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Napier's ideal construction of the logarithms
Abstract
This report explains the construction of logarithms by Napier, and provides reconstructions of the 1614 and 1616 tables.
... Of course, our position has seemed heretic to some, and we were perhaps even considered a revisionist, 1 but we trust that anybody who studies carefully both Napier's and Bürgi's works, as well as all the secondary literature (see a number of references in [56,55]), will without any doubt concur with us. 2 But alas, those who write on Napier often don't know about Bürgi, and those who write on Bürgi often don't know about Napier! 3 ...
... For that world, it was Ibn Hamza who should be credited, having published a treatise on progressions in 1591, and therefore surely had thought of his subject many years before[2].4 Parts of this introduction are borrowed from our earlier analysis of Napier's logarithms[56]. ...
This reports reviews cylindrical slide rules and introduces the first 7-8 places logarithmic slide cylinder, having an unwound length of 2000 meters.
... Les nombres tabulés dans les tables imaginées par Napier permettent de remplacer la multiplication par le sinus 6 d'un angle (lors de la résolution de triangles y compris sphériques) par une addition. La table de Napier n'était donc pas une table de « pures » logarithmes, mais les logarithmes de sinus d'angles, valeurs que Napier avait repris en partie d'autres tables (Roegel, 2010). ...
... Du point de vue historique encore, si le lien entre progressions géométriques et arithmétiques a été abordé par de nombreux mathématiciens (Roegel, 2010), les premières tables de correspondance sont dues à Napier et Jost Bürgi. Ce dernier, astronome suisse qui a travaillé aux côtés de Kepler, élabore des tables trigonométriques et une table de logarithmes (en fait d'antilogarithmes) qui ne sera publiée qu'en 1620 bien que conçue entre 1603 et 1611 (dès 1588 selon Voellmy & Extermann (1972) et le dictionnaire historique de la Suisse 10 ). ...
L'article présente une table de logarithmes éditée en 1795. Cette présentation est accompagnée d'informations générales sur la fabrication et l'utilisation de telles tables.
... A more detailed analysis of the actual tables might reveal what were these " pivot " values, if any. By comparison, it is interesting to observe that Napier apparently did not apply some checks that he could have applied, and consequently his tables display a systematic error [143] ...
... Pitiscus, for instance, used separating dots in his 1608 and 1612 tables, but these dots were not exclusively separators for a decimal part. The first who seems to have used the dot systematically as a decimal point seems to be Napier [143]. Figure 3: Last page of Bürgi's tables (source: [105]). ...
This article analyzes Jost Bürgi's work (1620) and its place in the history of logarithms.
... ; Burton (2007); and Roegel (2010) indicate that the word logarithm was derived from Portmanteau word. Napier (1550 -1617), who coined the term bring it up has "reckoning number," which means the number of ratios used. ...
The four-figure of power of base numbers of tables needs to be created for easy synchronization
with the existing Logarithm table of power of base 10 which is working on four figures. Although the five-figure of power base numbers of tables are more accurate because it is having less approximation in its establishment. The four-figure table is easy to compute doing utilization
because it is having less digits to work with. This paper presents the four and five figures of Kifilideen (Power of base 11) and AntiKifilideen (Antipower of base 11) tables for the computation of mathematical problems. The four and five figures are both reliable to work with. However, there is a tradeoff between easy computation as related to the four-figure table and more accuracy as related to the five-figure table in their utilization.
... 26 We express the log functions in general as log a for base a, and in particular we use ln = log e to denote the "natural" logarithm, and log = log 10 for decimal logs. 27 See [Roegel, 2010a]. lim r→0 r a r −1 = log a e, implicating e in logarithms to any base a > 1 and similarly, with modification for 0 < a < 1. 28 As far as we can tell, Leibniz was the first to record an explicit value for the base of the natural logarithm or connect it with the exponential series. ...
Leibniz published his Euclidean construction of a catenary in Acta Eruditorum of June 1691, but he was silent about the methods used to discover it. He explained how he used his differential calculus only in a private letter to Rudolph Christian von Bodenhausen and specified a number that was key to his construction, 2.7182818, with no clue about how he calculated it. Apparently, the calculations were never divulged to anyone but were discovered later among his personal papers. They may be the earliest record of an accurate approximation of the number we label e and a demonstration of its role as the base of the natural logarithm and exponential function.
