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Sources for a history of the ternary diagram

Published online by Cambridge University Press:  05 January 2009

Richard J. Howarth
Richard J. Howarth
Affiliation:
Research School for Geological and Geophysical Sciences, Birkbeck College and University College London, Gower Street, London WC1E 6BT.
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Extract

Anyone reading the literature on the history of graphs will soon realize that the use of graphie displays of any type was really quite unusual until the mid-ninetenth century and that those scientists who did make use of them are often familiar to us as creative thinkers in their own fields of endeavour. A ternary diagram (also known as a triangular diagram) is a particular type of graph which consists of an equilateral triangle in which a given plotted point represents the relative proportions (a, b, c) of three end-members (A, B and C), generally expressed as percentages and constrained by a + b + c = 100%. It has long been used to portray sample composition in terms of three constituents, or an observed colour in terms of three primary colours, because it is a convenient means of representing a three-component System in a planar projection, rather than as an isometric, or similar, view of a three-dimensional space. Recent papers suggest that its use is not as familiar to some statisticians as are other commonly used forms of graph. For example, although it was cited by Peddle in 1910 and more recently by Dickinson, it is not discussed in modern texts on statistical graphies nor in the key papers on the history of graphs. However, beginning with studies of colour-mixing in the eighteenth century, it has subsequently become widely used, particularly in geology, physical chemistry and metallurgy. In this paper, I attempt to document its gradual uptake as a standard method of data display and some of the scientific advances which its use has facilitated.

Type
Research Article
Information
The British Journal for the History of Science , Volume 29 , Issue 3 , September 1996 , pp. 337 - 356
Copyright
Copyright © British Society for the History of Science 1996

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References

1 Mead, R., ‘Statistical games 2 – medical diagnosis’, Teaching Statistics (1992), 14, 12–16CrossRefGoogle Scholar; Digby, P. K. W., ‘Do we need trilemmas?’, Teaching Statistics (1993), 15, 66–7.CrossRefGoogle Scholar

2 Peddle, J. B., The Construction of Graphical Charts, 2nd edn, New York, 1910, figs. 57, 58.Google Scholar

3 Dickinson, G. C., Statistical Mapping and the Presentation of Statistics, London, 1973, figs. 11, 27, 50.Google Scholar

4 Riggleman, J. R., Graphic Methods for Presenting Business Statistics, 2nd edn, New York, 1936Google Scholar; Chambers, J. M., Cleveland, W. S., Kleiner, B. and Tukey, P. A., Graphical Methods for Data Analysis, Belmont, CA, 1983Google Scholar; Schmid, C. F., Statistical Graphics. Design Principles and Practices, New York, 1983Google Scholar; Tufte, E. R., The Visual Display of Quantitative Information, Cheshire, CO, 1983Google Scholar; Cleveland, W. S., The Elements of Graphing Data, Monterey, CA, 1985Google Scholar; Tufte, E. R., Envisioning Information, Cheshire, CO, 1990.Google Scholar

5 Funkhouser, H. G., ‘Historical development of thegraphical representation of statistical data’, Osiris (1938), 3, 269–404CrossRefGoogle Scholar; Royston, E., ‘A note on the history of the graphical presentation of data’, Biometrika (1956), 43, 241–7Google Scholar; Beniger, J. R. and Robyn, D. L., ‘Quantitative graphies in statistics: a brief history’, American Statistician (1978), 32, 1–11.Google Scholar

6 Newton, I., Opticks or A Treatise of the Reflections, Refractions, Inflections and Colours of Light, 2nd edn, London, 1718, prop. VI, prob. II, 134–7Google Scholar; book I, part II, plate III, fig. 11. See also 4th edn, reprinted New York 1952, 155.

7 Hall, A. R., All Was Light. An Introduction to Newton's Opticks, Oxford, 1993.Google Scholar

8 Forbes, E. G., Tobias Mayer's Opera Inedita. The First Translation of the Lichtenberg Edition of 1775, London, 1971, 84.Google Scholar

9 Forbes, , op. cit. (8), 86.Google Scholar

10 Lambert, J. H., ‘Mémoire sur la partie photométrique de l'art du peintre’, Histoire de L'Académie Royale des Sciences et Belles-Lettres (1770), 80–108, especially unnumbered figure on 98. (Read 1768.)Google Scholar

11 Tilling, L., ‘Early experimental graphs’, BJHS (1975), 8, 193–213.CrossRefGoogle Scholar

12 Diderot, D. and d'Alembert, J., Encyclopédie ou dictionnaire raisonné des sciences, des arts et des métiers par une société de gens de lettres, 35 vols., Paris, 1751–1780, supplement I (1776), 663.Google Scholar

