Chilias logarithmorum ad totidem numerous rotundos, praemissa demonstration legitima ortus logarithmorum eorumque usus ... [with:] Supplementum chiliadis logarithmorum, continens praecepta de eorum usu

Marburg: Caspar Chemlin, 1625. First edition. the first theoretical work on the construction of logarithms.

First edition, the Macclesfield copy, of Kepler's logarithmic tables, constructed by means of his own original method. Of the greatest rarity, especially when complete with the correction leaf and the second part, which gives examples of the application of logarithms and details of their construction. It was through the use of these tables that Kepler was able to complete his monumental Tabulae Rudolphinae (1627), the superiority of which "constituted a strong endorsement of the Copernican system, and insured the tables' dominance in the field of astronomy throughout the seventeenth century" (Norman). Kepler indicated the importance of logarithms allegorically on the frontispiece to the Tabulae Rudolphinae. On the top of the temple stand six goddesses. The third from the left represents logarithms: in her hands she holds rods of the ratio of one to two, and the number around her head shows the Keplerian natural logarithm of 1/2: 0.6931472. But logarithms played another important role in Kepler's astronomical work, since without them he may never have discovered his third law of planetary motion. Kepler discovered this law early in 1618, at the same time that he first had access to tables of logarithms (see below). Moreover, his initial formulation of the third law was (to use modern terminology) in terms of a log-log plot, rather than the more familiar terms of squared periods and cubed distances: "The proportion between the periodic times of any two planets is precisely one and a half times the proportion of the mean distances" (Werke VI, 302). "In a sense, logarithms played a role in Kepler's formulation of the Third Law analogous to the role of Apollonius' conics in his discovery of the First Law, and with the role that tensor analysis and Riemannian geometry played in Einstein's development of the field equations of general relativity. In each of these cases we could ask whether the mathematical structure provided the tool with which the scientist was able to describe some particular phenomenon, or whether the mathematical structure effectively selected an aspect of the phenomena for the scientist to discern" (Brown, p. 555).

Provenance: The Earls of Macclesfield, Shirburn Castle, with engraved bookplate, shelf-mark on front pastedown, and blind-stamped Macclesfield crest on blank margins of first three leaves.

After painstakingly extracting from the observational data of Tycho Brahe his first two laws of planetary motion around 1605 (first published in Astronomia nova, 1609), there followed a period of more than twelve years during which Kepler searched for further patterns or regularities in the data. "Then, as Kepler later recalled, on the 8th of March in the year 1618, something marvelous 'appeared in my head'. He suddenly realized that The proportion between the periodic times of any two planets is precisely one and a half times the proportion of the mean distances... why, after twelve years of struggle, [did] this way of viewing the data suddenly 'appear in his head' early in 1618? ... It seems as if a purely mathematical invention, namely logarithms, whose intent was simply to ease the burden of manual arithmetical computations, may have led directly to the discovery/formulation of an important physical law, i.e., Kepler's third law of planetary motion ... Kepler announced his Third Law in Harmonices Mundi, published in 1619, and also included it in his Ephemerides of 1620. The latter was actually dedicated to Napier, who had died in 1617. The cover illustration showed one of Galileo's telescopes, the figure of an elliptical orbit, and an allegorical female (Nature?) crowned with a wreath consisting of the Napierian logarithm of half the radius of a circle. It has usually been supposed that this work was dedicated to Napier in gratitude for the 'shortening of the calculations', but Kepler obviously recognized that it went deeper than this, i.e., that the Third Law is purely a logarithmic harmony" (Brown, p. 555). Kepler further illustrated the importance he attached to logarithms in the famous frontispiece to Tabulae Rudolphinae (which he designed himself): one of the muses standing on the temple is 'Logarithmica', and in her halo shines the number 69314.72 (100,000 times the natural logarithm of the number 2).

