Maxwell and the science of colour

BY MALCOLM S. LONGAIR*

Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, UK

This non-technical review of Maxwell’s contributions to the quantitative theory ofcolour was presented at a symposium in Aberdeen to celebrate the 150th anniversaryof his appointment as professor of natural philosophy at Marischal College. Maxwellmaintained his interest in the science of light and colour from his childhood to the lastdecade of his life. He lavished the same care and imagination on these studies as he did onhis epochal contributions to electromagnetism and statistical physics.

Keywords: James Clerk Maxwell; colour; light box; history

On

*m

1. Introduction

James Clerk Maxwell’s contributions to electromagnetism, the kinetic theory ofgases, thermodynamics and statistical mechanics are rightly recognized as amongthe greatest contributions to science, providing the essential link betweenNewtonian physics and the revolutions in physics that were to take place in theearly decades of the twentieth century. At the same time, he had a deep interestin many other topics. In particular, his interest in the science of colour remainedwith him throughout his life, from early childhood to his last decade. In thispaper, Maxwell’s achievements in the science of colour are set in their historicalcontext. The story will begin with the history of light and colour from Snell toYoung. Then, Maxwell’s early interest in colour phenomena will be discussed,leading to his development of the quantitative theory of colour vision.

2. From Snell to Young

In 1621, Willebrord van Snel van Royen discovered the law of refraction when alight ray crosses the interface between two media. In modern notation, thewell-known form of what is known as Snell’s law is written

n1 sin q1 Zn 2 sin q2; ð2:1Þwhere n1 and n2 are the refractive indices of the two media and q1 and q2 are theangles of incidence and refraction. This is the law which is at the heart of optics,lenses, microscopes and telescopes. One of the first applications of Snell’s law wasmade by Rene Descartes to account for the formation of rainbows. He studied theinternal reflection of light within spherical raindrops and showed that there is aminimum angle of deflection of the light rays amounting to 428 for a single

Phil. Trans. R. Soc. A (2008) 366, 1685–1696

doi:10.1098/rsta.2007.2178

Published online 25 January 2008

e contribution of 20 to a Theme Issue ‘James Clerk Maxwell 150 years on’.

sl1000@cam.ac.uk

1685 This journal is q 2008 The Royal Society

a

b D

E

B

d

F

O

Figure 1. A sketch of the origin of single and double rainbows. (Adapted from Descartes’ LesMeteores of 1637.)

M. S. Longair1686

reflection within the raindrop. If there are two internal reflections, the minimumangle is 518. This theory predicted correctly the observed angles of single anddouble rainbows with respect to a line from the Sun through the observer’s headto the centre of the rainbow. A sketch of the origin of single and double rainbowsappeared in Descartes’ Les Meteores of 1637 (figure 1). There was, however, nounderstanding of the origin of the colours in the rainbow. This was to beelucidated by Isaac Newton.

In 1665–1666, Newton was aged 22 and in these years he discovered thebinomial theorem, the integral and differential calculuses, the theory of colour inoptics and unified celestial mechanics and the theory of gravity. These werestaggering achievements. In the field of optics and colour, the key experiment ofthese years was Newton’s ‘experimentum crucis’. He passed sunlight though ahigh quality prism and observed that white light was split up into the colours ofthe spectrum. He then passed this spectrum through a hole that allowed only oneof the colours to be selected. On passing this single colour through a secondprism, he found that the light was not split up into any further colours. InNewton’s words, ‘Light itself is a heterogeneous mixture of differently refrangiblerays’ (Turnbull et al. 1959–1977).

This result gave rise to one of Newton’s many disputes with other scientists.The French experimenters could not reproduce the result of the experimentumcrucis, but found instead that the separate colours could be split up into furthercolours. It turned out that Newton had found the correct answer owing to his useof higher quality prisms than those available to the French scientists, but it wasmany years before Newton’s view prevailed (Shaffer 1989).

