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The history of the rainbow is as old as that of science. The ancient Greek philosophers tried to describe the rainbow, and Aristotle was the first to fully include it among the phenomena studied by physicists. Sunlight reflected in the clouds, the incidence of light rays, the reason for the rainbow’s circular shape, the optical effect of an infinite depth are aspects that have for centuries intrigued scholars, who studied the rainbow with a mixture science and alchemy, sense and sensibility. In the 17th century the rainbow became a strictly physical phenomenon, the object of rigorous investigations according to the law of reflection and refraction. Here we survey this often forgotten history, from ancient Greeks to modern scientists, the rainbow’s colours belonging to the world of physics but also—as Thomas Young wrote in 1803—to the world of speculation and imagination.
1 23
Lettera Matematica
International edition
ISSN 2281-6917
Lett Mat Int
DOI 10.1007/s40329-016-0127-3
A short history of the rainbow
Massimo Corradi
1 23
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A short history of the rainbow
Massimo Corradi
Centro P.RI.ST.EM, Universita
`Commerciale Luigi Bocconi 2016
Abstract The history of the rainbow is as old as that of
science. The ancient Greek philosophers tried to describe
the rainbow, and Aristotle was the first to fully include it
among the phenomena studied by physicists. Sunlight
reflected in the clouds, the incidence of light rays, the
reason for the rainbow’s circular shape, the optical effect of
an infinite depth are aspects that have for centuries intri-
gued scholars, who studied the rainbow with a mixture
science and alchemy, sense and sensibility. In the 17th
century the rainbow became a strictly physical phe-
nomenon, the object of rigorous investigations according to
the law of reflection and refraction. Here we survey this
often forgotten history, from ancient Greeks to modern
scientists, the rainbow’s colours belonging to the world of
physics but also—as Thomas Young wrote in 1803—to the
world of speculation and imagination.
Keywords Optics Theory of colours Rainbow
‘Hic ubi sol radiis tempestatem inter opacam
adversa fulsit nimborum aspargine contra
tum color in nigris existit nubibus arqui’’.
(Lucretius, De rerum natura, VI, 524–526).
1 Introduction
The history of the rainbow is as old as the history of
science itself. As long ago as in the 3rd-2nd centuries
BCE, Alexander of Aphrodisias tried to describe the
rainbow as a phenomenon involving light and colour; he
is regarded as the discoverer of the darker region
between the primary and the secondary rainbows
(Fig. 1).
However, Aristotle (384/383-322 BCE) was the first one
to give a complete description of the optical phenomenon,
in Book III of his Meteorology (here given in the transla-
tion by E.W. Webster):
The rainbow never forms a full circle, nor any
segment greater than a semicircle. At sunset and
sunrise the circle is smallest and the segment lar-
gest: as the sun rises higher the circle is larger and
the segment smaller. After the autumn equinox in
the shorter days it is seen at every hour of the day,
in the summer not about midday. There are never
more than two rainbows at one time. Each of them
is three-coloured; the colours are the same in both
and their number is the same, but in the outer
rainbow they are fainter and their position is
reversed. In the inner rainbow the first and largest
band is red; in the outer rainbow the band that is
nearest to this one and smallest is of the same col-
our: the other bands correspond on the same prin-
ciple. These are almost the only colours which
painters cannot manufacture: for there are colours
give red, green, or purple. These are the colours of
the rainbow, though between the red and the green
an orange colour is often seen.
&Massimo Corradi
Dipartimento di Scienze per l’Architettura, Scuola
Politecnica, Universita
`degli Studi di Genova, Stradone di
Sant’Agostino 37, 16121 Genoa, Italy
Lett Mat Int
DOI 10.1007/s40329-016-0127-3
Author's personal copy
At that point the rainbow fully became one of the phe-
nomena investigated by physicists. However, according to
Lee and Fraser [31]:
Despite its many flaws and its appeal to Pytha-
gorean numerology, Aristotle’s qualitative expla-
nation showed an inventiveness and relative
consistency that was unmatched for centuries.
After Aristotle’s death, much rainbow theory
consisted of reaction to his work, although not all
of this was uncritical.
