History of logarithms

"All computation was greatly simplified early in the seventeenth century by the invention of s."

"Johannes Kepler dedicated his 1620 Ephemerides to Napier, stating that the invention of logarithms was the central idea that enabled him to discover the third law of planetary motion."

"As the Indian figures are on infinite service in all branches of mixed Mathematics, and particularly in Astronomy... the next considerable improvement in this science was by the introduction of DECIMAL ARITHMETIC. This, according to Dr. Wallis, in his Preface to his Algebra was first done by ', about the year 1450. But the greatest improvement of all was made by the introduction ofLOGARITHMS.For, by their means, numbers almost infinite, and such as are otherwise impracticable, are managed with ease and expedition. They are the incontestable invention of the Lord Neper, a Scotchman, about the year 1614."

"Although the has largely given way to the pocket calculator... the pedagogical value of making one's own [slide or other] rules of various kinds remains. Napier's rods are easily made by schoolchildren and the historical route by which the modern slide rule evolved can be followed through with advantage. If has been taught at some stage, the principles of Napier's rods will already be understood. Many different kinds of graduated rule can be experimented with, including those of arithmetic scales (to be used for addition and subtraction) and those with geometric scales (for multiplication and division). It is not necessary to mention the word 'logarithm'; it is sufficient to introduce arithmetic and and to utilize the rules themselves in order to introduce the principles of logarithms."

"I find very few of those, who make constant use of logarithms, have attain'd an adequate Notion of them, or to understand the Extent of Use of them: contenting themselves with the Tables of them, as they find them, without daring to question them, or caring to know how to rectify them, should they be found amiss, being, I suppose, under the Apprehension of some great Difficulty therein, &c."

"[I]n the year 1543... Arithmetica Integra, of Stifelius... contained several curious things, some ascribed to a much later date. He treats... fully and ably, of pregressional and figurate numbers, and in particular of the... table for constructing them and the coefficients of all powers of a binomial so often used since... and... more than a century later was , by Pascal... called the arithmetic triangle... [T]he same table was used... by Cardan and Stevin, and other writers or arithmetic. Cardan's Opus Novum de Proportionibus... quotes it, and extracts the table and its use from Stifelius's book. ...Stifelius, at fol. 35... of the same book, treats of the nature and use of logarithms, though not under the same name, but under the idea of a series of arithmeticals, adapted to a series of geometricals. He there explains all their uses; such as that the addition of them, answers to the multiplication of their geometricals; subtraction to division; multiplication of exponents, to involution; and dividing of exponents, to evolution. And he exemplifies the use of them in cases of the Rule-of-Three, and in finding mean proportionals between given terms, and such like, exactly as is done in logarithms. So that he seems to have been in the full possession of the idea of logarithms, and wanted only the necessity of troubleſome calculations to induce him to make a table of such numbers."

"The learned calculators, about the close of the 16th, and beginning of the 17th century, finding the operations of multiplication and division by very long numbers, of 7 or 8 places of figures, which they had frequently occasion to perform, in resolving problems relating to geography and astronomy, to be exceedingly troublesome, set themselves to consider, whether it was not possible to find some method of lessening this labour, by substituting other easier operations in their stead. In pursuit of this object, they reflected, that since, in every multiplication by a whole number, the ratio, or proportion, of the product to the multiplicand, is the same as the ratio of the multiplier to unity, it will follow that the ratio of the product to unity (which, according to Euclid's definition of compound ratios, is compounded of the ratios of the said product to the multiplicand and of the multiplicand to unity) must be equal to the sum of the two ratios of the multiplier to unity and of the multiplicand to unity. ... And therefore they thought these artificial numbers, which thus represent, or are proportional to, the magnitudes of the ratios of the natural numbers to unity, might not improperly be called the Logarithms of those ratios, since they express the numbers of smaller ratios of which they are composed. And then, for the sake of brevity, they called them the Logarithms of the said natural numbers themselves, which are the antecedents of the said ratios to unity, of which they are in truth the representatives."

"Jost Burgi, a Swiss clockmaker and mathematician, invented logarithms independently of Napier and Briggs, although it is not clear when he started work on them. Some historians have suggested that Burgi may have invented logarithms earlier than Napier, but his work was not published until 1620, when the German mathematician and astronomer Johannes Kepler asked him to do so. ...six years after the publication of Napier's work."

