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Isaac Newton's TONOMETER and COLORIMETER1 Volume 22, Number 1, Spring, 2013 41 “In Experiments, where Sense is Judge” Isaac Newton’s TONOMETER and COLORIMETER1 Charles R. Adams In 1631, William Oughtred (1573-1660) published a small tones and colors, hearing and seeing. Newton was, of course, book on algebra, Clavis Mathematicæ (The Key of Mathe- no stranger to most fields of applied mathematics (such as matics), that was to become, through numerous editions and astronomy, mechanics, and navigation). But that he would commentaries, one the most popular and influential intro- also create significant uses for circular rules in the realm of ductory texts of the 17th century. He wrote the book, he said: sensory perception and the neurophysiological workings of audition and vision might seem unusual. However, he did “… to extend to students of mathematics, as it succeed in making technical advances in just those areas in were, Ariadne’s thread, by the help of which they part by incorporating a matho-musical model into his new may be led to the innermost secrets of this knowl- theory of light and colors, in effect tuning the rainbow. edge [such as Euclid’s mathematical demonstra- tions], and directed towards an easier and deeper Not only were discourses “on the consent and dissent of understanding of the most ancient and favoured visibles and audibles” (Francis Bacon) ubiquitous through- authors”. out all the sciences of the 17th century, but attention to the senses was a significant aspect of Newton’s own philosophy “Oughtred’s purpose was to use the method of analysis to of nature. He considered “the motion of light and sound … understand and recreate the work of ancient writers; Ari- [to be] topics that are most general and most fundamental for adne’s thread was to be a guide not to the future but to the natural philosophy”. He also emphasized that the basic ac- past. Oughtred went on to describe how this investigation tivities of scientific investigation required comfortable and of ancient writings was to be done by interpretation, com- productive relationships between curious perceivers and na- parison, and reduction of equations, and in symbols, which ture’s perceiveds. In The Mathematical Principles of Natu- rendered these matters ‘clearer to the eyes’.” [1] ral Philosophy (1687), Newton differentiated absolute time and space from relative time and space, restricting the latter In 1632, Oughtred published his creative and future-oriented to situated and sensate experience. To paraphrase: in respect work on circular and rectilinear slide rules, Circles of Pro- to absolute, true, and mathematical time and space, in and of portion and the Horizontal Instrument, incorporating mul- themselves, and without reference to anything external, time tidimensional logarithmic scaling into practical instruments flows uniformly and space always remains homogeneous of calculation. “All such questions in arithmetic, geometry, and immovable. Relative, apparent, and common time is any astronomy, and navigation, as depended upon simple and sensible and external measure of duration by means of mo- compound proportion, might be wrought by [the Circles of tion, while relative space is any movable measure or dimen- Proportion]; and it was the first sliding rule that was pro- sion of this absolute space; and such a measure or dimension jected for those uses, as well as that of gauging.” [2] is determined by our senses from the situation of the space with respect to bodies. Moreover, in order to not confound Isaac Newton [1642-1727] was among the many distin- human perceptions (audition and vision) with whatever is guished beneficiaries of Oughtred’s mathematical legacy, perceived (sounds and lights) Newton understood the ac- learning from him — “that very good and judicious man, tivities and phenomena and sensations to occur in a special Mr. Oughtred, a man whose judgment (if any man’s) may be place called the sensorium. “The place of the whole is the safely relyed upon” — techniques of calculating with loga- sum of the places of the parts and therefore is internal and rithms as well as the construction and uses of slide rules. in the whole body.” [4] Progress in natural philosophy re- Newton was a “mathematician to his toe tips [preeminently quired a correct understanding how the phenomena of na- a geometer]. From the first he regarded everything in his ture become present to mind, even those originating at great view with an outward eye attuned to arithmetical and geo- distances from observers. A widespread topic of the times, metrical niceties, and with an inner vision which sensed the perceptual acquaintance, [5] addressed both the biological mathematicisable under-structure in all things”. [3] Newton mechanisms of sensation and the intellectual questions of incorporated Oughtred’s ideas into the new and experimen- the nature of sensory information. Precise measurements of tal mathematical physics, and he extended them to original perceptual appearances were especially crucial for pursuing designs of specialized instruments, a circular TONOM- mathematical reasoning