In Aristotle’s description, the colours of the rainbow are
just three; this interpretation was accepted for a long time,
with subtle numerological differences associating the three
colours to the Trinity or, in other cases, four colours to the
four elements of Empedocles’ tradition. The way sunlight
gets reflected among the clouds, the analysis of the angle of
incidence of light rays, an explanation of the circular shape
of the rainbow, the optical effect of an infinite depth with
respect to the origin of the phenomenon are all questions
that for centuries excited the curiosity of scholars working
in various fields.
In his Naturales Quaestiones (ca. 65 CE), Lucius
Annaeus Seneca (ca. 4 BCE-65 CE), devotes some chap-
ters of Book I to explaining this phenomenon. He suggests
that the rainbow, which always appears opposite the sun, is
created by the reflection of sunrays on water droplets, as
well as by their reflection in a hollow cloud. He describes
how it is possible to observe a rainbow when a light ray
passes through a glass cylinder, thus anticipating the
experiments by Isaac Newton (1642–1727) with the optical
Roger Bacon (1214–1294), Theodoric of Freiberg
(Meister Dietrich, Theodoricus Teutonicus de Vriberg,
ca. 1250–ca. 1310),
and Rene
´Descartes (1596–1650)—
to mention just a few—investigated the phenomenon in a
speculative way, mixing science and alchemy, sense and
sensibility: the rainbow’s colours arrive to the eye by
way of effects that are physical and sensory, interpreta-
tive and experiential. We owe to Willebrord Snell
(Willebrordus Snellius, 1580–1626) the insight of 1621
that the rainbow is a strictly physical phenomenon and,
as such, has to be studied rigorously, according to the
physico-mathematical laws of reflection and refraction.
Later, in 1666, Newton understood that the refractive
index depends on the wavelength: hence, each sunray
generates its own rainbow.
In this short note, we intend to retrace an often forgotten
history which, from the first insights by the Greek
philosophers to modern science, has characterised research
in a field of physics where the rainbow’s colours belong
both to the natural world—in 1803 Thomas Young
(1773–1829) gave through a simple experiment, midway
between investigation and speculation, ‘‘so simple and so
demonstrative a proof of the general law of the interference
of two portions of light’’ [47], that two light rays from a
single source passing through two slits may interfere with
each other and produce on a screen alternate light and dark
stripes—and hence the theoretical world, and to the world
of imagination.
2 The beginnings of our history
The first scientific—in the modern meaning of the
word—studies date back to the Arabic Middle Ages: the
Persian astronomer and mathematician Qutb Al-Dı¯n al-
Shira¯zı¯ (1236–1311) and his pupil al-Fa¯ri¯, also known
Fig. 1 Primary and secondary rainbows with, between them,
‘Alexander’s dark band’’ (a reference to Alexander of Aphrodisias):
The Aegean Sea (c. 1877) by Frederic Edwin Church (1826–1900),
Metropolitan Museum of Art (public domain image)
In his treatise De iride et de radialibus impressionibus, Theodoric
provided a first scientific explanation, still considered phenomeno-
logically valid, of the rainbow, a phenomenon that had already
attracted the interest of such scholars as Robert Grosseteste (ca.
1175–1253), Roger Bacon and Witelo (Erazmus Ciolek Witelo, also
known as Vitellio, (ca. 1230-post 1280/ante 1314). The German
Dominican gave an interpretation of the rainbow as a result of
refraction of light in its spectrum of colours, even though he was not
actually a scientist, nor in particular an experimentalist, and as a
consequence he did not master the experimental method; neverthe-
less, he showed an attitude to research with a properly scientific
object. So the German scholar is a natural interpreter of studies ‘‘in
the tradition of Albertus Magnus’’ [see Elisa Chiti, article ‘‘Teodorico
di Freiberg’’ in online Manuale di Filosofia Medievale published by
the University of Siena, Faculty of Letters and Philosophy]. As
regards Theodoric’s work, see his Opera Omnia [44]; it includes the
writings De coloribus,De elementis corporum naturalium,De iride et
de radialibus impressionibus,De luce et eius origine,De miscibilibus
in mixto,De tempore. See also Venturi [45], part III: ‘‘Dell’Iride’’,
pp. 149–246 and Krebs [30].
See, in this regard, Corradi [14].
