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Blaise Pascal's Mechanical Calculator machines Article Blaise Pascal’s Mechanical Calculator: Geometric Modelling and Virtual Reconstruction José Ignacio Rojas-Sola 1,* , Gloria del Río-Cidoncha 2 , Arturo Fernández-de la Puente Sarriá 3 and Verónica Galiano-Delgado 4 1 Department of Engineering Graphics, Design and Projects, University of Jaén, 23071 Jaén, Spain 2 Department of Engineering Graphics, University of Seville, 41092 Seville, Spain; [email protected] 3 Department of Design Engineering, University of Seville, 41011 Seville, Spain; [email protected] 4 University of Seville, 41092 Seville, Spain; [email protected] * Correspondence: [email protected]; Tel.: +34-9-5321-2452 Abstract: This article shows the three-dimensional (3D) modelling and virtual reconstruction of the first mechanical calculating machine used for accounting purposes designed by Blaise Pascal in 1642. To obtain the 3D CAD (computer-aided design) model and the geometric documentation of said invention, CATIA V5 R20 software has been used. The starting materials for this research, mainly the plans of this arithmetic machine, are collected in the volumes Oeuvres de Blaise Pascal published in 1779. Sketches of said machine are found therein that lack scale, are not dimensioned and certain details are absent; that is, they were not drawn with precision in terms of their measurements and proportions, but they do provide qualitative information on the shape and mechanism of the machine. Thanks to the three-dimensional modelling carried out; it has been possible to explain in detail both its operation and the final assembly of the invention, made from the assemblies of its different Citation: Rojas-Sola, J.I.; del subsets. In this way, the reader of the manuscript is brought closer to the perfect understanding of Río-Cidoncha, G.; Fernández-de la the workings of a machine that constituted a major milestone in the technological development of Puente Sarriá, A.; Galiano-Delgado, V. the time. Blaise Pascal’s Mechanical Calculator: Geometric Modelling and Virtual Keywords: Blaise Pascal; mechanical calculator; CATIA; geometric modelling; virtual reconstruction Reconstruction. Machines 2021, 9, 136. https://doi.org/10.3390/ machines9070136 1. Introduction Academic Editor: Marco Ceccarelli Blaise Pascal was a seventeenth-century French mathematician, physicist, theologian and philosopher who made a notable contribution to mathematics and natural history. In Received: 3 July 2021 particular, he designed and built mechanical calculators, one of which being the first me- Accepted: 15 July 2021 chanical calculator used for accounting purposes, called Pascaline [1,2]. Pascal developed Published: 16 July 2021 this mechanical calculator to facilitate the laborious arithmetic calculations of his father Étienne Pascal’s job as a tax inspector in Rouen (France). The purpose of the machine was Publisher’s Note: MDPI stays neutral to add and subtract two numbers directly and to carry out multiplication and division with regard to jurisdictional claims in published maps and institutional affil- by repeated addition or subtraction. It is, therefore, an automatic computing system that iations. returns the result of an arithmetic operation based on a mechanical principle [3,4]. In the seventeenth century, the currency in circulation in France was the livre, which was equivalent to the franc. The division and subdivision of this currency were the sous (sols) and the denarii (deniers), whereby one livre was equivalent to 20 sous, and the sou to 12 denarii. This monetary system came from medieval times and was adopted by the Copyright: © 2021 by the authors. Carolingian Empire and spread throughout Western Europe. It remained in France until Licensee MDPI, Basel, Switzerland. the 1789 revolution [5]. Therefore, Pascal designed his accounting calculator taking into This article is an open access article distributed under the terms and account these divisions and subdivisions, and did not limit to the decimal system, although conditions of the Creative Commons he also subsequently manufactured machines with only decimal divisions. Attribution (CC BY) license (https:// Pascal built more than 50 original machines of which only 9 have been preserved, most creativecommons.org/licenses/by/ of which are on display to the public in museums. Four of them can be found in the Musée 4.0/). des Arts et Métiers in Paris: two eight-digit accounting calculators, a six-digit decimal Machines 2021, 9, 136. https://doi.org/10.3390/machines9070136 https://www.mdpi.com/journal/machines Machines 2021, 9, 136 2 of 22 calculator and a six-digit accounting calculator that was assembled in the eighteenth century from original parts; two original calculators are exhibited at the Musée d’Historie Naturelle Henri-Lecoq in Clermont-Ferrand: an eight-digit decimal and a five-digit countable, which is the smallest of all preserved machines and the only one that has a box that facilitates its transport. In the Mathematisch-Physikalischer Salon in Dresden (Germany), there is an eight-digit accounting calculator. Finally, the two remaining calculators are in private