ESHS 2018: Saturday 15 September, 16:00-18:00

With what prisms can ’s optical experiments be reproduced? Yoshimi TAKUWA* [email protected]

Good afternoon everyone. My name is Yoshimi TAKUWA from the Tokyo Institute of Technology. I have a background in laboratory laser spectroscopy, and now I conduct estimations and replications of ’s optical experiments. Newton often omitted some experimental information in his books and manuscripts, yet I believe that estimations and replications can enable us to pick up on the subtext.

Introduction For many years, historians tried to search for authentic prisms of Isaac Newton. For example, about sixty years ago, I. Bernard Cohen said as follows:

As to the question of whether Newton ever made measurements of flint glass prisms, we can only say that he seems to have possessed a flint glass prism, because, as we have seen, the specimen of the British Museum is of flint glass. Its existence alone, however, does not prove that Newton ever made any measurements, but only that he had the opportunity to do it. What a pity that he did not do many!1

Cohen was disappointed because all the data that Newton left behind seemed to be those of crown glass. However, a surviving so-called “Newton’s prism” was of flint glass. Cohen got to know that there were three prisms called “Newton’s” in Treviso in Italy, and with the help of Vasco Ronchi, he found that one of them was of crown glass. However, he was disappointed again because the dimensions of one in Treviso were different from Newton’s own data.

“Newton’s prisms” At this moment there are six so-called “Newton’s prisms” known to the world: one in the British Museum donated by Newton’s family in 1927, two in the Whipple Museum in Cambridge, one being donated by Trinity College and the other by the Cavendish Laboratory. The three prisms in the UK were examined by A. A. Mills in 1981.2 The other three in the Treviso Museum were donated in 1937 by a doctor from Treviso and were examined by Vasco Ronchi in 1957.3

* Institute for Liberal Arts / School of Environment and Society, Tokyo Institute of Technology. 1 I. B. Cohen, “I prismi del Newton e i prismi dell’Algarotti,” Publ. dell’Istituto Nazionale di Ottica, 4 (1957): 1-11. 2 A. A. Mills, “Newton’s Prisms and His Experiments on the Spectrum,” Notes and Records of the Royal Society of London, 36(1981): 13-36. 3 Vasco Ronchi, “I ‘prismi del Newton’ del Museo Civico di Treviso,” Publ. dell’Istituto Nazionale di Ottica, 4 (1957): 12-28.

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Now, I have to explain to you why “Newton’s prisms” exist in Italy. These prisms are thought to have been preserved by the family of Francesco Algarotti. Algarotti was the author of the popular book, Il newtonianismo per le dame (1737), and he visited England in 1736. Although Newton was already dead, there is a possibility that Catherine Bartron Conduitt, Newton’s niece, gave prisms as a present to the Italian scholar who struggled to spread Newton’s theory. Here is the data of surviving prisms and prisms mentioned by Newton in his works. As we see in the tables, there is no surviving prism that fits Newton’s data. The most problematic thing is that many of the surviving prisms have high refraction indexes and high dispersion (i.e. small ν퐷) but all the written prisms have low refraction indexes and low dispersion (i.e. big ν퐷). Today we divide the types 4 of glass with the border of Abbe number ν퐷 around 50. The glass which has a high refraction index and high dispersion is called “flint glass” and it mainly contains lead oxide (PbO). This was relatively new technology which emerged in the middle of the seventeenth century. The glass which has low refraction index and low dispersion is called “crown glass” and it mainly contains sodium carbonate

(Na2CO3), and it was traditional technology before the eighteenth century. Because of these differences, now we can understand the reason why Cohen and other historians cared about whether the prisms were of flint or crown.

Table 1. Optical characteristics of six surviving “Newton’s prisms”

location material angles refraction dispersion

British Museum flint glass 59°30′ 60° 10′ 60° 25′ n퐷 = 1.5898 ν퐷 = 40.0

Whipple Museum flint glass 59°00′ 60°20′ 60°35′ n퐷 = 1.5792 ν퐷 = 37.4

flint glass 59°15′ 59°20′ 61°20′ n퐷 = 1.5805 ν퐷 = 35.8

Treviso Museum crown glass 58°30′ 60°30′ 61°00′ n퐷 = 1.5149 ν퐷 = 57.2 quartz crystal 59°30′ 60°00′ 60°30′ quartz crystal 52°00′ 55°00′ 73°00′ * Based on the examination by A. A. Mills in 1981 and by Vasco Ronchi in 1957.

