Jost Bürgi

Jost Bürgi (also Joost, Jobst; Latinized surname Burgius or Byrgius; 28 February 1552 – 31 January 1632), active primarily at the courts in Kassel and Prague, was a Swiss clockmaker, mathematician, and writer. Bürgi has been credited, independently of John Napier, as a co-inventor of logarithms.

Quotes

[D]ue to a lack of languages, the door to... authors has not always been open to me, as... to others, I have had to follow my own thoughts a little more than the learned and well-read, and seek new paths.
(Und weil mir auß mangel der sprachen die thür zu den authoribus nit alzeitt offen gestanden, wie andern, hab jch etwas mehr, als etwa die glehrte vnd belesene meinen eigenen gedanckhen nachhengen vnd newe wege suechen müessen.)
- Jost Bürgi, Einleitung zur Coss or Introduction to Coss As quoted by M. List & V. Bialas: Nova Kepleriana: Die Coss von Jost Bürgi in der Redaktion von Johannes Kepler. (1973) Verlag der Bayerischen Akademie der Wissenschaften, Munich. English translation with the aid of Google translate.

Divide a right angle in as many parts as you want and construct herefrom the sine table.
(Einen rechten Winckell in also viel theile theilenn alß man will, vnnd aus demselben den Canonen Sinuum vermachenn.)
- As quoted by Menso Folkerts, "Eine bisher unbekannte Schrift von Jost Bürgi zur Trigonometrie" pp. 107–114. Arithmetik, Geometrie und Algebra in der frühen Neuzeit ed. R. Gebhardt (2014) Adam-Ries-Bund, Annaberg-Buchholz. Ref: Grégoire Nicollier, "How Bürgi computed the sines of all integer angles simultaneously in 1586" (2018) Math Semesterber (2018) Vol. 65, pp. 15–34.

In such a way, with much trouble and labor, the whole Canon has been established. For many hundreds of years, up to now, our ancestors have been using this method because they were not able to invent a better one. However, this method is uncertain and dilapidated as well as cumbersome and laborious. Therefore we want to perform this in a different, better, more correct, easier and more cheerful way. And we want to point out now how all sines can be found without the troublesome inscription [of polygons], namely by dividing a right angle into as many parts as one desires.
- Jost Bürgi, Fundamentum Astronomiae (1592) Chapter 10, on fol. 34r-v. As quoted by Menso Folkerts, Dieter Launert, Andreas Thom, "Jost Bürgi’s Method for Calculating Sines" (Feb 2, 2016) Version 2, p. 8, arXiv:1510.03180.

Quotes about Bürgi

It would seem that J. Bürgi, independently of Napier, had constructed before 1611 a table of antilogarithms of a series of natural numbers: this was published in 1620. ...Bürgi also employed decimal franctions ...
- W. W. Rouse Ball, A Short Account of the History of Mathematics (1888)

[I]n 1575 Western Europe had recovered most of the major mathematical works of antiquity now extant. Arabic algebra had been... mastered and improved... through the solution of the cubic and quartic and through... partial... symbolism; and trigonometry had become an independent discipline. The time was almost ripe for rapid strides... The transition from the Renaissance to the modern world was... made through... intermediate figures, a few of the more important... Galileo Galilei... and Bonaventura Cavalieri... from Italy; several... as Henry Briggs.., Thomas Harriot.., and William Oughtred... were English; two... Simon Stevin... and Albert Girard... were Flemish; others came from varied lands—John Napier... from Scotland, Jobst Bürgi... from Switzerland, and Johann Kepler... from Germany.
- Carl Benjamin Boyer, A History of Mathematics (1968, 1991)

Napier was... the first... to publish.., but...it is possible that the idea of logarithms had occured to Bürgi as early as 1588... half a dozen years before Napier began work... However, Bürgi printed... in 1620, half a dozen years after Napier published... Descriptio. Bürgi's... book... Arithmetische und geometrische Progress-Tabulen... indicates... the influences were similar... to Napier. Both... proceeded from the properties of arithmetic and geometric sequences, spurred, probably by the method of prosthaphaeresis. The differences... lie chiefly in... terminology and... numerical values..; the fundamental principles were the same. Instead of proceeding from a number a little less than one (...Napier used $1-10^{-7}$ $1-10^{-7}$ ), Bürgi... a little greater than one... $1+10^{-4}$ $1+10^{-4}$ ; and instead of multiplying powers of this number by $10^{7}$ $10^{7}$ , Bürgi multiplied... $10^{8}$ $10^{8}$ . ...[O]ne other minor difference: Bürgi multiplied all... power indices by ten... [I]f $N=10^{8}(1+10^{-4})^{L}$ $N=10^{8}(1+10^{-4})^{L}$ , Bürgi called $10L$ $10L$ the "red"... corresponding to the "black"... $N$ $N$ . If... we were to divide all black[s]... by $10^{8}$ $10^{8}$ and all red[s]... by $10^{5}$ $10^{5}$ , we should have... a system of natural logarithms. ...Bürgi gave for the black...1,000,000,000 the red...230,270.022, which on shifting decimal points, is equivalent to... $\ln 10=2.3027022$ $\ln 10=2.3027022$ ... not a bad approximation... especially when... $(1-10^{-4})^{10^{4}}$ $(1-10^{-4})^{10^{4}}$ is not quite the same as $\lim _{n\to \infty }(1+{\frac {1}{n}})^{n}$ $\lim _{n\to \infty }(1+{\frac {1}{n}})^{n}$ although... values agree to four significant figures.
In publishing... he had... an antilogarithmic table... The essence of the principle is there... Bürgi must be regarded as an independent discoverer who lost credit... because of Napier's priority in publication. In one respect his logarithms come closer to ours than Napier's, for as... black[s] increase, so do the red[s]..; but the two systems share the disadvantage that the logarithm of the product or quotient is not the sum or difference of the logarithms.
- Carl Benjamin Boyer, A History of Mathematics (1968, 1991)