"In the seventeenth century, perhaps the greatest of all for the development of mathematics, there appeared a work which in the history of British science can be place second only to Sir Isaac Newton's monumental Pincipia. In 1614, John Napier of Merchiston issued his Mirifici Logarithmorum Canonis Descriptio, ("A Description of the Admirable Table of Logarithms"), the first treatise on logarithms. To Napier, who also invented the decimal point, we are indebted for an invention which is as important to mathematics as Arabic numerals, the concept of zero, and the principle of positional notation. Without these, mathematics would probably not have advanced much beyond the stage to which it had been brought two thousand years ago. Without logarithms the computations accomplished daily with ease by every mathematical tyro would tax the energies of the greatest mathematicians."

"Johannes Kepler provided more accurate values for the Napier series with the aid of successive proportions between two given terms. In the Tabulae Rudolphinae (1627), he was the first to divide a table of logarithms into numerical and trigonometric parts."

"If we really desire to advance to a full understanding of the theory of logarithms, it is best to follow in broad outline the history of its creation."

"As contrasted with an absolute number, a logistic number represented measurement. ...Kepler welcomed the invention of logarithms as an ingenious device to facilitate laborious computations. Since he was concerned principally with astronomical computations involving sexagesimal fractions of the degree and of the hour, logistic logarithms were of prime importance to him. ...Kepler's logarithms were based on proportion, as he made clear in the following definition of a logarithm in his Thousand Logarithms: "Express the measurement of every proportion between 1000 and a number smaller than 1000... by a number which is placed alongside this smaller number in the Thousand and which is called its logarithm, that is, the number (arithmos) indicating the proportion (logos) which that number, to which the logarithm is attached, bears to 1000.""

"It [the Rudolphine Tables] was only the third new set of planetary tables in European history. And whereas Copernicus's and Ptolemy's tables were more or less equally accurate, Kepler's were some 50 times more so. Within a few years, it was possible to pinpoint the time of transit of Mercury across the face of the sun so that it was possible to observe it in transit for the first time in human history. Of course, Kepler's theories were more difficult, especially since he had incorporated logarithms, which had only been invented a few years earlier. Much of the book, therefore, was made up of explanatory text that told the reader how to use the tables."

"The next improvement in mathematics, which we have to mention, is the introduction of logarithms, those numbers so important by diminishing the labour of tedious calculations, and which play so conspicuous a part in the transcendental analysis. For this admirable discovery we are indebted to John Napier, Baron of Merchiston, near Edinburgh. ...Napier seems to have turned the bent of his genius towards the discovery of methods to facilitate and abridge trigonometrical calculations; and various contrivances were proposed by him in succession, all remarkable for their ingenuity. The last and most memorable of all was his discovery of logarithms."

"There is a story told by Mr. Wood, but it does not appear entitled to any attention, that one Dr. Craig, a Scotchman, coming out of Denmark into his own country, called upon John Napier... and told him of a new invention in Denmark, by Longomontanus, to save tedious multiplications and divisions in astronomical calculations. Napier being solicitous to know further of him of this matter, he could give no other account of it than that it was by proportional numbers; which hint Napier taking, desired him, at his return, to call upon him again. Craig, after an interval of some weeks, did so, and Napier then showed him a rude draught of what he called canon mirabilis logarithmorum. Had there been any truth in such a story, we may be sure that Longomontanus and the Danes would not have abstained from laying their claim to so admirable a discovery."

"Napier has also been considered as having been anticipated in his invention by Stifels, and by Juste Byrge, two German mathematicians; but these allegations originating from jealousy, or from national partiality, are entitled to no attention whatever, and Napier's claims have for many years been allowed by the universal consent of all mankind."

"The logarithms which first presented themselves to Napier were those at present known by the name of hyperbolic logarithms. But it afterwards occurred to him that logarithms, similar to those in our modern tables, in which the logarithm of 1 is 0; that of 10, 1; that of 100, 2; &c., would be more convenient. But he died, in 1618, before he had time to put his new plan in execution; but not till he had explained its nature to Mr. Henry Briggs, Gresham Professor of Mathematics, who had seen at once all the importance of logarithms, and had early devoted himself to bring them to perfection."

"Henry Briggs... applied himself chiefly to the study of mathematics. ...As soon as the Napierian discovery of logarithms was announced, he made two successive journeys into Scotland, to confer with the discoverer himself, and settle plans for the calculation and construction of logarithmic tables. An account of the nature and properties of logarithms was published at Edinburgh, in 1618, by Robert Napier, the son of the great discoverer, under the following title: Mirifici Logarithmorum Canonis constructio et eorum ad Naturales ipsorum Numeros Habitudines una cum Appendice de alia caque prestantiori Logarithmorum Specie condenda, &c. &c. This book had been written, and was ready for the press, when John Napier, the inventor of logarithms, was prevented from publishing it by his death. The same year Briggs published a table of the logarithms of the first 1,000 natural numbers, under the title of Logarithmorum Chilias prima. In 1624, he published, under the title of Arithmetica Logarithmica, the logarithms of all numbers from 1 to 20,000 and from 90,000 to 100,000, calculated to 14 decimal places."