This, at that time, was a remarkably precise estimate for e, accomplished more than 22 years before Roger Cotes published e to 12 significant digits, and some 57 years before Euler's treatment of the logarithm in his Introductio in Analysin Infinitorum. The Leibniz construction reveals a hyperbolic cosine built on an exponential curve based on his estimated value, which implies that he understood the number as the base of his logarithmic curve. The sheets of arithmetic used by Leibniz preserved at the Gottfried Wilhelm Leibniz Bibliothek (GWLB) in Hannover, confirm this.
Those sheets show how Leibniz calculated e and applied it to his catenary construction. The data actually yield e to 12 significant figures: 2.71828182845, missed by Leibniz because of a misplaced decimal point. We summarize the construction and examine the worksheets. The unpublished methods seem entirely modern to us and could serve as enrichening examples in modern calculus texts.
... In his "Rabdology" published in 1617 under the title Rabdologiae, seu numerationis per virgulas libri duo [26], John Napier (1550-1617) introduced his famous "bones." Three years after the seminal work on logarithms [25,33], here was a much more practical tool for everyday calculations. ...
This note presents a survey of Napier's bones and of their more " automatic " evolution, the Genaille-Lucas rods. Generalizations of both to other bases are also shown.
... In 1614, John Napier (1550–1617) published his Mirifici logarithmorum canonis descriptio, the description of his table of logarithms [46, 62]. It is through this work that Briggs was early exposed to the theory of logarithms. 4 After Napier's publication, Briggs went to visit him in Scotland in the summers of 1615 and 1616 and they agreed on the need to reformulate 1 Briggs was baptized on February 23, 1560 (old style), which is 1561 new style. ...
This document is a reconstruction of Henry Briggs' first book on logarithms, printed privately in 1617.
Eines der frühen Rechenhilfsmittel vor der Einführung der Logarithmen war die sog. ProsthaphaeresisProsthaphaeresis (πρόσθεσιϛ = Hinzugabe; ἀϕαίρεσιϛ = Wegnahme). Sie bestand darin, ein Produkt mithilfe von trigonometrischen Formeln in eine Summe bzw. eine Differenz zu verwandeln.
Several German- and French-language resources contain brief biographies of Jost Bürgi (e.g., Cantor 1900; Lutstorf 2005; Montucla 1758; Naux 1966; Wolf 1858). No substantial personal information on Jost Bürgi exists in the English language, other than the short (just over one page) account by Nový (1970) in the Dictionary of Scientific Biography. We can, however, construct a decent timeline of Bürgi’s life from German-language resources (see Appendix A), particularly when it is situated with respect to Bürgi’s contemporaries who were engaged in or aided in the development of scientific work dependent upon the logarithmic relationship. Staudacher (2014) published (in German) a quite extensive account of Bürgi’s life, which included content on his mathematical and scientific achievements and contributions, as well as accompanying obstacles, family relationships, and other personal attributes. Using translations of Staudacher’s text, as well as more traditional sources of biographical information on Bürgi, the major aspects of Bürgi’s professional life are highlighted in the brief biography presented here.
In school mathematics, the logarithmic function is defined as the inverse function of an exponential function. And the natural logarithm is defined by the integral of the fractional function 1/x. But historically, Napier had already used the concept of logarithm in 1614 before the use of exponential function or integral. The calculation of the logarithm was a hard work. So mathematicians with arithmetic ability made the tables of values of logarithms and people used the tables for the estimation of data. In this paper, we first take a look at the mathematicians and mathematical principles related to the appearance and the developments of the logarithmic tables. And then we deal with the confusions between mathematicians, raised by the estimation data which were known as proportional parts or mean differences in common logarithmic tables.
This document is a reconstruction of the tables of Henry Briggs' ''Trigonometria britannica'' (1633).
This document is a reconstruction of the tables of Henry Briggs' ''Arithmetica logarithmica'' (1624). In addition, an ideal version of Briggs' table is provided.
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