Cursus Mathematici Practici Volumen Primum [all published]: continens Illustr. & Generosi Dn. Dn. Johannis Neperi Baronis Merchistonii &c. Scoti. Trigonometriam Logarithmicam.

Cölln an der Spree: Martin Guthius, 1618.

First edition, exceptionally rare, of the book that introduced logarithms to Continental Europe; in particular, it was through this work that Johannes Kepler, in ‘a happy calamity,’ as he called it, became aware of Napier’s epoch-making work, a discovery that enabled him to complete his great Rudolphine Tables (1627), “the foundation of all planetary calculations for over a century” (Sparrow). “The earliest publication of Napier’s logarithms on the Continent was in 1618, when Benjamin Ursinus included an excerpt from the canon, shortened by two places, in his Cursus mathematici practici. Through this work Kepler became aware of the importance of Napier’s discovery and expressed his enthusiasm in a letter to Napier dated 28 July 1619, printed in the dedication of his Ephemerides (1620)” (DSB, under Napier). Ursinus assisted Kepler with the computations for the Rudolphine Tables, and Kepler presented and inscribed a copy to Ursinus (Honeyman 1800 – this copy is now held by the Adler Planetarium in Chicago); in the inscription, Kepler calls Ursinus and Tycho Brahe the scientific fathers of the tables. “The [Rudolphine] Tables was far more accurate than its predecessors – its margin of error staying within 10 seconds compared to up to 5 degrees with earlier tables. Instead of providing a sequence of planetary positions for specified days (which Kepler did in his Ephemerides), the Rudolphine Tables were set up to allow calculations of planetary positions for any time in the past or future. The finding of the longitude of a given planet at a given time was based on Kepler’s equation and he exploited logarithms for this tabulation. The precise geocentric positions had to be worked out from combining the heliocentric positions of the planets and the earth that were calculated separately. Logarithmic tabulations were used again to facilitate calculation” ( Ursinus was for several years Kepler’s assistant: in Prague, he made observations with Kepler of the newly discovered satellites of Jupiter, published in his Narratio (1611), and later, after Kepler had moved to Linz, lived there in Kepler’s house for a year (1613/1614). A second issue of the Cursus Mathematici Practici was published in 1619 (same place and publisher). OCLC lists four copies (British Library, Chicago, Columbia, Göttingen); KVK adds no further copies outside Germany. As far as we can determine ours is the only copy of the first issue to have appeared in commerce; the Macclesfield copy of the second issue (Sotheby’s, October 26, 2005, lot 2027), in an 18th century binding, realised £9600 ($16942).

Provenance: Patrick Hume, 1st Earl of Marchmont (engraved armorial bookplate on front paste-down).Sir Patrick Hume (1641-1724) was a Scottish Presbyterian statesman and a supporter of William of Orange. He began his long political career in opposition during the reigns of Charles II and James VII and II. Because of his involvement in the 1685 anti-Catholic rebellion, Hume spent several years in exile in the Netherlands. He returned after the revolution of 1688 when he accompanied the Protestant William of Orange to Britain. His forfeited estates were returned to him and in 1696 he was appointed Lord Chancellor. Created Earl of Marchmont in 1697, he opposed the claims of the Jacobites and voted for Parliamentary union between Scotland and England.

Benjamin Ursinus (originally Benjamin Behr, Latinized Ursinus), was born on July 15, 1587 in Sprottau in Silesia (now in Poland). Ursinus was a private tutor in Prague and then high school teacher at the Gymnasium of the Unity of the Bohemian brothers in Sobieslau and in Beuthen. From 1615 he taught at the Elector of Brandenburg’s Gymnasium in Joachimsthal near Berlin, a school for gifted boys founded in 1607. From 1630 he was mathematics professor at the University of Frankfurt an der Oder, where he died in 1633 or 1634.

We do not know exactly when Ursinus first came into contact with Kepler (some sources suggest that Ursinus was Kepler’s student), but certainly by 1610 Ursinus was acting as Kepler’s assistant. Following the publication of Sidereus nuncius (1610), Kepler, then Imperial Court Astronomer to Rudolph II, was keen to test Galileo’s observations. At the end of August, “the Elector of Cologne passed through Prague and lent Kepler the very instrument earlier sent to him by Galileo. Consequently, in just over one week (from August 30 to September 8), Kepler was able to observe what he now called for the first time the ‘satellites’ of Jupiter, and he was careful to do so with the testimony of various named and carefully described witnesses. Presumably, these were the kind of testimonials that Kepler had expected from Galileo. The first was Benjamin Ursinus, ‘a diligent student of astronomy who, from the start, because he loves this art and has decided to practice philosophizing in it, never dreams of ruining the credit necessary to a future astronomer by false witness.’ But there was more to Ursinus’s reliability than concern for his future reputation. Kepler explained: ‘We adopted the following method: with a piece of chalk and out of sight of each other, each of us drew on a wall what he had been able to observe; afterward, each of us went at the same time to see the other’s picture to see if it was in agreement. This [method] is also to be understood for the following [observations]’ … From August 30 to September 5, Benjamin Ursinus was Kepler’s principal co-witness” (Westman, p. 480). Kepler acknowledged Ursinus’s assistance in the preface of his Narratio De Observatis a se quatuor Iouis satellitibus erronibus (1611).