Table 2. Optical characteristics of prisms mentioned by Newton

Newton’s work material angles refraction dispersion

“New Theory” (crown) glass 63°12′ n푔푒푛푒푟푎푙 = 1.55 (ν퐷 = 55 − 70) n = 1.5412 Lectiones opticae (crown) glass 63°12′ 표푟푎푛푔푒−푦푒푙푙표푤 (ν퐷 = 55 − 70) (crown) glass c. 60° c. 60° c. 60° (crown) glass 62°30′ n표푟푎푛푔푒−푦푒푙푙표푤 = 1.5440 (ν퐷 = 55 − 70) (crown) glass 63°30′ (crown) glass 64°00′ * The items in ( ) are estimated data by the author.

4 푛퐷−1 The Abbe number is calculated as: 휈퐷 = . 푛퐹−푛퐶 2

ESHS 2018: Saturday 15 September, 16:00-18:00

The problem of “flint or crown” is more relevant than just the concerns of Newton enthusiasts who search for authentic “Newton’s prisms.” I will explain two critical issues based on the assumption that “Newton used not only crown glass but also flint glass” by doing estimations and replications. In my estimations and replications, I asked OHARA Inc. to make new prisms of flint and crown glass where n퐷 and ν퐷 are almost the same as the surviving prisms.

First issue: the demonstration of the immutability of colors The point of Newton’s optical theory is that color is a property of light and it cannot be changed by refraction or other external factors. His new theory went against the modification theory of light, which was dominant from the times of ancient Greece to the seventeenth century. In this traditional understanding, colors are generated when light rays are modified by an external cause. Thus the colors of rays can be changed by refraction or another cause. To overcome the modification theory, Newton had to show that colors are immutable by refractions. On the vignette in the first page of the French second edition of Opticks (1722), Newton summarized his theory in a motto: Nec variat lux fracta colorem (refracted light does not change color).5 To demonstrate the immutability of colors, Newton utilized the experiment with two prisms, or so- called experimentum crucis.6 Roughly speaking, Newton’s two-prism experiment was conducted as follows: Newton set up two prisms and two boards with one small hole in each. The first prism was rotatable, and the second prism was fixed, so that by rotating the first prism he could project different parts of the spectrum through the hole. If the final image projected on the wall kept its color without any changes, the color of the rays was immutable. Although Newton wrote that the final image kept its color in his early manuscripts (c. 1666)7, in later years he realized that a certain amount of other colors emerged in the image. For example, I will show you the final image of the replication of the two-prism experiment. At the edges of the yellow image, orange and green are visible. This is because the color separation in the first prism was not enough and adjoining colors were mixed in the image. The prismatic spectrum is seen as a sequence of the circular image corresponding to the shape of the small hole on the board. Newton realized that if he could make the circular image narrower, the mixture of adjoining colors would diminish. I made a calculation and determined how narrow the spectrum should be to avoid the mixture of other colors: The length of the spectrum should be 14 times longer than the width. This ratio of length

5 Isaac Newton, Traité d’optique, 2nd ed., translated by Pierre Coste (Paris, 1722), p. 1. 6 Newton used the term experimentum crucis only in his “New Theory” but not in Opticks and Lectiones opticae. Isaac Newton, “A Letter of Mr. Isaac Newton, Professor of the Mathematicks in the University of Cambridge; containing his New Theory about Light and Colours: sent by the Author to the Publisher from Cambridge, Febr. 6. 1671/72; in order to be communicated to the R. Society,” Philosophical Transactions, 6 (1671/2): 3078-79. Isaac Newton, Opticks: Or, a Treatise of the Reflections, Refractions, Inflections and Colours of Light (London, 1704). Isaac Newton, Isaaci Newtoni, Eq. Aur. in Academia Cantabrigiensi Matheseos olim Professoris Lucasiani Lectiones Opticae (London, 1729). 7 “Of Colours.” MS. Add. 3975 (Cambridge University Library), ff. 1-21.

