Abstract
The late nineteenth century gradually witnessed a liberalization of the kinds of mathematical object and forms of mathematical reasoning permissible in physical argumentation. The construction of theories of units illustrates the slow and difficult spread of new “algebraic” modes of mathematical intelligibility, developed by leading mathematicians from the 1830s onwards, into elementary arithmetical pedagogy, experimental physics, and fields of physical practice like telegraphic engineering. A watershed event in this process was a clash that took place during 1878 between J. D. Everett and James Thomson over the meaning and algebraic manipulation of dimensional formulae. This precipitated the emergence of rival “Maxwellian” and “Thomsonian” approaches towards interpreting and applying “dimensional” equations, which expressed the relationship between derived and fundamental units in an absolute system of measurement. What at first looks like a dispute over a seemingly esoteric mathematical tool for unit conversion turns out to concern Everett’s break with arithmetical algebra in the representation and manipulation of physical quantities. This move prompted a vigorous rebuttal from Thomsonian defenders of an orthodox “arithmetical empiricism” on epistemological, semantic, or pedagogical grounds. Their resistance in Victorian Britain to a shift in mathematical intelligibility is suggestive of the difficult birth of theoretical physics, in which the intermediate steps of a mathematical argument need have no direct physical meaning.
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Source: Wikimedia Commons
Source: Collected Papers in Physics and Engineering by James Thomson, D.Sc., LL.D, F.R.S., selected and arranged by Sir Joseph Larmor and James Thomson, M.A. (Cambridge: Cambridge University Press, 1912), ii
Source: Thomson 1880, 110
Source: Sundell 1882, 94
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Notes
For several centuries prior, quantitative experimental relations were expressed algebraically in abbreviated ratio equations as proportionalities. In 1788, Coulomb could articulate the inverse square variation of electrostatic force with distance without needing to define a unit of electrical “quantity.” “The repulsion between the two small similarly electrified spheres,” he wrote, “varies inversely as the square of the distance apart of their centers” (quoted in Roche 1998, p. 140).
For a longue durée history of these extensions, see Corry (2015).
I have adopted the term “mathematical intelligibility” rather than “mathematical truth” because it seemed to me that the conceptual difficulties discussed in this paper, and the debates to which they gave rise, revolved primarily around the specific physical basis of mathematical meaning. This is particularly apparent in Sect. 6, where semantic issues in arithmetic come to the fore. Cf. Richards (1980), esp. pp. 344–6, 360–4, who relates mathematicians’ conceptions of mathematical truth to a unitary norm of truth in the early Victorian period. Future work might probe, both conceptually and historically, the distinction between intelligibility and truth, which seems to me to map neatly onto de Morgan’s distinction between algebra as an art and algebra as a science (see Sect. 2).
The epistemology of the modern quantity calculus relies upon a clear distinction between quantity and magnitude, which was spelled out clearly by Bertrand Russell around the turn of the twentieth century. The results of mathematical operations among magnitudes are established by fiat using empirical relations between physical quantities as a guide. See de Courtenay (2015), pp. 63–4 and Darrigol (2003, pp. 559–61). Note, however, that the modern usage of the terminology is far from uniform: what Kyburg (1997, p. 381) means by “quantity,” for instance, corresponds roughly to “magnitude” in the present sense. Likewise, his use of “magnitude” refers to a concrete “quantity” such as (to use his own examples) 6.5 feet or 25 °C.
To be fully explicit, by “Maxwellian,” here I mean a conception of the relationship between mathematical and physical intelligibility derived from Maxwell and instantiated through some form of physical algebra of concrete quantities. This definition clearly suits the present focus on novel mathematical practices associated with measurement and metrology, but, as such, inevitably raises the question of what relationship it bears to the senses given to the term by Jed Buchwald and Bruce Hunt. In simple terms, they took it to refer to extensions of Maxwell’s field-theoretic approach to electrodynamics as laid out in the Treatise, but diverged in fine albeit substantive ways with regard to Maxwellians’ specific physico-mathematical commitments. Buchwald (1985, pp. 22–3, 60–4, cf. pp. 73–4) primarily (but not exclusively) based his conception upon a group of Cambridge-trained theorists committed to Hamilton’s principle of least action as their central tool, whereas Hunt (2005 [1991]) cached out the term as it was used by Oliver Lodge, George Fitzgerald, and Oliver Heaviside, three self-identifying non-Cambridge “Maxwellians.” [For a full discussion and Buchwald’s most recent position, see Buchwald and Hong (2003, pp. 177–80).] Since a proper response warrants much more detailed development and analysis than can be given in this paper, I shall simply note that recent work by myself (Mitchell 2017a, §5, 2017b, esp. pp. 89–92) and Sybil de Clark (2016, 2017, pp. 312–19) shows that the connecting thread between my notion and theirs lies in theoretical electrical metrology. See also Sect. 8.
For the theoretical development of an absolute system of units in relation to the BAAS Committee on Electrical Standards, see Mitchell (Mitchell 2017a, §§2–3), Smith (1998, chapter 13) and Smith and Wise (1989, pp. 684–92). The Committee was set up 3 years after the failure of the first Atlantic Telegraph cable in 1858. On this, in addition to the works just cited, see especially Hunt (1994).
In particular, they aimed to convey the experimental foundations of the empirical laws that were used to define the absolute electrostatic and electromagnetic systems of electrical units, and, owing to the importance of the measurement of electrical resistance to telegraphic practice, explain how this quantity (in absolute electromagnetic units) could be measured and represented in terms of the unit of velocity. See Mitchell (2017a, pp. 72–5, 2017b, §4), and Hunt (2015, pp. 319–28).