in research on acoustics and optics, ETER and a circular COLORIMETER, for measurements that is, “in Experiments, where Sense is Judge”. [6] and calculations pertaining to the senses: sounds and lights, 42 Journal of the Oughtred Society While he was still a young student, and close to the time of progressing in continuous proportions” (John studying Oughtred’s works, Newton wrote out “a discourse Napier, The Description … and The Construction of ye motion of strings sounding … & of ye Logarithms of … of the Wonderful Canon of Logarithms (1614, these strings, or distances of ye notes.” [7] He adopted a sys- 1619). [12] tem of musical tuning nearly identical to the one constructed by Johannes Kepler (in The Harmony of the World, 1619) Among Newton’s more innovative applications of and proceeded to analyze it systematically using numerous logarithmic thinking to the phenomena of sounds was different mathematical techniques. Newton’s musicologi- a (somewhat sketchy) design for a specially calibrated cal studies were modern and abstract exercises in canonics, circular TONOMETER (see Figure 1), a paper instrument an ancient discipline whose techniques had been developed for measuring the intervals of musical tones by logarithmic even before Euclid wrote a treatise on the subject: katatom scales. The inspiration for this device derived rather kanonoi (Sectio canonis), The Division [Cutting] of the directly from the basic concepts of William Oughtred’s Canon or monochord (Gk. kanōn, L. regula). [8] Canonics, logarithmically calibrated Circles of Proportion (see Figure essentially an applied metrical geometry of the tensioned 2). string, was ubiquitously practiced throughout the 17th cen- tury and was well known to mathematicians and musicians alike. During that time the ancient and limited arithmologi- cal techniques of expressing the magnitudes of musical in- tervals, that is, the distances of the notes, became supplanted by advances in mathematical thinking, such as logarithms, which emphasized the measurement of continuous variables and the infinite divisibility of geometrical objects. [9] [10] Newton argued that proper mathematical quantities were geometrical objects generated by motions, a point moving to a line (straight or curved), a line to a plane, a plane to a solid, and so on. Such geneses, he said, really did take place natu- rally (as in the elliptical shapes of planetary orbits) and that was what made a mathematical physics both possible and productive. His own manner of expressing those quantities did not usually invoke metrical units but compared masses with masses, distances with distances, and forces with forc- es; in other words, Newton preferred to use the language of physical ratios and proportions to the fullest. The metrical geometry of stretched strings, that is, musical canonics, was, for all its antiquity, thoroughly compatible with this mod- Figure 1 ern philosophy of nature. The whole length of a tensioned Isaac Newton’s Circular TONOMETER (redrawn after Newton, CUL, Add. Ms. 4000, fol. 109r,v) string (resounding a characteristic tone) could be cut (sev- ered, divided, partitioned) into partial lengths (each with its characteristic tone), and when all of them were considered together they made up a system or mutual attunement of self-referential components. The historical convention was to express the relationships of the various string lengths (and related tones) in a tuning system as simple number ratios de- noting their various magnitudes, that is, as the differences of the lengths or the distances (intervals) of the corresponding tones. The logarithms developed by John Napier and Henry Briggs, and incorporated into rules by William Oughtred, provided a more felicitous way to measure (numerically quantify) those distances, a logarithm (logos+arithmos) be- ing a ‘number of a ratio’ and ratios signifying geometrical magnitudes (on the general topic see [11]). “A logarithmic table is a small table by the use of which we obtain a knowledge of all geometrical Figure 2 William Oughtred’s Circles of Proportion divisions and motions in space; by a very easy (courtesy of Chris Sangwin, 2003) [13] calculation it is picked out from numbers Volume 22, Number 1, Spring, 2013 43 Newton was interested in general mathematical methods bodied a linear law of prismatic dispersion” [16] that could much more than in specific applications. So in his notebooks readily be restated as the continuous variation in the sine of he wrote out numerous equal-ratio divisions (x) of the refer- refraction relative to the sine of incidence of light through a ence quantum of an octave interval (ratio 2:1), all of them prism. The oblong visible spectrum of light is “a continual conforming to the expression (2:1)n/x. The TONOMETER, series … of lights perpetually varying” from the least to the as drawn, indicated two of those divisions in the outermost most refracted, but it also appears systematically (“in Ex- perimeter, an equal-12 and an equal-53 measurement of a periments, where Sense is Judge”) to consist of a natural and musically partitioned octave.
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