“In 1611 the political situation in Prague took an abrupt turn, ending Kepler’s exhilarating atmosphere of intellectual freedom. The gathering storm of the Counter-Reformation reached the capital, and brought about the abdication of Rudolph II. As warfare and bloodshed surged around him, Kepler sought refuge in Linz, where he was appointed provincial mathematician … The Linz authorities charged him first of all to ‘complete the astronomical tables in honor of the Emperor and the worshipful Austrian House, for the profit of … the entire land as well as also for his own fame and praise.’ After Rudolph’s death in 1612, his successor Matthias confirmed Kepler as court mathematician and agreed to his new residence away from Prague. But Kepler realized that as long as the Rudolphine Tables were unfinished, he would be tied to Linz. Thus the work on the tables became part of his fate” (Gingerich).

Despite Kepler’s move, Ursinus evidently remained close to him – indeed, he lived in Kepler’s house in Linz at Hofgasse 7 from October 1613 until autumn 1614 (Meyer, p. 4). It was presumably during this period that Ursinus assisted Kepler with the computation of the tables.

At about the same time as Ursinus left Kepler’s house in Linz, Napier’s great work was published. There is evidence that the Scottish polymath John Napier of Merchiston (1550-1617) started working on logarithms in 1594, but his completed work, Mirifici logarithmorum canonis descriptio, was not published until 1614. “This is one of the most influential mathematical books ever published. It introduced the world to the concept of logarithms and their use. By simplifying arduous calculation, that is, by reducing multiplication and division to addition and subtraction, logarithms became the fundamental principle behind most of the methods of, and aides to, computation prior to the invention of the electronic computer. They also proved to be a fundamental component of many mathematical systems” (Tomash and Williams). The importance of Napier’s work was immediately recognized, but it was not until 1618, in the offered work, that any account of logarithms was published outside Great Britain.

In his dedication of the Cursus Mathematici Practici, after singing the praises of Tycho Brahe, Kepler, and Galileo, whose telescope has made it possible for us to see what he and Kepler describe in their works, the author goes on to say that his ‘versatile friend’ Georg Vechner (1590–1647) had communicated to him Napier’s book, and that he had not ceased until he had made it public to his pupils in the Gymnasium in Joachimsthal. Although primarily a theologian and philosopher, the versatile Vechner published in 1613 a posthumous biography of Bartholomaeus Pitiscus (1561-1613), the German mathematician and astronomer who first coined the word trigonometry. Ursinus’s work was published at Cölln, the twin city of Old Berlin (Altberlin) from the 13th century to the 18th century, and now part of modern Berlin (and not at Cologne, as some sources would have it).

The structure of Ursinus’s book is similar to that of Napier’s Descriptio, although it is by no means simply a translation. After giving a general description of the logarithmic tables, and explaining that multiplying or dividing numbers corresponds to adding or subtracting their logarithms, Ursinus, like Napier, goes on to give examples of the application of the tables to solving various trigonometrical problems (a few of the examples are repeats of examples given by Napier, but most are not). He begins with right-angled plane triangles, before treating oblique-angled triangles in the plane, and then spherical triangles. This last case receives the most detailed treatment, as befits its importance to astronomical calculations. The text is followed by two tables. First, a Tabula Proportionalis giving the decimal equivalents of simple fractions, together with examples illustrating its use; and finally J. Neperi … Mirificus Canon Logarithmorum, Napier’s table of logarithms from the Descriptio, but with the number of decimal places reduced by two. (DSB describes Ursinus’s table as an ‘excerpt’ from that of Napier, but in fact both tables occupy the same 90 pages; the only difference is the number of decimal places.) The book concludes with several pages of errata.

Kepler became aware of the invention of logarithms a few months before the publication of Ursinus’s book, as we learn from a letter dated 11 March 1618 to his friend Wilhelm Schickard (1592-1635). “After detailing the various difficulties and resources of trigonometry, he stated:

‘A Scottish baron has appeared, whose name escapes me, but he has proposed some wonderful method by which all necessity of multiplications and divisions are commuted to mere additions and subtractions; nor does he make any use of a table of sines. However, he still requires a table of tangents and in some cases the variety, frequency and difficulty of additions and subtractions exceed the labour of multiplication and division’ [Kepler, Epistolae (1718), p. 672].