If these procedures were those employed by Maxwell to derive the dimensional formula of the coefficient of gaseous friction, µ, in a letter to Thomson written in 1865, then they are exactly those we would sensibly infer. Maxwell defined µ as the tangential pressure per unit area, due to a gas moving at unit velocity, at unit distance from a fixed plane, and then performed the following calculation:
\( \displaystyle\frac{{\displaystyle\frac{\hbox{Force}}{\hbox{area}}}}{{\displaystyle\frac{\hbox{velocity}}{\hbox{distance}}}} = \frac{{\displaystyle\frac{ML}{{L^{2} T^{2} }}}}{{\displaystyle\frac{L}{TL}}} = \displaystyle\frac{M}{LT} \). See Maxwell (1990 [1865]), p. 218.
De Morgan also used the terms “technical” and “logical” algebra to capture this distinction (Koppelman 1971, pp. 218–9; Richards 1980, pp. 354–5). Although his position underwent substantial shifts, he neither embraced a purely formal approach nor limited algebra to a generalisation of arithmetic (Pycior 1983). A fully formal conception of algebra that entirely severed the connection between physical things and processes on the one hand, and their symbolic representation on the other, went too far even for the architects of symbolic algebra. See, for example, Richards (1980, 1987, pp. 12–17, 21–23).
A number in front of the letter “w” for “wrangler” indicates placement in the Cambridge mathematical tripos, followed by the year of examination.
Dodgson’s dismissive attitude to symbolic algebra is described in Pycior (1989, pp. 124–45, esp. pp. 137–45). The instance of dividing a loaf by a knife is mentioned on page 140.
The Electrical Standards Committee was reconvened in 1880, a year in advance of the 1881 Congrès International des Electriciens in Paris (Anonymous 1880, p. lx).
The notion of a “North British” science of energy is due to Crosbie Smith (1998, esp. pp. 1–3 and chapter 9).
Adams and Stewart were appointed to the Committee in 1873 following the death of the Edinburgh engineer and professor Macquorn Rankine at the end of 1872 (Anonymous 1873, p. lvii).
A purely numerical “dimensional algebra” would have taken the same form if the symbols stood for ratios between different units of the same kind. As I describe in Sect. 8, however, this would have required setting up the derivation quite differently, and in any case, there is no explicit suggestion in the Illustrations that L, M, T, and so on, ought to be conceived as ratios of like quantities.
Kuhn (2012 [1962], pp. 5, 46–7, quotation at p. 47). See also Kuhn (1977, pp. xviii–xix). In his reply to critics (“Postscript–1969”), Kuhn (2012 [1962], pp. 182–3, 186–90) emphasized how the articulation of symbolic generalisations—illustrated by the mathematical adaptation of Newton’s second law to a given physical situation—mediated this process. The role of exemplars (and pedagogy more generally) in Kuhn’s thinking has been subjected to critical scrutiny and developed much further in Kaiser (2005, pp. 2–3), Warwick (2003, pp. 172–3, 229–31), and Warwick and Kaiser (2005, pp. 394–7), but mostly in terms of how students or practitioners acquire the tacit skills required to define and solve puzzles, rather than how they come to assign meaning to physical concepts, which is what primarily interests us here.
This task was further confused by the subtle changes that found their way into the bound volume of the reports of the Electrical Standards Committee, which was edited by Jenkin and published in 1873. The version of OTER printed there now introduced Gauss’s mode of representing physical quantities and used Maxwell’s square-bracket notation to designate concrete units. The derived dimensional formulae were also enclosed in square brackets, like in the Treatise, and referred to as the dimensions of derived units and not of derived quantities. See Maxwell and Jenkin (1873) and Mitchell (2017a, §4).
Wormell consequently placed himself in the vanguard of a campaign undertaken by proponents of a modern education to supplant Euclidean deduction with a “practical-heuristic” method of teaching geometry (Brock 1975, pp. 23–7; Wormell 1871b). His pedagogy seems to have been a great success with his students, who on one occasion greeted public praise for his teaching with loud cheers (Bryant 1986, p. 245).
Everett’s “Essay on Mathematical Study,” (1860) written during his professorship at King’s College, Windsor, provides some insight into his conception of mathematics circa 1860, but not enough to determine whether he favoured algebraic approaches to the mathematics of measurement at that time. “Mathematics may be defined as the science of magnitude,” he wrote. Of its two branches, pure mathematics dealt with “abstract magnitude” because it “takes no account of the particular substance or thing whose magnitude is in question,” unlike mixed or applied mathematics. “Under this head fall Mechanics, Physical Astronomy, and considerable portions of the sciences of Heat, Light, and Electricity… Its function is, to deduce accurate numerical conclusions from data furnished by experiment, to show the precise amount of the effect which will follow from the operation of any force whose law is known” (quotations at pp. 53–4). The distinction between pure and mixed mathematics was standard by the second half of the eighteenth century (Heilbron 1993, p. 109).
This may have been why Maxwell avoided computing dimensional formulae using a magnitude calculus in the Treatise whenever the dimensions of a derived electrical unit were straightforwardly those of a dynamical quantity, in which case combinations of geometrical, physical, and operational reasoning provided alternative pathways to the same result. See Maxwell (1873, Vol. 1, pp. 278–9 §226, 281, 332–3).
This did not appear to catch the attention of Buchwald or Hunt’s Maxwellians (see note 5), who managed to prosecute an entire debate on his means of calculating the dimensional formula of unit magnetic pole in the electrostatic system of measurement without even mentioning quaternions (see de Clark 2016).
Maxwell actually said little that directly addressed the meaningfulness of multiplication and division in quantity equations independently of a vector calculus, which makes it difficult to interpret examples such as this. In this respect his article 1877 article “On dimensions” for the Encyclopaedia Britannica (cited below) stands out.