Since different colours are refracted through different angles, white light is notfocused at a single point by a lens, the phenomenon known as chromaticaberration. To get round his problem, Newton invented the all-reflecting, or

Phil. Trans. R. Soc. A (2008)

P

purpleA Sol

Laindigo

E

G

H

I

K

M

blue

green

yellow

orange

red

Fa

Sol

La

Mi

Fa

B Sol

C

D

T

(a) (b)

Figure 2. (a) Newton’s all-reflecting telescope. The sketches above the telescope show a comparisonof a crown observed with Newton’s telescope (fig. 2) and a refracting telescope (fig. 3; Newton1672). (b) Newton’s division of the spectrum of white light into seven colours, with the sevenseparate tones of the harmonic scale shown to the right of the spectrum. (Adapted from P. Cook inFauvel et al. (1988, p. 118).)

1687Maxwell and the science of colour

Newtonian, telescope (figure 2a). He built the telescope himself, including thegrinding of the mirrors. The resulting telescope had superior imaging qualitieswhen compared with the Galilean refracting telescope, as he demonstrated in hispaper published in the Phil. Trans. R. Soc. in 1672.

Newton divided the spectrum into seven colours, the famous VIBGYORsequence of violet, indigo, blue, green, yellow, orange and red (figure 2b). In hischoice of seven colours, he was influenced by the ideas of harmonic proportion,which had been set out in Kepler’s great treatise The harmony of the world of1618 and which had influenced all aspects of the arts and sciences in theseventeenth and eighteenth centuries. The treatise contained the discovery ofKepler’s third law of planetary motion, which was crucial in Newton’s discoveryof the law of gravity, and so there is no doubt that Newton knew the treatisewell. There are seven different notes in the just harmonic scale, their frequenciesall being in the ratios of small whole numbers, and so Newton divided thespectrum into seven colours. Newton’s seven colour theory was elaborated byVoltaire in his Elemens de la Philosophie de Neuton of 1738.

The next notable figure in the story is Thomas Young, whose brilliantresearches of 1801 not only put the wave theory of light on a firm physicalfoundation but also resulted in the theory of three-colour vision. His most famousexperiment was the double-slit experiment in which he found that, when sunlightis passed through a single narrow slit and then through a pair of similar slits, acharacteristic pattern of bright and dark bands is observed on a screen. He

Phil. Trans. R. Soc. A (2008)

M. S. Longair1688

interpreted the appearance of this pattern of light rays in terms of theconstructive and destructive interference of light waves. This was conclusiveevidence in favour of the wave theory of light.

In the same paper, he also discussed how the eye would perceive lights ofdifferent wavelengths or colours. In his picture, the light receptors in the retina ofthe eye act as resonators which are excited by the incoming light waves. Herealized, however, that there was a problem with this picture. In his words:

Phil. T

As it is almost impossible to conceive each sensitive point of the retina to contain an infinitenumber of particles, each capable of vibrating in perfect unison with every possible undulation,it becomes necessary to suppose the number limited: for instance to the three principal colours,red, yellow and blue. and that each of the particles is capable of being put intomotionmore orless forcibly by undulations differing less or more from perfect unison. . each sensitivefilament of the nerve may consist of three portions, one for each principal colour.

(Young 1802, pp. 20–21)

There are several interesting points about Young’s proposal. First, he took thethree principal colours to be red, yellow and blue. Second, there was nodistinction between the mixing of lights and the mixing of pigments. Third, thetheory was qualitative rather than quantitative.

3. The young Maxwell

James Clerk Maxwell came from a distinguished Scottish family. Although bornin Edinburgh, he spent most of his childhood at the family estate at Glenlair inthe Dumfries and Galloway region. By great good fortune, James’s cousin,Jemima Wedderburn, who was 8 years his senior, was a brilliant artist whopainted scenes from family life almost every day. Her watercolours, a number ofwhich can be viewed on the James Clerk Maxwell Foundation’s website (http://www.clerkmaxwellfoundation.org), are a wonderful record of life at Glenlair andEdinburgh. Among my favourites are the paintings of James and Jemima‘tubbing’, meaning using washing tubs as coracles in a nearby duck pond. Jameshad an insatiable curiosity about everything. According to the biography byCampbell & Garnett (1882), from his earliest years, he would continually ask,‘Show me how it doos’, or ‘What’s the go o’that?’ If he did not receive asatisfactory answer to the latter question, he would ask, ‘But what’s theparticular go o’that?’

At home, James’s father, John Maxwell, James and Jemima were fascinatedby the new range of optical toys which became available during the 1830s. Theserelied upon the phenomenon of the persistence of vision by which the eye andbrain preserve the image for approximately 1/20th of a second. As in acinematographic film, the appearance of movement is obtained by viewing theimages faster than the eye can track individual frames.

