John 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."

"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"

"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."

"These properties of the indices of numbers were taken notice of by Stifelius, and even by Archimedes in his work on the numbering of the sands; but it is to Baron Napier, of Merchiston, in Scotland, that we are indebted for the happy idea of applying such numbers to the purposes of arithmetical and trigonometrical calculation, which first appeared in his "Mirisici Logarithmorum Canonis Descriptio," published at Edinburgh in 1614. This work was translated by Mr. Edward Wright, and published by his son in 1616. The method of constructing the table was reserved by the ingenious author till the sense of the learned, upon his invention, should be known; nevertheless Kepler, in his "Chilias Logarithmorum," &c. published in 1624; [John] Speidell, in his "New Logarithms," published in 1619; [Benjamin] Ursinius, in his "Table of Logarithms," 1625; and many other mathematicians constructed small tables conformably to the plan of Lord Napier. But of all those who assisted in the construction of logarithmic tables, Briggs is the most conspicuous; it was he who first suggested our present system, the advantages of which are incalculably greater than those first constructed by Napier, at the same time that he laboured more than anyone in the construction of them."

"Napier, in 1614, ...employed the idea of the fluxion of a quantity to picture by means of lines the relation between logarithms and numbers."

"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."

"Many computing devices have been used since the invention of the abacus. These include , sector compasses, slide rules, calculators, and computers."

"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."

"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?"

"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."

"Napier considered the synchronized motion of two points, each moving on a straight line, the one with constant velocity, and the other with a decreasing velocity proportional to the distance remaining to a fixed point, the initial velocity being the same. In modern notation his model may be written as"

"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."

"Napier also invented a calculating device known as Napier's bones in 1617 and made common use of decimal multiplications and divisions. The device itself does not use logarithms, but rather is a convenient tool to reduce multiplication and division to a sequence of simple addition and subtraction operations. The method employed by Napier's bones was based on Arab mathematics and Fibonacci's Liber Abaci."

"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."

"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."

"Napier presented his concept of logarithms as a boon to mankind for, as he pointed out, working with logarithms reduces the drudgery of numerical computations enormously."

"Napier's book required the intensity of concentration that had allowed Newton to conceptualize his remarkable vision of a universe governed by a pervasive law of gravitation; yet, while Newton's book was packed with tedious geometric axioms, it was probably much more enjoyable to compose than Napier's mind-numbing treatise which consisted of thousands of laborious calculations of the logarithms of numbers."

"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"

"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."

"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."

"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."

"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."

"Rather curiously, his works of greatest scientific interest, the Descriptio and Constructio have been most neglected. The former was reprinted in 1620, and also in Scriptores Logarithmici, besides being translated into English. The latter was reprinted in 1620 only. This neglect is no doubt largely accounted for by the advantage for practical purposes of tables computed to the base 10, an advantage which Napier seems to have been aware of even before he had made public his invention in 1614."

"Several years ago (Reader, Lover of the Mathematics) my Father, of memory always to be revered, made public the use of the Wonderful Canon of Logarithms; but, as he himself mentioned on the seventh and on the last pages of the Logarithms, he was decidedly against committing to types the theory and method of its creation, until he had ascertained the opinion and criticism on the Canon of those who are versed in this kind of learning. But, since his departure from this life, it has been made plain to me by unmistakable proofs, that the most skilled in the mathematical sciences consider this new invention of very great importance, and that nothing more agreeable to them could happen, than if the construction of this Wonderful Canon, or at least so much as might suffice to explain it, go forth into the light for the public benefit."

"You have then (kind Reader) in this little book most amply unfolded the theory of the construction of logarithms, (here called by him artificial numbers, for he had this treatise written out beside him several years before the word Logarithm was invented,) in which their nature, characteristics, and various relations to their natural numbers, are clearly demonstrated."

"We have also taken care to have printed some Studies on the above-mentioned Propositions, and on the new kind of Logarithms, by that most excellent Mathematician Henry Briggs, public Professor at London, who for the singular friendship which subsisted between him and my father of illustrious memory, took upon himself, in the most willing spirit, the very heavy labour of computing this new Canon, the method of its creation and the explanation of its use being left to the Inventor. Now, however, as he has been called away from this life, the burden of the whole business would appear to rest on the shoulders of the most learned Briggs, on whom, too, would appear by some chance to have fallen the task of adorning this Sparta. Meanwhile (Reader) enjoy the fruits of these labours such as they are, and receive them in good part according to your culture. Farewell, Robert Napier, Son"

"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?"

"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?"

"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."

"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."

"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."

"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)})"

"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."

"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."

"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, 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."

"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."

"Whence a geometrically moving point approaching a fixed one has its velocities proportionate to its distances from the fixed one."

"From the Radical table completed in this way, you will find with great exactness the logarithms of all sines between radius and the sine 45 degrees; from the arc of 45 degrees doubled, you will find the logarithm of half radius; having obtained all these, you will find the other logarithms. Arrange all these results as described, and you will produce a Table, certainly the most excellent of all Mathematical tables, and prepared for the most important uses."

"Therefore, Sir, let it be your Majesty's continuall study (as called and charged thereunto by God) to reforme the universall enormities of your country, and first (taking example of the princely prophet David) to begin at your Majesty's owne house, familie and court, and purge the same of all suspicion of Papists and Atheists and Newtrals, whereof this Revelation foretelleth that the number shall greatly increase in these latter daies. For shall any Prince be able to be one of the destroyers of that great seate, and a purger of the world from Antichristianisme, who purgeth not his owne countrie? Shall he purge his whole countrie who purgeth not his owne house? Or shall he purge his house, who is not purged himself by private meditations with his God?"

"And if any number of equals to a first sine be multiplied together producing a second, just so many equals to the Logarithm of the first added together produce the Logarithm of the second."