"Kepler first saw Napier's tables [Mifirici logarithmorum canonis descriptio, 1614] in the spring of 1617, but he examined them only superficially at that time. Not until 1619 did Kepler have a copy of Napier's tables, but by then he was more familiar with the logarithms in a book of 1618 by Benjamin Ursinus [Trigonometria logarithmica], his former assistant at Prague and Linz, who had adapted Napier's logarithms, abbreviating the tabular data to two places. The value and significance of the new tables now became clear to Kepler" (Belyi, p. 655). "However, he was not content simply to accept the new mechanical aid as he found it. Napier, in his work, had simply presented the tables of numbers without stating how his logarithms were to be computed. So in the first instance his "wonderful canon" must have operated like a magic trick. In fact, in the beginning, mathematicians as serious as Maestlin mistrusted the new aid to calculation. Was it permissible for a rigorous mathematician to utilize numerical tables about whose construction he knew nothing? Was there not danger that employing them might lead to false conclusions, even if the calculation was proved to agree in many cases? When Kepler, during his visit to Württemberg in 1621, discussed these questions with Maestlin, the latter even ventured so far as to observe "it is not seemly for a professor of mathematics to be childishly pleased about any shortening of the calculations." Kepler differed. He wanted to prove and interpret the new aid to calculation by solid methods and subsequently calculate logarithms himself.

"In the winter of 1621-1622 he carried out his plan and composed a book about the subject in which he again demonstrated his fine mathematical instinct. The work was an achievement completely independent of Napier's" (Caspar, Kepler, pp. 308-9). By this time Kepler had received a copy of the work in which Napier described how to construct his logarithms, Mirifici logarithmorum canonis constructio (1619), but he had decided on his own method of construction. "Whereas Napier had approached the logarithms of numbers on the basis of geometrical or, so to speak, kinematical considerations (two points are in linear and parallel motion, one moving uniformly from zero and the other slowing down at a rate proportional to its distance from the end point of the motion), Kepler starts from a purely arithmetic basis ... Kepler writes that for him logarithms were not associated "inherently with categories of trajectories, or lines of flow, or any other perceptible qualities, but (if one may say so) with categories of relationship and qualities of thought" (Belyi, p. 656). As Gronau (p. 8) states, Kepler was the first to introduce the natural logarithm as a function in the modern sense. He started with the functional equation for the logarithm, L(xy) = L(x) + L(y), and solved it in a constructive way to compute the values of the logarithm.

"The printing of the logarithm book has an unusual history. Kepler sent the completed manuscript to Maestlin to have it printed in Tübingen. But Maestlin was not interested and postponed the matter. It took considerable effort on the part of Kepler's friend, William Schickard, to get the manuscript back from Maestlin. When this was finally successful in September 1623, Kepler had just been requested by Landgrave Philip of Hesse-Butzbach to remove certain objections in the carrying out of logarithmic calculation. Therefore, he felt obliged to leave the printing of the work to this prince, to whom it is dedicated, and to leave it to him whether he wanted to order it printed in Tübingen under Schickard's guidance or whether he had some fit person in Frankfurt, who would correct carefully, "because there are lovely types there." Thereupon, Kepler heard nothing further about his opus until, to his great astonishment, he read in the catalogue of the 1624 autumn fair that it had appeared. The Landgrave had had it printed in Marburg without getting in touch with Kepler again" (Caspar, pp. 309-10). Although paginated continuously, the Supplement was published independently the following year, and is not always present. Both Caspar and Zinner give separate references.

With the help of his logarithms Kepler quickly completed the calculation of the Rudolphine Tables. They were ready in December 1623, but publication took a further four years.

Caspar 74 & 75; Cinti 75 (first part only and with the note: 'a quest'opera doveva sequire un supplement nel 1625'); Parkinson, Breakthroughs, p. 72; Zinner 4983 & 5007. Belyi, 'Johannes Kepler and the development of mathematics', pp. 643-660 in: Beer and Beer (eds), Kepler: Four Hundred Years, 1975. Brown, Reflections on Relativity, 2015. Gronau, 'The logarithms - from calculation to functional equations,' pp. 1-8 in II Österreichiches Symposium zur Geschichte der Mathematik, Neuhofen an der Ybbs, 1989.

4to (190 x 148 mm), pp. [1-2] 3-55, [56-108]; [2], 113-116, [2], 121-216, with one folding table and numerous woodcut diagrams in text; text; some browning throughout as often, seventeenth-century speckled calf with blind fillets, flat spine blind-tooled then gilt in the eighteenth century, morocco lettering-piece (joints repaired).