Sempel (Latin: Sempilius) included a music treatise in his De mathematicis disciplinis libri duodecim (Antwerp, 1635), dedicated to his patron Philip IV, in which he considers the use of the pantometer in music, although without any detailed explanation (pp.56, 102-15).
Richard corresponded with Marin Mersenne, and they held each other in mutual respect.(53) Mersenne recommended the use of the pantometer for equal temperament, explaining in detail its function on the basis of a `harmonic' line drawn by Jean de Beaugrand with this octave division.
In addition, in the Cursus mathematicus, ii (Sant Angelo, 1668), Caramuel describes the use of the musical line on the pantometer in tuning instruments, although his explanation is confused, for a diatonic monochord with Pythagorean intonation is included in the text, and the musical line in the illustration of the pantometer is divided chromatically according to a system that could correspond only to just intonation or a mean-tone temperament.(56)
He also outlines the use of the pantometer, providing a drawing with three musical lines according to the intervallic measurements of the diatonic, chromatic and enharmonic genera.
Although their approach and aims were different, Caramuel and Kircher were the first to apply both logarithms and the pantometer to music, even if they treat these as separate elements.
He also deals with the pantometer and the reductional compasses (pp.770-84), saying of the latter that `it is used for the division of the tetrachord and the monochord'.
The second volume (`Acustica, in VII libros digesta') of Caspar Schott's Magia universalis naturae et artis (Wurzburg, 1657-9) is entirely dedicated to music; he describes and illustrates a monochord and two `monochords' of three strings with moveable bridges for measuring intervals in just intonation (pp.277-87); he also deals with the pantometer, providing an engraving with three musical lines corresponding with the diatonic, chromatic and enharmonic genera, following Kircher's system (pp.288-92).
Corachan allots space to music in his Arithmetica demonstrada theorico-practica para lo mathematico y mercantil (Valencia, 1699, pp.485-94), while Tosca mentions the possibility of engraving musical lines on the pantometer (Compendio mathematico, i, Valencia, 1707, pp.359-80) and also wrote an extended musical treatise based on Dechales, but with many passages taken literally from Zaragoza (Compendio mathematico, ii, Valencia, 1709, pp.329-482).
One of Zaragoza's successors, the Burgundian Jean Francois Petrey, left a number of jottings on the use of the pantometer.(66) Another anonymous manuscript from the Colegio, dating from c.1700, includes two synoptic charts on `theoretical music' and `practical music' respectively, intended for a musical treatise.(67) A later professor, Pedro de Ulloa, also wrote a musical treatise in which logarithmic calculations are completely accepted and in which he openly recommends the equal temperament of the guitar for the organ, harpsichord and double-strung harp.(68) A further anonymous treatise on the pantometer from about 1760 also survives.(69) On the fringes of the Colegio Imperial, Sebastian Fernandez de Medrano describes the proportional ruler (El ingeniero...
(8) On the function of the pantometer (and its forerunner the compass of reduction) and its use in instrument-making, see R.
Remy Gug shows convincingly that the first illustration of what appears to be a pantometer is to be found in Petrus Ramus, Arithmeticae libri duo: geometriae septem et viginti (Basel, 1569), p.2, but this appears to be an isolated example and is not explained or referred to by Ramus.
All three are closely linked with the Spanish treatise on the pantometer in Naples (see n.37 above).