I hope this goes some way towards reconciling the divergent scholarly views on Maxwell’s attitude to the mathematical expression of physical quantities and his historical role in developing the quantity calculus, which deserve even more consideration than I have given them here. John Roche (1998, p. 226) considered that “Maxwell did not rule out the quantity interpretation,” and indeed assumed it in many of his writings on dimensions, whereas Jan de Boer (1994/5, pp. 405–6) judged Maxwell reluctant to use the notion of physical quantity. In contrast, Nadine de Courtenay (2015, p. 62) argued that Maxwell subscribed to a “double interpretation” of the equations of physics, having “appended to the numerical interpretation… an intrinsic interpretation in terms of quantity equations.” The former were more suited for experimental purposes, and the latter theoretical ones. To support their positions, Roche (1998, pp. 225–6) provides the full quotation except for the last sentence, de Boer (1994/5, p. 406) cites only the last sentence, whereas de Courtenay (2015, p. 53) omits the last paragraph.
As indirect evidence for this claim, I refer to Alfred Lodge’s aggressive lobbying for the incorporation of a quantity representation of physical laws into a syllabus for elementary dynamics being drawn up by the Association for the Improvement of Geometrical Teaching (AIGT) during the late 1880s. Lodge argued that by solving physical equations, “in all cases the student would very soon perceive that the standards involved… might be treated exactly like numbers, and he would also learn from the resulting expressions (e.g. \( \frac{\hbox{foot}}{{{\hbox{sec}}.}} \), \( \frac{\hbox{foot}}{{\left( {{\hbox{sec}}.} \right)^{2} }} \)) to appreciate the meaning of the dimensions of quantities with a thoroughness unattainable in any other way” (Lodge 1888, pp. 57–8). Around that time Everett likewise explicitly advocated the forms \( \frac{{{\text{ft}}.}}{\text{sec}} \) and \( \frac{{{\text{ft}}.}}{{{\text{sec}}^{2} }} \) over a proposal to give foot-(pound)-second units of velocity and acceleration special names (Anonymous 1890, p. 15). See also the reaction to Lodge’s move in “Letters” (1888).
The ideological basis for James Thomson’s arithmetical empiricism lay most likely in his father’s liberal Presbyterianism, which, as Smith and Wise (1989, p. 229) have described, sought to avoid “doctrinal disputes of every kind while seeking the fruits of shared truth.” In addition, the strong semantic component to his philosophy of arithmetic displays clear parallels with late-eighteenth-century Unitarian mathematicians like William Frend, whose religious convictions led them to treat the misuse of words as a crime against reason (Richards 2011, pp. 508–9; see also Pycior 1981, pp. 27–9, 1983, pp. 213–4). Indeed, James would eventually renounce his Presbyterianism to become a Unitarian himself (Smith 2004).
On the influence on de Morgan of the progressive philosophy of arithmetic of his mentor, George Peacock, see Richards (1987, pp. 17–18).
The brothers’ preface on page v states that “considerable changes and additional explanations have been introduced by Professor James Thomson in the earlier chapters of the work.”
Brook-Smith (1910 [1881], pp. 1–2 §§1–5, 55 §98); de Morgan (1840, pp. 1–2 §§1–3, 51–2 §§104–5). Consider also de Morgan’s insistence (ibid., p. 14 §28) that “there is no process in arithmetic which does not consist entirely in the increase or diminution of numbers. There is then nothing which might not be done with collections of pebbles.” Brook-Smith (1910 [1881] p. 2 §6) simply observed that “arithmetic is the science of numbers.”
Cf. James Bryce (1872, p. 1), who stated in his Treatise on Algebra: “Writers on Algebra generally confine the word number to those numerical expressions in which the Arabic figures alone occur; while they apply the term quantity to any symbol representing a number, whether consisting of figures, or of letters, or of both combined: thus, 5, 19, are called numbers, while 5, 19, 4a, may all be called quantities.”
Cf. William Thomson’s protégé and Maxwell’s friend and collaborator Fleeming Jenkin (1887, p. 267): “when it is said that by scientific measurement things, not sensations, are compared, the word things must be interpreted as bearing a very wide meaning. Time, space, matter, energy, are included in the term.”
Note in pencil inscribed by James Thomson on Everett (1874a).
Everett (1875b) nonetheless stated his intention “to adopt as far as possible as I conveniently can [in the Illustrations] your idea of favouring the singular form rather than the plural” for the abbreviations of unit names, i.e. “cm.” for “centimetre or centimetres” and “gm.” for “gramme or grammes.”
Again, cf. Anonymous (1860 [“Multiple, Submultiple, Multiplication”], pp. 820–1), which reads here as if it were written by James Thomson himself: “The very definition of multiplication requires that every question should contain a number of times which another number, abstract or concrete, is to be repeated; and this number of times or repetitions cannot be a number of anything else. Thus, to talk of multiplying 10 feet by 7 feet is a contradiction in terms” (also quoted in Roche 1998, p. 225).
Brook-Smith (1910 [1881], p. 138 §193*) notes that ratios can only be formed between numbers or quantities of the same kind, but does not go on to define the notion of a “rate,” which features frequently in many of the exercises from p. 180 (§255) onwards. James Thomson’s detailed theoretical handling of rates, which de Morgan (1840, e.g. p. 144 §233) and Brook-Smith merely address through practical examples as part of commercial arithmetic, may well be a direct response to Everett’s physical algebra.
This usage is not as distant as it may seem. In Britain, automobile fuel economy is typically cited in miles per gallon, whereas the preferred European measure is litres per 100 km.
On Thomson’s friendship with Purser and Thomas Andrews, also Professor of Natural Philosophy at Queen’s College, see Purser’s recollections of their scientific and mathematical exchanges in Larmor and Bottomley (1912, pp. lix–lx).
Towards the end of the letter, James told William that part of the reason for writing was to see whether he “may be perhaps inserting some teaching on these [dimensional] relations between units, in T and T′ [i.e. the second edition of Thomson and Tait’s Treatise on Natural Philosophy, which was published the following year]” (Thomson 1878, p. 7). That Thomson and Tait (1879, see chapter 4) did not introduce such teaching is not surprising given their unresolved difference of opinion over quaternions. Notwithstanding Tait’s advocacy, these never featured either. See Smith and Wise (1989, pp. 185–8, 363–5).
