Jost 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.”

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

“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.”

“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”

“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.”

“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.”

“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.”

“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 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.”

“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.”

“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 {{w|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...”

“[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.”

“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.”

“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.”

“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.”

“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.”

“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 {{w|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.”

“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...”

“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.”

“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.”