Volume 22, Number 1, Spring, 2013 41

“In Experiments, where Sense is Judge” ’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 “ 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 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 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 and Henry Briggs, and incorporated into rules by William Oughtred, provided a more felicitous way to measure (numerically quantify) those distances, a (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. Specific values signifying the orderly succession of differentiated colors. Newton’s second sizes of the intervals contained within that octave were ob- major paper on light and colors made most explicit the anal- tainable by calculation with roots and powers. For example, ogy between a prism’s action in severing sunlight into its the size of an interval ratio 3:2 (ut-sol in the diagram), which “several more eminent” colorific constituents and that of is one of the smaller parts of a whole octave, is approxi- canonically cutting a sonorous string into a scale of distinct mately (2:1)31/53 or (53√2:1)31, that is, it measures about 31 tones. parts of the whole divided into 53 equal-ratio parts. More exactly, the calculation … (53/log2)(log3−log2) = 31.00301254… “Whence if the rayes [of light] which come pro- … gives the number of the interval ratio 3:2 as its size or rel- miscuously from the Sunn, be refracted by a ative magnitude measured on an equal-ratio scale of 53. The Prism, … those of severall sorts being variously measure of the same interval (3:2) on an equal-ratio scale of refracted must go to several places on an oppo- 12 is approximately (2:1)7/12, or calculated … site paper or wall & so parted, exhibit every one (12/log2)(log3−log2) = 7.01955001… . of their owne colours, which they could not do while blended together. And because refraction The accompanying JOS Plus essay provides an extensive onely severs them, & changes not the bigness or discussion of Newton’s tuning system and his tonometric strength of the ray, thence it is, that after they are constructions and calibrations. once well severed, refraction cannot make any further changes in their colour. … And possibly The principal idea of musical canonics, measuring the sec- colour may be distinguished into its principal De- tioned parts of a tensioned string (= geometrical line), was grees, Red, Orange, Yellow, Green, Blew, Indigo, incorporated into Newton’s early research on optics. His and deep violett, on the same ground that Sound central concern was to correctly define the nature of pris- within an eighth is graduated into tones.” [17 matic refraction and thereby explain the apparent shape of the spectrum as well as the particular distribution of its vivid and intense colors. In his first published paper on the subject (New Theory about Light and Colours, 1672) he wrote, “I became surprised to see [the colors dispersed] in an oblong form; which, according to the received laws of refraction, I expected should have been circular.” Newton proceeded empirically and analyzed the phenomenon geometrically. Diligently measuring the perceived extents and boundaries of the various colors distributed along the spectral continu- um, he described its elongated shape in a distinctly Euclidian idiom: “taking the whole length of the visible spectrum XY bisected at H and trisected at G and I, and XI trisected at E, and taking KY to be the fifth and MY the eighth part of the whole” [14] (see Figure 3).

Furthermore, Newton noticed that the geometry of the vis- ible spectrum coincided exactly with the metrical geometry of a particular arrangement of musical tones, such that the “several more eminent species of prismatick Colours, red, orange, yellow, green, blue, indigo, violet [are] in propor- tion as the Differences of the Lengths of a Monochord which sound the Tones in an Eight [‘octave’], sol, la, fa, sol, la, mi, fa, sol, … [that is, they are] proportional to the Numbers 1, 8/9, 5/6, 3/4, 2/3, 3/5, 9/16, 1/2, which express the Lengths of a Monochord sounding the Notes in an Eighth.” [15] The Figure 3 musically measured distribution of spectral colors “em- Several Sorts of the Sun’s Promiscuous Rays 44 Journal of the Oughtred Society