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.
- Yoshihide Igarashi, Tom Altman, Mariko Funada, Barbara Kamiyama, Computing: A Historical and Technical Perspective (2014) Ch. 10 Decimal Fractions and Logarithms

The idea of the logarithm probably had its source in the use of... trigonometric formulas that transformed multiplication into addition and subtraction. ...[I]f one needed to solve a triangle using the law of sines, a multiplication and division were required. ...[C]alculations were long and errors... made. Astronomers realized... multiplication and division could be replaced by additions and subtractions. To accomplish this... sixteenth century astronomers used formulas... as $2\sin \alpha \sin \beta =cos(\alpha -\beta )-\cos(\alpha +\beta )$ $2\sin \alpha \sin \beta =cos(\alpha -\beta )-\cos(\alpha +\beta )$ . ...A second source of the... logarithm was probably found in... algebraists as Stifel and Chuquet, who both displayed tables relating the powers of 2 to the exponents and showed that multiplication in one table corresponded to addition in the other. But because these tables had large gaps, they could not be used for necessary calculations. ...[T]wo men... independently, the Scot John Napier... and the Swiss Jobst Bürgi... came up with the idea of producing an extensive table... to multiply any... numbers... (not just powers of 2)... Napier published... first.
- Victor J. Katz, A History of Mathematics: An Introduction (1993, 1998)

Joost Bürgi... a Swiss watch and instrument maker and an assisitant to Kepler in Prague was... interested in facilitating astronomical calculations; he invented logarithms independently of Napier about 1600 but did not publish his work, Progress Tabulen, until 1620. Bürgi too was stimulated by Stifel's remarks that multiplication and division of terms in a geometric progression can be performed by adding and subtracting the exponents. His arithmetical work was similar to Napier's.
- Morris Kline, Mathematical Thought from Ancient to Modern Times (1972)

I do not have to explain to which level of comprehensibility this extremely deep and nebulous theory has been corrected and improved by the tireless study of my dear teacher, Justus Bürgi... by assiduous considerations and daily thought. ...Therefore neither I nor my dear teacher, the inventor and innovator of this hidden science, will ever regret the trouble and the labor which we have spent.
- Nicolaus Reimers Ursus, Fundamentum astronomicum (1588) fol. 4. As quoted by Menso Folkerts, Dieter Launert, Andreas Thom, "Jost Bürgi’s Method for Calculating Sines" (Feb 2, 2016) Version 2, p. 8, arXiv:1510.03180.

The calculation of the Canon Sinuum can be done... in the usual way, by inscribing the sides of a regular polygon into a circle... geometrically. Or... by a special way,.. dividing a right angle into as many parts as one wants... arithmetically. This has been found by Justus Bürgi... the skilful technician...
- Nicolaus Reimers Ursus, Fundamentum astronomicum (1588) as quoted by Menso Folkerts, Dieter Launert, Andreas Thom, "Jost Bürgi’s Method for Calculating Sines" (Feb 2, 2016) Version 2, p. 8, arXiv:1510.03180.

It is surprising that Kepler did not consider Jost Bürgi, who from around 1580 to 1592 already constructed planetary globes that were considered awesome works of art. ...In the letter 42 dated May 28th 1598, to the Duke Friederich I von Württemberg (1557–1608), Kepler writes that he will construct a globus with a planetarium. It is possible and likely that the Duke had in mind a sky globus similar to those already constructed by Eberhard Baldewein (1525-1593), Gerhard Emmoser (1556-1584) or Jost Bürgi (1552-1631), but none of these authors were cited in Kepler’s overview. These machines were designed and constructed to show the overall motion of the sky, to identify the position of the stars, to show the motion either of the Sun, the Moon or both.
- Daniele L. R. Marini, "The Planetary Machine Designed by Johannes Kepler" (August 2022) pp. 10 (footnote 30) to 12. Creative Commons 4.0 International License.

Memoirs of John Napier of Merchiston, (1834)

His Lineage, Life and Times, with a History of the Invention of Logarithms by Mark Napier. A source. @archive.org

That fine old gossip, Anthony Wood, picked up a story of Napier, Dr Craig, and the Logarithms, which he thus recorded in the Athenæ Oxonienses.
"It must be now known, that one Dr Craig, a Scotchman, perhaps the same mentioned in the Fasti, under the year 1605, among the incorporation, coming out of Denmark into his own country, called upon Joh. Neper, Baron of Mercheston, near Edinburgh, and told him, among other discourses, of a new invention in Denmark (by Longomontanus, as 'tis said,) to save the tedious multiplication and division in astronomical calculations. Neper being solicitous to know farther of him concerning this matter, he could give no other account of it than that it was by proportional numbers. Which hint Neper taking, he desired him at his return to call upon him again. Craig, after some weeks had passed, did so, and Neper then showed him a rude draught of what he called Canon mirabilis logarithmorum. Which draught, with some alterations, he printing in 1614, it came forthwith into the hands of our author Briggs, and into those of Will. Oughtred, from whom the relation of this matter came."
- Ref: Anthony A. Wood, Athenae Oxonienses: an Exact History of all the Writers and Bishops who have had their Education in the University of Oxford to which are added the Fasti, or Annals of the said University (1815) under "Henry Briggs" (Brigius) Vol. 2, pp. 491-492.