The thaumatrope was invented in 1825 by the London physician John Paris,who made the toy popular. Two images are printed on either side of a circularcard which was then rotated rapidly about a diameter by twisted stringsattached at opposite ends of this diameter. Owing to the persistence of vision, theimages on either side of the disc are observed to be superimposed.

In 1832, Belgian physicist Joseph Plateau and his sons introduced thephenakistoscope, meaning ‘spindle viewer’. About a dozen images were drawn on

rans. R. Soc. A (2008)

Figure 3. Maxwell’s improved zoetrope of 1861. By inserting concave lenses instead of slits, thevirtual image appears on axis while the drum is rotated and the field of view is considerablyincreased. (Courtesy of the Cavendish Laboratory, Cambridge.)

1689Maxwell and the science of colour

the disc which was then rotated about its axis and the images viewed in a mirrorby looking through radial slits cut in the disc. The viewer sees the images veryrapidly one after the other, producing the effect of a moving image.

The phenakistoscope was soon overtaken by the zoetrope which was inventedin 1834 by William Horner. Now, the images were painted on a strip which waspasted to the inside of a rotating cylinder. The images were viewed through theslits in the rotating drum. Zoetropes became popular, a 1905 supplement of theNew York Sunday American and Journal newspaper including a cardboard cut-out entitled ‘Make Your Own Zoetrope’ (this can be downloaded from the web-site http://brightbytes.com/collection/zoetrope.pdf).

James and Jemima delighted in these scientific toys. As expressed byCampbell & Garnett:

Phil. T

This was a source of endless amusement to the two cousins, the younger generally contriving,

and in part executing, the elder giving life and spirit to the creatures represented. The cow

jumping over the waxing and waning moon, the dog pursuing the rat in and out of his hole,

the circus horse, on which the man is jumping through the hoop, have the firmness and truth

of touch, the fullness of life, familiar to the many admirers of [Jemima Wedderburn]—the

tadpole that wriggles from the egg and changes gradually into a swimming frog; the cog-

wheels moved by the pendulum, and acting with the precision of clockwork.

(Campbell & Garnett 1882, p. 23)

James was not yet 10 when they invented many of their own designs. Jamesremained fond of the zoetrope and in 1861 improved its performance by insertingconcave lenses instead of slits on the drum so that the virtual image appeared on

rans. R. Soc. A (2008)

Figure 4. Maxwell aged 24 with his colour top. (Courtesy of the James Clerk Maxwell Foundation.)

M. S. Longair1690

axis while it rotated (figure 3). The resulting image was much improved and awider field of view observable. He set this invention as a problem in theCambridge 1869 tripos examinations.

James attended Edinburgh Academy from 1841 to 1847 and lived at the houseof his aunt Isabella Wedderburn. He was regarded as somewhat eccentric by hisschoolmates. As his biographers wrote, ‘Some eccentricity of behaviour earnedhim the name Dafty’ (Campbell & Garnett 1882). But he took this all in goodpart and his originality was soon recognized by his school friends. This was alsothe period of his first scientific papers. In 1846, his paper On the description ofoval curves was read to the Royal Society of Edinburgh by Prof. James DavidForbes.1

After high school, Maxwell proceeded to Edinburgh University where amongthe books he borrowed from the library were Newton’s Optics and Poisson’sMechanics. During his years at Edinburgh University, he published papers On

1Many more details of Maxwell’s life and scientific works can be found in the books by Everitt(1976), Harman (1990, 1995, 1998, 2002) and Niven (1890).

Phil. Trans. R. Soc. A (2008)

painted paper discs

colour sample

wood dowel

wood disc

nut and washer

1.00.9 0.9

1.0

0.8 0.80.7 0.7

0.6 0.60.5 0.5

0.2

0.1

0

0.5

0.6

0.7

0.8

0.9

1.0

0.20.1

0

0.20.1

0

0.40.3 0.3

0.3 0.4

0.4

blue red

green

(a)

(c)

680630

610600

590

580

570

560

550

540530

520 CIE 1931 chromaticity diagram

1931 2-degree observer

efg's computer labwww efg2.com/lab

510

490

480

4700.1

0.2

0.4

0.5

0.6

0.7

0.8

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

460420

(b)