He might have done this, for example, in the manner of George Peacock, who recognized that the algebraic generalization of arithmetical signs like “−” (minus) could never follow from the operations to which these signs corresponded in arithmetic. See Pycior (1981, pp. 34–5).
Letter from the Rev. E. F. M. MacCarthy (Letters 1888, p. 65).
James had told William that he “would hesitate to bring the matter before the Brit. Association unless I had your approval or approval of good judges” (Thomson 1878, pp. 6–7).
Cf. the philosopher and experimental physicist Percy Bridgman (1951 [1922], pp. 28–30), who upheld Thomson’s interpretation but allowed for dimensional formulae to be manipulated in a purely symbolic way, which presupposed a clear distinction between magnitude and quantity.
Interestingly, this translation appeared in the midst of a debate, initiated unwittingly by Everett, over the correctness of Maxwell’s statement in the Treatise of the dimensions of magnetic pole in electrostatic units. What had started off as an innocuous point of detail regarding a unit never used in practice turned into a showdown between continental action-at-a-distance and Maxwellian field conceptions of electrical action. The details need not concern us, suffice to note that, via a piece of ingenious physico-mathematical reasoning, Oliver Lodge ultimately reconciled the result of the German mathematical physicist Rudolph Clausius with the various rebuttals of J. J. Thomson, Joseph Larmor, and George Fitzgerald (de Clark 2016). What is significant here is that neither the physical interpretation nor the mathematical nature of the dimensional formula entered into the discussion.
Maxwell (2002 [1877], p. 518) had also given a similar form in terms of fundamental units for a physical quantity in his 1877 article “On Dimensions” for the Encyclopaedia Britannica. In the 1884 edition of his Absolute Measurements, Gray accidentally left off the C in the equation for Q′, a mistake that was rectified in later editions.
For a modern reader, Glazebrook and Shaw’s formalism is easier to follow because the Cambridge pair explicitly allowed the Gaussian forms q[Q] for physical quantities. It is unclear whether they worked independently from the Glasgow school; the absence of the term “change-ratio” from their definitions may be significant.
Among Maxwellian theoreticians there was of course another crucial reason for deriving the dimensional formulae of electrical units, namely to explain how the ratio of the electrostatic to the electromagnetic units of electrical quantity provided evidence for Maxwell’s electromagnetic theory of light. In his two-volume Physical Treatise of Electricity and Magnetism (1880), James Gordon argued from the dimensional formula of the ratio, LT−1, that it was physically a velocity, and that experiment had shown this velocity to be equal to the speed of light. This was the only case in the Physical Treatise, however, where the physical interpretability of dimensional formulae as concrete units really mattered (Bowers 2004; Falconer 2014, pp. 74, 94; Gordon 1880, Vol. 2, pp. 180–90). I have previously argued that Maxwell himself did not reason from the dimensional formula for the ratio to its physical nature (Mitchell 2017b). Gordon’s theory of units owed much to the Units and Physical Constants and Everett’s oversight (Gordon 1880, Vol. 1, p. vi).
A full study of these Maxwellian projects would provide important historical context for what is known as the “strong” view of dimensional formulae, which asserts that the dimensional formula of a given quantity in some way indicates its appropriate physical interpretation. See Bridgman (1951 [1922], pp. 23–7).
In fact, in electrical engineering this process arguably began during the late nineteenth century with Oliver Heaviside’s development of an operational calculus suitable for describing the behaviour of electrical circuits, as well as his more famous vector reformulation and rationalization of “Maxwell’s equations.” Heaviside was a Maxwellian (see note 5) whose work began to gain theoretical recognition and find practical application from roughly the beginning of the 1890 s. He failed, however, to impress mathematicians. See Hunt (2005 [1991]), chapter 3, Nahin 1988, chapters 7, 9, and 10, and pp. 179–80, 275–7, and Crowe 1985 [1967], pp. 162–77.
References
1860. Multiple, submultiple, multiplication. The English Cyclopaedia: A new dictionary of universal knowledge, arts and sciences—volume 5 (8 Vols, 1859–61), ed. Charles Knight, 820–3. London: Bradbury and Evans.
1871. New works on mechanics. Nature 5(107): 41–2.
1872a. Applications for reports and researches not involving grants of money. In Report of the Forty-First Meeting of the British Association for the Advancement of Science; held at Edinburgh in August 1871, lxii–lxxiv. London: John Murray.
1872b. Sir William Thomson, LL.D., F.R.S., examined. In Royal Commission on Scientific Instruction and the Advancement of Science, Vol. 1, First, Supplementary and Second Reports, with Minutes of Evidence and Appendices, 160–71, §§2652–2873. London: Her Majesty’s Stationery Office.
1873. Applications for reports and researches not involving grants of money. In Report of the Forty-Second Meeting of the British Association for the Advancement of Science; held at Brighton in August 1872, lvi–lviii. London: John Murray.
1876. Proceedings of the Physical Society of London. Vol. 1, from 21st March, 1874, to 26th June, 1875. London: Taylor and Francis.
1879. The British Association Reports: Section F—Economic Science and Statistics. Nature 20(516): 492–3.
1880. Recommendations of the General Committee for Additional Reports and Researches in Science. In Report of the Fiftieth Meeting of the British Association for the Advancement of Science, Held at Swansea in August and September 1880, lx–lxv. London: John Murray.
1890. Mechanical units [Suggestions by Professor Everett, November 27th, 1889]. The Engineer 70: 15–16.
A. R. 1925. Andrew Gray—1847–1925. Proceedings of the Royal Society of London A 110: xvi–xix.
A. R. 1927. James Thomson Bottomley—1845–1926. Proceedings of the Royal Society of London A 113: xii–xiii.
A. W. B. 1871. Rodwell’s Dictionary of Science. Nature 3(69): 325–6.
B. L. 1869. Our book shelf: Theoretical and Applied Mechanics by R. Wormell. Nature 1(2): 53–4.
Bishop, Roy L. 1978. Joseph Everett and the King’s College observatory. Journal of the Royal Astronomical Society of Canada 72: 138–148.
