Poleni's instruments to trace transcendental "by an astonishingly pure and simple method" Pietro Milici, [email protected]

1. Introduction

During the 17th century, mathematicians radically modified their idea of a , transitioning from the trace of a machine to the solution of an equation. With that in mind, Descartes and Leibniz introduced theoretical machines (i.e. sketches to be considered mentally but not practically realized) to legitimate the geometrical status of algebraic and transcendental curves obtained as solutions of analytical equations. The general method to construct transcendental curves, in particular, was the solution of the inverse problem by the so-called "tractional motion." At the end of the 19th century, integraphs practically implemented such ideas to mechanically plot the of a graphically defined .

A theoretical machine (left, Descartes' 17th c. mesolabe - cf. https://www.youtube.com/watch?v=jhwRBoOK40E) and a geometric-mechanical instrument (right, a 19th c. integraph designed by Abdank-Abakanowicz and produced by the Coradi firm - cf. https://www.historyofinformation.com/detail.php?id=1267).

We are interested in the intermediate 18th-century perspective. In particular, we analyse the geometric instruments designed by Giovanni Poleni (1683-1761), polymath and Professor at the University of Padua in Italy. These instruments, which were technically similar to posterior integraphs1, had been described with an engineer's precision in a letter to Jacob Hermann who was Poleni's predecessor in the chair of Mathematics. The letter was included in the Epistolarum Mathematicarum Fasciculus [Poleni (1729)].

2. The exactness of transcendental curves

At the beginning of the 17th century, the development of algebra deeply shocked the mainstream geometrical setting. From this perspective, the groundbreaking Descartes' Géométrie (1637) proposed some criteria to define ideal machines to trace curves (synthetic part, construction by continuous motion) and from the other, it proposed polynomial equations as a suitable language to study such curves (analytic part, algebraic formulas).2 The description of curves by equations introduced the possibility of considering

1 The theoretical components of both instruments are linkages and wheels. While a general theory for linkages has been developed by Kempe with the Universality Theorem (every finite branch of a planar algebraic curve can be traced by an ideal linkage, cf. [Kempe (1876)]), limits of constructions by linkages and wheels have only recently been given by a Differential Universality Theorem (in this case the constructible functions are Differentially Algebraic ones, cf. [Milici (2020)]). 2 Cf. [Bos (2001)]. 1 curves thanks to their symbolic definitions, without an explicit mechanical procedure. That opened the road to an infinity of new curves that, with the mediation of algebra, can be studied with standardized methods. Adopting today's nomenclature, Descartes considered as acceptable curves the algebraic ones (i.e. zeros of a polynomial in two variables), marginalizing the non-algebraic ones, called transcendental.

The Cartesian method was not able to solve two geometrical problems, a classical and a new one: to find the area of a curve and to solve the inverse tangent problem. Differently from the direct tangent problem (i.e. to find the tangent to a given curve), such an inverse problem consists in finding the curve whose tangent has to satisfy certain given conditions. The first example of an inverse tangent problem was posed by Florimond de Beaune to Descartes and consisted in finding the curve with constant subtangent (such a solution is an exponential curve). However, while this problem and Descartes' solution were purely theoretical geometrical exercises, the first concrete continuous construction of a curve by inverse tangent conditions was the tractrix, traced by Claude Perrault in Paris (in the 1670s) by dragging an extremity of the chain of a pocket watch: that marked the beginning of a new method for geometrical constructions, the "tractional motion." The cause of such a motion is that, for the friction of the clock on the base plane, the chain becomes in traction and always remains tangent to the curve defined by the watch. Beyond Huygens' preliminary studies, it was Leibniz who used this kind of constructions to justify the geometrical constructability of transcendental curves overcoming Cartesian canon. While for the analytical part he developed the calculus beyond finite algebraic equations, for the synthetic part he used the tractional motion.3

Perrault's construction of a tractrix by dragging a pocket watch. The image on the left comes from Poleni's Letter to Hermann; on the right, there is a representation of the behaviour when the chain extremity A is pushed to the left starting from the position A0.