collections, IBM’s and Léon Parcé’s [6]. The eight-digit accounting calculator found in the Mathematisch-Physikalischer Salon in Dresden is the most important reference of the model presented in this manuscript, measuring 10 cm in height, 44.5 cm in width and 14.7 cm in depth, and the materials used are brass, wood and paper as are most of the specimens exhibited today [7]. During the attempted commercialisation of the Pascaline machine, Pascal became aware of the existence of counterfeits. To remedy this situation, all manufactured copies henceforth included his signature and the inscription “Esto probati instrumenti symbolum hoc.; Blasius Pascal; arvernus, inventor, 20 Mai 1652“, which means” Let this signature be the sign of a verified machine. Blaise Pascal of Auvergne, Inventor, 20 May, 1652” [3]. Today, some of the replicas of these originals are currently exhibited in museums [8], although there are also versions made manually by artisans in modern workshops using the same techniques and materials of the time, such as brass and mahogany, which faithfully reproduce the principles of the scientific Pascaline [9]. The present investigation pursues the study of historical machines that have made a notable contribution to the technological development of computing [6], such as the Pascaline, designed by Blaise Pascal in 1642 at the tender age of 19, and the Antikythera Mechanism, an ancient analogue calculator [10]. To this end, said historical invention is analysed from the point of view of engineering graphics to obtain its three-dimensional (3D) geometric modelling and virtual reconstruc- tion by following a methodology established in previous work by the authors [11–17]. Likewise, as a first step in the study and recovery of the technical historical heritage, the 3D model is required. To this end, CAD (computer-aided design) techniques have provided fundamental support in numerous investigations related to various machines [18–21] that follow the objectives set out in the document Principles of Seville [22] on virtual archaeology, which in turn cites the London Charter [23] regarding the computer-based visualisation of cultural heritage. The objective pursued by this research has been to obtain a reliable 3D CAD model that allows the reader to understand the importance of this invention from an engineering point of view. Likewise, this model can help in the conservation of this example of the technical historical heritage since it will allow having a model to carry out future studies and develop applications in a virtual environment. The innovation presented in this research is due to the fact that there is no 3D CAD model with this degree of detail, which will help to fully understand the operation of said historical invention. As future uses of this 3D CAD model, the following stand out: • The possibility of developing user interaction applications that allow knowing the operation of the invention, detailed knowledge of each of its components or the materials from which they were manufactured. • The possibility of incorporating its WebGL model into a website. • The possibility of obtaining realistic virtual recreations for public display or printing said 3D CAD model using additive manufacturing techniques. The remainder of this paper is as follows: in Section2, the materials and methods are presented, and in Section3, the results of the modelling process of said historical invention are presented, explaining in detail its operation and graphically showing its main subsets and how they have been assembled. Finally, in Section4, concluding remarks are presented. Machines 2021, 9, 136 3 of 22 2. Materials and Methods The initial materials for this research, mainly in the form of the plans of this arithmetic machine, are collected in the Oeuvres de Blaise Pascal volumes published in 1779 [24]. Sketches of said machine are found therein that lack scale, are not dimensioned and certain details are absent; that is, they were not drawn with precision in terms of their measurements and proportions, but they do provide qualitative information on the shape and mechanism of the machine. These sketches consist of six projections (views) of the Pascaline: the first includes the top view of the machine; the second shows a perspective of the interior; in the third, the left view is shown with a transparency effect, thereby visualising the mechanism of internal gears; and the last three are front or perspective views of certain subsets of the mechanism on which the operation is based. The methodology followed has been supported by parametric 3D CAD modelling, based on the knowledge of descriptive geometry and on empirical techniques of direct measurement. To this end, known measurements of the machines currently preserved have been taken that have enabled the graphic scale of the plans to be ascertained in order to model each of its elements. Therefore, in order to compose the 3D CAD model of Pascaline, the sketches were studied for the completion of the analysis of the form and function of all its components, together with the observation of images and descriptions of the preserved specimens.
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