(d )

Figure 5. (a) Illustrating the three coloured discs used in Maxwell’s colour top (see http://www.handprint.com/HP/WCL/colortop.html). (b) Maxwell’s original version of the colour triangle(Campbell & Garnett 1882). (c) Comparison with a modern version, showing quantitatively theproportions of different primary colours needed to synthesize those within the triangle. The pictureis rotated 1208 clockwise with respect to (b) (see http://www.handprint.com/HP/WCL/colortop.html). (d ) The CIE diagram. The pure colours in the white light spectrum are shown around theperimeter of the diagram from 420 to 680 nm.

1691Maxwell and the science of colour

the theory of rolling curves (1849) and The equilibrium of elastic solids (1850) inthe Transactions of the Royal Society of Edinburgh.

In 1847, Maxwell was taken by his uncle John Cay to visit the opticallaboratory of William Nicol, the distinguished optician and inventor of the Nicolprism. Maxwell later recalled that:

Phil. T

I was taken to see [William Nicol], and so, with the help of ‘Brewster’s Optics’ and a glazier’s

diamond, I worked at polarisation of light, cutting crystals, tempering glass, etc.

In 1848, he undertook a series of experiments on the chromatic effects of polarizedlight in doubly refracting materials, crystals and mechanically strained glasses.He remarked on the ‘gorgeous entanglements of colours’ in strained glasses.

Maxwell went up to Trinity College, Cambridge, in 1850 and was to remainthere until his appointment to the Chair of Natural Philosophy at Marischal

rans. R. Soc. A (2008)

M. S. Longair1692

College, Aberdeen in 1856. In his letter of recommendation, Forbes wrote toWilliam Whewell, Master of Trinity College:

Phil. T

Pray do not suppose that . I am not aware of his exceeding uncouthness, as wellMathematical as in other respects . I thought the Society and Drill of Cambridge the onlychance of taming him and much advised his going.

In his obituary of Maxwell, his great friend Peter Guthrie Tate wrote:

. he brought to Cambridge in the autumn of 1850, a mass of knowledge which was reallyimmense for so young aman, but in a state of disorder appalling to hismethodical private tutor.

4. Maxwell’s quantitative theory of colour mixing

In 1855, Maxwell was awarded a fellowship at Trinity College. Among his manyinterests, he studied the composition of light by means of his colour top, which wascentral to his first major assault upon the quantitative theory of colour (figure 4).

The three coloured paper discs could be clamped to the wooden disc of the topin such a way as to allow different amounts of the primary colours to be mixedwhen the top was spun (figure 5a). A smaller central disc contained the coloursample which was to be matched by adding together different amount of theprimary colours. To eliminate the effect of the different brightnesses of thecentral disc and the outer colours, different amounts of black could be added tothe central disc. Maxwell demonstrated that all colours could be synthesized bydifferent combinations of the three primary lights, red, green and blue. He alsodistinguished clearly between the results of mixing lights of different colours andmixing pigments. For example, he confirmed Helmholtz’s discovery that mixingblue and yellow light does not produce green, but rather ‘a pinkish tint’.

By 1855, Maxwell had adopted the three-colour receptor theory of Youngwith primary lights red, green and blue, and determined experimentally colourequations which quantified how much of each primary colour was necessary tocreate any particular colour. All colours could then be represented on a colourtriangle in which the distance from the corners indicated how much of eachprimary colour has to be mixed (figure 5b). Note that these colour diagramsare projections onto a two-dimensional plane of three-dimensional colourspace. He carried out his colour matching experiments with many independentobservers and found little variation between them. These experiments alsosuggested an explanation for colour blindness if one of the three sets ofreceptors in the eye was not present. Figure 5c shows a modern version ofMaxwell’s colour triangle.

5. Maxwell at Marischal College, Aberdeen

Maxwell accepted the post of professor of natural philosophy at MarischalCollege, Aberdeen in 1856 to be closer to his family and the estate at Glenlair.His father died in 1856 and in 1859 he married Katherine Mary Dewar, daughterof the principal of Marischal College. He was made redundant when Marischaland King’s colleges combined in 1860.

rans. R. Soc. A (2008)

1693Maxwell and the science of colour

Maxwell’s 5 years at Aberdeen were among the most innovative and importantperiods of his life. In 1857, he won the prestigious Adams Prize for his work on Themotion of Saturn’s rings. This was also the period of his papers onThe dynamical top(1857) andThe theory of colours (1857, 1859–1860). Hewas also developing his ideason electromagnetism which were to reach their culmination over the succeeding 5years with the formulation of what we now call Maxwell’s equations.

