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  1 THE LEIBNIZ CATENARY AND APPROXIMATION OF AN ANALYSIS OF HIS UNPUBLISHED CALCULATIONS 1   MICHAEL RAUGH AND SIEGMUND PROBST A BSTRACT . 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 records. This, at that time, was a remarkably precise estimate for the number we label e  , accomplished 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 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 The unpublished methods seem entirely modern to us and could serve as enrichening examples in modern calculus texts. They are a milestone in the early development of analysis. 1 I NTRODUCTION   Leibniz’s estimation of e   and its use in drafting a catenary were remarkable achievements in the early development of analysis. The work has become known to the public in three stages: (1) a publication in which Leibniz prescribed a Euclidean construction for a catenary, manifestly based on a natural logarithm and exhibited as a curve we know as the hyperbolic cosine, 2  (2) a private letter in which he disclosed that he had used 2.7182818 as a key factor in his construction, and a separate page in Latin explaining his analysis (differential calculus) used to derive the formula for the curve, but nothing more about that number, 3  and (3) undisclosed records written in his hand showing the arithmetic he used in computing the number we know as e   and its 1   Date: April 30, 2018. This study srcinated in an invitation to Raugh by Adrian Rice to present work-in-progress about the Leibniz catenary at JMM 2015, and subsequent emails and conversations with Eberhard Knobloch (2015 – 16) at the Berlin-Brandenburg Academy, Berlin. The work has continued with collaboration between the authors after Probst sent Raugh a copy of the private letter of 1691 from Leibniz to von Bodenhausen explaining his analysis. Raugh is a retired mathematician ( Probst is research fellow and deputy department head at the Leibniz Editorial Center (Leibniz Archive) at the Gottfried Wilhelm Leibniz Bibliothek (GWLB), Hannover (   2   [Leibniz 1691b], published in German translation in [Leibniz 2011], 115 – 124.   3  Letter to Rudolf Christian von Bodenhausen,10/20 August, 1691 and attached “Analysis problematis catenarii”, in  [Leibniz 1923-]: series III, volume 5, 2003, 143-155. In his letter from 12/22 June 1691, ibid, 118, Leibniz writes to von Bodenhausen: “The only thing I withheld [from the publication] is the proportion to of 33  to 33…  The three lines 33 ,  , 33  are to each other like these three numbers 0.3678794, 1.0000000, 2.7182818.” See  Figure 1 on page 3 and Figure 2 on page 4.  Leibniz’s approximate value for e   in the letter to Bodenhausen is rarely mentioned in the literature, although the part of the text containing the estimate for e   was already published by Gerhardt in [Leibniz 1849-1863], vol. 7, 360-362, the values for e   and for 1/ e are printed on page 361). An exception are two papers by Marcel Erné, [Erné 2006] and [Erné 2017]. The solution given to von Bodenhausen has been explained in talks at    2 reciprocal, and the application of these two numbers to plotting a catenary. 4  The work itself must have advanced in a different order, as will become apparent. Presenting the catenary as a Euclidean construction was a singular demonstration of the power of analysis, because without it, it could not be done. He had to have accomplished the analysis first, and his silence about his method of discovery would strike most readers as a challenge. This is an interesting story, but our attention is aimed more narrowly   at the accurate eight-digit estimate for e   that Leibniz communicated privately to his friend Rudolph Christian von Bodenhausen and his use of it for plotting the logarithmic curve. That accuracy may have been without precedent, prompting questions about whether Leibniz was in fact the first to compute e to such high accuracy, to tie a value for e   to the definition of the natural logarithm, and to reveal this number as the base of the exponential function. We shall see below that the Leibniz data yield e   to 12 significant figures (2.71828182845), but Leibniz missed this because of a simple arithmetic error discussed in Section 2.1 on page 5. In his correspondence with Christiaan Huygens, Leibniz used the letter b   to denote the basis of the natural logarithm :   “ b   is a magnitude whose logarithm is the unity” . 5  The notation e   made its first appearance in a manuscript of Leonhard Euler in the late 1720s and then in a letter to Goldbach in 1731. 