The Birth of Computing: Torres Quevedo's Calculating Machines

Torres Quevedo pioneered analog calculators that could solve complex equations.

By Torres Quevedo Museum

Museo Torres Quevedo

Colmar Arithmometer -Upper coverTorres Quevedo Museum

Background: Thomas de Colmar's Arithmometer

The Torres Quevedo Museum owns a version of Thomas de Colmar's arithmometer dating from 1820.

This machine uses a simple method to solve basic addition, subtraction, multiplication, and division processes.

It was used throughout the 19th century, until the development of more complex machines that were capable of more sophisticated calculations.

Colmar Arithmometer -Complet deviceTorres Quevedo Museum

Leonardo Torres Quevedo presented his electromechanical arithmometer in Paris in 1920, to celebrate the centenary of Thomas de Colmar's arithmometer.

The machine was one of Torres Quevedo's greatest contributions to the scientific community.

In the words of Leonardo himself, "My apparatus is based on the same principles as that of Thomas de Colmar, but it works in an entirely different way. In mine, all of the movements are automatic and, today, I am going to talk to you about how they are automated."

Essai 'Algebraic Machines, by Leonardo Torres Quevedo'Torres Quevedo Museum

Report on Algebraic Machines

Torres Quevedo traveled abroad frequently, especially to France, where he was in touch with leading scientists.

As a result, his work displayed a technical rigor that could be seen in the quality of his inventions.

In 1893 he presented his "Report on Algebraic Machines" at the General Directorate of Public Works in Spain, and later also presented it in Paris.

Report concerning the algebriac machines. Civil Works MagazineTorres Quevedo Museum

The final version of the report was published in Bilbao in June 1895, while he worked on the first model of a calculating machine.

The image shows the published report in the Public Works Journal.

Essay 'Concerning the construction of algebraic machines, by Leonardo Torres Quevedo'Torres Quevedo Museum

Torres Quevedo's work on analog machines meant that his report, "Calculating Machines," was well-received by the Paris Academy of Sciences in 1900.

As a result, in 1901 he became a member of the Academy of Sciences in Madrid, where he gave a speech on algebraic machines.

Read discourses before the Royal Academy of ScienceTorres Quevedo Museum

Torres Quevedo's writings ushered in a new era in mathematical theory based on new concepts. His contributions led to the development of innovative new machines.

He built a series of analog calculating machines to complement his theoretical work, all of which were mechanical.

The arithmometers and the equations they solveTorres Quevedo Museum

Mechanical Machines

Mechanical machines used continuous variables and physical mechanisms to solve mathematical problems.

They were calculating machines in which the numbers were represented by different kinds of physical magnitudes.

Mathematical equations were transformed into various operative processes in which a physical problem would be solved, with the numerical result providing the solution to the mathematical equation.

Machine for integrating primary differential equationsTorres Quevedo Museum

Torres Quevedo's theoretical ideas and inventions were based on kinematics, establishing relationships between the values of particular movements. The machine established the mathematical formulae that connect these variables.

The machine that can calculate polynomial roots is now one of the Torres Quevedo Museum's most important objects.

Machine for solving mechanically the equation x^2 - px + q = 0, with imaginary coefficients an rootsTorres Quevedo Museum

As the first machine to solve equations, it allowed the user to find the roots of quadratic equations involving real and complex numbers. It could be used to solve any quadratic equation.

The device works by generating a series of values, both integer and rational, for which a polynomial function is continuously reevaluated as the variable is increased or decreased.

Machine for solving algebraic equations - Complet deviceTorres Quevedo Museum

This device finds the numerical values of the roots of an 8-degree polynomial. Its development started in 1910 and finished in 1920, and its success is proof of the persistence and genius of Torres-Quevedo.

The goal to be achieved with the device was obtaining values of a polynomial in a continuous and automatic way. Since the mechanism is analog, and not digital, it can take any value, without jumps between discrete values.

It shows two important improvements over what existed up to date: using a logarithmic scale, thus reducing to additions the evaluation of monomial expressions, and the use of the "husillos sin fin" or endless spindles, invented by Torres-Quevedo.

Diagram of the machine's relay circuits and the equations that it solves.

Machine for solving algebraic equations - Detail view of the spindlesTorres Quevedo Museum

In a polynomial equation, the wheels representing the unknown quantity spin round and the values of the sum of the variables are given as a final result.

Machine for solving algebraic equations - Upper detail view of the spindlesTorres Quevedo Museum

When this sum coincides with the value of the second member, the wheel of the unknown quantity shows a root.

View from above of the wheels that spin to give the root results, using endless spindles.