Bowers, Brian. 2004. Gordon, James Edward Henry (1852–1893). Oxford dictionary of national biography. http://www.oxforddnb.com/view/article/11057. Accessed 9 Jan 2017.
Bowler, Peter. 2008. James Thomson and the culture of a Victorian engineer. In Kelvin: Life, labours, and legacy, ed. Flood Raymond, Mark McCartney, and Andrew Whitaker, 56–63. Oxford: Oxford University Press.
Bridgman, Percy. 1951 [1922]. Dimensional analysis. New Haven and London: Yale University Press.
Brock, William. 1975. Geometry and the universities: Euclid and his modern rivals 1860–1901. History of Education 4(2): 21–35.
Brock, William. 2000. Who were they? Richard Wormell (1838–1914). School Science Review 81(297): 93–97.
Brock, William. 2004. Wormell, Richard (1838–1914). Oxford dictionary of national biography. http://www.oxforddnb.com/view/article/40970. Accessed 9 Jan 2017.
Brook-Smith, J. 1910 [1881]. Arithmetic in theory and practice, 6th edn. Macmillan and Co.
Bryant, Margaret. 1986. The London experience of secondary education. London: The Athlone Press.
Bryce, James. 1872. Treatise on Algebra, 4th ed. Edinburgh: Adam and Charles Black.
Buchwald, Jed Z. 1985. From Maxwell to microphysics: Aspects of electromagnetic theory in the last quarter of the nineteenth century. Chicago: University of Chicago Press.
Buchwald, Jed Z., and Sungook Hong. 2003. Physics. In From natural philosophy to the sciences, ed. David Cahan, 163–195. London: University of Chicago Press.
Cajori, Florian. 1917. A history of elementary mathematics, with hints on methods of teaching. Revised and enlarged edition. New York: The Macmillan Company.
Carroll, Lewis [Dodgson, Charles]. 1998 [1865/1872]. Alice’s adventures in Wonderland, and Through the looking-glass. London: Penguin Books.
Clifford, William Kingdon. 1878. Elements of dynamic: An introduction to the study of motion and rest in solid and fluid bodies. London: Macmillan & Co.
Corry, Leo. 2015. A brief history of numbers. Oxford: Oxford University Press.
Crowe, Michael. 1985 [1967]. A history of vector analysis: The evolution of the idea of a vectorial system. New York: Dover.
Cumming, Linneaus. 1876. An introduction to the theory of electricity, with numerous examples. London: Macmillan and Co.
Cumming, Linneaus. 1885. An introduction to the theory of electricity, with numerous examples, 3rd ed. London: Macmillan and Co.
de Boer, Jan. 1994/5. On the history of quantity calculus and the international system. Metrologia 31: 405–29.
de Clark, Sybil Gertrude. 2010. L’analyse dimensionnelle en France et en Grande-Bretagne au XIXe siècle. Ph.D. dissertation, Université Paris Diderot (Paris 7).
de Clark, Sybil Gertrude. 2016. The dimensions of the magnetic pole: A controversy at the heart of early dimensional analysis. Archive for History of Exact Sciences 70: 293–324.
de Clark, Sybil Gertrude. 2017. Qualitative vs quantitative conceptions of homogeneity in nineteenth century dimensional analysis. Annals of Science 74(4): 299–325.
de Courtenay, Nadine. 2015. The double interpretation of the equations of physics and the quest for common meanings. In Standardization in measurement, ed. Oliver Schlaudt and Lara Huber, 53–68. London: Pickering and Chatto.
de Morgan, Augustus. 1840. The elements of arithmetic, 4th ed. London: Taylor and Walton.
Denniss, John. 2008. Learning arithmetic: Textbooks and their users in England 1500–1900. In The Oxford handbook of the history of mathematics, ed. Eleanor Robson and Jacqueline Stedall, 448–467. Oxford: Oxford University Press.
Despeaux, Sloan Evans. 2002. The development of a publication community: Nineteenth-century mathematics in British scientific journals. Ph.D. dissertation, University of Virginia.
Despeaux, Sloan Evans. 2011. A voice for mathematics: Victorian mathematical journals and societies. In Mathematics in Victorian Britain, ed. Raymond Flood, Adrian Rice, and Robin Wilson, 155–174. Oxford: Oxford University Press.
Elwick, James. 2014. Economies of scales: Evolutionary naturalists and the Victorian examination system. In Victorian scientific naturalism: Community, identity, continuity, 131–56. Chicago: University of Chicago Press.
Everett, Joseph David. 1860. Essay on mathematical study. The Calendar of King’s College, Windsor, Nova Scotia, 51–6. Halifax, N.S.: James Bowes and Sons.
Everett, Joseph David. 1872. On units of force and energy [Abstract]. Report of the Forty-First Meeting of the British Association for the Advancement of Science; held at Edinburgh in August 1871, 29. London: John Murray.
Everett, Joseph David. 1874a. Letter to James Thomson, 15th Dec 1874. MS.13.M.6.b. Queen’s College Archives, Belfast.
Everett, Joseph David. 1874b. Letter to James Thomson, 21st Dec 1874. MS.13.M.6.c. Queen’s College Archives, Belfast.
Everett, Joseph David. 1874c. First report of the Committee for the Selection and Nomenclature of Dynamical and Electrical Units. In Report of the Forty-Third Meeting of the British Association for the Advancement of Science; held at Bradford in September 1873, 222–5. London: John Murray.
Everett, Joseph David. 1875a. Illustrations of the centimetre-gramme-second system of units. London: Taylor and Francis.
Everett, Joseph David. 1875b. Letter to James Thomson, 25th March 1875. MS.13.K.1.a. Queen’s College Archives, Belfast.
Everett, Joseph David. 1875c. Second report of the Committee for the Selection and Nomenclature of Dynamical and Electrical Units. In Report of the Forty-Fourth Meeting of the British Association for the Advancement of Science; held at Belfast in August 1874, 255. London: John Murray.