13 Forbes, , op. cit. (8), 122–3.Google Scholar

14 Young, T. (ed.), A Course of Lectures on Natural Philosophy and the Mechanical Arts, 2 vols., London, 1845, i, plate XXIX, fig. 427.Google Scholar

15 Helmholtz, H., ‘On the theory of compound colours’, Philosophical Magazine, series 4 (1852), 4, 519–34Google Scholar; and ‘Ueber die Theorie der zusammengesetzen Farben’, Annalen der Physik (1852), 87, 45–66.Google Scholar

16 Forbes, J. D., ‘Hints towards a classification of colours’, Philosophical Magazine, series 3 (1849), 34, 161–78, especially fig. 1, on 168.Google Scholar

17 Harman, P. M. (ed.), The Scientific Letters and Papers of James Clerk Maxwell. 1. 1846–1862, Cambridge, 1990, 300.Google Scholar

18 Harman, , op. cit. (17), 302 n9.Google Scholar

19 Maxwell, J. C., ‘Experiments on colour, as perceived by the eye, with remarks on colour-blindness’, Transactions of the Royal Society of Edinburgh (1857), 21, 275–98CrossRefGoogle Scholar, reprinted in Niven, W. D. (ed.), The Scientific Papers of James Clerk Maxwell, New York, 1890, 126–54Google Scholar. A colour photograph of Maxwell's colour-mixing apparatus will be found in Campbell, F. W., ‘Prologue: Cambridge colour contributions’, in Colour Vision, Physiology and Psychophysics (ed. Mollon, J. D. and Sharpe, L. T.), London, 1983, plate 3, opposite p. xxiii.Google Scholar

20 Maxwell, , op. cit. (19), 130.Google Scholar

21 Maxwell, , op. cit. (19), 132.Google Scholar

22 Maxwell, , op. cit. (19), 127–8.Google Scholar

23 Maxwell, , op. cit. (19), 131–2.Google Scholar

24 Maxwell, , op. cit. (19), 133–4.Google Scholar

25 Campbell, L. and Garnett, W., The Life of James Clerk Maxwell, London, 1882, plates I–II.Google Scholar

26 Maxwell, J. C., ‘On the theory of compound colours, and the relations of the colours of the spectrum’, Philosophical Transactions of the Royal Society (1860), 150, 57–84, and fig. 4.CrossRefGoogle Scholar

27 Digby, , op. cit. (1).Google Scholar

28 Wright, W. D., ‘A re-determination of the trichromatic coefficients of the spectral colours’, Transactions of the Optical Society (1929), 30, 141–64CrossRefGoogle Scholar; The Measurement of Colour, 2nd edn, London, 1958Google Scholar; and ‘The origins of the 1931 CIE System’, in Human Colour Vision (ed. Boynton, R. M.), New York, 1979, 397–403Google Scholar. It is interesting that both Boynton (figs. 5.1, 5.12, 5.14, 5.15) and Werner, A. in From Pigments to Perception (ed. Valberg, A. and Lee, B. B.), New York, 1991, 392Google Scholar, have reverted from use of the CIE System to that of ternary diagrams. Boynton (on 133) notes ‘use of a triangular chart instantly clarifies relationships that are otherwise obscure’.

29 Brewster, D., ‘On the laws which regulate the distribution of the polarising force in plates, tubes, and cylinders of glass, that have received the polarising structure’, Transactions of the Royal Society of Edinburgh (1818), 108, 199–273Google Scholar; and ‘Optics’, in The Edinburgh Encyclopaedia (ed. Brewster, D.), Edinburgh, 1830, 460–662.Google Scholar

30 Campbell, and Garnett, , op. cit. (25), 487 and plate III.Google Scholar

31 Maxwell, J. C., ‘To find the form of the central bars seen by polarised light in pieces of an unannealed glass’ (Cambridge, 1848)Google Scholar, in Harman, , op. cit. (17), 101–3, and fig. 14.2.Google Scholar

32 Maxwell, J. C., ‘On the equilibrium of elastic solids’, Transactions of the Royal Society of Edinburgh (1853), 20, 87–120, and figs. 2–4.CrossRefGoogle Scholar

33 Gibbs, J. W., ‘Graphical methods in the thermodynamics of fluids’, Transactions of the Connecticut Academy (1873), 2, 309–42.Google Scholar

34 Rukeyser, M. (ed.), Willard Gibbs, Garden City, NY, 1942Google Scholar; unnumbered figure opposite 202; Wheeler, L. P., Josiah Willard Gibbs. The History of a Great Mind, New Haven, 1962, unnumbered figure opposite 86.Google Scholar

35 For a System in equilibrium, the number of phases present (p) cannot be more than (n + 2), where n is the number of independently variable components in the whole System. If p = n + 2, equilibrium can exist only at a single temperature and a single pressure (for example ice–water–steam); if p = n + 1, either pressure or temperature can be arbitrarily chosen (for example a liquid and its vapour).

