages to admit of complete emancipation from mysticism even among scientists. Scholars like Kepler, Napier, Albrecht Duerer, while in the van of progress and planting one foot upon the firm ground of truly scientific inquiry, were still resting with the other foot upon the scholastic ideas of preceding ages.
  • Florian Cajori, A History of Mathematics (1893)

• It is no exaggeration to say that the invention of logarithms "by shortening the labours doubled the life of the astronomer." Logarithms were invented by John Napier, Baron of Merchiston, in Scotland. It is one of the greatest curiosities of the history of science that Napier constructed logarithms before exponents were used. To be sure, Stifel and Stevin made some attempts to denote powers by indices, but this notation was not generally known,—not even to Harriot, whose algebra appeared long after Napier's death. That logarithms flow naturally from the exponential symbol was not observed until much later. It was Euler who first considered logarithms as being indices of powers. What then was Napier's line of thought?
  • Florian Cajori, A History of Mathematics (1893)

• Early in the 1660s, a pair of mathematicians from the British Isles, John Napier and Henry Briggs, jointly introduced, perfected, and exploited the "logarithm," a concept having tremendous practical and theoretical significance. Logarithms have the remarkable property of simplifying such otherwise tedious computations as multiplication, division, and the extraction of roots so that no scientist of sound mind would thereafter go about finding ${\sqrt[{7}]{234.65}}$ without the benefit of logarithms.
  • William Dunham, Journey Through Genius: The Great Theorems of Mathematics (1990)

• Among other persons of distinction, who united themselves to him [the earl of Montrose, in support of the royalists and Charles I of England], was Lord Napier of Merchiston, son of the famous inventor of the logarithms, the person to whom the title of GREAT MAN is more justly due, than to any other whom his country ever produced.
  • David Hume, The History of England, from the Invasion of Julius Cæsar to the Revolution in 1688 (1812) Vol.7, Ch.58

• The biggest improvement in arithmetic during the sixteenth and seventeenth centuries was the invention of logarithms. The basic idea was noted by Stifel. In Arithmetica Integra [1544] he observed that the terms of the geometric progression 1, r, r², r³, ... correspond to the terms in the arithmetic progression 0, 1, 2, 3, ... . Multiplication of two terms in the geometric progression yields a term whose exponent is the sum of the corresponding terms in the arithmetic progression. Division of two terms in the geometric progression yields a term whose exponent is the difference of the corresponding terms in the arithmetic progression. This observation had also been made by Chuquet in Le Triparty en la science des nombres (1484). Stifel extended this connection between the two progressions to negative and fractional exponents. Thus the division of r² by r³ yields r^-1, which corresponds to the term -1 in the arithmetic progression. Stifel, however, did not make use of this connection between the two progressions to introduce logarithms. John Napier, the Scotsman who did develop logarithms about 1594, was guided by this correspondence between the terms of a geometric progression and those of the corresponding arithmetic progression. Napier was interested in facilitating calculations in spherical trigonometry that were being made on behalf of astronomical problems.
  • Morris Kline, Mathematical Thought from Ancient to Modern Times (1972)

• It will be admitted... that... artificial helps may prove useful in laborious and protracted multiplications, by sparing the exercise of memory, and preventing the attention from being overstrained. Of this description are the Rods or Bones, which we owe to the early studies of the great Napier, whose life, devoted to the improvement of the science of calculation, was crowned by the invention of logarithms, the noblest conquest ever achieved by man.
  • Sir John Leslie, The Philosophy of Arithmetic: Exhibiting a Progressive View of the Theory and Practice of Calculation, with an Enlarged Table of the Products of Numbers under One Hundred (1817)

• It may seem extraordinary to quote Lilly the astrologer, with respect to so great a man as Napier; yet as the passage I propose to transcribe from Lilly's Life, gives a picturesque view of the meeting betwixt Briggs and the Inventor of the Logarithms, at Merchiston near Edinburgh, I shall set it down in the original words of that mountebank knave: "I will acquaint you with one memorable story related unto me by John Marr, an excellent mathematician and geometrician... When Merchiston first published his Logarithms, Mr. Briggs, then reader of the Astronomy Lectures 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 whose only invention they were: He acquaints John Marr therewith who went into Scotland before Mr. Briggs purposely to be there when these two so learned persons should meet... He brings Mr. Briggs up into My Lord's chamber, where almost one quarter of an hour was spent, each beholding other with admiration before one word was spoken: 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 found it out before, when now being known it appears so easy." He was nobly entertained by the Lord Napier, and every Summer after that, during the Laird's being alive, this venerable man Mr. Briggs went purposely to Scotland to visit him."
• David Stuart & John Minto in "Account of the Life of John Napier of Merchiston," The Edinburgh Magazine, or Literary Miscellany (1787) Vol.6

An Account of the Life, Writings, and Inventions of John Napier, of Merchiston (1787)

David Stewart Erskine Earl of Buchan, Walter Minto

• SIR, AS the writings of Archimedes were addressed of Sicily, who had perused and relished them, so I do myself the honour, to address to Your Majesty, the following account of the Life, Writings, and Inventions of our British Archimedes...