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as Kama¯l al-¯n (1260–1320),
tried to give a first
mathematical explanation of the rainbow, which was
quite accurate for its age, since it was based on the
phenomenon of refraction as described in the Book of
Optics by Alhazen (965–1039), a Persian mathematician
born in Basra, and on the studies by Avicenna
(980–1037) (Fig. 2a). Alhazen suggested that, in order to
be able to form a rainbow, sunlight had to be reflected by
clouds before reaching human eyes. Thus the water drops
composing the clouds reflect the light ray and create the
rainbow’s colours through a refraction and two or more
reflections, like an image forming in a reflecting,
smooth, concave, spherical mirror consisting of dense,
wet air.
It is undoubtedly difficult to verify this
experimentally, but, according to the Persian scholar, it
could be done by studying the phenomenon of refraction
of a light ray through a transparent glass sphere full of
water, a large-scale experimental model of a raindrop. In
order to evaluate the colour spectrum, it is necessary to
study the model in a dark room having a controlled
opening to let a sunray in. Both Alhazen and the poly-
math Avicenna further imagined that the rainbow forms,
not in the dark cloud itself, but in the narrow layer of fog
between the cloud, the sun and the observer. The cloud
acts as a background to this subtle substance, just like the
lining of quicksilver applied to the back of a mirror.
Several trials with accurate observations about reflection
and refraction allowed al-Fa¯ri¯toestablishthatthe
rainbow’s colours result from a decomposition of light,
as he reported in his treatise on optics (Kita
¯b Tanqı
¯zir, ca. 1319). How exactly colours are formed is
much harder to explain; al-Fa¯risı¯ imagines that they are
generated by a superposition of different versions of an
image on a dark background, mixed by light. In this he
relays a theory stated by his mentor al-Shira¯zı¯.
In Chinese science, at the time of Song dynasty
(960–1279), a versatile scholar named Shen Kuo
(1031–1095) suggested, like Sun Sikong (1015–1076)
before him, that a rainbow was formed by sunrays meeting
rain droplets suspended in the air,
according to the prin-
ciples underlying the modern scientific explanation.
In Europe, Albertus Magnus (1206–1280), in Chap-
ter XXVIII of his Libri quattuor meteororum (Cologne,
attributed to single raindrops, rather than to the
whole cloud, the formation of the rainbow, which hap-
pened not just by virtue of a simple reflection on a convex
surface, but by refraction too. So, in order to explain the
phenomenon, he introduced ‘‘reflection’’ and ‘‘refraction’
of light on drops, even though only reflection on the inner
surface of drops was supposed to create the coloured arc.
At the same time, the English philosopher Roger Bacon
(also remembered as the Doctor mirabilis), expanding on
the studies on light by Robert Grosseteste (ca.
1175–1253)—the founder of the tradition of scientific
thinking in medieval Oxford, according to Alistair
Cameron Crombie (1915–1996)—devoted himself to the
study of this phenomenon. Using an astrolabe, he attemp-
ted to measure the angle formed between the incident
sunlight and the light diffused by the primary and secondar
Fig. 2 a Above Title page of Opticae Thesaurus (Basileae, Friedrich
Risner, 1572), which includes Alhazen’s treatise on optics (Kita¯b al-
Mana¯zir), where rainbows, parabolic mirrors, images bent by
refraction of light in water and other optical effects are featured.
bBelow The refraction of light through a spherical bowl full of water,
from Roger Bacon or possibly Robert Grosseteste: Opus Majus
[British Library, manuscript: Royal 7 F. VIII, Page Folio Number:
f.25. Crombie [16] describes the image as follows: ‘‘Diagram illus-
trating Grosseteste’s theory, in De natura locorum (see pp. 122, 149)
of the focusing of the sun’s rays by a spherical lens; from Roger
Bacon’s Opus Maius, iv. ii. 2, MS Roy. 7. F. viii, f. 25v’’]
Kama¯l al-Dı¯n Hasan ibn Ali ibn Hasan al-Fa¯risı¯, or Abu Hasan
Muhammad ibn Hasan, a mathematician born in Tabriz (Iran), gave
important contributions to number theory and to the mathematical
theory of light, with interesting insights about colours and rainbows
(see Roshdi Rashed, s.v. al-Fa¯risı¯, in [48]).
See Gazi Topdemir [23].
See Boyer [7].
See Boyer [8], pp. 127–129.
See Sivin [43], p. 24.
See Albertus Magnus [2], Meteora, Liber III, Meteororum, Trac.
IV, Caput X ‘‘De causa efficiente et materiali colorum iridis in
communi’’, pp. 678–679.
See Hackett [27].
