Editing Leonardo Torres Quevedo (section)

===Analogue calculating machines===

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Since the middle of the 19th century, several mechanical devices were known, including [[integrator]]s, multipliers, etc. The work of Torres in this matter is framed within this tradition, which began in 1893 with the presentation of the "Memória sobre las máquinas algébricas" ("Memory about algebraic machines") at the [[Spanish Royal Academy of Sciences]] in Madrid.<ref name="MaquinasAlgebricasLTQ">Leonardo Torres. ''[https://books.google.com/books?id=Eo0NAQAAIAAJ Memoria sobre las máquinas algébricas: con un informe de la Real academia de ciencias exactas, fisicas y naturales]'', Misericordia, 1895.</ref> This paper was commented in a report by [[Eduardo Saavedra]] in 1894 and published in the {{Interlanguage link|Revista de Obras Públicas|es|Revista_de_Obras_Públicas}}.<ref>[https://quickclick.es/rop/pdf/publico/1895/1895_tomoI_28_01.pdf Memória sobre las Máquinas Algébricas, (pdf)], pp. 217–222, Revista de Obras Públicas, 10 October 1895.</ref> Saavedra, who considered Torres' calculating machine as "an extraordinary event in the course of Spanish scientific production",<ref>{{cite web |url = https://dbe.rah.es/biografias/9042/leonardo-torres-quevedo |title = Leonardo Torres Quevedo|publisher = [[Royal Academy of History|Real Academia de la Historia]]|date = |language= es|access-date = 8 August 2024 }}</ref> recommended that the final project of the device be financed.<ref name="-.bio-.-itefi.-."/>

In 1895 Torres presented "Sur les machines algébriques", accompanied by a demonstration model, at the [[Bordeaux]] Congress of the ''Association pour l'Avancement des Sciences'', and in Paris in the ''[[Comptes rendus de l'Académie des Sciences]]''.<ref name="Redondo2019">Francisco A. González Redondo. ''[https://www.aend.org/Revistas%20digitales/RevAEND88.pdf Leonardo Torres Quevedo: Ingeniero, matemático, inventor],''(pdf) pp. 29–30, Revista de la Asociación Española de Ensayos No Destructivos, 2019.</ref> Later on, in 1900, he presented a more detailed work, "Machines à calculer" ("Calculating machines") at the Paris [[Académie des Sciences|Academy of Sciences]].<ref>{{cite journal|date=1901|first=Leonardo|language=fr|last=Torres Quevedo|publisher=Impr. nationale (París)|title=Machines à calculer|url=https://gallica.bnf.fr/ark:/12148/bpt6k840139b?rk=21459;2 |journal=Mémoires Présentés par Divers Savants à l'Académie des Scienes de l'Institut de France |volume=XXXII}}<!-- auto-translated by Module:CS1 translator --></ref> The commission formed by [[Marcel Deprez]], [[Henri Poincaré]] and [[Paul Émile Appell|Paul Appell]], asked the academy for its publication,<ref name="-.bio-.-itefi.-."/> where they reported favorably: "In [[Mécanique analytique]], [[Joseph-Louis Lagrange]] considers material systems whose connections are expressed by relationships between the coordinates or parameters used to define the position of the system. We can, and this is what Mr. Torres does, take the opposite point of view." Concluding: "In short, Mr. Torres has given a theoretical, general and complete solution to the problem of the construction of algebraic and transcendental relations by means of machines; moreover, he has effectively constructed machines that are easy to use for the solution of certains types of algebraic equations that are frequently encountered in applications."<ref>Académie des sciences (France). [https://gallica.bnf.fr/ark:/12148/bpt6k3086n/f874n3.capture# ''Mécanique Appliquée. – Rapport sur un Mèmoire de M. Torres intitulè Machines à calculer présenté à l'Académie dans la séance du 19 de février 1900''] pp. 874–876, Commissaires: MM. Marcel Deprez, Poincaré, Appell, rapporteur.</ref><ref>Rafael López-García. Marco Ceccarelli. [https://books.google.com/books?id=dBvHEAAAQBAJ&dq=leonardo+torres+quevedo+innovator&pg=PA330 Distinguished Figures in Mechanical Engineering in Spain and Ibero-America.], Springer Nature, p. 330, 2023. ISBN 978-3031310751</ref>