• I Have undertaken to write the Life of John Napier, of Merchiston, a man famous all the world over, for his great and fortunate discovery of Logarithms in Trigonometry, by which the ease and expedition in calculation, have so wonderfully assisted the Science of Astronomy, and the arts of practical Geometry and Navigation. Elevated above the age in which he lived, and a benefactor to the world in general, he deserves the epithet of Great. Napier lived in a country of proud Barons, where barbarous hospitality, hunting, the military art, and religious controversy, occupied the time and attention of his contemporaries, and where he had no learned society to assist him in his researches.

• It is fit, that men should be taught to aim at higher and more permanent glory than wealth, office, titles or parade can afford; and I like the task, of making such great men look little, by comparing them with men who resemble the subject of my present enquiry.

Dissertation, exhibiting a general view of the progress of mathematical and physical science (1822)

..."since the revival of letters in Europe" in The Works of John Playfair (1822) Vol.2

• As the accuracy of astronomical observation had been continually advancing, it was necessary that the correctness of trigonometrical calculation, and of course its difficulty, should advance in the same proportion. The sines and tangents of angles could not be expressed with sufficient correctness without decimal fractions, extending to five or six places below unity, and when to three such numbers a fourth proportional was to be found, the work of multiplication and division became extremely laborious. Accordingly, in the end of the sixteenth century, the time and labour consumed in such calculations had become excessive, and were felt as extremely burdensome by the mathematicians and astronomers all over Europe. Napier of Merchiston, whose mind seems to have been peculiarly turned to arithmetical researches, and who was also devoted to the study of astronomy, had early sought for the means of relieving himself and others from this difficulty. He had viewed the subject in a variety of lights, and a number of ingenious devices had occurred to him, by which the tediousness of arithmetical operations might, more or less completely, be avoided. In the course of these attempts, he did not fail to observe, that whenever the numbers to be multiplied or divided were terms of a geometrical progression, the product or the quotient must also be a term of that progression, and must occupy a place in it pointed out by the places of the given numbers, so that it might be found from mere inspection, if the progression were far enough continued. If, for instance, the third term of the progression were to be multiplied by the seventh, the product must be the tenth, and if the twelfth were to be divided by the fourth, the quotient must be the eighth; so that the multiplication and division of such terms was reduced to the addition and subtraction of the numbers which indicated their places in the progression. This observation or one very similar to it was made by Archimedes...

• The discovery might certainly have been made by men much inferior either to Napier or Archimedes. What remained to be done, what Archimedes did not attempt, and what Napier completely performed, involved two great difficulties. It is plain that the resource of the geometrical progression was sufficient, when the given numbers were terms of that progression; but if they were not, it did not seem that any advantage could be derived from it. Napier, however, perceived, and it was by no means obvious, that all numbers whatsoever might be inserted in the progression, and have their places assigned in it.

• It is probable... that the greatest inventor in science was never able to do more than to accelerate the progress of discovery, and to anticipate what time, "the author of authors," would have gradually brought to light. Though logarithms had not been invented by Napier, they would have been discovered in the progress of the algebraic analysis, when the arithmetic of powers and exponents, both integral and fractional, came to be fully understood. The idea of considering all numbers, as powers of one given number, would then have readily occurred, and the doctrine of series would have greatly facilitated the calculations which it was necessary to undertake. Napier had none of these advantages, and they were all supplied by the resources of his own mind. Indeed, as there never was any invention for which the state of knowledge had less prepared the way, there never was any where more merit fell to the share of the inventor.

• Even the sagacity of their author did not see the immense fertility of the principle he had discovered; he calculated his tables merely to facilitate arithmetical, and chiefly trigonometrical computation, and little imagined that he was at the same time constructing a scale whereon to measure the density of the strata of the atmosphere, and the heights of mountains; that he was actually computing the areas and the lengths of innumerable curves, and was preparing for a calculus which was yet to be discovered, many of the most refined and most valuable of its resources. Of Napier, therefore, if of any man, it may safely be pronounced, that his name will never be eclipsed by any one more conspicuous, or his invention superseded by any thing more valuable.