Automatism of one calculating machineTorres Quevedo Museum

Thanks to his theories on automation and machines, Torres Quevedo was able to overcome the difficulties of creating machines that carried out calculations using exclusively mechanical methods.

This automaton is part of the calculating machine.

Endless Spindle - Front viewTorres Quevedo Museum

The endless spindle was a fundamental part of the calculation machine that could solve 8-term equations. It generated sums to solve the core equation of the algorithm. It was the most interesting and original of Torres Quevedo's inventions, and its mechanism was the first of its kind.

Endless Spindle - Back viewTorres Quevedo Museum

Torres Quevedo's calculating machine represented a significant theoretical and practical breakthrough. The endless spindle is a mechanical device that evaluates the logarithm of an expression as the sum of several logarithms, thus solving the problem of achieving enough precision with a mechanical set of parts.

Spindle DrawingTorres Quevedo Museum

The problem log(u + v) = log(u/v + 1) = log v + log(u/v + 1) is resolved, from a mechanical point of view, using 2 wheels whose angular movements are linked by way of a curve and its asymptotes.

Paintings of the Sorolla Museum Detail of the spindle in the portrait painted by SorollaTorres Quevedo Museum

This invention was so important that the endless spindle appears in the portrait that Joaquín Sorolla painted of Torres Quevedo in 1917.

Integrator - General viewTorres Quevedo Museum

In 1911, having already begun working on his electromechanical calculating machines, he presented his report "Mechanical Construction of the Relationship Expressed in the Formula y' = dy/dx" to the Academy of Sciences in Paris.

Integrator - Back viewTorres Quevedo Museum

In this report, he presented a new machine: the integrator. It was further clear evidence of his superb imagination.

As well as carrying out calculations using the formula y' = dy/dx, the integrator produced a drawing of the result.

Aritmómetro electromecánico - Complet deviceTorres Quevedo Museum

Electromechanical Machines

Leonardo Torres Quevedo became a pioneer of modern automation thanks to his inventions of new calculating machines that used electromechanical systems.

His machines are considered to be the first digital calculators in history, using discrete variables to do calculations.

Aritmómetro electromecánico pequeño - Complet deviceTorres Quevedo Museum

Torres Quevedo came to the conclusion that a machine could work independently, carrying out actions, and responding to orders and its surroundings. He applied this concept in the development of his calculating machines.

In his "Essays on Automation," he set out the theory behind what would become his arithmometers: electromechanical machines capable of carrying out calculations independently.

Aritmómetro electromecánico pequeño - Detail view of the mechanical systemTorres Quevedo Museum

In this text, Torres Quevedo explores the idea of a machine that works sequentially to carry out calculations and floating-point arithmetic, enabling it to handle very large numbers.

Calculating machines videoTorres Quevedo Museum

His invention of a digital calculator made him a pioneer of automation as we understand it today.

An arithmometer was a device capable of recording numerical values, completing different processes, carrying out any calculations, printing the results, and confirming when it was finished.

Aritmómetro electromecánico - Complet deviceTorres Quevedo Museum

With the arithmometers, Torres-Quevedo introduced new original automatic mechanical devices.

His greatest contribution, in terms of originality and novelty, was to conceive the device in such a way that it was capable of calculating and comparing without human interaction.

As Santesmases has said, “he was the first to succeed in developing an automaton which compared numbers with several figures”.

These arithmometers used a type writer as an input device.

The processing and registering unit is achieved by a system using slats, pulleys, needles, pointers, brushes, electromagnets and switches.

Aritmómetro electromecánico - Logic unit detailTorres Quevedo Museum

Detail of the inside of the processing unit.

Aritmómetro electromecánico - Keyboard detailTorres Quevedo Museum

The output device - also a typewriter - was a spectacular innovation that (as early as 1920) was the precursor to modern-day computers.

Credits: Story

Torres Quevedo Museum (Madrid)
School of Civil Engineering
Technical University of Madrid (UPM)

Director: Francisco Javier Martín Carrasco
Secretary: Felipe Gabaldón Castillo
Museum Manager: Manuel G. Romana
Editing: Miriam Guerrero Pérez
Texts: Miriam Guerrero Pérez and Consuelo Durán Cermeño
Advisors: Francisco González Redondo, Antonio López Vega, and María Pascual Nicolás
Documentation: Manuel Romana García, Consuelo Durán Cermeño, Miriam Guerrero Pérez
Image Sources: Museum collection, Francisco González Redondo Collection, Manuel Romana Collection, National Newspaper Library, Public Works Journal, Sorolla Museum
Video Source: YouTube

Credits: All media
The story featured may in some cases have been created by an independent third party and may not always represent the views of the institutions, listed below, who have supplied the content.
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