If up to the end of the 17th century the attention to tractional construction was mainly focused on foundational facets, it was at the beginning of the 18th century that some tractional machines overcame the realm of theoretical ideas and became working instruments, mainly thanks to the introduction of a wheel instead of the dragged heavy load (the watch in Perrault's construction). 4 Such a technical improvement allowed the construction of quite precise machines tracing curves by solving inverse tangent problems, as Perks' and Poleni's machines for the tractrix and the quadrature of hyperbolas (i.e. the

3 Cf. [Bos (1988)], [Tournès (2009)] and [Blasjo (2017)]. 4 While the passage from dragged load to wheel is usually considered as a purely technological modification, it also implies different mathematical principles: while the direction of the wheel directly imposes the direction of the tangent, the geometrical condition posed when dragging a point is a bit more complicated, as visible in the preprint [Milici and Plantevin (2020)]. 2 exponential or related curves). We have to note that today it is believed that these machines were developed independently (differently from the machines of Suardi, a pupil of Poleni).5

3. The "Letter to Hermann"

Mathematician, physician, astronomer and also interested in engineering studies, Poleni inaugurated in 1739 a unanimously praised laboratory of experimental physics at the University of Padua ("Teatro di Filosofia Sperimentale"6). However, even before the creation of the laboratory, he was always interested in machines: here we deal with a very particular kind of machines, the mathematical ones, specifically with the ones designed by himself.

In his earliest paper [Poleni (1709)], Poleni proposed an operative arithmetic machine based on the ideas of Pascal and Leibniz, that he constructed and that was widely considered simple, easy to use and reliable. But when in 1727, by Marinoni's correspondence, Poleni knew of an improved version of the arithmetic machine by the Viennese Portrait of Giovanni Poleni mechanician Brauer, he destroyed his own and never rebuilt it.7

It is quite interesting that his only other work on mathematical machines followed quite shortly the destruction of the arithmetical one (we propose below an interpretation for this behaviour). In any case, the new machines introduced in the Letter to Hermann belong to a different field of mathematics, moving from arithmetics to geometry. As already introduced, these machines are the ones to trace the tractrix and the logarithmic curve (or exponential, as curves they coincide). They have been realized with particular attention on the precision of the traces and the easiness of the usability: this is well underlined by three recognized mathematicians close to Poleni (the comments of Antonio Conti, Gabriello Manfredi and Jacopo Riccati have been published with the Letter). Specifically, the title of this presentation comes from a comment of Riccati, that was amazed at how Poleni's machines permit to trace transcendental curves "by an astonishingly pure and simple method."

From Poleni's foundational perspective, these machines had to prove the geometricity of the tractrix and logarithmic curves. While these curves were well known and generally accepted, for him a curve could only be considered geometrical if constructible by the continuous trace of a simple instrument, guaranteeing adequate usability and accuracy. Anyway, despite initial enthusiastic feedback from contemporaries such as Leonard Euler, such machines were soon forgotten, probably because geometric constructions fell out of favour in mainstream mathematics.

While Poleni listed these machines among the artefacts in his cabinet of Experimental Philosophy, the model present today in the Paduan University collection doesn't fit with the description, as we show below.

5 To deepen these machines and also for the sketch of a new machine unifying them, cf. [Crippa and Milici (2019)]. 6 Cf. [Talas (2013)]. 7 Cf. [Hénin (2012)] or https://mathshistory.st-andrews.ac.uk/DSB/Poleni.pdf 3

Poleni's machines for the tractrix (left) and the logarithmic curve (right) in the Letter to Hermann.