Maxwell was not fully satisfied with the results of his experiments with the colourtop and so devised a series of ‘light boxes’ which enabled different amounts of thethree primary lights to be mixed more precisely. The perfected light box,constructed by the firm of Smith and Ramage of Aberdeen, was portable and socould be used as a tool to study the colour sensitivities of many subjects (figure 6).

To understand how the light box worked, let us first use it backwards(figure 7a). If white light is shone through the eyepiece at A, on passing throughthe prisms and mirror, the normal spectrum of colours is produced at B. At B,there are three adjustable slits which allow different amounts of blue, green andred light to be transmitted.

If we now shine white light onto the slits at B, only the blue, green and redcolours are combined at A (figure 7b). The amount of each colour could beprecisely measured from the width of the slits at B. In addition, a comparisonwhite light beam could be viewed at A. To obtain an intense enough light source,bright sunlight was shone onto white paper to provide a white light source.

Maxwell used the light box to study many different aspects of colour vision forlarge samples of subjects. The experiments gave precise information about the com-positionofdifferent colours andprovided the forerunnersof themodernchromaticity,or Commission Internationale d’Eclairage (CIE), diagrams which define howdifferent colours can be synthesized from chosen primary colours. These conveyinformation about the different variables of colour vision, namely, the spectral colourand the degree of saturation. An example of such a diagram is shown in figure 5d.

Maxwell was awarded the Rumford Medal of the Royal Society of London in1860 for his researches in colour vision. He continued using light boxes after hemoved toKing’s College London in 1860. He designed and had built an 8 ft light boxwhich was housed in a large garret which ran the whole length of his house at 8Palace Gardens Terrace, Kensington in London. Campbell & Garnett noted that:

Phil. T

When experimenting at the window with the colour-box ., he excited the wonder of hisneighbours, who thought him mad to spend so many hours in staring into a coffin.

(Campbell & Garnett 1882, p. 223)

Also in 1861,Maxwell took the first coloured photograph. A small piece of tartanribbon was photographed by a professional photographer on three plates throughred, green and blue-violet filters. Three positive plates were produced and thesewere projected through the same filters onto a screen. When these images werecombined, a reasonably fully coloured image was produced. This additive three-colour separation technique is employed nowadays in colour astrophotography.

6. Maxwell’s legacy

Maxwell’s studies provided the basis for the present understanding of the processesof colour vision and also for the quantitative measurement of colour whichunderpins a huge variety of modern manufacturing and service industries. The light

rans. R. Soc. A (2008)

Figure 6. Maxwell’s light box constructed by the firm of Smith and Ramage of Aberdeen.(Courtesy of the Cavendish Laboratory.)

bluegreenred

bluegreenred

B

A

prisms

mirror

white light

whitelight

A

C

comparison beamof white light

B

working in reverse, only the rays reachingthe eyepiece are shown

(a)

(b)

Figure 7. (a) Maxwell’s light box working backwards. The diagram illustrates the dispersion ofwhite light when it is shone through the eyepiece onto the reflecting mirror and through the pair ofdispersing prisms. (b) Maxwell’s light box working forwards. The blue, green and red regions of thespectrum are focused at A. The amount of each of these lights is precisely varied by changing thewidth of the entrance slits.

M. S. Longair1694

sensors in the retina are the rods and cones. The rods are very sensitive and canregister the arrival of individual photons of light but have no colour discrimination.The cones are less sensitive but are sensitive to light centred on red, green and bluewavelengths. The sensitivity curves shown in figure 8 illustrate versions on normalthree-colour vision for different species and different forms of colour blindness.