Memoirs of John Napier of Merchiston (1834)

His Lineage, Life, and Times, with a History of the Invention of Logarithms by Mark Napier, Esq.

• That his invention was the greatest boon genius could bestow upon a Maritime Empire is a truth universally felt, and which no person is better qualified to appreciate than your Majesty. It is a proud reflection for Britain, that she does not owe to a stranger the creation of that intellectual aid which renders your Majesty's Fleets as free and fearless in Navigation as they have ever been in battle.

• From every line of his descent talent seems to have flowed in upon John Napier.
  • Mark Napier, Memoirs of John Napier of Merchiston" (1834)

• He was born in the year 1550, at Merchiston, the seat of his forefathers, near Edinburgh; four years after the birth of Tycho, fourteen before Galileo, and twenty one before Kepler.

• He was distant and isolated from the great arena of letters; cooped up within the narrow limits of desolate Scotland, and encircled with savage sights and sounds of civil discord, above which the name of God was howled by those whose hands were red with murder. When we regard his times, and observe the influence that for so long a period of his life, the war of religion exercised over his intellectual exertions, the wonder is, not that his great contemporaries of the continent became distinguished before him, but that after all he should have extricated his mind from so many toils, and have placed himself by a single effort—though one like the spring of a roused lion—at the side of the astonished demi-gods of science, who had been unconscious of their rival.

• The Church of Scotland was planted by such noblemen as Argyle and Glencairn; such barons as Tullibardine and Grange. It was rendered popular, and thus greatly aided, by such preachers as Knox and Goodman; and it became dignified in the eyes of Protestant Europe by its first and greatest theologian, John Napier.

• It may surprise the reader to find this honour claimed for the Inventor of Logarithms, who has hitherto been regarded only on his throne of science, and that by the limited number capable of appreciating his genius. The celebrated historian and philosopher [David Hume] who pronounced him to be the greatest man his country ever produced, founded, probably, none of that estimate upon his theological merits; and more recent authors, ranking high among the historians of Christianity and theological learning in Scotland, have omitted to illustrate their subject with the most efficient example they could have found.

Napier Tercentenary Memorial Volume (1915)

edited by Cargill Gilston Knott

• The invention of logarithms came on the world as a bolt from the blue. No previous work had led up to it, nothing had foreshadowed it or heralded its arrival. It stands isolated, breaking in upon human thought abruptly without borrowing from the work of other intellects or following known lines of mathematical thought. It reminds me of those islands in the ocean which rise up suddenly from great depths and which stand solitary with deep water close around all their shores. In such cases we may believe that some cataclysm has thrust them up suddenly with earth-rending force. But can it be so with human thought? Did this discovery come as a revelation to Napier, bursting on him as a light from Heaven, or was it the result of slow growth, the evidences of which are now obliterated, like those rocks whose abrupt sides are due, not to sudden and isolated disruption, but to the denudation which has carried away the neighbouring rocks, which, while they remained, testified to the gradual upheaval of the whole?
  • Inaugural Address by Lord Moulton: The Invention of Logarithms its Genesis and Growth

• The undoubted fact that Napier worked for some twenty years at the invention of logarithms before he published his first book relating to them is, to my mind, decisive upon this point. It must have been a slow and gradual evolution, even though that which remains furnishes so few traces of the earlier efforts. Is it then possible, out of what he has left us and out of the circumstances of the times, to read the history of this evolution to reconstitute the process of discovery by deciphering the half-effaced records of its growth?
  • Inaugural Address by Lord Moulton: The Invention of Logarithms its Genesis and Growth

• All that Napier has left us on the subject of logarithms is contained in two short books, the one known as the Descriptio, published in 1614, and the other known as the Constructio, published after his death in 1619. Internal evidence as well as the distinct statement of his son, who published the Constructio, make it clear that it was in fact written many years before the Descriptio, and it represents in many passages an earlier stratum of thought. ...Napier saw and approved of a translation into English of the Descriptio, and about twenty-five years ago an excellent translation of the Constructio was published in Edinburgh.
  • Inaugural Address by Lord Moulton: The Invention of Logarithms its Genesis and Growth