The machine previously classified as the one for the tractrix and the logarithmic curve (by courtesy of Sofia Talas, curator of the Physics Cabinet of the University of Padua). 4

Indeed, differences between the artefact in the collection and the designs of the Letter are visible since the first glance (e.g. the curved component in the right part). Furthermore, reading the description in Poleni's index,8 the machine differs in materials (wood pieces in the frame, rods in metals that are not brass). In any case, also because of missing pieces, we have no idea of the purposes of such an artefact.9

Thus, for at our knowledge no copy of such tractional machines is available, together with Frédérique Plantevin we decided to realize their reconstruction for the Cabinet of Curiosity in Brest (a little informal museum mainly for educational activities). Unfortunately, the lockdown prevented us to realize a model in metal, and what we introduce below are just 3D models and an "almost operative" prototype realized with 3D-printed pieces and a bit of DIY.

4. Reconstruction of the machines

In the Letter, Poleni provided not only the drawing of the machines but also precise dimensions.10 If above you saw just figures that can be quite easily found (online or in printed editions), here follow some images for your eyes only: two 3D models and the relative prototypes spread for the first time.

The main idea behind Poleni's machines is that, considering the trace of a wheel rolling (without slipping) on a plane, in any position the projection of the wheel direction on the plane is tangent to the traced curve. In both the proposed machines the wheel is a sharp disk (more precisely, a truncated cone) with many little teeth on its perimeter to both improve the grip and leave a trace on the paper fixed on the base plane. The wheel is inserted in a box that, on its top, has a handle to be pressed by a finger.

Details of the reconstructed wheel (with many little teeth) and its "boxes" in the case of the tractrix (dimensions are in mm).

8 Cf. the machine n. 204 in the index of Poleni's machines (pp. 65-66, machine visible in Fig. 46) included in [Pancino and Salandin (1987)]. The machine is described as "An instrument with its base in wood and iron and two brass plates over which there are two sliding brass devices: one describes the tractrix curve, the other the logarithmic curve. This instruments serves to show the use of mechanics in the description of curves." Such a description fits with the machines in the Letter to Hermann but not with the model present today (and represented in the Fig. 46 of the index). 9 Such a machine could partially fit with a third machine described in the Letter (but not illustrated) to trace by tractional motion a quarter of a circle (Poleni wrote many letters to Gabriello Manfredi about this machine for the formal resolution of the related differential equation). The right part of the machine could be somehow related to the construction of a circular tractrix (a tractrix with the extremity of the fixed-length rod moving along a circle and not on a straight line), as presented in the correspondence between Euler and Poleni. 10 He adopted Paris inches: although the length of the machines is not given, the width is about 20 cm. We can note that the machine in the Paduan collection of the previous section has a length of 125 cm.

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The direction of the wheel is posed by a rod linked to a frame that slides between two external guides. In the case of the tractrix, the curve is traced because of the property of keeping constant the tangent length; in the case of the exponential, the property is to keep constant the subtangent.11

To see the prototypes in action, and so to understand their way of use, see: https://ubotube.univ-brest.fr/videostream/video/polenitractrix/ and https://ubotube.univ-brest.fr/videostream/video/polenilogarithmic/

3D models of the machines for the tractrix (top) and the logarithmic curve (bottom)

11 For "tangent" and "subtangent" we intend the historical meaning, cf. https://en.wikipedia.org/wiki/Subtangent. 6

Prototypes of the machines for the tractrix and the logarithmic curve.

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5. The wheel for tractional constructions

The first published works introducing a wheel to impose the direction of the tangent (i.e. to solve an inverse tangent problem) belong to John Perks and have been published in Philosophical Transactions.12 Even though the proposed machines are much less accurately designed than Poleni's, we can find the same underlying ideas.13

Perk's machines for the quadrature of the hyperbola (left) and the tractrix (right).

As already said, today's historians believe that there was no influence of Perks on Poleni's Letter to Hermann: this probably follows from Poleni's accurate historical reconstruction of tractional motion, in which there is no mention to the English mathematician. But what if such a lack was an intentional purpose to avoid problems with the paternity of the idea? Even though we cannot prove it, we proceed with the evidence gathered. First of all, we want to show that Poleni could have read Perks' works (at least the last one on the tractrix).