Everett, Joseph David. 1878. Letter to James Thomson, 24th July 1878. MS.13.M.6.g. Queen’s College Archives, Belfast.
Everett, Joseph David. 1879. Units and physical constants. London: Macmillan and Co.
Everett, Joseph David. 1886. Units and physical constants, 2nd ed. London: Macmillan and Co.
Falconer, Isobel. 2014. Cambridge and building the Cavendish laboratory. In James Clerk Maxwell: Perspectives on his life and work, ed. Raymond Flood, Mark McCartney, and Andrew Whitaker, 67–99. Oxford: Oxford University Press.
Ferguson, Robert M. 1882. Electricity. New edition revised and extended by James Blyth, M.A., F.R.S.E. London and Edinburgh: W & R. Chambers.
G. A. G. and R. A. H. 1926. Andrew Gray. Proceedings of the Royal Society of Edinburgh 45(4): 373–7.
Gauss, Charles Frédéric [Carl Friedrich]. 1834. Mesure absolue de l’intensité du magnetisme terrestre. Annales de Physique et de Chimie, 57: 5–69.
Gladstone, John Hall. 1876. Report of the Council. In Proceedings of the Physical Society of London. Vol. 1, from 21st March, 1874, to 26th June, 1875, 13–6. London: Taylor and Francis.
Glazebrook, Richard Tetley, and William Napier Shaw. 1885. Practical physics. New York: D. Appleton and Co.
Glazebrook, Richard Tetley and Shaw, William Napier. 1904 [1893]. Practical physics. New impression [of 4th edition]. London, New York, and Bombay: Longmans, Green, and Co.
Gooday, Graeme. 2000. “Lies, damned lies and declinism: Lyon Playfair, the Paris 1867 Exhibition and the contested rhetorics of scientific education and industrial performance. In The golden age: Essays in British social and economic history, 1850–1870, ed. Ian Inkster, Colin Griffin, Jeff Hill, and Judith Rowbotham, 105–120. Aldershot: Ashgate.
Gooday, Graeme. 2004. The morals of measurement. Accuracy, irony, and trust in Late Victorian electrical practice. Cambridge: Cambridge University Press.
Gordon, James. 1880. A physical treatise on electricity and magnetism. London: Sampson Low et al.
Gray, Andrew. 1884. Absolute measurements in electricity and magnetism. London: Macmillan and Co.
Gray, Andrew. 1888–93. The theory and practice of absolute measurement in electricity and magnetism, 2nd edn (2 Vols). London: Macmillan and Co.
Gray, Andrew. 1921. Absolute measurements in electricity and magnetism, 2nd ed. London: Macmillan and Co.
Holmberg, Peter. 1988. August Fredrik Sundell—Fysiker, Matematiker och Astronom. Arkhimedes 40: 19–30.
Hunger Parshall, Karen. 2011. Victorian algebra: The freedom to create new mathematical entities. In Mathematics in Victorian Britain, ed. Raymond Flood, Adrian Rice, and Robin Wilson, 339–358. Oxford: Oxford University Press.
Hunt, Bruce J. 1994. The ohm is where the art is: British telegraph engineers and the development of electrical standards. Osiris 9: 48–63.
Hunt, Bruce J. 2005 [1991]. The Maxwellians. Ithaca and London: Cornell University Press.
Hunt, Bruce J. 2015. Maxwell, measurement, and the modes of electromagnetic theory. Historical Studies in the Natural Sciences 45(2): 303–339.
J[ohn], P[erry]. 1904. Obituary notices of fellows deceased. Proceedings of the Royal Society of London 75: 377–380.
Jenkin, Fleeming. 1887. A fragment of truth. In Papers literary, scientific, &c. by the late Fleeming Jenkin, Vol. 1 (2 Vols), ed. Sidney Colvin and J. A. Ewing. London: Longmans, Green, and Co.
Heilbron, John L. 1993. A mathematicians’ mutiny, with morals. In World changes: Thomas Kuhn and the nature of science, ed. Paul Horwich, 81–129. Cambridge: MIT Press.
Kaiser, David. 2005. Introduction: Moving pedagogy from the periphery to the center. In Pedagogy and the practice of science: Historical and contemporary perspectives, ed. David Kaiser, 1–8. Cambridge: MIT Press.
Kim, Dong-Won. 2002. Leadership and creativity: A history of the Cavendish laboratory 1871–1919 [Archimedes 5]. Dordrecht/Boston/London: Kluwer.
King Bidwell, James. 2013. The teaching of arithmetic in England from 1550 until 1800 as influenced by social change. In The European mathematical awakening: A journey through the history of mathematics from 1000 to 1800, ed. Frank J. Swetz, 76–81. Mineola, New York: Dover.
Kohlrausch, Friedrich. 1874. An introduction to physical measurements, with appendices on absolute electrical measurement, etc., trans. Thomas Hutchinson Waller and Henry Richardson Proctor. New York: D. Appleton and Co.
Kuhn, Thomas. 1977. Preface. The essential tension. Selected studies in scientific tradition and change, ix–xxiii. Chicago: University of Chicago Press.
Kuhn, Thomas. 2012 [1962]. The structure of scientific revolutions, 4th edn. Chicago: University of Chicago Press.
Koppelman, Elaine. 1971. The calculus of operations and the rise of abstract algebra. Archive for History of Exact Sciences 8(3): 155–242.
Kyburg Jr., Henry E. 1997. Quantities, magnitudes, and numbers. Philosophy of Science 64(3): 377–410.
[“Larmor and Bottomley”]. 1912. Biographical sketch. In Collected papers in physics and engineering by James Thomson, D.Sc., LL.D, F.R.S., selected and arranged by Sir Joseph Larmor and James Thomson, M.A., xiii–xci. Cambridge: Cambridge University Press.
Lees, Charles Herbert. 1912. Everett, Joseph David. In Dictionary of national biography: Supplement, January 1901–December 1911, ed. Sir Sidney Lee, 638–9. [S.I.]: Smith, Elder, and Co.