36 Gibbs, J. W., ‘On the equilibrium of heterogenous substances I’, Transactions of the Connecticut Academy (1875), 3, 108–248.Google Scholar

37 Letter from Maxwell, to Stokes, , 8 10 1859Google Scholar, in Harman, , op. cit. (17), 619–22.Google Scholar

38 Wheeler, , op. cit. (34), 238.Google Scholar

39 Stokes, G. G., ‘On a graphical representation of the results of Dr Alder Wright's experiments on ternary alloys’, Proceedings of the Royal Society of London (1891), 49, 174–8, and fig. 1.Google Scholar

40 Taylor, S. F., ‘Mass law studies, III’, Journal of Physical Chemistry (1897), 1, 542–6.CrossRefGoogle Scholar

41 Donnan, F. G. and Haas, A. (eds.), A Commentary on the Sdentific Writings of J. Willard Gibbs. I. Thermodynamics, New Haven, 1936.Google Scholar

42 Roozeboom, H. W. B., ‘Die Gleichgewichte von Lösungen zweier oder dreier bestandteile mit festen Phasen: Komponenten, binäre und ternäre Verbindungen, in ihrem Zusammenhang dargestellt’, Zeitschrift für physikalische Chemie (1893), 12, 359–89, and figs. 7, 8, 9, 11, 16.Google Scholar

43 Bancroft, W. D., ‘A triangular diagram’, Journal of Physical Chemistry (1897), 1, 403–10, and 404, fig. 1.CrossRefGoogle Scholar

44 Roozeboom, H. W. B., ‘Graphische Darstellung der heterogenen Systeme aus ein bis vier Stoffen, mit Einschluss der chemischen Umsetzung’, Zeitschrift fur physikalische Chemie (1894), 15, 145–58, and fig. 3.CrossRefGoogle Scholar

45 Bancroft, , op. cit. (43), 405.Google Scholar

46 Roozeboom, H. W. B. and Aten, A. H. W., ‘Gleichgewichte zwischen festen und flüssigen Phasen in ternären Systemen, welche pseudo-binär sind, mit Anwendung zur Erklärung anomaler Schemlz- und Lösungserscheinungen’, Zeitschrift für Physikalische Chemie (1905), 55, 449–501, and figs. 2, 5, 36.Google Scholar

47 Taylor, , op. cit. (40), fig. 1.Google Scholar

48 Taylor, , op. cit. (40), 542.Google Scholar

49 Taylor, S. F., ‘Mass law studies, II’, Journal of Physical Chemistry (1897), 1, 461–73.CrossRefGoogle Scholar

50 Guthrie, F., ‘On eutexia’, Philosophical Magazine, series 5 (1884), 17, 462–82.Google Scholar

51 Bancroft, , op. cit. (43), 408–9, and fig. 2.Google Scholar

52 Rankin, G. A., ‘The ternary System CaO–Al2O3–SiO2’, American Journal of Science, series 4 (1915), 34, 1–79.CrossRefGoogle Scholar

53 Mead, W. J., ‘Redistribution of elements in the formation of sedimentaiy rocks’, Journal of Geology (1907), 15, 238–56, figs. 3, 6CrossRefGoogle Scholar; Reid, A., ‘The igneous rocks near Pajaro’, Bulletin of the Department of Geology, University of California (1902), 3, 173–90, and plate 18.Google Scholar

54 van Philipsborn, H., ‘Zur graphischen Behandlung quaternärer Systeme’, Neues Jahrbuch für Mineralogie, Geologie und Paläontologie, Mügge-Festband (1928), supplementary volume 57A, 973–1012.Google Scholar

55 Iddings, J. P., Igneous Rocks. I. Composition, Texture and Classification, 2 vols., New York, 1909, i, fig. 3Google Scholar; Holmes, A., Petrographie Methods and Calculations with some Examples of Results Achieved, London, 1921, figs. 48, 72–78Google Scholar; Niggli, P. and Beger, P. J., Gesteins-und mineralprovinzen. Band I. Einführung. Zielsetzung. Chemismus der Eruptivgestine, insbesondere der Lamprophyre, Berlin, 1923, figs. 6, 7, 9aGoogle Scholar; Rosenbusch, H. and Osann, A., Elemente der Gesteinslehre, 4th edn, Stuttgart, 1923, plates I–IIIGoogle Scholar; Niggli, P., Lehrbuch der Mineralogie. I. Allgemeine Mineralogie, Berlin, 1924, figs. 143, 494–8Google Scholar, and Tabellen zur allgemeinen und speziellen Mineralogie, Berlin, 1927, part II, figs. 9, 19.Google Scholar