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to width “14:1” fits Newton’s own observation. When the ratio was “9:1” he still observed some gradation of other colors (c. 1670)8, however when the ratio reached “40:1” he confirmed that there was no visible change (1704)9. To realize the ratio such as “40:1”, Newton added a lens to the setup of the two-prism experiment so that he could narrow the light beams. With the simple setup of the experiment without a lens, Newton reported the spectrum’s ratio “4:1” or at most “5:1”.10 With this ratio, it is impossible to show the immutability of colors and it seems that a lens is indispensable in the demonstration. However, a strange thing happened: Although Newton admitted that the demonstration of the immutability of colors could not be done without a lens, in the middle of the eighteenth century, Newtonian writers, such as William Whiston, J. T. Desaguliers, Willem ‘s Gravesande, Henry Pemberton, Voltaire and Francesco Algarotti, explained that the demonstration can be done with the simple setup of the two-prism experiment (i.e. without a lens).11 At the beginning I thought that the Newtonian writers misunderstood the two-prism experiment. But just to be sure, I checked whether it is possible to demonstrate the immutability of colors with the simple setup of the experiment without a lens. There are five probable factors that may affect the ratio of the spectrum: the quality of the prism, the angle of the prism, the distance from the prism to the screen, and the widths of the first and second holes on the boards. I have already discussed these five factors in the last ESHS Conference in 2016, so I will omit details. My conclusion is that only two factors, the dispersion and the angle of the prism, affect the ratio. But others have little effect on the image. Therefore, the only way to separate the colors well is using a prism of high dispersion with a broad angle. Here is the result of my estimation and replication when the prism is of flint glass (S-FTM16) with the angle of 70°. The ratio of the length to the width of the spectrum reaches almost “14:1”. However, there is no record that Newtonian writers tried the experiments with a flint glass prism having a broad angle. There are still two potential explanations: Newtonian writers accomplished the experimental condition that even Newton could not have reached, or they just misunderstood the experiment.

Second issue: the division of a spectrum into seven colors Another point of Newton’s optical theory is that colors become mathematical. The famous example of the mathematical colors is the seven colors in the spectrum. Newton divided the prismatic spectrum into seven colors, making an analogy between the spectrum and the seven-note musical scale.12

8 Newton, op. cit. (1729), p. 167. 9 Newton, op. cit. (1704), pp. 88-89. 10 Newton, op. cit. (1704), pp. 18-22. 11 See Yoshimi Takuwa, “The Historical Transformation of Newton’s experimentum crucis: Pursuit of the Demonstration of Color Immutability,” Historia Scientiarum, 23 (2013): 113-140. It can be viewed online at: http://researchmap.jp/takuwa.y/?lang=english 12 Newton introduced the observation of the seven-color spectrum three times: in a letter called “An Hypothesis Explaining the Properties of Light” (1675), in Lectiones opticae and in Opticks. See The Correspondence of Isaac Newton, eds., H. W. Turnbull, J. F. Scott, A. Rupert Hall, and Laura Tilling (Cambridge: Cambridge University Press,

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Newton claimed that each color band of a spectrum corresponds to musical chord lengths: 1, 8/9, 5/6, 3/4, 2/3, 3/5, 9/16, 1/2. This proportion of numbers indicates “re, mi, fa, sol, la, si, do, re” of the dorian mode scale. Although Newton insisted that this theory of seven-color spectrum agrees with the phenomena very well, I doubt that this quantitative regularity is replicable, especially when the prism is of flint glass. I estimated the position of the 13 spectral lines in the spectrum made by a typical flint and crown glass (S-FTM16 and S-NSL3). I calculated them with the condition that the extreme red was the A’ line (768 nm) and extreme violet was the h line (405 nm). The positions of the lines separated at a distance of up to 4% of the whole length of the spectrum. I consider this displacement to be rather large, because each color band occupies more or less 10% of the whole length. As a result, it is apparent that Newton’s proportions of the color bands in a spectrum are not replicable when the prism is made of flint glass, and consequently, his theorem was wrong. What is more, this 4% difference indicates that Newton’s statement that his musical scale (1, 8/9, 5/6, 3/4, 2/3, 3/5, 9/16, 1/2) should be selected based on the results of this experiment, is incorrect. This is because other types of scales, such as the equal temperament scale, give similar proportions to Newton’s scale, and the differences between them are only 2% of the whole length. Newton selected the scale “not only because it agrees with the phenomena very well, but also because it perhaps involves something related to the harmonies of colors,” 13 however, the harmonies that Newton thought to have found did not appear with flint glass.