If a hint could have urged any human mind thus rapidly upon the theory of the Logarithms, there was a hint which arose in the school of Alexandria, which was submerged in the middle ages, and rose again with the letters of Greece; which Tycho had — which Stifellius, Byrgius [Bürgi], Longomontanus, and above all which Kepler had—and all made no more of it than Archimedes had done.

But where were all the " learned calculators of the 16th and 17th centuries," whom Dr Hutton pictures as evolving the Logarithms by profound reasonings upon the doctrine of progressions? And who were they? Not Kepler, who, when he first heard of Napier's method, could hardly form an accurate idea of its meaning. Not Tycho, nor Longomontanus, nor Galileo, nor any one of Kepler's numerous correspondents, including... nearly all the learned calculators of the period. ...Kepler, who to his dying day never ceased to marvel at the achievement, seems a little excited by discovering that one other person had actually approached the theory without being aware of it. In his Rudolphine Tables... 1627, he remarks,
"the accents in calculation led Justus Byrgius on the way to these very Logarithms many years before Napier's system appeared; but being an indolent man, and very uncommunicative, instead of rearing up his child for the public benefit, he deserted it in the birth."
This was the result of Kepler's indefatigable inquiries, for nine years... and... it amounts to this, that Byrgius had made some observations upon the adaptation of an arithmetical to a geometrical progression, very naturally occurring to him in trigonometrical calculations. The Apices Logistici ["accents in calculation"], to which Kepler alludes, are those accents which the Greeks used... to change the value or mark the order of a symbol, as we use the cypher; and this is... exemplified in their sexagesimal division of the circle still in use, where the accents ′, ″, ′″, ″″, &c. of minutes, seconds, thirds, fourths, &c. are an arithmetical progression denoting the fractional orders, the values of which descend in a ratio of 60, and form the corresponding geometrical progression.
- Kepler Quote & Alt. Translation: "Apices Logistici, Justo Byrgio, multis annis ante editionem Nepeiranam, viam præiverunt ad hos ipsissimos logarithmos, etsi homo cunctator, et scretorum suorura custos, fcetum in partu destituit, non ad usos publicos educavi." [The Apices Logistici, led Jost Bürgi, many years before the edition of Napier, to these very logarithms, but being a hesitant man and keeper the of nun's secrets, abandoned the fetus in childbirth, did not rear it for public benefit.]

Kepler meant no honour to his friend to the prejudice of Napier. On the contrary, the spirit... is, that Byrgius had substantially failed to perceive that a chapter of algebra might be composed in which that property of progressions would be reared into vast importance; an importance never felt until Napier demonstrated it by a method far more nearly allied to the profound algebraic views of Newton, than those easy progressions,—so obvious in the Arabic scale itself, and through which, perhaps, Byrgius had been unwittingly on a tract to Logarithms,—are to Napier's system.

The mathematician whose claim we are considering ranked not meanly in science; he was instrument-maker and astronomer to the Landgrave of Hesse, and must have been well known to Kepler; he may have been "homo cunctator," [an indolent, or hesitant man] but he was not so foolish as to have cast aside his own immortality had he really extended the Archimedean principle in any remarkable manner; he was a public astronomer, under high patronage, in a country teeming with rivals in science, and where a great mathematical discovery was the means of obtaining rank, wealth, and adoration; it is absolutely impossible, therefore, that...[he] could have calculated tables of Logarithms... and then have cast them aside; there was the gulf of ignorance betwixt him and Logarithms, and so we must construe the expressions of Kepler, "fœtum in partu destituit, non ad usos publicos educavit [instead of rearing up his child for the public benefit, he deserted it in the birth]." Supposing him even to have observed all the curious properties of a corresponding series, under the fertile and flexible Arabic notation,—the parent of progressions,—he would not have been singular in thus obtaining a glimpse of Logarithms without knowing them; and there would still be this distinction betwixt Byrgius and Napier, that the former, neither seeking nor dreaming of such a power, stumbled upon a natural tract in the system of notation, which might have led him, but did not, to an imperfect and accidental developement of Logarithms; whereas the latter saw that the power was wanted, that calculation was impeded, and, to use his own words, "began therefore to consider in my mind by what certain and ready art I might remove those hindrances," and in doing so sought no easy path pointed out to him by the progressive power of cyphers, but, plunging at once into the algebraic depth of his own original fluxionary system, took the very path which Newton and Leibnitz would have taken, and returned leading the whole system of Numbers captive to the properties of progressions.

Justus Byrgius is the solitary mathematician for whom any thing like an independent claim to the invention has been set up betwixt the time of Archimedes and Napier. Not that it has ever been said that our philosopher borrowed any thing from the German; for the priority of Napier's publication, and the surpassing beauty of his algebraic method, has never met with contradiction. But there is a story that Kepler's friend had actually computed tables of Logarithms years before Napier published his canon, and, consequently, that the German stands nearly in the same relation to this great discovery that Newton himself does to the infinitesmal calculus, in the celebrated competition with Leibnitz. It would, indeed, be singular, if this public astronomer had computed such tables without giving them to the world, or ever himself pretending to the discovery.

Yet the facts have been imposingly detailed by Montucla in his great history of Mathematics, and hitherto without any refutation. If Dr Hutton, instead of confusing the history of Logarithms to the further detriment of Napier's intellectual rights, by appearing to assume that the conquest, which our philosopher alone imagined and accomplished, was the work of many, had refuted the false claim we are about to expose, he would thereby have only done justice to his country.
- Ref: Jean Étienne Montucla, Histoire des mathématiques See pp. 9-12 & p. 20.

