On an Integrating Machine having a new Kinematic Principle (James Thompson) WITH On an instrument for calculating the integral of the product of two given functions (William Thomson) WITH Mechanical Integration of the Linear Differential Equations of the Second Order with Variable Coefficients (William Thomson) WITH Mechanical Integration of the general Linear Differential Equation of Any order with Variable Coefficients (William Thomson) in Proceedings of the Royal Society of London 24 pp. 262-275, 1876 WITH (2nd Volume): Harmonic Analyzer (William Thomson) in Proceedings of the Royal Society of London 27 pp. 181-189, 371-373, 1878; WITH (3rd Volume): On a Machine for the Solution of Simultaneous Linear Equations (William Thomson) in Proceedings of the Royal Society of London 28 pp.111-113, 190-195, 1879

London: Harrison & Sons. 1st Edition. FIRST EDITIONS (IN THREE VOLUMES) OF THE ONLY PUBLISHED DESCRIPTION (& ILLUSTRATIONS) BY WILLIAM THOMSON (Baron Kelvin) OF HIS HARMONIC ANALYZER, THE FIRST OPERATIONAL TIDE PREDICTOR. Six papers appear in three separate volumes. "Thomson's system [was] used regularly to collect tide information at a given location, analyze the amplitude and phase of a set of known frequencies to be found in the tide and then, based on the collected information to predict the tide at the point in future times...The machine was very successful and was adopted by many countries. The United States was using the Tide Predictor well into the 20th century" (Otnes, notes on Mechanical Fourier Analyzers, JOS, 34). Lord Kelvin noted "that four hours of crank turning would produce tidal predictions for one harbor for a full year" (ibid 37).

Harmonic analysis, is a "mathematical procedure for describing and analyzing phenomena of a periodically recurrent nature. Many complex problems have been reduced to manageable terms by the technique of breaking complicated mathematical curves into sums of comparatively simple components" (Encyclopedia Britannica). The ‘harmonic analyzer' was the first analog computer designed for general purpose applications.

"Many physical phenomena are periodically recurrent in nature "such as sound waves, alternating electric currents, tides, and machine motions and vibrations... Such motions can be measured at a number of successive values of the independent variable, usually the time, and these data or a curve plotted from them will represent a function of that independent variable. Generally, the mathematical expression for the function will be unknown. However, with the periodic functions found in nature, the function can be expressed as the sum of a number of sine and cosine terms. Such a sum is known as a Fourier series (Joseph Fourier) and the determination of the coefficients of these terms is called harmonic analysis (EB).

The large number of calculations needed were "best done by machines called harmonic (or spectrum) analyzers; [those capable of measuring] the relative amplitudes of sinusoidal components of a periodically recurrent function"; existent ones were incapable of doing so (EB). ‘"The first step came in the early 1870s when James Thomson (William's brother) designed a ‘mechanical integrator' capable of solving first order differential equations. In 1873, William Thomson (1st Baron Kelvin) then determined how to link several integrators together to solve equations of second degree or higher. With that knowledge, Thomson designed the first harmonic analyzer. "Based on the disk-globe-and-cylinder integrator" with "11 sets of mechanical integrators one for each harmonic to be measured" Thomson designed his harmonic analyzer (EB; Origins of Cyberspace 382).

While invented in 1873, it was quickly followed in 1876 and 1879 by two larger machines built on similar principles - those are the machines described and illustrated in these three papers - and the machines used to collect tidal information for decades to come. CONDITION & DETAILS: London: Harrison & Sons. Complete. 3 vol. 4to. 8.75 x 5.5 inches. [4], ix, [566], 4. Stamp on title, first page of table of contents and verso; no exterior. In text illustrations throughout. Handsomely rebound in aged half-calf over marbled paper boards; 5 gilt-ruled bands at the spine with compartments gilt-tooled and the title, etc. in gilt as well. Very tightly and solidly bound. Clean and bright inside and out. Fine condition.