# Plans for a trip to the future

By Torres Quevedo Museum

Museo Torres Quevedo

Take a tour of Leonardo Torres Quevedo's beautiful plans and illustrations for a calculating machine that would achieve something never done before - solving complex mathematical equations.

Machine for solving algebraic equations Machine for solving algebraic equationsTorres Quevedo Museum

Torres Quevedo's reputation and international recognition began with his 1893 lecture, 'Report on Algebraic Machines'.

Alongside this theoretical work, he built a series of analog algebraic machines that were able to solve mathematical equations.

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

Torres Quevedo's 'algebraic' machines used a physical model to solve mathematical problems.

The numbers on them were represented by quantities of a particular physical magnitude such as length, movement, rotation of an axis, etc.

The machine included 2 innovations: the use of a logarithmic scale (enabling the calculation of a monomial to be reduced to a sum) and Torres Quevedo's endless spindles.

Endless Spindle Drawings Endless Spindle Ensemble No 1 by Leonardo Torres QuevedoTorres Quevedo Museum

The endless spindle

The endless spindle was a fundamental part of the calculating machine that could solve 8-term equations.

It was the most interesting and original of Torres Quevedo's inventions, and its mechanism was the first of its kind.

Endless Spindle Drawings Endless Spindle Ensemble Diagram by Leonardo Torres QuevedoTorres Quevedo Museum

The collection of plans for the endless spindle consists of a series of black-and-white pen and ink drawings.

Although the drawings are excellent, they do not give a complete breakdown of the mechanism with details, scale, or dimensions.

Endless Spindle Drawings Endless Spindle Ensemble No 2 by Leonardo Torres QuevedoTorres Quevedo Museum

From a mechanical point of view, this machine enables the problem log(u + v) = log(u/v + 1) = log v + log(u/v + 1) to be solved using 2 wheels, whose angular movements are linked by a curve and its asymptotes.

Endless Spindle Drawings Endless Spindle Ensemble No 3 by Leonardo Torres QuevedoTorres Quevedo Museum

This set of plans dates from before 1905, the date that the machine was built.

Endless Spindle Drawings Endless Spindle Ensemble Sections by Leonardo Torres QuevedoTorres Quevedo Museum

Some of the plans include handwritten annotations, possibly to make them easier to understand when it came to manufacturing the machine.

Equations Machine Drawings Work No 1 - D shaft and details (I) by Leonardo Torres QuevedoTorres Quevedo Museum

The Equation Machine

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

The Torres Quevedo Museum houses a collection of plans for the equation machine which were drawn in 1908 and 1909, before it was built.

Equations Machine Drawings Work No 1 - Exponential Trains by Leonardo Torres QuevedoTorres Quevedo Museum

The plans were drawn by a professional draftsman.

They contain several detailed drawings of the parts needed to build the machine.

Equations Machine Drawings Order 171 - Ensemble of Drawing 10 - Drawing 11 by Leonardo Torres QuevedoTorres Quevedo Museum

They were drawn in pencil and ink pen, with added watercolor details to give the impression of volume.

Equations Machine Drawings Order 171 - Ensembles and Partial Quarterings - Drawing 7 by Leonardo Torres QuevedoTorres Quevedo Museum

This collection helps us to understand the inner workings of the equation machine.

Equations Machine Drawings Order 171 - Partial Quartering - Drawing 10 by Leonardo Torres QuevedoTorres Quevedo Museum

The breakdown of the parts, as seen here, is essential for constructing the machine.

Equations Machine Drawings Order 171 - Ensembles and Partial Quarterings - Drawing 9 by Leonardo Torres QuevedoTorres Quevedo Museum

It would be very difficult to build the machine today without it.

Equations Machine Drawings Partial Ensemble No 2 by Leonardo Torres QuevedoTorres Quevedo Museum

The accuracy of the details and the care with which the diagram was drawn are evident in this picture.

Equations Machine Drawings Work No 1 - C, C' shafts and details (II) by Leonardo Torres QuevedoTorres Quevedo Museum

Almost all of the plans show the date that they were drawn, between 1908 and 1910.

Equations Machine Drawings Work No 1 - Left Side of the Box by Leonardo Torres QuevedoTorres Quevedo Museum

The drawings also allow us to visualize the inside of the machine and understand how it works.

Equations Machine Drawings Work No 1 - Developed ensemble of shafts E and C by Leonardo Torres QuevedoTorres Quevedo Museum

Every item in the plans is accurately marked out.

Equations Machine Drawings Work No 1 - F shafts and details by Leonardo Torres QuevedoTorres Quevedo Museum

This diagram shows how the tables are linked to the cogwheels as they turn.

If you zoom in, you can take a closer look at the spindles inside the mechanism of the calculating machine.

Equations Machine Drawings Work No 1 - Mechanism for the spindles of the shafts F3 and F4 by Leonardo Torres QuevedoTorres Quevedo Museum

This set of plans shows the ideas developed in the Essays on Algebraic Machines.

Torres Quevedo's writings ushered in a new era in mathematical theory based on new concepts, which led to a new understanding of machines.

Credits: Story

museotorresquevedo.caminos@upm.es
School of Civil Engineering

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