Lees, Charles Herbert [revised by Graeme J. N. Gooday]. 2004. Everett, Joseph David (1831–1904). Oxford dictionary of national biography. http://www.oxforddnb.com/view/article/33049. Accessed 9 Jan 2017.
[“Letters”]. 1888. Letters referring to the proposed emendation. In Association for the Improvement of Geometrical Teaching. Fourteenth General Report, January 1888, 60–70. Bedford: W. J. Robinson.
Lindley, David. 2004. Degrees Kelvin: A tale of genius, invention, and tragedy. Washington: Joseph Henry Press.
Lodge, Alfred. 1888. The multiplication and division of concrete quantities. In Association for the Improvement of Geometrical Teaching. Fourteenth General Report, January 1888, 47–60. Bedford: W. J. Robinson.
Longair, Malcolm. 2016. Maxwell’s enduring legacy: A scientific history of the Cavendish laboratory. Cambridge: Cambridge University Press.
Macleod, Roy M. 1971. The support of Victorian science: The endowment of research movement in Great Britain, 1868–1900. Minerva 9(2): 197–230.
Martins, Roberto de A. 1981. The origin of dimensional analysis. Journal of the Franklin Institute 311: 331–337.
Maxwell, James Clerk. 1871. Presidential address. In Report of the Fortieth Meeting of the British Association for the Advancement of Science; Held at Liverpool in September 1870. London: John Murray.
Maxwell, James Clerk. 1872. Theory of heat, 3rd ed. London: Longmans, Green, and Co.
Maxwell, James Clerk. 1873. A treatise on electricity and magnetism, 2 Vols. Cambridge: Cambridge University Press.
Maxwell, James Clerk. 1890 [1871]. Remarks on the mathematical classification of physical quantities. In The scientific papers of James Clerk Maxwell, Vol. 2 (2 Vols), ed. W. D. Niven, 257–66. Cambridge: Cambridge University Press.
Maxwell, James Clerk. 1990 [1865]. Letter to William Thomson, 17 and 18 April 1865. In The scientific letters and papers of James Clerk Maxwell, Vol. 1 (3 Vols, 1990–2002), ed. Peter Harman, 218–20. Cambridge: Cambridge University Press.
Maxwell, James Clerk. 2002 [1877]. On dimensions. In The scientific letters and papers of James Clerk Maxwell, Vol. 3 (3 Vols, 1990–2002), ed. Peter Harman, 517–20. Cambridge: Cambridge University Press.
Maxwell, James Clerk and Jenkin, Fleeming. 1864. Appendix C—On the elementary relations between electrical measurements, In Report of the thirty-third meeting of the British Association for the Advancement of Science; held at Newcastle-upon-Tyne in August and September 1863, 59–96. London: John Murray.
Maxwell, James Clerk and Jenkin, Fleeming. 1865. Appendix C—On the elementary relations between electrical measurements. Philosophical Magazine, 4th series 4(29): 436–60, 507–25.
Maxwell, James Clerk, and Fleeming Jenkin. 1873. Appendix C—On the elementary relations between electrical measurements. In Reports of the Committee on Electrical Standards appointed by the British Association for the Advancement of Science, ed. Fleeming Jenkin, 59–96. London: E. & F. N. Spon.
Mitchell, Daniel Jon. 2017a. Making sense of absolute measurement. James Clerk Maxwell, Fleeming Jenkin, William Thomson, and the construction of the dimensional formula. Studies in History and Philosophy of Modern Physics 58: 63–79.
Mitchell, Daniel Jon. 2017b. What’s nu? A re-examination of Maxwell’s “ratio-of-units” argument, from the mechanical theory of the electromagnetic field to “On the elementary relations between electrical measurements”. Studies in History and Philosophy of Science 65–66: 87–98.
Mitchell, Daniel Jon. Forthcoming. The dynamics of the masses: Absolute measurement and elementary physics pedagogy in Victorian Britain, 1863–1879. History of Science.
Nahin, Paul J. 1988. Oliver Heaviside: Sage in solitude. The life, work and times of an electrical genius of the Victorian age. New York: IEEE Press.
Olesko, Kathryn. 2005. The foundations of a canon: Kohlrausch’s Practical Physics. In Pedagogy and the practice of science. Historical and contemporary perspectives, ed. David Kaiser, 324–356. Cambridge: MIT Press.
Price, Michael. 1986. The Perry movement in school mathematics. In The development of the secondary curriculum, ed. Michael Price, 103–155. London: Croom Helm.
Pycior, Helena M. 1981. George Peacock and the British origins of symbolical algebra. Historica Mathematica 8: 23–45.
Pycior, Helena M. 1982. Early criticism of the symbolical approach to algebra. Historica Mathematica 9: 392–412.
Pycior, Helena M. 1983. Augustus de Morgan’s algebraic work: The three stages. Isis 74(2): 211–226.
Pycior, Helena M. 1989. At the intersection of mathematics and humor: Lewis Carroll’s Alices and symbolical algebra. In Energy and entropy: Science and culture in Victorian Britain, ed. Patrick Brantlinger. Bloomington: Indiana University Press.
Richards, Joan L. 1980. The art and science of British algebra: A study in the perception of mathematical truth. Historica Mathematica 7: 343–365.
Richards, Joan L. 1987. Augustus de Morgan, the history of mathematics, and the foundations of algebra. Isis 78(1): 6–30.
Richards, Joan L. 2011. “This compendious language:” Mathematics in the world of Augustus de Morgan. Isis 102(3): 506–510.
Roche, John. 1998. The mathematics of measurement: A critical history. London: The Athlone Press.
Schaffer, Simon. 1992. Late Victorian metrology and its instrumentation: A manufactory of Ohms. In Invisible Connections: Instruments, institutions, and science [Proceedings of the Society of Photo-Optical Instrumentation Engineers (SPIE), Vol. 10309], ed. Robert Bud and Susan E. Cozzens, 23–51. Bellingham, WA: SPIE Optical Engineering Press.