Many years later, Newton diagrammed a circular COLOR- colorimetric rule, stated as a problem to be solved, or a cal- IMETER (see Figure 4). The new instrument was designed culation to be made, by means of the COLORIMETER. He not only for continuing to measure the dimensions of the asked, “what will be the hue and saturation of a prismatic separate colors in the prismatic spectrum, but for the uni- color compounded of 10 parts red, 6 parts orange, 5 parts versal calculation and prediction of their mixtures or com- yellow, 3 parts green, 2 parts blue, 1 part indigo, and 1 part pounds, where “the Quantity and Quality of each constituent violet?” A complete solution to that problem, indeed for any was given.” The device encapsulated not only the precise other prismatic color mixture (following Newton’s specifi- metrical geometry of colors expressed in the proportion- cations for the geometrical construction and musical metri- alities of Newton’s earlier musical calibrations, but it also cal calibration of the COLORIMETER), is explained in the accommodated a law operating within the sensorium and accompanying JOS Plus essay. governing the “powers or dispositions of the various rays of light to stir up sensations of this or that color”. The circular COLORIMETER was, for Newton, not only imbued with the cachet of ancient wisdom — Ariadne’s “With the Center O and Radius OD describe a Cir- thread acknowledged by way of its noble canonical design cle ADF, and distinguish its Circumference into and echoes of the long-standing harmonic principles of sys- seven Parts DE, EF, FG, GA, AB, BC, CD, pro- tem and symmetry — but it also promised to be a practical portional to the seven Musical Tones or Intervals instrument in entirely modern contexts. of the eight Sounds, Sol, la, fa, sol, la, mi, fa, sol, contained in an eight, that is proportional to the “This Rule I conceive accurate enough for prac- Number 1/9, 1/16, 1/10, 1/9, 1/10, 1/16, 1/9. Let tice, …; and the truth of it may be sufficiently the first Part DE represent a red Colour, the sec- proved to Sense …” ond EF orange, the third FG yellow, the fourth GA “All the Colours in the Universe which are made green, the fifth AB blue, the sixth BC indigo, and by Light, … are either the Colours of homogeneal the seventh CD violet. And conceive that these are Lights, or compounded of these, according to the all the Colours of uncompounded Light gradually Rule …” passing into one another, as they do when made by “… Thus it is by the computation: and they that Prisms; the Circumference representing the whole please to view the Colours made by a Prism will Series of Colours from one end of the Sun’s co- find it also in Nature.” [19] loured Image to the other.” [18] Newton’s confidence in his rule was indeed justified. The theory, experimentally derived and confirmed, embedded in the musically calibrated color-mixing diagram became “the foundation of color science as now accepted, [and it was laid down] 300 years ago by the great mathematician and physi- cal scientist Isaac Newton” [20], although his original and sophisticated techniques of canonical measurement have long since been neglected or forgotten.

Note

1. JOS Plus indicates that supplemental material for this article is available at www.oughtred.org. The material consists of an essay on the technical details of Isaac Newton’s TONOMETER and COLORIMETER, including discussion of their relationships, historical contexts, and philosophical rationales.

References

1. Stedall, Jacqueline Anne, Ariadne’s Thread: The Life and Times of Oughtred’s Clavis, Annals of Science 57:1, 2000, pp. 33-34. Figure 4 2. A New and General Biographical Dictionary…new edi- Isaac Newton’s Circular COLORIMETER tion in 12 vols., London, 1794; quoted in Nature Vol. (color enhanced) 40, 1889, p. 458; re-quoted IN Cajori, Florian, A History of the Logarithmic and Allied Instruments Newton provided one sample case of the operation of this [1910] and On the History of Gunter’s Scale and the Volume 22, Number 1, Spring, 2013 45

