History of logarithms

"The word "" is of Greek structure and signifies "number of ratios." The idea is this: v(1 - \frac{1}{v})^n is gotten from v by n successive applications of the ratio (1 - \frac{1}{v}) . Hence n, which is the logarithm, indicates "the number of ratios." Napier restates the definition in 1614 as follows: ...Logarithms may be called equidifferent companions to proportional numbers."

"Not only is Napier's definition of a logarithm different from the modern definitions, but the notion of a "base" is inapplicable to his system. To force the concept of a "base" upon his system we must modify it somewhat. If each number of the two progressions of Napier is divided by 10^7, so that 0 becomes the logarithm of 1, then 1 is the logarithm of (1 - \frac{1}{10^7})^{10^7} , which is nearly equal to e^{-1}, where e is the base of the natural system. Hence the base of Napier's logarithms, when modified as here indicated, is very nearly e^{-1}."

"Joost Bürgi invented logarithms independently of Napier, but he lost all rights of priority by failure to publish until the praises of Napier's book began to resound throughout Europe."

"In 1620 appeared in Prag the Progress-Tabulen, containing Bürgi's logarithmic tables, but omitting the explanations of them that were promised on the title-page. Hence his logarithms were unintelligible to the ordinary reader."

"Common to Bürgi and Napier was the use of progressions in defining logarithms. In Bürgi's tables the numbers in the were printed in red, the numbers in the were in black. The relation between Bürgi's logarithms, 10n, and their antilogarithms is expressed in modern notation by the equation 10n = \log[10^8(1 + \frac{1}{10^4})^n], \qquad n = 1, 2, 3, \cdots ."

"The notion of a "base" can no more be forced upon Bürgi's logarithms than it can be upon the logarithms in Napier's tables. In neither system is \log 1 = 0. Their logarithmic concepts were more general than those of the present day in... that by sliding one progression past the other they could select any positive number at random as the one whose logarithm is zero. We have seen that Napier originally chose \log 10^7 = 0 while Bürgi chose \log 10^8= 0. The logarithms in their tables were integral numbers. More than this, the terms of the two series could be made to increase in the same direction or in opposite directions, at pleasure. That is, if m > n, one can make \log m < \log n , or \log m > \log n , just as one may choose. Napier originally chose the first alternative, Bürgi the second."

"Napier and Briggs conferred with each other and agreed to modify the original logarithms of Napier. In the Appendix to Napier's posthumous work, the Constructio, an improvement is suggested, "which adopts a cypher as the Logarithm of unity, and 10,000,000,000 as the Logarithm of either one tenth of unity or ten times unity." The subsequent use of decimal fractions in logarithmic tables led to the common logarithm proper, in which \log 1 = 0 and \log 10 = 1. A readjustment of Napier's original logarithms was made in 's New Logarithmes, published in 1619 in London, whereby the logarithms virtually became the so-called "s" of to-day."

"The word "logarithm" means "ratio number" and was an afterthought with Napier. He first used the expression "artificial number," but before he announced his discovery he adopted the name by which it is now known."

"Briggs introduced (1624) the word "mantissa." ...originally meaning an addition, a makeweight, or something of minor value, and was written mantisa. In the 16th century it came to be written mantissa and to mean "appendix"... The term "characteristic" was suggested by Briggs (1624) and is used in the 1628 edition of Vlacq."

"Napier worked at least twenty years upon the theory. His idea was to simplify multiplications involving sines, and it was a later thought that included other operations, applying logarithms to numbers in general. He may have been led to his discovery by the relationsin A\;sin B = \frac{1}{2}(cos\overline{A - B} - cos\overline{A + B})for, as Lord Moulton says, in no other way can we "conceive that the man to whom so bold an idea occurred should we have so needlessly and so aimlessly restricted himself to sines in his work, instead of regarding it as applicable to numbers in general.""

"Napier published his Descriptio of the table of logarithms in 1614. This was at once translated into English by Edward Wright, but with the logarithms contracted by one figure."

"In Napier's time sin &phi; was a line, not a ratio. The radius was called the sinus totus, and when this was equal to unity the length of the sine was simply stated as sin&phi;. If r was not unity, the length was r sin&phi;. With this statement we may consider Napier's definition of a logarithm:The logarithme therefore of any sine is a number very neerely expressing the line, which increased equally in the meane time, whiles the line of the whole sine decreased proportionally into that sine, both motions being equal-timed, and the beginning equally swift.From this it follows that the logarithm of the sinus totus is zero. Napier saw later that it was better to take log 1 = 0."

"Napier's logarithms are not those of the so-called Naperian, or hyperbolic, system, but are connected with this system by the relation\log_{n} a = 10^7\cdot\log_{e} 10^7 - 10^7\cdot\log_{e} a.The relation between the sine and its logarithm in Napier's system issin \phi = 10^7\cdot e^\frac{-\log_{n} sin \phi}{10^7}so that the sine increases as its logarithm decreases."