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arcs of a rainbow, obtaining values in the range 130–138.
The colours are light embedded within matter, stimulated
by an external light (from the sun); the more the matter is
stimulated, the brighter the light shines. In his experiments
he analysed the passage of light rays through crystals and
water droplets (Opus Majus, Part V: Optics and Part VI:
Experimental Science, 1628)
(Fig. 2b). Let us remark
that the treatise about optics by ibn al-Haytham had been
translated into Latin by Robert Grosseteste and almost
surely Roger Bacon knew it.
At the beginning of the 14th century, Theodoric of
Freiberg, a Dominican, suggested that the rainbow phe-
nomenon is due to sunlight being reflected through water
drops suspended in mid-air. He performed some experi-
ments by using spherical bowls filled with water and gave
an accurate description of both the primary and the sec-
ondary rainbows (Fig. 3a). According to the Saxon scholar,
the primary rainbow forms.
When sunlight falls on individual drops of moisture
[and] the rays undergo two refractions (upon ingress
and egress) and one reflection (at the back side of the
drop) before transmission to the eye of the observer
while the secondary rainbow is a phenomenon that,
analogously, implies two refractions and two reflections.
In Theodoric’s opinion, the colours are a mixture of two
qualities: brightness and darkness. The same ideas recur in
the 1571 treatise by Johann Fleischer (1539–1593) on
Aristotle’s theory of rainbows [21].
Gregor Reisch (ca. 1467–1523), in his Margarita
philosophica (Freiburg im Breisgau: Johannes Schott,
1503) claimed that a primary rainbow appears when light
reflects on a concave surface, and that the way it looks
depends on the multitude of drops. Girolamo Cardano
(1501–1576) elaborated on the theories by Albertus Mag-
nus, Roger Bacon, and Witelo. Francesco Maurolico
(1494–1575), in his Photismi de lumine et umbra [37],
probably one of the best works about optics in the 16th
claimed that light reflects both on the convex
external and on the concave internal surfaces of the water
drop, and that the rainbow was formed through a large
number of reflections (Fig. 3b). Giambattista Della Porta
(1535–1615), in his De refractione optices (Neapoli:
Iacobum Carlino, 1593), imagined the two arcs as gener-
ated by refractions through different clouds.
Dalmatian scholar Marco Antonio de Dominis
(1560–1624) published in 1611 the first work tackling rain-
bows from a physico-geometrical viewpoint. His book, titled
Tractatus de radiis visus et lucis in vitris, perspectivis et iride
[17], provided a very convincing explanation of rainbows. He
showed that sunrays crossing a glass sphere full of water
formed a rainbow on his laboratory’s walls. By observing the
light rays’ path, he observed that they were reflected by the
bottom of the sphere, which behaved as a concave mirror, and
on exiting it they underwent a new refraction. His treatise
followed a series of studies performed through experiments
similar to Theodoric of Freiberg’s; his studies demonstrate a
profound knowledge of the literature on the subject.
In 1604, Johannes Kepler (1571–1630) followed up on
the studies by Polish monk, mathematician, physicist,
philosopher and theologian Witelo, but he was unable to
establish the law of refraction.
Later, Sicilian astronomer Giovan Battista Hodierna
(1597–1660) studied the phenomenon of light passing
through a prism, anticipating Newton’s research, as his
Thaumantiae miraculum (Palermo: Nicolai Bua, 1652),
shows. He also gave a vague explanation for rainbows.
Moreover, he introduced a distinction between ‘‘strong’
and ‘‘weak’’ colours, separated by white. In his text, the
Fig. 3 a Left A representation of the secondary rainbow, according to Theodoric of Freiberg. bright The phenomenon of multiple reflections
according to Maurolico (Photismi, Theorema XXIX, p. 54, Neapoli: Tarquinij Longi, 1611)
See vol. II, pp. 172–201 of Bridges [12]. Moreover, it is necessary
to remark that, at the end of 13th century, the Polish philosopher and
physicist Witelo, building on Alhazen’s hypoteses, had claimed that
the bending of light by refraction was larger the denser the medium
through which light had passed. Witelo’s essay, Vitellonis Thuringo-
poloni opticæ libri decem, is contained in Opticæ Thesaurus by
Friedrich Risner (1533-1580), published in Basel in 1572. See also El-
Bizri [20].
The writings appearing in this treatise were composed between
1521 and 1552.
