Source on Jost Bürgi

“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.”

“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.”

“"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.”

“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.”

“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.”

“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."”

“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 , or log m > log n , just as one may choose. Napier originally chose the first alternative, Bürgi the second.”

“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.”

“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.”

“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: {| class="wikitable" style="text-align:right" ! !!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 {| border="0" style="text-align:right" cellspacing="25" |- |||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: {| border="0" style="text-align:right" cellspacing="25" |- | || ||20||21||22||23||24||25||26||27 |}From this it is evident that2^3 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 expresseda^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.”

“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.”

“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 CUsing 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.”

“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.”

“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.”

“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 identitysin 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.”

“[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.”

“[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.”