"Briggs was assisted in his calculations by Gunter... the contriver of the graduated rule which passes under his name. He calculated the logarithms of the sines and tangents, and published a table of them in 1620, entitled, Canon of Triangles. Briggs had made considerable progress in a table of sines and tangents, calculated to 100 parts of a degree, (for he wished to introduce the decimal notation into trigonometry) but died, in 1630, before he had completed it. It was finished by Henry Gellibrand... and he published it in 1633, under the title of Trigonometria Britannica."

"One of the first persons on the Continent who properly appreciated the importance of logarithms, was Kepler. He published a work on the subject in 1624, in which he simplified the theory considerably, and developed the views of Napier with great sagacity and simplicity."

"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. ...he had privately communicated a summary of his results to Tycho Brahe as early as 1594. ...Napier explains the nature of logarithms by comparison between corresponding terms of an arithmetical and geometrical progression. ...it is the first valuable contribution to the progress of mathematics which was made by any British writer. The method by which logarithms were calculated was explained in the Constructio, a posthumous work issued in 1619... 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 was 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..."

"J. Bürgi, independently of Napier, had constructed before 1611 a table of antilogarithms of a series of natural numbers... published in 1620."

"In [1620] a table of the logarithms... of sines and tangents of angles in the first quadrant was brought out by Edmund Gunter... Four years later [he] introduced a "line of numbers," which provided a mechanical method for finding the product of two numbers: this was the precursor of the slide-rule, first described by Oughtred in 1632."

"In 1624, Briggs published tables of the logarithms of some additional numbers and of various trigonometrical functions. ...The calculation of 70,000 numbers which had been omitted by Briggs was performed by Adrian Vlacq and published in 1628: with this addition the table gave logarithms of numbers from 1 to 101,000."

"The Arithmetica Logarithmica of Briggs and Vlacq are substantially the same as existing tables: parts have at different times been recalculated but no tables of an equal range and fulness entirely founded on fresh computations have been published since. These tables were supplemented by Brigg's Trigonometrica Britannica, which contains tables not only of the logarithms of the trigonometrical functions, but also of their natural values... published posthumously in 1633."

"A table of logarithms to the base e... and of the sines, tangents, and secants of angles in the first quadrant was published by John Speidell... as early as 1619, but... these were not as useful in practical calculations as those to the base 10."

"By 1630 tables of logarithms were in general use."

"The miraculous powers of modern calculation are due to three inventions: the Arabic Notation, Decimal Fractions, and Logarithms. The invention of logarithms in the first quarter of the seventeenth century was admirably timed, for Kepler was then examining planetary orbits, and Galileo had just turned the telescope to the stars. During the Renaissance German mathematicians had constructed trigonometrical tables of great accuracy, but this greater precision enormously increased the work of the calculator. 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... 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... was Napier's line of thought? ...Napier's process is so unique and so different from all other modes of presenting the subject that there cannot be the shadow of a doubt that this invention is entirely his own; it is the result of unaided, isolated speculation. He first sought the logarithms only of sines..."

"\text{Nap. log y} = 10^7\;\text{nat. log} \frac{10^7}{y}It is evident from this formula that Napier's logarithms are not the same as the natural logarithms. Napier's logarithms increase as the number itself decreases."

"Napier's genesis of logarithms from the conception of two flowing points reminds us of Newton's doctrine of fluxions. The relation between geometric and arithmetical progressions, so skilfully utilised by Napier, had been observed by Archimedes Stifel and others. Napier did not determine the base to his system of logarithms. The notion of a "base" in fact never suggested itself to him. The one demanded by his reasoning is the reciprocal of that of the natural system, but such a base would not reproduce accurately all of Napier's figures, owing to slight inaccuracies in the calculation of the tables."

"Napier's great invention was given to the world in 1614 in a work entitled Mirifici logarithmorum canonis descriptio In it he explained the nature of his logarithms, and gave a logarithmic table of the natural sines of a quadrant from minute to minute."