"There is a geometer," says Montucla,
"to whom we must here give a place, and that is, Juste Byrge. That which chiefly renders him worthy of notice is the fact, that he invented and constructed tables of Logarithms simultaneously with Napier. Kepler represents him to us as a man of considerable genius, but thinking so modestly of his own inventions, and so indifferent about them, as to suffer them to be buried in the dust of his study; and, says Kepler, for that reason he never gave any thing to the public through the medium of the press.
But Kepler was in error when he said so, and we shall proceed to unfold a tale... Notwithstanding what Kepler says of J. Byrge, Benjamin Bramer bears witness to the fact, that... Byrge... did publish something relative to Logarithms. That author in a German work... Description of an Instrument very useful for perspective and drawing plans, (...1630, 4to,) says...
"It was upon these principles that my dear brother-in-law and master, Juste Byrge, constructed, more than twenty years ago, a beautiful table of progressions, with their differences from 10 to 10, calculated to 9 places, and which he caused to be printed at Prague in 1620, so that the invention of Logarithms is not Neper's, but was made by Juste Byrge long before him."

Montucla continues...
"But the work of this geometer was nowhere to be found, and probably would never have been discovered had not the passage led M. Kästner to recognize these tables among some old mathematical works which he had purchased. They bore this title in German: Tables of Arithmetical and Geometrical Progressions, with an introduction explanatory of their meaning and use in all manner of Calculations, by J. B. printed in the ancient city of Prague, 1620. The tables contain seven leaves and a-half, printed in folio, but the introduction announced is awanting, which leads to the conjecture, that some peculiar circumstances had stopped the progress of the work; and, indeed, Bramer informs us in another of his own works, that Juste Byrge contemplated the publication of several of his inventions, and, for that purpose, had his portrait engraved in the year 1619, but the thirty years' war, which unhappily desolated Germany, opposed an obstacle to his design.

Montucla then proceeds to give a specimen of the fragment of Byrgius taken from M. Kastner, and concludes...
We must remark at the same time, that it would be unjust to conclude, from the work printed in 1620, that Byrge had invented Logarithms before Neper; for the work of Neper appeared in 1614, and it is the priority of dates of works which determines at the bar of public opinion the anteriority of the invention. How then does Bramer from that date, 1620, arrive at the conclusion, that his brother-in-law had made the discovery long before Napier? It is well known, that the date of an invention requiring much calculation is necessarily anterior to that of publication, and Neper is equally entitled to the assumption, that his invention existed in his head for several years before he published it; and besides, in a court of law itself, Byrge would lose his suit, for, according to the strictest administration of justice, a date of publication anterior by six years must be held to have afforded an opportunity of becoming acquainted with the discovery, and disguising it under another form. Let us be contented, therefore, with associating at a distance, and to a certain extent only, Byrge with the honour of that ingenious invention; but the glory must always belong to Neper."

The value of Byrgius's share of any honour in the matter may be expressed by that ghostly symbol which is the soul of Arabic notation, 0. We might say so upon the evidence adduced in his favour, which is totally inadequate to sustain his claim. His brother-in-law is, under the circumstances, not competent evidence; for the peremptory manner in which he springs from so vague a statement to the astounding conclusion, that Byrgius, and not Napier, is the Inventor of Logarithms, proves Bramer to have been either an idiot or a false witness.

The miserable fragment of miscalculated tables discovered by Kästner proves nothing, for there is neither description nor claim attached to them, and their date is 1620; and any support which the claim attempted to be reared upon that fragment may seem to obtain from the notice of Kepler (also very vague) is more than neutralized by Kepler himself.

According to Bramer, his kinsman had calculated tables... more than twenty years before 1630. As he has not fixed the date, we take the assumption as referring to the year 1609. "But," says Kepler, writing in... 1624, and without the slightest notice of Byrgius, "a certain Scotchman, so early as the year 1594, wrote to Tycho a promise of that wonderful canon." According to Bramer, his kinsman, the "homo cunctator," [a hesitant man] did so far bestir himself as to have his portrait engraved, in the year 1619, for a frontispiece to his great discoveries, among which, and probably the least, were the Logarithms! In 1620 the fragment of his tables was printed at Prague, but without frontispiece or anything else.

[T]hough Montucla was not aware of the fact... the... place where Kepler himself first saw a copy of John Napier's Canon Mirificus was THE ANCIENT CITY OF PRAGUE, and this was in the year 1617.

Our authority is the letter from Kepler to Napier, with which these Memoirs conclude, and which Montucla had never seen. So the "homo cunctator" calculated tables of Logarithms in 1609, and then cast them among the rubbish of his study; in the year 1617 a copy of Napier's Canon is laid, as the wonder of the day, before Kepler himself, the oracle of European science, in the city of Prague; from that moment Kepler's whole existence is identified with his love of Logarithms, and all that he ever says for his friend Byrgius is, that he did not make the discovery; in 1619 (two years after Napier's death,) the "homo cunctator" has his portrait engraved; in 1620 he is said to have printed at Prague some isolated and useless fragment of a table, but it is not even pretended that he put forth any claim; ten years afterwards, namely, in 1630, Bramer, brother-in-law to the "homo cunctator," has the effrontery to announce, and without so much as a detailed or explicit account in support of his allegation, that Justus Byrgius, and not John Napier, is the inventor of Logarithms.