As I have explained so far, flint glass may give experimental results that differ from those of crown glass. If Newton had used some flint glass prisms, he might have noticed the defects of his strategy to convince the readers. However, he did not notice them, and furthermore, he often mentioned the dispersion of glass as if it was always equal. Therefore, I assume that he solely used crown glass prisms and not flint ones in his experiments.

Rough examination Recently, I happened to come upon another prism called “Newton’s” in the Wren Library at Trinity College. It was donated by an alumnus in the 1940s and has probably never received any modern examinations. Fortunately, the Library permitted me to cast low power laser beams on the prism. So I thought up a simple examination that gives rough optical characteristics of prisms without touching or moving them. If a prism has the angles of approximately 60°, this simple examination may enable us to distinguish flint glass from crown glass. The examination is conducted as follows: keep the beam of the laser pointer fixed horizontally and cast it through a laying prism on the desk. The light beam is refracted twice and seems to go out with a certain emergence angle. However if the glass has high

1959-77), 7 vols., vol. 1, pp. 376-377. Newton, op. cit. (1729), pp. 239-247. Newton, op. cit. (1704), pp. 91-96. 13 Newton, op. cit. (1729), p. 246.

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refraction index such as flint glass, the beam does not exit. Here is a table that shows the emergence angles of a light beam when it receives a certain amount of refractions. Refraction indexes depend on both types of glass and wavelengths of light. As you see in the table, when the glass is a typical crown glass (S-NSL3), visible light beams can go out. However, when the glass is a typical flint glass (S- FTM16), visible light beams cannot exit but are wholly reflected inside. When a prism has the angles of approximately 60°, the boundary of such difference exists at the refraction index of about 1.55. Because of this characteristic, I assume that we can distinguish flint glass from crown glass by just casting the laser light on the historical prism. At this moment I have no other idea to distinguish flint from crown easily. If some of you know other ways, please let me know after my talk. First, I tested this rough examination on my modern glass prisms of flint and crown. Then, I tested the three historical prisms in the British Museum and in the Whipple Museum. I could confirm that this rough examination works to some extent. Finally, I tested the relatively unknown prism in the Wren Library. My judgement is that it is of flint glass. So, if Cohen or other Newton enthusiasts know this result, they may be disappointed again.

Table 3. Optical characteristics of seven surviving “Newton’s prisms”

location material angles refraction dispersion

British Museum flint glass 59°30′ 60° 10′ 60° 25′ n퐷 = 1.5898 ν퐷 = 40.0

Whipple Museum flint glass 59°00′ 60°20′ 60°35′ n퐷 = 1.5792 ν퐷 = 37.4

flint glass 59°15′ 59°20′ 61°20′ n퐷 = 1.5805 ν퐷 = 35.8

Treviso Museum crown glass 58°30′ 60°30′ 61°00′ n퐷 = 1.5149 ν퐷 = 57.2 quartz crystal 59°30′ 60°00′ 60°30′ quartz crystal 52°00′ 55°00′ 73°00′

Wren Library flint glass 58°00′ 61°00′ 61°00′ (n퐷~ 1.6) (ν퐷 ~ 40)

Prospects of research

My estimations and replications are based on the data of modern glass where n퐷 and ν퐷 are the same as those of “Newton’s prisms”. If the refraction indexes of Newton’s prism with extreme colors

(such as n퐴′ and nh) are measured, the result of my estimations would be more precise. Many decades have passed since the examination by Ronchi and Mills. Furthermore, the prism in the Wren Library has never received any examination. I hope that the museums will accept me examining them for further measurements. I have a plan to ask ATAGO Co., Ltd. to make a custom-made Abbe refractometer which can measure the refraction indexes for a wide range of wavelengths without any damage to the historical prisms. I would appreciate it if some of you could give me advice on examining historical prisms.

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