"The theory of natural ("hyperbolic") logarithms apparently first suggested itself to mathematicians engaged in the mensuration of spaces between the hyperbola and its asymptotes. About a quarter of a century later, in 1695, Edmund Halley discarded geometrical figures and published a remarkable article containing a purely arithmetical theory of logarithms. In this original and meritorious investigation he lays great stress upon what we now call the "modulus". By Napier's logarithms Halley understands those which give Briggs's logarithms when divided by 2.302 585 or when multiplied by 0.43429448. From this statement it appears that Halley considered Napier's logarithms to be identical with natural logarithms, and we must look upon him as one of the first (perhaps the first) to commit this error. That the two systems are not identical is shown by the following formula:\log_{N} x = 10^7 \log_{e} \frac{10^7}{x},During the eighteenth century this misunderstanding regarding the two systems does not appear to have been as wide-spread as it was later."

"The confusion marked in the writings of Halley and Saverien spread among French writers. Montuclu, the great mathematical historian of the eighteenth century, made the same mistake; Bossut helped to perpetuate the error."

"In England Charles Hutton, who in 1785, published the first edition of his Mathematical Tables (which includes an elaborate and in many respects excellent history of logarithms) describes Napier's logarithms correctly, but subsequently he speaks of "the right-angled hyperbola, the side of whose square inscribed at the vertex is 1, gives "Napier's logarithms"."

"De Morgan carefully explains the difference between Napier's and natural logarithms in the article "Tables" in the English Cyclopaedia but in De Morgan's Budget of Paradoxes (р. 70) Günther has found a passage which is inaccurate."

"It is a pleasure to find that Kästner presents the subject in a way free of error. In his Geschichte he refers to an article, which he had written, setting forth the exact relation between the two systems. Nevertheless the misconception became prevalent in Germany also."

"Proceeding to... the earliest publication of tables of natural logarithms... John Speidell... in 1619 brought out his New Logarithmes, only five years after Napier's publication of the Descriptio. Speidell's book received little attention, either during his life-time or since. It would seem as if the earliest publication of a table of natural logarithms should be mentioned in histories of mathematics, but so far as I know, no general history by a German, French, or British author, takes notice of Speidell. ..However, Speidell's New Logarithmes has been described in at least three special historical articles. Hutton speaks of it in the "Introduction" to his Tables; Augustus De Morgan makes a careful study of his book in the article "Tables" in the English Cyclopaedia; a J. W. L. Glaisher gives a brief account of Speidell's work in the report on "Tables" in the British Association Report, 1873..."

"Speidell's... sole object was to simplify matters for persons unacquainted with the use of negative quantities. ...Speidell did not advance a new theory. He simply aimed to make all the logarithms in his table positive."

"Logarithms were invented before our modern exponential notation, a^n, was introduced into algebra. ...[A]lgebraic symbols... to indicate powers and roots of a number had been suggested before the advent of the logarithm, but these suggestions... remained unheeded; the fact is that the inventors of logarithms did not use the modern exponential notation and were not familiar with the exponential concept which now plays such a fundamental rôle..."

"What... were the basic considerations in the development of logarithms... by their inventors, John Napier and Joost Bürgi?"

"John Napier's Mirifici logarithmorum canonis descriptio appeared in 1614 in Edinburgh, and his Mirifici logarithmorum canonis constructio appeared there... posthumous... in 1619, though written as early... or earlier than, the Descriptio."

"Napier based his explanations upon... (1) The geometrico-mechanical concept of flowing points, (2) the relations which exist between arithmetic and geometric series. Several writers before the time of Napier called attention to... relations between the terms of a geometric series and... arithmetic series, which... involve the logarithmic idea... [b]ut did not realize the possibilities... nor... conceive and execute... computing a pair of corresponding series sufficiently dense for practical use..."

"From certain passages in authors like Stifel one might be tempted to say that the logarithmic concept really existed before the time of Napier and Bürgi. Yet how much of a novelty the logarithms of Napier really were to the foremost mathematicians of his day can be realized by the enthusiasm with which Briggs and Kepler took up the new topic."

"Briggs addressed... [Napier], "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 in astronomy viz., the logarithms.""

"In the language of Napier, the definition of a logarithm is... The logarithm of a given sine is that number which has increased arithmetically with the same velocity throughout as that with which radius began to decrease geometrically, and in the same time as radius has decreased to the given sine."

"Letting v = 10^7, the geometric and arithmetic series of Napier may be exhibited in modern notation as follows:"

"In the Descriptio logarithms are defined as follows: ...Logarithms are numbers which correspond to proportional numbers and have equal differences. The proportional numbers are the terms of the geometric progression; the numbers having equal differences are the terms of the arithmetic progression."