Poleni's personal library was very rich and international, as his wide recognition.14 Specifically, in his collection of Philosophical Transactions,15 there is the vol. 29, No. 345 (the one with Perks' work on the tractrix). At the moment we have no references indicating when Poleni acquired the volume but, from his

12 Cf. [Perks (1706)] and [Perks (1714)]. John Perks is an almost forgotten English mathematician, rediscovered thanks to [Pedersen (1963)]. Besides the two papers on mechanical curves, Perks published (always in the Transactions) only another work, on the squaring of lunulae [Wallis, Gregory and Caswell (1699)]. It is curious that also Poleni dealt with lunulae [Poleni (1712)]. 13 In the paper on the quadrature of the hyperbola, Perks introduced a machine that does not trace a logarithmic curve, but he also noted that, by a little modification, the machine can trace the logarithmic curve. 14 As readable in https://history-computer.com/People/PoleniBio.html: In 1710 Poleni was elected a fellow of the Royal Society of London, upon proposal by Sir Isaac Newton. In 1715 he was honoured by being proposed by Gottfried Wilhelm von Leibniz, a correspondent of him, for election to the Berlin Academy of Science in 1715. In 1723 he was admitted to the Academy of the Institute of Sciences of Bologna. In 1724 he was appointed a partner of the newly founded Saint Petersburg Imperial Academy, starting his correspondence with Leonard Euler. In 1739, the prestigious Académie Royale des Sciences of Paris admitted him among its members. 15 After Giovanni Poleni's death, his library has been acquired by the S. Giustinia Library of Padua; now Poleni's Philosophical Transactions are in the University Library of Padua, "Atti Accademici" n. 76. 8 correspondences, we can hypothesize that, at that time, Poleni's collection of Philosophical Transactions was kept very up to date.16

Furthermore, Poleni taught at Padua University for more than 50 years, introducing quite innovative courses, like the ones of applied mathematics.17 We are interested in the course of Mathematical geography (1721-22): that proves Poleni's interest in cartography. Adding this interest to the holding of Perks' article (An easy mechanical way to divide the nautical meridian line in Mercator's projection...), we think that it is reasonable to hypothesize that Poleni could have read Perks' ideas.

But, assuming that Poleni read Perks, that should have happened long before 1729, the year of the Letter to Hermann. As already said, it is quite singular the timing between the destruction of the arithmetic machine and the conception of the geometric ones. From the behaviour of destroying an operative and well- recognized instrument, we may suppose that for Poleni the arithmetic machine was a way to be remembered in the history of mathematics. No second places: to keep staying in the Olympus he had to invent something new and groundbreaking. For this reason, he could have exhumed the idea of improving Perks' machines, also taking profit of the fact that Perks was quite unknown.

6. Conclusions

Even though Poleni could have been inspired by Perks for the idea of using wheels and could have deliberately tried to obscure his role, Poleni has to be remembered for designing the first effective machine solving inverse tangent problems, anticipating the same constructive methods of late 19th-century integraphs. Furthermore, he linked such machines to the important (although short) tradition of tractional motion. In any case, in addition to being considered as one of the grandparents of digital computation for his arithmetic machine, Poleni has to be remembered as a father of analogue computation: with the distinction of today, more than from a mathematical perspective, we should consider his contributions as the prehistory of computer sciences in a very wide way.