"Henry Briggs... was one of the first to appreciate the work of Napier. Upon reading the Descriptio he wroteNaper, lord of Markinston, 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 which pleased me better, and made me more wonder."

"Briggs's Aritmetica Logarithmica the preface... contains the following statement by the author...That these logarithms differ from those which that illustrious man, the Baron of Merchiston published in his Canon Mirificus must not surprise you. For I myself, when expounding their doctrine publicly in London to my auditors in Gresham College, remarked that it would be much more convenient that 0 should be kept for the logarithm of the whole sine (as in the Canon Mirificus)... And concerning that matter I wrote immediately to the author himself; and as soon as... permitted I journeyed to Edinburgh, where, being most hospitably received by him, I lingered for a whole month. But as we talked over the change in logarithms he said that he had for some time been of the same opinion and had wished to accomplish it. ...He was of the opinion that... 0 should be the logarithm of unity."

"The real value of the proposition made by Briggs at this time was that he considered the values of log 10n a, for all values of n. The relation between the two systems as they first stood were as follows:Napier, log\;y = r(log_{e} r - log_{e} y), where r = 10^7; Briggs, log\;y = 10^{10}(10 - log_{10}\;y); Napier (later suggestion), log\;y = 10^9 log_{10}\;y."

"The first table of logarithms of trigonometric functions to the base 10 was made by Gunter."

"In the 1618 edition of Edward Wright's translation of the Descriptio there is printed an appendix, probably written by Oughtred, in which there is an equivalent of the statement that loge10 = 2.302584, thus recognizing the base e. Two years later John Speidell published his New Logarithmes, also using this base. He stated substantially thatlog\;n = 10^{-1} (nap\;log\;1 - nap\;log\;n), or, log\;n = 10^5 (10 + log_{e} 10^{-5} x)."

"By the middle of the seventeenth century, logarithms had found their way into elementary arithmetics, as seen in Hartwell's (1646) edition of Recorde's Ground of Artes, where it is said that "for the extraction of all roots, the table of Logarithms set forth by M. Briggs are most excellent, and ready.""

"It is evident that"

"Stifel... in the Arithmetica Integra of 1544 ...refers several times to the laws of exponents. At first he uses the series"

"The theory was again given by [Pierre] Forcadel (1565), with a statement that the idea was due to Archimedes, that it was to be found in Euclid, and that Gemma Frisius had written upon it."

"When... Schoner came to write his commentary on the work of Ramus, in 1586, a decided advance was made, for not only did he give the usual series for positive exponents, but, like Stifel, he used the geometric progressions with fractions as well, although... not with negative exponents."

"In general the German writers were in the lead. ...particularly ...Simon Jacob (1565), who followed Stifel closely, recognizing all four laws, and... influencing Jobst Bürgi. These writers did not use the general exponents essential to logarithms, but the recognition of the four laws is significant."

"In 1620 Jobst Bürgi published his Progress Tabulen... he was influenced by Simon Jacob's work. The tables... are simply lists of antilogarithms with base 1.0001. ...none seem to be later than 1610, so that he probably developed his theory independently of Napier. ...he approached the subject algebraically, as Napier approached it geometrically."

"The... invention of logarithms was to reduce all such human labor as Kepler's to more manageable proportions. The history of logarithms is another epic of performance second only to Kepler's. ...Napier ...in the leisure ...as landlord, and his unavailing labors to prove that the reigning pope was Antichrist, invented logarithms."

"When remembered... Napier died before Descrtes introduced the notation n, n^2, n^3, \cdots for powers, we cease to wonder why it took him twenty years to reason out... logarithms."

"The fundamental idea of the correspondence between two series of numbers, one in arithmetic, the other in , ...was explained by Napier through the conception of two points moving on separate straight lines, the one with uniform, the other with accelerated velocity. If the reader... will attempt to obtain... in this way a demonstration of the fundamental rules of logarithmic calculation, he will rise from the exercise with an adequate conception of the penetrating genius of the inventor of logarithms."

"Napier's of n would be our 10^7 \log_{e}(10^7 n^{-1})..."

"After the invention of the calculus, investigation of the logarithmic function... followed... from the simple differential equation dy = y\;dx. ...The only facts concerning logarithms of any importance for the development of mathematics..."

"Napier gave Tycho a forecast of his invention in 1594, and in 1614 published his Descrioptio. In 1624 a usable table by H. Briggs... was published, as also... one by Kepler. Other tables quickly appeared, and by 1630 logarithms were in the equipment of every computing astronomer."

"[L]ogarithms are one of the most disorderly battlegrounds in mathematical history. ... [A]s adjudicated in 1914... Napier's priority ...is undisputed; J. Bürgi ...independently invented logarithms and constructed a table between 1603 and 1611, while "Napier worked on logarithms probably as early as 1594 ...; therefore, Napier began working on logarithms probably much earlier than Bürgi.""

"Disputes like this and the other over the calculus have made more than one man of science envy his successors of ten thousand years hence, to whom Newton and Leibniz, Napier and Bürgi, and scores of lesser contestants for individual fame will be semimythical figures as indistinct as Pythagoras."