We... add the name... of another distinguished historian of science... carried by this groundless pretension, which was probably a villanous though weak attempt to wrest the laurels from the grave of a foreigner. M. Kluegel, in his philosophical dictionary, a work of great ability, records, that
"Neper in Scotland, and Jobst Byrg in Germany, were the first who, without any intercommunication, calculated tables of Logarithms."
...But how happened it, we would ask M. Kluegel, that Kepler gave all the glory to Napier, and none to his own countryman? This same author expresses most graphically the enthusiastic zeal with which the legislator of the stars rushed upon the Logarithms; "Kepler ergriff Nepers Erfindung mit Eifer,"—[translation] Kepler seized Napier's discovery with enthusiasm,—now Kepler expressly regards the speculation of Byrgius with contempt.

Had it appeared a century before Napier, would not physical astronomy have been as far advanced in his time as it was a century after, and would not NAPIER have been NEWTON?
But there were many persons having thoughts of such a table of numbers besides the few who are said to have attempted it! Dr Hutton, in support of this assertion... clings to Byrgius;
"Kepler also says, that one Juste Byrge, assistant astronomer to the Landgrave of Hesse, invented or projected Logarithms long before Neper did, but that they had never come abroad on account of the great reservedness of their author with regard to his own compositions."
But Hutton, though he suppresses what... qualifies the words of Kepler, and ventures not into the slightest examination of the pretension for Byrgius (who never made it for himself) is fond of the story, and does what he can to fix it upon the legislator of the stars as an unqualified assertion of his; for, speaking of the Rudolphine Tables, our author takes occasion to repeat,
"and here it is that he (Kepler) mentions Justus Byrgius as having had Logarithms before Napier published them."
- Ref: Charles Hutton, Mathematical Tables (1801) History of Logarithms p. 24.

Longomontanus and Byrgius, are all whom Dr Hutton can find to represent his learned calculators of the sixteenth and seventeenth centuries, who anticipated or coincided with Napier in the discovery. ...But he is contradicted by the history of science, ancient and modern, and by every philosopher of greatest name, both in Napier's time and ours. Among the finest characteristics of our philosopher's invention was the unhoped-for manner in which it removed a pressure, long and severely felt, and which might have crushed the temple of science, had that not possessed such a pillar as Kepler. To use the expressions of a distinguished writer, " What all mathematicians were now wishing for, the genius of Neper enabled him to discover; and the invention of Logarithms introduced into the calculations of trigonometry a degree of simplicity and ease, which no man had been so sanguine as to expect." Kepler, Ursine, Speidell, Gunter, Briggs, Vlacq, [Petrus] Cugerus, Cavalieri, Wolff, Wallis, Halley, Keill, and a host of others, all bear witness against Dr Hutton, in the honourable and enthusiastic manner they acknowledge Napier as the only author of that revolution in science.
- Quote Ref: Review of Woodhouse's Trigonometry, Edin. Review (1810) Vol. XVII. p. 124.

Histoire des Sciences Mathématiques et Physiques (1883)

by Maximilian Marie, Tome III: De Viète a Descartes, pp. 85-86. Google Translate assisted translation.

Byrge... was one of the most skilled builders of mathematical instruments of his time, and employed in this capacity by the Landgrave of Hesse, William IV, then by the Emperor. He is considered to be the inventor of the reduction compass. He published, in Prague, in 1620, a table of logarithms more judiciously arranged than those we... use today, in that he made the logarithms increase in arithmetic progression, whereas, in our tables, it is the numbers which vary in arithmetic progression...

These are the hyperbolic logarithms that Byrge had entered in his table; it would be difficult to know whether he had been aware of Néper's invention.

The work in which Néper develops this invention is dated 1614, therefore six years before that of Byrge, which gives Néper priority. But it is unlikely that in six years Byrge could have learned of the existence of the work of the Scottish geometer, studied this work, prepared to carry out the invention that it indicated, actually calculated 33,000 numbers corresponding to 33,000 logarithms in arithmetic progression and had the table containing all of this printed on seven and a half sheets.

A History of Mathematics (1893, 1906)

by Florian Cajori

To Simon Stevin we owe the first systematic treatment of decimal fractions. In his La Disme (1585) he describes... the advantages... What he lacked was a suitable notation... In place of our decimal point, he used... indices... designating powers... After Stevin, decimals were used by Joost Bürgi, a Swiss by birth, who prepared a manuscript on arithmetic soon after 1592...

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"... 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... Mirifici logarithmorum canonis descriptio. In it he explained... his logarithms, and gave a logarithmic table of the natural sines of a quadrant from minute to minute. ...The only possible rival of John Napier in the invention of logarithms was the Swiss Justus Byrgius (Joost Bürgi). He published a rude table of logarithms six years after the appearance of the Canon Mirificus, but it appears that he conceived the idea and constructed that table as early, if not earlier, than Napier did his. But he neglected to have the results published until Napier's logarithms were known and admired throughout Europe.

A Brief History of Mathematics (1903)

by Karl Fink, Tr. Wooster Woodruff Beman & David Eugene Smith, 2nd revised edn.

The tables of the numerical values of the trigonometric functions had now attained a high degree of accuracy, but their real significance and usefulness were first shown by the introduction of logarithms.

Napier is usually regarded as the inventor of logarithms, although Cantor's review of the evidence leaves no room for doubt that Bürgi was an independent discoverer. His Progress Tabulen, computed between 1603 and 1611 but not published until 1620 is really a table of antilogarithms. Bürgi's more general point of view should also be mentioned. He desired to simplify all calculations by means of logarithms while Napier used only the logarithms of the trigonometric functions.