Schaffer, Simon. 1997. Metrology, metrication, and Victorian values. In Victorian science in context, ed. Bernard Lightman, 438–474. Chicago: University of Chicago Press.
[“Second Report”]. 1864. Report of the committee on standards of electrical resistance. In Report of the thirty-third meeting of the British Association for the Advancement of Science; held at Newcastle-upon-Tyne in August and September 1863, 111–176. London: John Murray.
Simon, Josep. 2012. Secondary matters: Textbooks and the making of physics in nineteenth-century France and England. History of Science 1: 339–374.
Smith, Crosbie. 1998. The science of energy. Chicago: University of Chicago Press.
Smith, Crosbie. 2004. Thomson, James (1822–1892). Oxford dictionary of national biography. http://www.oxforddnb.com/view/article/27312. Accessed 9 Jan 2017.
Smith, Crosbie, and Norton Wise. 1989. Energy and empire: A biographical study of Lord Kelvin. Cambridge: Cambridge University Press.
Sundell, August Fredrick. 1882. Remarks on absolute systems of physical units. Philosophical Magazine 14(86): 81–109.
T[hompson], S[ilvanus] P. 1881. Our book shelf: Natural Philosophy for London University Matriculation by Edward B. Aveling. Nature 25(630): 76–7.
Thomson, James [Sr]. 1825. A treatise on arithmetic in theory and practice, 2nd edn. Belfast: Simms and McIntyre.
Thomson James [Sr]. 1880. A treatise on arithmetic in theory and practice, 72nd edn edited by his sons James Thomson and Sir William Thomson. London: Longmans, Green, and Co.
Thomson, James. Undated manuscript. MS.13.K.1.e. Queen’s College Archives, Belfast.
Thomson, James. 1878. Letter to William Thomson, 19th July 1878. MS.13.M.41.f. Queen’s College Archives, Belfast.
Thomson, James. 1912 [1878]. On dimensional equations, and on some verbal expressions in numerical science. In Collected papers in physics and engineering by James Thomson, D.Sc., LL.D, F.R.S., selected and arranged by Sir Joseph Larmor and James Thomson, M.A., 375–9. Cambridge: Cambridge University Press.
Thomson, William. 1889 [1883]. The six gateways of knowledge. In Popular Lectures and Addresses, Vol. 1 (3 Vols, 1889–94), 253–99. London: Macmillan and Co.
Thomson, William, and Peter Guthrie Tait. 1867. Treatise on Natural Philosophy. Oxford: Clarendon Press.
Thomson, William, and Peter Guthrie Tait. 1879. Treatise on natural philosophy, 2nd ed. Cambridge: Cambridge University Press.
Thomson Bottomley, James. 1912 [1893]. Obituary notice. In Collected papers in physics and engineering by James Thomson, D.Sc., LL.D, F.R.S., selected and arranged by Sir Joseph Larmor and James Thomson, M.A., xcii–cii. Cambridge: Cambridge University Press.
Timmons, George. 2001. Science and science education in schools after the Great Exhibition. Endeavour 25(3): 109–120.
Warwick, Andrew. 2003. Masters of theory: Cambridge and the rise of mathematical physics. Cambridge: Cambridge University Press.
Warwick, Andrew, and David Kaiser. 2005. Conclusion: Kuhn, Foucault, and the power of pedagogy. In Pedagogy and the practice of science: Historical and contemporary perspectives, ed. David Kaiser, 393–409. Cambridge: MIT Press.
Williams, W. 1892. On the relation of the dimensions of physical quantities to directions in space. Philosophical Magazine 22(146): 234–271.
Wormell, Richard. 1871a. Elementary geometry. Nature 4(100): 425.
Wormell, Richard. 1871b. Wormell’s mechanics. Nature 5(108): 63.
Wormell, Richard. 1876. The principles of dynamics: An elementary textbook for science students. London: Rivingtons.
Wormell, Richard. 1887. The principles of dynamics: An elementary text-book for science students, 2nd ed. London: Rivingtons.
Acknowledgements
I would like to acknowledge the financial support of the Leverhulme Trust (Grant ECF-2013-460) and, through the collaborative project “The Epistemology of the Large Hadron Collider,” the Deutsche Forschungsgemeinschaft (DFG grant FOR 2063). I thank Hasok Chang and members of the Philosophy and History of Physics reading group at the University of Cambridge for detailed discussions of an early draft of this paper, and Sybil de Clark and especially Olivier Darrigol for criticism of the penultimate draft. (The key terminology employed in fact derives from a proposal of Darrigol’s.) I am also grateful to him and to Nadine de Courtenay for inviting me to speak at their Séminaire d’Histoire et Philosophie de Physique (laboratoire SPHERE) in Paris in 2017, and likewise to Jim Grozier for his invitation to present at the workshop “A History of Units 1791–2018” (Teddington, UK, 2016). Audiences on those occasions and at the following conferences or workshops provided constructive feedback: “Measurement at the Crossroads” (Paris, France, 2018), the Seventh International Conference on Integrated History and Philosophy of Science (Hannover, Germany, 2018), and “Beyond the Academy: The Practice of Mathematics from the Renaissance to the Nineteenth Century” (York, UK, 2017). Finally, I am indebted to the archivists at Queen’s College, Belfast, for reproducing selected manuscripts in the James Thomson collection; Janet Hathaway at King’s College, Nova Scotia, for materials on Everett; Peter Holmberg, for supplying his essay on Sundell; and Sloan Evans Despeaux for directing me towards some relevant work in the history of mathematics.
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Mitchell, D.J. “The Etherealization of Common Sense?” Arithmetical and Algebraic Modes of Intelligibility in Late Victorian Mathematics of Measurement. Arch. Hist. Exact Sci. 73, 125–180 (2019). https://doi.org/10.1007/s00407-018-0218-y
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DOI: https://doi.org/10.1007/s00407-018-0218-y