Slide Rule during the Seventeenth Century [1920], The structio, Edinburgh: Andrew Hart, 1619] (with an Ap- Astragal Press, Mendham, New Jersey, 1994, p. 13. pendix On the Construction of another and better kind 3. Whiteside, Derek Thomas, Newton the Mathematician, IN of Logarithms, namely one in which the logarithm of I. Bernard Cohen and Richard S. Westfall (eds.), New- unity is 0: along with Some Remarks by the Learned ton: Texts, Backgrounds, Commentaries, W. W. Norton Henry Briggs on the relations of Logarithms and their (Critical Edition), New York & London, 1995, p. 412. natural numbers to each other, when the Logarithm of 4. Newton, Isaac, The Principia: Mathematical Principles unity is made 0). Translated from the Latin by William of Natural Philosophy, A New Translation, I. Bernard Rae Macdonald, with notes and A Catalogue of the Vari- Cohen and Anne Whitman (trans.), assisted by Julia ous Editions of Napier’s Works, 1888, and reprinted for Budenz, University of California Press, Berkeley, 1999, Dawsons of Pall Mall, London, 1966. pp. 408-409. 13. http://web.mat.bham.ac.uk/C.J.Sangwin/Sliderules/ocir- 5. Yolton, John, Perceptual Acquaintance from Descartes cle.jpg to Reid, University of Minnesota Press, Minneapolis, 14. Newton, Isaac, The Optical Papers of Isaac Newton. Vol 1984. I: The Optical Lectures, 1670-1672, Alan E. Shapiro 6. Newton, Isaac, Opticks: or a Treatise of the Reflections, (ed.), Cambridge University Press, Cambridge, 1984, Refractions, Inflections, and Colours of Light [London, pp. 548-549. 1704], Dover, New York, 1979, p. 123. 15. Newton, Isaac, Opticks, [1704]1979, pp. 295, 305. 7. Newton, Isaac, [manuscript] Of Musick, Cambridge Uni- 16. Shapiro, Alan E., Newton’s ‘Achromatic’ Dispersion versity Library, Add. Ms. 4000, c. 1665. Law: Theoretical Background and Experimental Evi- 8. Creese, David E., The Monochord in Ancient Greek dence, Archive for History of Exact Sciences 21, 1979, Harmonic Science, Cambridge University Press, Cam- pp. 91-128. bridge, 2010. 17. Newton. Isaac, An Hypothesis Explaining the Properties 9. Roche, John J., The Mathematics of Measurement: A of Light, Discoursed of in my Several Papers [1675], Critical History, The Athlone Press, London, 1998. IN I. Bernard Cohen and Robert E. Schofield (eds.), 10. Neal, Katherine, From Discrete to Continuous: The Isaac Newton’s Papers and Letters on Natural Philoso- Broadening of Number Concepts in Early Modern Eng- phy and Related Documents, Harvard University Press, land, Kluwer Academic Publishers, Boston, 2002. Cambridge, MA, 1958 (2nd ed., 1978), pp. 177-199. 11. Kuehn, Klaus, and Rodger Shepherd, Calculating with 18. Newton, Isaac, Opticks, [1704]1979, pp.154-155. Tones: The Logarithmic Logic of Music, The Oughtred 19. Newton, Isaac, Opticks, [1704]1979, pp. 158, 164. Society, Pleasanton, CA, 2009. 20. Williamson, Samuel J., and Herman Z. Cummins, Light 12. Napier, John, The Construction of the Wonderful Canon and Color in Nature and Art, John Wiley & Sons, New of Logarithms [Mirifici Logarithmorum Canonis Con- York, 1983, p. 20.

Suggestions for Dating pre-1920 Faber-Castell Slide Rules: an Update

Trevor Catlow I would like to bring to our readers’ attention a few miscel- cases in other documents. Therefore, clearly when Faber de- laneous findings regarding Faber slide rules in the pre-1920 scribed an addition to its range as “new”, the addition could time period, which have come to my attention since my ear- have actually appeared six or more years beforehand. I doubt lier article [1] was published. As in [1], I will use the name that Faber is alone in using such confusing language in its Faber to refer to the company, being mindful of the fact that sales literature, but this does mean that several of my tenta- the company name changed from A.W.Faber to A.W.Faber- tive dates have become less certain than I originally believed Castell during the period in question. and may need to be adjusted backwards in time. I leave the details of actual cases as an exercise for the interested reader. The first of these findings concerns the language style used by the company in its literature. In my original article, I re- My second alert relates to slide rule boxes. When I wrote ferred to a Faber publication dated 1901 [2], which contains my earlier article I did notice that one or two of the Faber an addendum referring to recent additions to the product slide rule boxes in my collection had curiously blackened la- range. In my naivety I deduced from this that the additions bels, but I thought these were aberrations, perhaps caused by had been made just a year or so before 1901. I have since dis- unsuitable storage conditions. Now that my collection has covered that a 1906 version of the publication [3] contains grown, I have deduced that these blackened labels resulted an addendum in virtually the same form as the 1901 version, from a feature of the Faber production line. I have noticed also describing the additions as “new”. I have found similar that, in my collection, blackened labels appear on all the