Bürgi was led to this method of procedure by comparison of the two series 0, 1, 2, 3, .... and 1, 2, 4, 8, ... or 2⁰, 2¹, 2², 2³, ... He observed that for purposes of calculation it was most convenient to select 10 as the base of the second series, and from this standpoint he computed the logarithms of ordinary numbers, though he first decided on publication when Napier's renown began to spread in Germany by reason of Kepler's favorable reports.

Bürgi's Geometrische Progress Tabulen appeared at Prague in 1620, and contained the logarithms of numbers from 10⁸ to 10⁹ by tens. Bürgi did not use the term logarithmus, but by reason of the way in which they were printed he called the logarithms "red numbers," the numbers

corresponding, "black numbers."

Bürgi, Joost (Jobst). Born at Lichtensteig, St. Gall, Switzerland, 1552; died at Cassel in 1632. One of the first to suggest a system of logarithms. The first to recognize the value of making the second member of an equation zero.
- Biographical Notes

"History of the Exponential and Logarithmic Concepts" (1913)

by Florian Cajori, The American Mathematical Monthly (Jan, 1913) I. From Napier to Leibniz and John Bernoulli I. 1614-1712. Vol. 20, No. 1, pp. 5-35. @jstor.org Open Access article.

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

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.

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 arithmetic progression were printed in red, the numbers in the geometric progression 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.

History of Mathematics (1923)

by David Eugene Smith, Vol. 1.

Simon Jacob... wrote two commercial arithmetics. Bürgi mentions Jacob’s treatment of series, and apparently the... table of antilogarithms, the Progress Tabulen, was suggested by the nature of exponents as laid down in these and similar books of the 16th century.

Two... Swiss mathematicians of the 17th century deserve mention,—one a genius, the other a plagiarist. The genius was Jobst Bürgi, from 1579 to 1603 court watchmaker to Landgraf Wilhelm IV of Hesse, and later (until 1622) to Kaiser Rudolph II. He wrote on the proportional compasses and on astronomy, but is best known for his invention of logarithms independently of Napier. He was led to the idea by an entirely different route from that taken by the latter, approaching it through the theory of exponents. He did not publish anything upon the subject until after Napier had made known his discovery, and when he finally concluded to print his work it was in the form of a small table of antilogarithms, issued anonymously at Prag in 1620. The book never attracted any attention and remained practically unknown except to historians of mathematics.
- Note: Born at Lichtensteig, February 28, 1552; died at Cassel, January 31, 1632. The first name also appears as Joost and Justus; the last as Burgi, Byrgi, Borgen, and Byrgius. See R. Wolf, “Zwei Kleine Notizen zur Geschichte der Mathematik,” Bibl. Math., III (2) p. 33. Ref: Arithmetische und Geometrische Progress Tabulen (1620) Prag.

The other Swiss writer was of a different character. He was a professor while Bürgi was a watchmaker; his name has been known for three centuries, while Bürgi’s has been almost forgotten; but he was a plagiarist, while Bürgi was a genius. Paul Guldin began his work as a goldsmith. He later entered the Jesuit order, lived for a long time in Rome, and became professor of mathematics at the University of Vienna and later at Gratz.

History of Mathematics (1925)

by David Eugene Smith, Vol. 2.

In 1616 Kepler wrote a work on mensuration in which he distinctly took up the decimal fraction, using both a decimal point (comma) and the parentheses to separate the fractional part. He stated it as his opinion that these fractions were due to Bürgi, although it seems strange that he was not familiar with the work of Stevin.
- Ref: Johannes Kepler, Ausszug auss der uralten Messe-Kunst Archimedis (1616). It appears in Vol. V, Frisch edition of Kepler’s works, 1864. Quote: "Dise Art der Bruchrechnung ist von Jost Bürgen zu der sinusrechnung erdacht." or "This type of fraction calculation was invented by Jost Bürgen for the sine calculation."

It is difficult to say who it was who first recognized the advantage of always equating to zero in the study of the general equation. It may... have been Napier, for he wrote... De Arte Logistica before 1594 (although... first printed... 1839), and in this there is evidence that he understood the advantage... Bürgi (c. 1619) also recognized the value of making the second member zero, Harriot (c. 1621) may have done the same, and the influence of Descartes (1637) was such that the usage became fairly general.

In 1620 Jobst Bürgi published his Progress Tabulen, a work conceived some years earlier. ...[H]e was influenced by Simon Jacob's work. The tables were printed at Prague and are simply lists of antilogarithms with base 1.0001. The logarithm is printed in red in the top line and the left-hand column, and the antilogarithms are in black, and hence Bürgi calls the logarithm Die Rothe Zahl [The Red Number]. The first part of his table is as follows:

	0	500	1000	1500	2000
0	100000000	100501227	101004966	101511230	102020032
10	....10000	....11277	....15067	....21381	....30234
20	....20001	....21328	....25168	....31534	....40437
30	....30003	....31380	....35271	....41687	....50641

...The manuscripts are at the Observatory at Pulkowa, but none seem to be... later than 1610, so that he probably developed his theory independently of Napier. ...[H]e approached the subject algebraically, as Napier approched it geometrically.
The only extensive table of logarithms is due to James Dodson (...1742).