6  Euler made various discoveries regarding e   in the following years, but it was not until 1748 that he published Introductio in Analysin Infinitorum   where he gave a classic treatment of the natural logarithm, exponential function and e  . He showed that lim →∞ 1+1⁄  1+11!⁄+12!⁄+13!⁄+⋯  and gave an approximation for e   to 24 decimal places: 2.71828182845904523536028 . 7   The date of Euler’s work can be compared with that of 1691, the date of Leibn iz’s letter to von Bodenhausen, or even 1690, when Jacob Bernoulli published a rough estimate for the sum of the exponential series, amounting to 2.5<<3,  in a solution to a problem he had posed in 1685. 8  For some other possibilities for dates prior to Leibniz, particularly Newton, see brief comments in our Conclusion page 17.   We will examine the worksheet held at the GWLB on which Leibniz   carried out calculations for e   and its reciprocal, numbers denoted by Leibniz as   and  . He used partial sums of the series of reciprocal factorials (both positive and alternating) for calculating both numbers. To establish context, we begin with an excursion to show how the construction implicates e  . 1 1 The catenary construction . Leibniz’s figure for the catenary  first appeared in print as a Euclidean construction in the Acta Eruditorum   article of June of 1691. Our Figure 1 on page 3 is taken from a modern German translation of th e Leibniz article “De la chainette”. 9  In the letter to von Bodenhausen mentioned above, Leibniz refers to   4   The workshe et is identified as manuscript 58610 [“Aufz. zur Logarithmenrechnung und zur Kettenlinie”] according to the online catalog of the Leibniz edition (, call number LH 35, 6, 11 fol. 3-4, in the holdings of the GWLB. High definition images are accessible online in the digital collections of the GWLB:   5  See letter to Christiaan Huygens from 3/13 October 1690, [Leibniz 1923-], series III, volume 4, 612; also in a letter to Christiaan Huygens from 27 January / 6 February 1691, [Leibniz 1923-], series III, volume 5, 45.  6   See [Euler 1911-], series IVA, volume 4, 2015, 131-134 and 632-636, especially note 4 on page 635.   7   See [Euler 1748], 69-93, [Euler 1911-], series I, volume 8, 103-132; for an English translation see [Euler 1988], 75-100.   8   [Bernoulli 1690]; see also [Bernoulli 1969-], volume 3, 1975, 91-97, and volume 4, 1993, 160-163, and [Bernoulli 1685], also printed in [Bernoulli 1969-], volume 3, 1975, 91.   9   [Leibniz 1692], published in German translation in [Leibniz 2011], 137-145, diagram on page 141.    3 segments and ; they appear as segments K   and D   at the left-hand side of  Figure 1. 10  The two segments are used to initiate the construction. Leibniz says nothing more about them in the publication, but the role of e   is implicit among the facts he explained about the figure. We mention two. Leibniz claimed that the length of the catenary between points C   and A  is equal to the constructed segment   , and he claimed that the segment   is parallel to the tangent at ( C  ). From these facts it can be inferred that the ratio of K   to D   must be e  , as the reader can verify. F IGURE 1. The “Catenary curve” between C   and A  is claimed equal in length to segment   . This can be true only if   has length e  . Scale is set by the “unit”    . The length of   is 1⁄ . See text. Figure 2 on page 4 illustrates how Leibniz constructed his “logarithmic” curve, where values on the abscissa represent logarithms of ordinates. We tend to think of it as an exponential curve when presented as in the figure. 10   [Leibniz 1923-]: series III, volume 5, 2003, 118. The use of and can be seen in [Leibniz 1691b].    4 F IGURE 2. In this schematic representation, the segments  ,   and   are of unit length. The ordinates over N  , O   and ( N  ) are in ratio ⁄:1:⁄.  Successive points on the curve are constructed from two previously constructed points by constructing an ordinate equal to the geometric mean of the flanking ordinates set above the midpoint between the two flanking abscissas. This algorithm can be carried out for arbitrary refinement. See text. The scale of the figure is set by the vertical “unit”     rising from the srcin O  . Then segments   and  , each of unit length, are drawn on the horizontal base line. Key segments K   and D   are given at the side of the figure. The ratio of K   to D   is used to construct the ordinate at ( N  ), and the ratio of D   to K   is used to construct the   ordinate at ( N  ). Other ordinates are constructed algorithmically: given any two ordinates constructed previously, an ordinate halfway between the two is constructed using the geometric mean of the previous two. To see the effect of this in modern terms, we may write that, given constructed coordinates   ,    and   ,   , then construct,   ,  (  +  2,     ).  