“This statement by Kepler is rather obscure since it seems to indicate that although he had seen a demonstration of logarithms before writing this letter to Schickard, Kepler had not fully understood how they were used (and therefore probably had not used them himself) since ‘a table of tangents’ is not required. Nor can Kepler’s concluding sentence be considered an accurate assessment of logarithms.

“This was Kepler’s first impression of Napier’s work but later in the same year he happened to read Cursus mathematici practici volumen primum, published in 1618 and written by his former assistant Benjamin Ursinus, in which Napier’s tables were reproduced for the first time on the Continent. Kepler soon recognized the potential of logarithms and he wrote a long letter to Napier in 1619, publishing it as a generous Dedication to Napier in his Ephemerides for 1620, shortly before he published Harmonices Mundi. Despite Kepler’s wide correspondence throughout Europe, it is clear that he was not aware that Napier had died two years previously” (Rice et al, p. 40).

“[Kepler] exploited the new logarithms to solve two problems introduced for the first time by the novel form of the Rudolphine Tables. The first arises in the solution of what is now called Kepler’s equation. For a planet moving in an ellipse, under Kepler’s law of areas, there is no elementary way to find explicitly the position angle corresponding to a given time. However, the converse is easily calculated. Therefore he solved his equation for a set of uniformly spaced angles, which determine a set of non-uniformly spaced times. Kepler tabulated the logarithms of these intervals as a convenient means for interpolating to the desired times. The second important use of logarithms arises from the thoroughly heliocentric nature of the book. In previous planetary tables, the motions of the sun and planets were combined into a single procedure. In the Rudolphine Tables we must find separately the heliocentric positions of the earth and planet in question. To find the geocentric position of the planet, these two positions must be combined – essentially a problem of vector addition. Kepler facilitated this maneuver by tabulating the logarithms of the radius vectors of earth and planet, and by providing a convenient double-entry table for combining them” (Gingerich).

A second issue of the Cursus Mathematici Practici appeared in 1619, and five years later Ursinus published an expanded set of logarithmic-trigonometric tables to eight decimal places and calculated to every ten-seconds of arc (compared with Napier’s seven decimal places, calculated to every minute of arc).

The Ursinus is here bound with two very rare philosophical works (Ursinus bound first).

KECKERMANN, Bartholomäus. Commentarius ad systema logicum majus posthumous. Berlin: Martin Guthius, 1619. 8vo, pp. [iv], 304. Not on OCLC.

Born in Danzig, Prussia, Keckermann (c. 1572-1608) was a German writer, Calvinist theologian, and philosopher. As a philosopher, Keckermann was the most prominent representative of an Aristotelian, anti-Ramist (Petrus Ramus) scholastic philosophy. As perhaps the most talented and independent Reformed theologian of orthodoxy, Keckermann with his ‘analytical method’ became the father of modern ‘systematic theology.’His numerous works were published towards the end of his short life, most of them posthumously.

TIMPLER, Clemens. Opticae systema methodicum per theoremata et problemata selecta concinnatum et duobus libris comprehensum. Cui subjecta est Physiognomia humana, itidem duobus libris breviter et perspicue pertractata. Hanover: Petrum Antonium, 1617. 8vo, pp. [xxiv], [8], 240.

Clemens Timpler (1563/4–1624) was a Reformed Protestant and professor of philosophy at the Gymnasium illustre Arnoldinum in Steinfurt (Westphalia). He is considered, together with Jakob Degen (1511-87), to be the most important Protestant metaphysician, establishing the Protestant Reformed Neuscholastik, and is sometimes considered the ‘Godfather of Ontology’. His textbook on metaphysics, first published in 1604 and reprinted at least eight times by 1616, was his most influential work.

Macclesfield 2027 (second issue, incorrectly stating that the book was published at Cologne); not in Tomash & Williams. Gingerich, ‘Johannes Kepler and the Rudolphine Tables,’ Sky and Telescope 42 (1971), pp. 328-333. Meyer, The mystery of Johannes Kepler’s place of residence in the Linzer Hofgasse ( Rice, González-Velasco & Corrigan, The Life and Works of John Napier, 2017. Westman, The Copernican Question, 2011.

8vo (158 x 97 mm), pp. [164], with geometric figures and tables in text (light dampstain in upper inner margin at the beginning of the volume). Contemporary German yapped vellum over boards, blue edges (soiled, extremities worn, front hinge cracked but firm, armorial bookplate on front paste-down, old shelf-mark pasted onto front free endpaper, later ownership signature on front paste-down and blind-stamp on front free endpaper), internally very good. A very good and genuine copy in original condition.

Item #4862

Price: $27,500.00