Forerunners of Bürgi. Napier approached logarithms from the standpoint of geometry, whereas... we approach the subject from the relation $a^{m}a^{n}=a^{a+m}$ . This relation was known to Archimedes and to various other writers.
More generally, if we take the two series

	0	1	2	3	4	5	6	7
and	1	2	4	8	16	32	64	128,

the first one being arithmetic and the second one being geometric, we see that the latter may be written as follows:

2⁰

2¹

2²

2³

2⁴

2⁵

2⁶

2⁷

From this it is evident that

2^{3}\cdot 2^{4}=2^{7},\qquad (2^{2})^{3}=2^{6}

2^{7}:2^{3}=2^{4},\qquad (2^{4})^{\frac {1}{2}}=2^{2}

which are the fundamental laws of logarithms.
Most writers refer to Stifel as the first to set forth these basal laws... but he was by no means the first... nor did they first appear even in his century. Probably the best... which appeared in the 15th century were... of Chuquet in Le Triparty en la Science des Nombres... 1484 ...Chuquet expressed

a^{m}a^{n}=a^{m+n}

(a^{m})^{n}=a^{mn}

In general the French writers... paid no attention to any of the laws except that of multiplication, while the German writers, following the lead of Stifel, took the broader view of the theory. ...[I]n general the German writers were in the lead. ...particularly This is particularly true of Simon Jacob (1565), who followed Stifel closely, recognizing all four [fundamental] laws [of logarithms], and... influencing Jobst Bürgi. These writers did not use the general exponents essential to logarithms, but the recognition of the four laws is significant.

The Development of Mathematics (1940)

by Eric Temple Bell

[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."
- Ref: Florian Cajori, Napier Tercentenary Memorial Volume (1914? 1915) Quote is split between pp. 103 & 105.

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.

Bürgi’s "Progress Tabulen" (1620): logarithmic tables without logarithms (2010, 2013)

by Denis Roegel (2010, last updated 2013) A source @LOCOMAT, The Loria Collection of Mathematical Tables

Tycho Brahe... [i]n his unprinted manual of trigonometry... expounded the prosthaphæretic method aimed at simplifying... trigonometric computations. This method first appeared in print in 1588, in Nicolas Reimerus (Ursus)' Fundamentum astronomicum. It was of great value... and was going to be a direct competitor to the method of logarithms.
- Ref: John Louis Emil Dreyer, "On Tycho Brahe’s manual of trigonometry" The Observatory (1916) Vol. 39, pp. 127–131. Ref: Nicolas Reimarus, Fundamentum astronomicum: id est, nova doctrina sinuum et triangulorum (1588) Strasbourg: Bernhard Jobin.

For a comparison between teaching manuals using prosthaphaeresis and those using logarithms, see Miura’s... article comparing Pitiscus' and Norwood's trigonometries.
- p.3. Ref: Nobuo Miura, "The applications of logarithms to trigonometry in Richard Norwood" Historia scientiarum: international journal of the History of Science Society of Japan (1989) Vol. 37 pp. 17–30.

Prosthaphaeresis... had been devised by Johannes Werner... and was likely brought to Tycho Brahe by... Paul Wittich... in 1580... This method was based on... $\sin A\sin B={\frac {1}{2}}[\cos(A-B)-\cos(A+B)]$ ,
$\cos A\cos B={\frac {1}{2}}[\cos(A-B)+\cos(A+B)]$ .With... a table of sines, these... could... replace multiplications by additions and subtractions, something...Wittich found out, but apparently Werner didn’t realize.
- Ref: Victor E. Thoren, The lord of Uraniborg: A biography of Tycho Brahe (1990) p. 237.

[I]t... seems... this method was brought to Napier by John Craig who obtained it from Wittich in Frankfurt at the end of the 1570s...
- Ref: Owen Gingerich, Robert S. Westman, The Wittich Connection: Conflict and Priority in Late Sixteenth-century Cosmology Transactions of the American Philosophical Society (1988) Vol. 78, Part 7, pp. 11–12. Held at Philadelphia for Promoting Useful Knowledge.

Thanks to Brahe’s manual of trigonometry, the fame of the method of prosthaphæresis spread abroad and it was brought by Wittich to Kassel in 1584... This is probably how Jost Bürgi... then an instrument maker working for the Landgrave of Kassel, learned of it.

Bürgi not only used this method, but... improved it. He found the second formula, for Brahe and Wittich only knew the first. In addition, he improved the computation of the spherical law of cosines... $\cos c=\cos a\cos b+\sin a\sin b\cos C$ Using the method of prosthaphæresis... cos a cos b and sin a sin b... could be computed but two new multiplications were... left... Bürgi realized that... prosthaphæresis could be used a second time, and... all multiplications could be replaced by additions or subtractions.

In 1588, when Ursus published the method of prosthaphæresis, he did not give any sources. But he acknowledged his debt to Wittich and Bürgi in... De astronomicis hypothesibus [1597]... Once Ursus... published the method... spread... and was improved by other mathematicians, in particular Clavius.

In 1597, in a letter to Kepler, Ursus wrote that Bürgi was on the "same level as Archimedes and Euclides"
- Ref: Rudolf Wolf, Biographien zur Kulturgeschichte der Schweiz (1858) Vol. 1, p. 58.

In 1603, Bürgi was called to the imperial court in Prague... There he... received the praise of Kepler who wrote that Bürgi would sometime be as famous in his art as Dürer is in painting...
- Ref: Johannes Kepler, Opera Omnia (1859) Vol. 2, p. 361.

The use of the prosthaphæretic method required a table of sines. This is likely... why Bürgi constructed a Canon sinuum... sine table... Bürgi... seems to have been reluctant at publishing it and in 1592, Brahe wrote that he did not understand why he was keeping the table hidden, after... a look at it.

"Jost Bürgi’s Method for Calculating Sines" (Feb 2, 2016)

by Menso Folkerts, Dieter Launert, Andreas Thom, Version 2, p. 8, arXiv:1510.03180.