These works examined mathematical and physical analogies that underlay analogue calculation or continuous quantities, and how to establish mechanically the relationships between them, expressed in mathematical formulae. The study included [[Complex analysis|complex variables]] and used the [[logarithmic scale]]. From a practical standpoint, it showed that mechanisms such as turning disks could be used endlessly with precision, so that changes in variables were unlimited in both directions.<ref>Jacob, L. [https://ia801407.us.archive.org/4/items/lecalculmecaniqu00doca/lecalculmecaniqu00doca.pdf Le Calcul mécanique. Appareils arithmétiques et algébriques intégrateurs (pdf)], pp. 165–187, Encyclopédie Scientifique, Bibliothèque de Mathémathiques Apliquées, 1911.</ref><ref>{{Cite book|chapter-url=https://archive.org/details/moderninstrument00horsuoft/page/263/mode/1up?view=theater&q=1906|title=Modern instruments and methods of calculation: a handbook of the Napier Tercentenary Exhibition|last1=Horsburg|first1=E. M. (Ellice Martin)|last2=Napier Tercentenary Exhibition|date=1914|publisher=London: G. Bell|others=Gerstein – University of Toronto|page=263|chapter='The Instrumental Solution of Numerical Equations' by D. Gibb, M.A.}}</ref><ref>Girvan, Ray. ''The revealed grace of the mechanism: computing after Babbage'', May 2003.</ref> Torres developed a whole series of analogue mechanical calculating machines that used certain elements known as ''arithmophores'', which consisted of a moving part and an index that made it possible to read the quantity according to the position shown thereon.<ref>{{Citation
  | last = Mehmke
  | first = R.
  | title = Encyclopédie des sciences mathematiques pures et appliquées
  | place = Paris
  | publisher = Gauthier-Villars
  | year = 1908
  | chapter = I23
  | pages = 351
}}</ref> The aforesaid moving part was a graduated disk or a drum turning on an axis. The angular movements were proportional to the [[logarithm]]s of the magnitudes to be represented. Between 1910 and 1920, using a number of such elements, Torres built a machine that was able to compute the roots of arbitrary [[polynomial]]s of order eight, including the complex ones, with a precision down to thousandths. This machine could calculated the equation:
<math>\alpha = \frac{A_1 X^a + A_2 X^b + A_3 X^c + A_4 X^d + A_5 X^e}{A_6 X^f + A_7 X^g + A_8 X^h} \,</math> where ''X'' is the variable and ''A<sub>1</sub>'' ... ''A<sub>8</sub>'' is the coefficient of each term. Considering the case of α = 1, it becomes the following formula, and the root of the algebraic equation can be obtained:
<math>A_1 X^a + A_2 X^b + A_3 X^c + A_4 X^d + A_5 X^e - A_6 X^f - A_7 X^g - A_8 X^h = 0 \,</math>

By calculated each term on a logarithmic scale, they can be calculated only by sums and products like ''A<sub>1</sub>'' + ''a'' × log(''X''), which can handle a very wide range of values, and the relative error during calculation is constant regardless of the size of the value. However, to calculate the sum of each term, it is necessary to accurately obtain log(u + v) from the calculated values log(u) and log(v) on a logarithmic scale. For this calculation, Torres invented a unique mechanism called the "endless spindle" ("''fusee sans fin''"), a complex differential [[gear]] using a [[helix|helical]] gear shaped like a wine bottle, which allowed the mechanical expression of the relation <math>y=\log(10^x+1)</math>. Putting log(u) – log(v) = log(u/v) = V, then u/v = 10 V, and the following formula is used to calculate log(u + v):  
<math>\log (u + v) = \log (v (u / v + 1)) = \log (v) + \log (u / v + 1) = \log (v) + \log(10^V + 1)\,</math>,<ref name="Thomas2008">{{Cite journal |last=Thomas |first=Federico |date=2008-08-01 |title=A short account on Leonardo Torres' endless spindle |url=https://www.sciencedirect.com/science/article/pii/S0094114X07001231 |journal=[[Mechanism and Machine Theory]] |publisher=[[International Federation for the Promotion of Mechanism and Machine Science|IFToMM]] |volume=43 |issue=8 |pages=1055–1063 |doi=10.1016/j.mechmachtheory.2007.07.003 |issn=0094-114X|hdl=10261/30460 |hdl-access=free }}</ref> the same technique which is the basis of the modern electronic [[logarithmic number system]].

Torres devised another machine around 1900 with a small computing using gears and [[Linkage (mechanical)|linkages]] to obtain the complex number solution of the [[quadratic equation]] X<sup>2</sup> – pX + q = 0.<ref name="Gomez-JaureguiGutierrez-GarciaGonzález-RedondoIglesiasManchadoOtero2022">{{Cite journal |last1=Gomez-Jauregui |first1=Valentin |last2=Gutierrez-Garcia |first2=Andres |last3=González-Redondo |first3=Francisco A. |last4=Iglesias |first4=Miguel |last5=Manchado |first5=Cristina |last6=Otero |first6=Cesar |date=2022-06-01 |title=Torres Quevedo's mechanical calculator for second-degree equations with complex coefficients|journal=[[Mechanism and Machine Theory]] |publisher=[[International Federation for the Promotion of Mechanism and Machine Science|IFToMM]] |volume=172 |issue=8|page=104830 |doi=10.1016/j.mechmachtheory.2022.104830|s2cid=247503677 |doi-access=free |hdl=10902/24391 |hdl-access=free }}</ref> Nowadays, all these machines are kept in the Torres Quevedo Museum at the  School of Civil Engineering of the [[Technical University of Madrid]].<ref>{{cite web | url=http://www.upm.es/UPM/MuseosUPM/MuseoTorresQuevedo | title=Museo Torres Quevedo}} en la [http://www.upm.es Universidad Politécnica de Madrid]</ref>