56 Rollinson, H., Using Geochemical Data: Evaluation, Presentation, Interpretation, Harlow, 1993.Google Scholar

57 Richard, L. R., Minpet Mineralogical and Petrological Data Processing System, vers. 2.0, Minpet Geological Software, Quebec, 1994.Google Scholar

58 Mead, J. W., ‘Occurrence and origin of the bauxite deposits of Arkansas’, Economic Geology (1915), 10, 28–54, and fig. 7CrossRefGoogle Scholar; Tomkeieff, S. I., ‘Clay minerais and bauxitic minerais. A review and classification based on a statistical method’, Mineralogical Magazine (1934), 23, 463–81, and figs. 1, 2, 7Google Scholar; Krumbein, W. C. and Pettijohn, F. J., Manual of Sedimentary Petrography, New York and London, 1938Google Scholar; Trefethen, J. M., ‘Classification of sediments’, American Journal of Science (1950), 248, 55–62CrossRefGoogle Scholar; Folk, R. L., ‘The distinction between grain size and mineral composition in sedimentary rock nomenclature’, Journal of Geology (1954), 62, 334–59CrossRefGoogle Scholar; Shepard, F. P., ‘Nomenclature based on sand-silt-clay ratios’, Journal of Sedimentary Petrology (1954), 24, 151–8Google Scholar; Folk, R. L., ‘Practical petrographic classification of limestones’, Bulletin of the American Association of Petroleum Geologists (1959), 43, 1–38Google Scholar; Dott, R. H., ‘Wacke, graywacke and matrix – what approach to immature sandstone classification?’, Journal of Sedimentary Petrology (1964), 34, 625–32Google Scholar; Robinson, G. W., Soils, Their Origin Constitution and Classification, 3rd edn, London, 1949Google Scholar; Avery, B. W., ‘Soil classification in the soil survey of England and Wales’, Journal of Soil Science (1973), 24, 324–38CrossRefGoogle Scholar; FitzPatrick, E. A., Soils. Their Formation, Classification and Distribution, Harlow, 1983Google Scholar; Hodge, E. T., ‘The composition of waters in mines of sulphide ores’, Economic Geology (1915), 10, 123–39CrossRefGoogle Scholar; Piper, A. M., ‘Graphic procedure in the geochemical interpretation of water-analyses’, Transactions of the American Geophysical Union (1944), 25, 914–28.CrossRefGoogle Scholar

59 Pearson, E. S., ‘“Student” as Statistician’, in Studies in the History of Statistics and Probability (ed. Pearson, E. S. and Kendall, M.), 2 vols., London, 1970, i, 382–8Google Scholar; Boland, P. J., ‘A biographical glimpse of William Sealey Gossett’, American Statistician (1984) 38, 179–83Google Scholar; Pearson, E. S., ‘Student’ A Statistical Biography of William Sealey Gossett (ed. Plackett, R. L. and Barnard, G. A.), Oxford, 1990Google Scholar; Box, J. F., R. A. Fisher. The Life of a Scientist, New York, 1978Google Scholar; Stigler, S. M., The History of Statistics: The Measurement of Uncertainty before 1900, Cambridge, MA, 1986, 243.Google Scholar

60 Claringbold, P. J., ‘Use of the simplex design in the study of the joint action of related hormones’, Biometries (1955), 11, 184–95, and fig. 1.CrossRefGoogle Scholar

61 Scheffé, H., ‘Experiments with mixtures’, Journal of the Royal Statistical Society, series B (1958), 20, 344–60, figs. 1–3Google Scholar; and ‘The simplex-centroid design for experiments with mixtures’, Journal of the Royal Statistical Society, series B (1963), 25, 235–63.Google Scholar

62 Cornell, J. A., Experiments with Mixtures, New York, 1981Google Scholar; Diamond, W. J., Practical Experimental Designs for Engineers and Scientists, Belmont, CA, 1981Google Scholar; Montgomery, D. C., Design and Analysis of Experiments, 3rd edn, New York, 1991.Google Scholar

63 Hare, L. B. and Brown, P. L., ‘Plotting response surface contours on a three-component mixture space’, Journal of Quality Technology (1977), 9, 193–7CrossRefGoogle Scholar; Koons, G. F. and Heasley, R. H., ‘Response surface contour plots for mixture problems’, Journal of Quality Technology (1981), 13, 207–14.CrossRefGoogle Scholar