"In 1576... Wittich met John Craig... who he later described as "knowledgable about mathematics and philosophy."... Wittich must have transmitted his ideas about... —for reducing the... labors of multiplying large numbers... Craig copied... [the] method into his... [copy] of De Revolutionibus and... when he returned to to become court physician to James VI, discussed it with John Napier... A few years after Wittich died, Craig praised him saying, "If you require mathematical demonstrations, when others do not suffice... I turn to those of Wittich.""

"In 1577 Wittich... observed the great comet... It is tempting to suppose he observed... with Bartholomew Scultetus... because Tycho groups Scultetus and Wittich... in discussing and criticizing the parallax method... they applied to the comet. ...Wittich never published his comet observations but lent them to... Thaddeus Hagecius. ...[H]e did not get high marks for his observing. Tycho remarked on the lack of accuracy in his lunar eclipse observations, although "in his treatment of geometry and trigonometry he was more agile and successful"... ... noted that Wittich had poor eyesight and should have stuck to geometry."

"Brahe and Wittich probably first met in 1566 or 1570 at ... Brahe recalled mentioning some trigonometrical notions, which then became the source of Wittich's ideas on... ... In 1580 Wittich paid a four-month visit to ... Tycho spoke of him... as an observing companion, as a "standard bearer for me in my astronomical studies." ...This was ...a time of intense joint work because Tycho ...spoke of the "sweat" of multiplying large numbers and the "tedium" that will be saved with the new method that Wittich is working on but has not yet perfected."

"In a letter a few years after Witten's death... "Certainly your Wittich... was a man very skilled in mathematics. In... 1580 when a... comet shone... he was... with me... observing... He noticed... it continued... several days along an exact on the sphere (which I... pointed out to him on... a globe), and he... conclude[d]... the comet had chosen... a path... in the highest aether and not at all in the elementary region.""

"Wittich left Hven around... November 1580. ...Tycho ...presented his Wratislavian friend with... a costly copy of Apianus' ', inscribed "to a friend and fellow lover of mathematics.""

"Wittich had gone to the court of Hesse-Cassel, where he acted less like a faithful standard bearer than as an agent of "technological espionage." Wilhelm of Cassel was, like Tycho, a practicing observer... Wilhelm also had in his entourage two skilled assistants... and . Wittich demonstrated... mechanical ingenuity by designing an ... [and] spoke freely about his mathematical methods and about principles of instrument design... learned on Hven. Many of these points were recorded by Rothmann..."

"Tycho... believed... Wittich was communicating crucial information about... his instruments... and... divulging "other things" at the Landgrave's observatory. To Bürgi he awarded a higher character reference: Bürgi "never presumed to claim his own things taken from Wittich." ...But for ...Nicholas Reymars Baer, and known ...as Ursus (the Bear), Tycho reserved ..."savage, inhuman, scurrilous, rotten and sycophantic.""

"Wittich... died... January 1586 without a publication to his name and... without a manuscript in preparation. ...His death ...changed the context of authorial credit; those who profited from his instruction could publicly praise the man... without reference to his work on the system of the world. ...[H]is ghost ...Tycho referred to it explicitly ...stalked the ...corridors of Uraniborg. ...Two years after Wittich's death, Tycho published the famous system... and Ursus came forth with... a new... similar "system of nature." Tycho... placed his diagram... labeling it "New Sketch of a World System lately invented by the Author; In which the Old Ptolemaic Gracelessness and Superfluity as well as the New Copernican Physical Absurdity of the Earth's Motion are eliminated and everything corresponds most fittingly to the Celestial Appearances.""

"In the ensuing battle... Tycho... vanquished Ursus' claims, and Wittich's name all but vanished into oblivion."

"To the end of his life remembered, with... euphoric nostalgia and... rage, the stimulating visit from in... 1580. Never before had he found such congenial intellectual companionship, nor... until the arrival of Johannes Kepler in 1600. To his Wratislavian guest Tycho opened all his secrets... so he wrote, intimating ruefully that he had learned his lesson and would never again display his inventions so freely. But by... October 1580... Wittich had resolved to go back home... [T]he next thing Tycho knew, Wittich was in the court of Wilhelm of Hesse, freely talking about all that was new in astronomy, whether his own or the great Dane's. Tycho remained obsessed by this outrage, and the volume of letters he published in 1596 is largely... designed to establish his... priorities... [and] depricate Wittich... referred to... obliquely as "a certain Wroclaw mathematician.""

"Tycho had learned from Wittich the first procedure... whereby trigonometric identities could replace tedious and division by simple and . He knew... Whittich had described the method in his... manuscript workbook and... annotated copies of Copernicus' De Revolutionibus. ...[W]hen he learned ...Wittich had died, he lost little time... taking steps to acquire Wittich's library."

"Wittich's annotations in the 1566 Copernicus, finally acquired by Tycho, eventually were attributed to Tycho..."