Abstract
One of the distinctive features of George Smith’s work on celestial mechanics is his emphasis on the role of what he calls “second-order phenomena” in the production of high-quality evidence. On Smith’s view, these gaps between theoretical predictions and observations can, under certain circumstances, be a source of evidence far stronger than that achievable through the hypothetico-deductive method. The practice of examining gaps between predictions and observations for the purposes of discovery and testing is commonplace in certain sciences such as seismology, and has played an important role in their development. I use the term reasoning from residuals as a general term for this practice. I think it is worth investigating examples of this set of practices from the history of science, in order to understand the different ways in which reasoning from residuals is done, under what situations it is done, and how it contributes to the growth of knowledge in certain sciences.
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Notes
- 1.
See, for example, ‘Closing the Loop’ (Smith, 2014).
- 2.
To be clear, this term is not intended to refer specifically to Smith’s view about the particular practice he identified in celestial mechanics. Rather, I intend it to encompass a diverse set of practices from a variety of different sciences, among which is the practice Smith describes.
- 3.
- 4.
There is some confusion about the dates for these articles. Many secondary sources, such as the biography of Herschel by Buttman (1970), give 1845 as their date of publication. I take the dates from Bolt (1998) to be more reliable. He lists the dates of publication as 1829 for “Physical Astronomy,” 1830 for “Light,” and 1830 for “Sound” (Bolt 1998, 417). Herschel himself, in an 1861 article for The Mathematical Monthly, gives the following as the dates of completion (not publication) of these articles: 1823 for “Physical Astronomy,” 1827 for “Light,” and 1829 for “Sound.” Throughout this chapter, I will use 1829 as the publication date for the “Physical Astronomy,” keeping in mind that it was probably written by 1823.
- 5.
See Buchwald (1989, 291) and Good (1982, 1). The article was an important resource for optical research among the younger generation of English-speaking natural philosophers in the 1830s, among them Baden Powell and William Rowan Hamilton. Powell wrote in 1832: “The publication of Sir J. Herschel’s Treatise on Light forms an epoch in the history of the science, and has given a material impulse to the study of it…” (Powell 1832, 433). Hamilton recounts in a letter that he kept the article under his pillow (along with Coleridge’s Aids to Reflection) in early 1832, the year he later discovered conical refraction (Graves 1882, 515).
- 6.
Many of the mathematical references in the article are to Lacroix’s An Elementary Treatise on Differential and Integral Calculus, which had been translated into English by Herschel and his Cambridge friends Charles Babbage and George Peacock in 1816.
- 7.
Herschel likely takes the idea of such ideal beings from the Scottish Common Sense philosopher Thomas Brown. Richard Olson (1975) has written extensively on the influence of Brown on the Preliminary Discourse, pointing out that many of the illustrative examples come from Brown’s Lectures on the Philosophy of the Human Mind (1824). The influence is already there in the earlier “Physical Astronomy.” In a discussion of the power of reasoning, and in mathematics in particular, Brown refers to “races of beings… …able to feel, in a single comprehensive thought, all those truths, of which the generations of mankind are able, by successive analysis, to discover only a few, that are, perhaps, to the great truths they contain, only as the flower which is blossoming before us, is to that infinity of future blossoms enveloped in it, with which, in ever renovated beauty it is to adorn the summers of other ages.” (Brown 1824 [vol 2], 198–9).
- 8.
This is a very brief reference to recent work on double refraction, about which Herschel gives a longer exposition in the Preliminary Discourse (Herschel 1830, 29–33). The “recent experiments” are a reference to work by Fresnel that showed that in double refraction the ordinary ray does not precisely follow Snell’s Law.
- 9.
There is a much more detailed account of the inference in Part I, pages 649–51.
- 10.
This account of Newton’s inference to the law of gravity is quite different from that given in Book 3 of the Principia. It appears to be based on a famous description of Newton’s thought leading up to the Principia that is given in the Preface to Henry Pemberton’s A View of Sir Isaac Newton’s Philosophy (1728). I have not found any references to Pemberton in Herschel’s published work, but the preface was well-known to Herschel’s contemporaries, and Herschel surely knew of it. See, e.g. Whewell (1837, 158).
- 11.
Herschel here is referring to attempts by Maskelyne to measure the attraction of Schiehallion, a Scottish mountain, and torsion pendulum experiments done by Cavendish, which could potentially decide the question, but they were limited by their use of an unreliable estimate of the mean density of the earth. Usually, in fact, these experiments are taken to be measurements of the earth’s mean density, under the assumption that the law of gravity holds exactly. See, e.g., Poynting (1894) and Bullen (1975); cf. also Allan Franklin’s chapter in this volume.
- 12.
More specifically, Herschel shows that the problem of determining the elements of a planet’s orbit from observations can be reduced to a system of nine equations in six unknowns (Herschel 1829, 664). He states: “The complication of the relations in question precludes all idea of a direct solution of the problem, and to apply them to any particular case indirect and approximative ones have been invented, but with every assistance from such simplification; and, after all the force of analysis has been exercised upon it, it still remains a very difficult problem, and one which our limits will by no means permit us to enter upon in its full extent” (1829, 665).
- 13.
Herschel provides a derivation of this formula on page 655.
- 14.
Herschel makes a slight error here. In the notation he uses, the primed letters always stand for parameters of the disturbing body, so he has switched n and n’.
- 15.
This is the case in the second example Herschel gives after this passage, on experiments Arago did on a magnetic needle suspended over a plate of copper.
- 16.
Good (1982, 1987) and Cobb (2012a, b) show how some of the methodological views Herschel presents in the Preliminary Discourse originate from his work on, respectively, optics and electromagnetism. I do not wish to underplay the role of Herschel’s work on sciences other than physical astronomy in the formation of his methodological views. What I do want to emphasize, however, is that many of the ideas in the Preliminary Discourse originate in his earlier work on the “Physical Astronomy”.
- 17.
The terminology here calls to mind Herschel’s reference to a “scale of generalization” in the Introduction to the “Physical Astronomy”: Natural philosophers are to “ascend as high as the imperfection of human means of observation, and the limited powers of the human intellect will allow us in the scale of generalization” (Herschel 1829, 647).
- 18.
My repeated use of the terms “details in the solar system” and “differences they make” in this chapter are, as the reader might surmise, a conscious attempt to draw connections to Smith (2014).
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Acknowledgment
The research for this chapter was supported by the Ministry of Education, Singapore, under its Academic Research Fund Tier 1 Grant, No. RG156/18-(NS).
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Miyake, T. (2023). ‘To Witness Facts with the Eyes of Reason’: Herschel on Physical Astronomy and the Method of Residual Phenomena. In: Stan, M., Smeenk, C. (eds) Theory, Evidence, Data: Themes from George E. Smith. Boston Studies in the Philosophy and History of Science, vol 343. Springer, Cham. https://doi.org/10.1007/978-3-031-41041-3_2
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