The procedure yields points on the curve, ()  ,  where x   is a constructible number on the closed interval [ ― 1; 1], i.e., any positive or negative number that can be expressed as a binary number with a finite number of unit digits to the right of the decimal point. 11  Because the ratio of K   to D   is e  , we arrive at the curve   .   11   Leibniz explains how to construct points of the curve outside the interval [ ― 1; 1], but the details are not necessary for our discussion.    5 The labelling of the curve as “logarithmic” calls attenti on to the defining property of a logarithm as the pairing of an arithmetic progression (successive values of x  ) with the corresponding powers of some base number, e   in this case. For example, the constructible points at {2  : 2  ≤≤2  }⁄  are the logarithms of the corresponding ordinates   ; the abscissa values increase arithmetically in correspondence with the geometric progression given by the powers of e  . 12  Now we turn to an overview of the worksheet. This will be followed by an explanation of the computations shown there for computing e   and its reciprocal; then in Section 3 on page 15, we explain how Leibniz used his estimates to plot a catenary. 2 T HE W ORKSHEET :   S CANS S1  AND S2 Leibniz’s calculations of an approximation for e   and its use for plotting a catenary can be viewed as scans of the two sides of an srcinal worksheet in the holdings of the Gottfried Wilhelm Leibniz Bibliothek (GWLB) in Hannover, Germany. We refer to these scans as S1 and S2, and display details from these scans where needed in the text, but for orientation we recommend that the reader inspect these scans in the magnified presentations available online while reading this overview. 13  The worksheet consists of one folio sheet, folded into a bifolium. Scans S1 and S2 show the two sides of the unfolded folio sheet, i.e. S1 shows from left to right fol. 4v + 3r, S2 shows fol. 3v + 4r.   When folded in its srcinal form, the worksheet looks like a “booklet” of four pages in  this order: 3r, 3v, 4r, 4v. 4v is nearly blank. The front page of the booklet, 3r, appears on top; turn the page (unfolding the booklet) and you will see pages 3v (on the left) and 4r (on the right). Turning page 4r (thus folding the booklet) brings you to 4v, and turning the folded booklet brings you back to 3r. Leibniz wrote first on 3r, where a prominent list of decimal representations of reciprocal factorials for integers from 2 to 14, along with arithmetic results, reveal that Leibniz used partial sums of ∑±1!⁄ =  to derive his estimate for e  , which he notated as  . 14  He started with the alternating series to compute an estimate of 1,⁄  notated as  , with correct figures to the 7-th decimal place (near the upper right hand corner), then an estimate of   obtained by adding 2 to the sum of the positive series. At the bottom of the page he noted that his estimate of   could also be obtained by dividing 1 by his estimate of   and started to compute it. We will discuss these calculations more fully below. These results were used on 3v, where two sketches of a catenary appear, along with more calculations. A list of nine ordinates, derived from the values for   and   calculated on 3r, are used to plot the sketches. S1 and S2 are discussed in more detail in Section 2.1  beginning next and Section 3 beginning on page 15.  2 1 Estimating using the data on 3r The reciprocals of the integers from 2 to 14 are the raw materials used by Leibniz. They are shown as Leibniz wrote them in Figure 3 on 12   The idea for a construction of this type is already implicit in the work of Napier and Briggs, and the connection with the logarithm was discovered by Gregory Saint-Vincent with subsequent development by De Sarasa and Mercator, all familiar to Leibniz. (On the history of logarithms, see [Naux 1966-1971], on Saint-Vincent and Sarasa, see [Burn 2001].) But the construction pushes the ideas to higher refinement. As noted above, Leibniz himself had already developed series expansions for the exponential function. 13  Manuscript images by courtesy of GWLB; for accessing S1 (fol. 4v + 3r) and S2 (fol. 3v + 4r) in the digital collections of the GWLB, first go to,  displaying folder LH 35, 6, 11; clicking forward on the right arrow until view [5], you’ll see S1, [6] shows S2.   14   Leibniz omits the reciprocals 1=0! and 1=1! from his list, but he is aware of them and includes their sum, 2, where appropriate; he does not need to account for their difference, 0.  
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