Considering the machine present in Padua, it is yet a mystery to understand what's for. Other unsolved points concern the adoption of tractional machines in education: we still miss documents clarifying how Poleni conducted his educational activities with these instruments. That use of tractional artefacts in education is also interesting to be developed for today's educational activities.18

16 In a letter (Avril 16th, 1719), the scholar Pietro Antonio Michelotti asked Poleni for a recent volume of the Philosophical Transactions (for a work of Jurin published in the same year on the motion of waters). In another letter (May 19th, 1719), Michelotti thanked Poleni for lending the requested volume. 17 Cf. [Pepe (2016), pp. 165-174]. 18 Some preliminary experimental activities have been analyzed in [Maschietto, Milici and Tournès (2019)]. Other activities have been proposed in University courses of Mathematics, as visible in this year F. Plantevin's SIC presentation (session on Historical Instruments and Education): "Historical machines, education and DIY in the Cabinet of Curiosity of Brest, France." 9

References

V. Blasjo. Transcendental Curves in the Leibnizian Calculus, Academic Press, 2017. H. J. Bos. Tractional motion and the legitimation of transcendental curves. Centaurus (1988), 31, pp. 9–62. H. J. Bos. Redefining Geometrical Exactness: Descartes' Transformation of the Early Modern Concept of Construction. Springer Science & Business Media, 2001. D. Crippa and P. Milici, A Relationship between the Tractrix and Logarithmic Curves with Mechanical Applications. Mathematical Intelligencer 41 (2019), pp. 29–34. S. Hénin, Early Italian computing machines and their inventors. In Reflections on the History of Computing (2012), Springer, Berlin, Heidelberg, pp. 204-230. A. B. Kempe. On a general method of describing plane curves of the nth degree by linkwork. Proc. London Math. Soc. 7 (1876), pp. 213–216. M. Maschietto, P. Milici and D. Tournès (2019) Semiotic potential of a tractional machine: a first analysis, In: U. T. Jankvist, M. van den Heuvel-Panhuizen, & M. Veldhuis (Eds.), Proceedings of the Eleventh Congress of the European Society for Research in Mathematics Education (CERME11, February 6 – 10, 2019), Utrecht University and ERME, pp. 2133-2140. P. Milici, A differential extension of Descartes' foundational approach: A new balance between symbolic and analog computation. Computability 9:1 (2020), pp. 51-83. P. Milici and F. Plantevin (preprint) The gardener's hyperbolas. 2020. hal-02889469, https://hal.archives- ouvertes.fr/hal-02889469 M. Pancino and G.A. Salandin, Il Teatro di Filosofia sperimentale di Giovanni Poleni (1683-1761), Trieste : Edizioni LINT, 1987. O. Pedersen, Master John Perks and his mechanical curves, Centaurus 8(1) (1963), pp. 1–18. L. Pepe. Insegnare matematica. Storia degli insegnamenti matematici in Italia. Vol. 27. Clueb, 2016 J. Perks, The construction and properties of a new quadratrix to the hyperbola, Philosophical Transactions 25 (1706), pp. 2253–2262. J. Perks, An easy mechanical way to divide the nautical meridian line in Mercator's projection, with an account of the relation of the same meridian line to the curva catenaria, Philosophical Transactions 29(345) (1714), pp. 331–339 G. Poleni. Miscellanea. Venetiis, apud Aloysium Pavinum, 1709. G. Poleni. De vorticibus coelestibus dialogus: Cui accedit Quadratura circuli Archimedis, et Hippocratis Chii analyticè expressa. Patavii, ex typographia Joannis Baptistae Conzatti, 1712. G. Poleni. Epistolarum mathematicarum fasciculus. Patavii, ex Typographia Seminarii, 1729. S. Talas , New Light on the Cabinet of Physics of Padua, In: Cabinets of Experimental Philosophy in Eighteenth-Century Europe, eds. Jim Bennett and Sofia Talas, pp. 49-67, Brill, Leiden-Boston, 2013. D. Tournès. La construction tractionnelle des équations différentielles, Paris: Blanchard, 2009. J. Wallis, D. Gregory and J. Caswell, II. A letter of Dr Wallis to Dr Sloan, concerning the quadrature of the parts of the lunula of Hippocrates Chius, performed by Mr John Perks; with the further improvements of the same, by Dr David Gregory, and Mr John Caswell, Philosophical Transactions 21 (1699), pp. 411–418.

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