Phil. Trans. R. Soc. A (2008)

1.0 (a)

(b)

(c)

0

1.0

0

1.0

0400

wavelength (nm)500 600 700

rela

tive

sens

itivi

ty

Figure 8. Light absorption curves for the pigments of cone cells. The curves show the relativesensitivities of the three types of light cone as a function of wavelength, normalized to unity atmaximum sensitivity. (a) Dichromicity (two-colour vision) that is found in non-primate mammalsand 2% of human males. (b) Anomalous trichromicity, found in 6% of human males. (c) Normaltrichromicity (three-colour vision) found in normal human vision, Old World monkeys and apes.(After J. Mollon, in Lamb & Bourrieau (1995, p. 129).)

1695Maxwell and the science of colour

What we actually perceive is determined by a large number of other factorsand this takes us into the realm of physiology and psychology (e.g. Lamb &Bourrieau 1995). But the important lesson from Maxwell’s colour mixingexperiments is that it is only when monochromatic light is used that there is aunique relationship between colour and wavelength. Any particular colour can beproduced in a number of different ways provided the same total signals areapplied to the three different cone types.

Phil. Trans. R. Soc. A (2008)

M. S. Longair1696

Maxwell understood that pigments are in a sense the opposite of lights. Theyabsorb some wavelengths and reflect others. For example, suppose we shinewhite light onto a pigment which absorbs strongly in the central region of theoptical spectrum. The light we see reflected consists of the red and blue ends ofthe spectrum, resulting in the colour magenta. This understanding of thedifference between mixing lights and pigments also solved the ‘green problem’.Blue and yellow pigments absorb the ends of the spectrum and so, when they aremixed, the residual reflected light is centred about green wavelengths.

What we actually perceive is determined by many other factors—is the surfacesmooth or rough, is it highly reflective or polarizing, and so on? Furthermore, theblue cones are rather sparsely distributed on the retina. This can fool the eye ifthe object is observed with low angular resolution.

Maxwell continued his studies of colour and optical phenomena during hissubsequent career at Kings’ College London, at his home at Glenlair and at theCavendish Laboratory, Cambridge. From ca 1860 onwards was the period of hisdevelopment of his deep insights into electromagnetism, the unification of lightand electromagnetism, thermodynamics and statistical mechanics, for which heis rightly best remembered. It was characteristic of him, however, that hebrought to the quantitative study of colour the same imagination andexperimental skill which he displayed in all his other researches. It is remarkablethat he was able to maintain his interests over such a vast range of the physicalsciences and that he continued to make fundamental contributions to all of themthroughout his career.

References

Campbell, L. & Garnett, W. 1882 The life of James Clerk Maxwell. London, UK: MacMillan and Co.Everitt, C. W. F. 1976 James Clerk Maxwell: physicist and natural philosopher. New York,

NY: Scribner.Fauvel, J., Flood, R., Shortland, M. & Wilson, R. (eds) 1988. Let Newton be: a new perspective on

his life and works. Oxford, UK: Oxford University Press.Harman, P. M. (ed.) 1990, 1995, 2002 The scientific letters and papers of James Clerk Maxwell,

vols 1–3 (1846–1862, 1862–1873, 1874–1879). Cambridge, UK: Cambridge University Press.Harman, P. M. 1998 The natural philosophy of James Clerk Maxwell. Cambridge, UK: Cambridge

University Press.Lamb, T. & Bourrieau, J. (eds) 1995 Light and colour. Darwin College Lectures. Cambridge, UK:

Cambridge University Press.Newton, I. 1672 An accompt of a new catadrioptrical telescope invented by Mr Newton. Phil.

Trans. 7, 4004–4010. (doi:10.1098/rstl.1672.0003)Niven, W. D. (ed.) 1890 The scientific papers of James Clerk Maxwell. Cambridge, UK: Cambridge

University Press. (Two volumes.) These are also available in one volume fromDover Publications,1965.

Schaffer, S. 1989 Glass works: Newton’s prisms and the uses of experiment. In The uses ofexperiment: studies in the natural sciences (eds D. Gooding, T. Pinch & S. Schaffer), pp. 67–104.Cambridge, UK: Cambridge University Press.

Turnbull, H. W., Scott, J. F., Hall, A. R. & Tilling, L. (eds) 1959–1977 The correspondence ofIsaac Newton, vol. 1, p. 95. Cambridge, UK: Cambridge University Press.

Young, T. 1802 The Bakerian lecture: on the theory of light and colours. Phil. Trans. R. Soc. Lond.92, 12–48. (doi:10.1098/rstl.1802.0004)

Phil. Trans. R. Soc. A (2008)