All [previous] procedures for calculating chords and sines... [were] based in principle on the method which Ptolemy had presented in his Almagest. Totally different... is a procedure which Jost Bürgi...invented... [H]e was able to compute the sine of each angle with any desired accuracy in a... short time... Bürgi explained his procedure in ...Fundamentum Astronomiae.

Jost Bürgi... invented logarithms independently of John Napier...
- Footnote: Bürgi invented the logarithms in the 1580s, but... published his table... not before 1620.

Another treatise of Bürgi is... his algebraic work Coss... Coss is not restricted to algebra, but... treats the division of an angle and the associated.... calculation of chords and sines.

In 1605 Bürgi went to Prague and lived there as a watchmaker at the Emperor’s Chamber until 1631... [and] was... involved in astronomical observations and interpretations. ...Christoph Rothmann... worked at the court. Among... temporary visitors... Paul Wittich... and Nicolaus Reimers Ursus... who had both worked with Tycho Brahe. Wittich... brought... knowledge of prosthaphaeresis... by which multiplications and divisions can be replaced by additions and subtractions of trigonometrical values... based on the identity $\sin \alpha \cdot \sin \beta ={\frac {1}{2}}[\sin(90^{\circ }-\alpha +\beta )-\sin(90^{\circ }-\alpha -\beta )]$ ,this... achieves the same as logarithms, which were invented... decades later.

Ursus... and Bürgi became good friends...

[A] manuscript has survived in which Bürgi... explains his method. From the preface... we know that Bürgi presented it to... Rudolf II... The manuscript consists of 95 folios and contains a[n]... extensive work on trigonometry by Bürgi, entitled Fundamentum Astronomiae.

Book 1 deals with logistic numbers, prosthaphaeresis and the calculation of sines. “Logistic numbers” are the sexagesimal numbers... used for astronomical calculations. ...Bürgi explains the four basic operations of arithmetic and the extraction of roots. ...The topic of Chapter 3 is prosthaphaeresis. ...The remaining 10 chapters ...are on sines and ...calculation of sine values. Chapters 11 and 12... explains here in detail his own method for computing... all sine values from 0 to 90 degrees... The result is a sine table for every minute with 5-7 sexagesimal places.

Book 2 deals with the calculation of triangles. The first four chapters are on plane triangles and chapters 5-11 on spherical ones.

Bürgi... found an arithmetic procedure for computing sine values with arbitrary accuracy. By dividing the right angle into 90 parts, Bürgi is able to calculate the sines of all degrees from sin 1° to sin 90°.

In Chapter 11 Bürgi also deals with... how to calculate sine values for every minute. ...He divides the known value for sin 1° by 60 to receive an approximation for sin 1′. This he improves by... two corrections and obtains a sufficiently exact value... With... trigonometric relations he then computes sin 2′, etc. ...[U]sing first and higher order differences of consecutive sine values he derives a simple relation for producing the further sines... The result is Bürgi’s sine table... 5400 entries... in his Fundamentum Astronomiae.

Bürgi’s algorithm... reverses the process of forming second differences, i.e., performs up to sign some form of two-fold discrete integration—with the right normalization at the start and end of the sequence. Of course, our Perron-Frobenius eigenvector v of M is also an eigenvector of M⁻¹, but now for the smallest eigenvalue. Bürgi’s insight must have been that the study of iterations of M is much more useful than those of M⁻¹ in order to approximate the entries of the critical eigenvector. ...[T]his process has unexpected stability properties leading to a quickly convergent sequence of vectors that approximate this eigenvector and hence the sine-values. The reason for its convergence is more subtle than just some geometric principle such as exhaustion, monotonicity, or Newton’s method, it rather relies on the equidistribution of a diffusion process over time—an idea which was later formalized as the Perron-Frobenius theorem and studied... in the theory of Markov chains. ...Bürgi’s insight anticipates some aspects of ideas and developments that came to full light only at the beginning of the 20th century.

"How Bürgi computed the sines of all integer angles simultaneously in 1586" (2018)

by Grégoire Nicollier, Math Semesterber (2018) Vol. 65, pp. 15–34. A source

[T]he autograph Fundamentum Astronomiæ... contained the author’s lost algorithm for computing sine tables...

Bürgi’s example of his skillful method, the Artificium, explains the calculation for the multiples of 10°, the ninth parts of the right angle.

The Artificium and the whole autograph were decrypted and edited by Dieter Launert... The first proof of convergence was given by Andreas Thom using the Perron–Frobenius theorem. Another proof with matrices by Jörg Waldvogel is more elementary and determines the rate of convergence. In this paper we present a new proof of convergence that encompasses a whole family of related methods...
- Ref: M. Folkerts, D. Launert, and A. Thom: "Jost Bürgi’s Method for Calculating Sines" Hist. Math (2016) 43(2) pp. 133–147. Ref: D. Launert: Nova Kepleriana: Bürgis Kunstweg im Fundamentum Astronomiæ, Entschlüsselung seines Rätsels (2015) Verlag der Bayerischen Akademie der Wissenschaften, Munich. Ref: J. Waldvogel, "Jost Bürgi’s Artificium of 1586 in Modern View, an Ingenious Algorithm for Calculating Tables of the Sine Function" Elem. Math. (2016) 71 pp. 89–99.

External links

Wikipedia

Wikipedia has an article about:

Jost Bürgi

Jost Bürgi @MacTutor History of Mathematics Archive

Bürgi, Jost bio from Oliver Knill @Harvard University

Bürgi’s “Progress Tabulen” (1620): logarithmic tables without logarithms (2010, 2013) by Denis Roegel.