• In the Descriptio the author published only so much of the reasoning on which his calculations rested as was necessary to enable the mathematical world to appreciate the nature and use of the tables which are to be found there. Indeed, we find Napier expressly stating in it that he does not propose to publish to the world the manner in which the tables were calculated until he finds that they have justified their existence by their acknowledged usefulness. The Descriptio therefore bears evidence of being written all at one time, to serve as an introduction and guide to the tables which were printed with it.
  • Inaugural Address by Lord Moulton: The Invention of Logarithms its Genesis and Growth

• The Constructio was evidently written at several different times. The order of its contents is peculiar, and there are to be seen in it evidences of different stages of the discovery. Its object was to explain fully the mode in which he had calculated the Tables and incidentally the reasoning on which they were based, but there are no historical references to the way in which he originally arrived either at the idea of a Table of Logarithms or at the method of constructing it.
  • Inaugural Address by Lord Moulton: The Invention of Logarithms its Genesis and Growth

• What set Napier to work on creating tables which were to enable multiplication to be performed by a process of addition? What first gave him the idea of any such thing? ...there is a peculiarity in the form of his investigations which gives us a useful clue. He usually frames his propositions as though they applied exclusively or at all events specially to sines. Now it is evident that all that concerns logarithms must relate to numbers generally, and that their being sines has no bearing on the matter. Hence his confining his work to sines must indicate that he set out with the idea of working on them only, and that it was only at a later stage and perhaps incidentally that he realised that his results could with like advantage be applied to numbers generally. I conclude from this that his original idea was only to construct tables that would enable the product of two sines to be readily ascertained. If I am right in this, the suggestion may well have come to him from his familiarity with the well known trigonometrical formula:—
  $\sin A\sin B={\frac {1}{2}}({\cos(A-B)-\cos(A+B)})$
  • Inaugural Address by Lord Moulton: The Invention of Logarithms its Genesis and Growth

• Writing about the middle of the eighteenth century, David Hume proclaimed John Napier of Merchiston as 'the person to whom the title of a great man is more justly due than to any other whom his country ever produced.' This judgment of Hume is the more remarkable, seeing he was himself naturally disposed to exalt literature above science. ...when he awarded the first place among his countrymen to Napier... it was doubtless from an enlivened conviction that his work had been of greater service to humanity.
  • Peter Hume Brown, "John Napier of Merchiston"

• Napier, the explorer of the secrets of nature, passed among his countrymen for a trafficker with Satan. Even to-day a certain mystery surrounds the figure of the Laird of Merchiston. ...In Scotland, as in other countries, the universities were exclusive centres of intellectual activity, and the studies at the universities were under the sole dominion of the Church, which naturally laid its ban on investigations that might imperil its own teaching. By his isolation Napier is thus wrapped in a certain mystery, and the mystery is deepened by the fact that we know so little of him, and what we know is at times strangely incongruous with the main preoccupations of his life.
  • Peter Hume Brown, "John Napier of Merchiston"

• The "marvellous Merchiston" (so he was known to the populace of his day) was born at Merchiston Castle in 1550. The period in which his birth and boyhood fell is the most momentous in the national history, and it determined and gave their peculiar character to his fundamental conceptions of human life and destiny. At the date of his birth the controversy had already begun which was eventually to cleave in twain the history of the Scottish people. The issue whether Roman Catholicism or Protestantism was to prevail was already joined. In 1546, four years before Napier was born, George Wishart was condemned by the Church and burned as a heretic, and in the same year Cardinal Beaton, the principal agent in his death, was assassinated. In 1547 John Knox began his mission which, after an interval, he was to see crowned with success. During the first ten years of Napier's life the struggle between the two religions was virtually settled. Between the years 1550 and 1560 the country was distracted by civil war, one party being for the old religion and alliance with France, the other for Protestantism and alliance with England. The contest ended in the victory of the Protestant party, and in 1560 a Convention of the Estates set up Protestantism as the national religion. It is in youth that the strongest and most permanent prepossessions and prejudices are formed, and we may trace the origin of Napier's abiding horror of the Church of Rome to the air which he breathed in the opening years of his life. ...it came to be his burning conviction that the salvation of mankind was bound up with the overthrow of the Papacy.
  • Peter Hume Brown, "John Napier of Merchiston"

• In 1563, the year of his mother's death, John was sent to the University of St. Andrews, the mother university of Scotland. He was only thirteen, but this was the usual age at which lads then entered the universities.
  • Peter Hume Brown, "John Napier of Merchiston"

External links

• Memoirs of John Napier of Merchiston: His Lineage, Life, and Times, with a History of the Invention of Logarithms (1834)
• Napier Tercentenary Memorial Volume by Cargill Gilston Knott (1915)