See Gedselman [24].
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geometrical, quantitative study of light phenomena has its
roots in what we could call a metaphysics of light, thus
justifying the primacy of optics over other sciences, on the
basis of the ontological primacy of light.
The beginning of the Eighth Discourse of Les Me´te´ores,
in which Descartes covers rainbows (Fig. 4a), marks a
paradigm shift in the investigations of the rainbow’s
physical phenomenon. Descartes begins his discourse with
this claim:
L’Arc-en-ciel est une merveille de la nature si
remarquable, & sa cause a este
´de tout tems si
curieusement recherche
´e par les bons esprits, & si
peu connue
¨, que i.e. ne sc¸aurois choisir de matiere
plus propre a faire voir comment par la methode dont
i.e. me sers on peut venir a des connoissances, que
ceux dont nous avons le escrits n’ont point eue
(The rainbow is such a remarkable phenomenon of
nature, and its cause has always been so carefully
sought after by good minds, yet so little understood,
that I could not choose anything better to show you
how, by means of the method I am using, we can
arrive at knowledge not possessed by any of those
whose writings we have [19]).
Descartes tackled the problem on a strictly scientific
and mathematical basis, starting from the laws of
refraction that now bear his name. He gave an interesting
proof of the reason why the rainbow has a semicircular
shape using the recent instruments provided by calculus,
which was being developed in those years. His goal was
to improve the scientific explanation of the phenomenon
through the use of mathematical tools and experimental
Descartes studied the passage of a light ray through a
large glass sphere filled with water. By measuring the
angles of the rays getting out, he concluded that the pri-
mary arc was created by a single reflection inside the drop,
while the secondary one may be caused by two inner
reflections. To reach this result, Descartes expounded the
theory of refraction—just as Snell
had done—and cor-
rectly computed the angles of incidence of light rays for
both arcs. He established that the ratio of the sines of the
incidence and the refraction angles was constant, for all
incidence angles. The light rays from the Sun, assumed to
be point-like, arrive along parallel directions at different
points of the water drop and intercept it with different
incidence angles. Those undergoing a single reflection
form the primary arc, while the secondary one, outward
with respect to the primary, consists of a double reflection
within the drop. However, not all incident rays are equally
effective, that is, visible to the observer: only those with an
angle of incidence close to 59for the primary arc and to
71for the secondary one are visible to the human eye.
On the other hand, Descartes’s explanation on how
colours form is not quite convincing, since it was based on
a mechanistic version of the traditional theory in which
colours are generated by a modification of white light. In
his 1637 treatise about Les Me´te´ores (Fig. 4a), he gave an
explanation of the physical phenomenon quite close to that
given by de Dominis, but without mentioning him, perhaps
not to be guilty of disobedience of the damnatio memoriae
pronounced by the Catholic Church towards the heretical
Descartes’s reasoning about the rainbow is of a geo-
metric nature, ridding the phenomenon of poetic or mys-
tical overtones—a reproach that John Keats (1795–1821)
Fig. 4 a Left Rene
Discours de la me
p. 251. bRight Newton’s
rainbow (illustration from Isaac
Newton’s treatise Opticks,
London: S. Smith and B.
Walford, 1704)
Snell’s law, also known as Snell-Descartes law, describes how a
light ray is refracted when passing from a medium to another one with
a different refractive index; in general, it is valid only for isotropic
substances, such as glass, and shows several similarities to Fermat’s
principle (due to Pierre de Fermat, 1601–1665) that ‘‘the path taken
between two points by a ray of light is the path that can be traversed
in the least time’’. A first formulation of Snell-Descartes law can be
found in a manuscript by the Arab mathematician Abu
¯Sa’d al-‘Ala¯’
ibn Sahl (X sec.) written in 984; it was later probably guessed in 1602
by Thomas Harriot (1560–1621), an astronomer and a mathematician,
who did not publish his work, though. It was rediscovered by
Willebrord Snell in 1621, in a form mathematically equivalent,
unpublished until his death, and, phinally, republished by Descartes in
terms of sine functions in his 1637 Discours de la me´thode, where
he used it to solve several problems in optics.
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would aim at Newton in his poem of 1820, Lamia—and
permits men to reason both as poets and as physicists, but
certainly no longer as theologians, abandoning a mystic
vision that, like Genesis 9:14 which describes the rainbow
as a sign of God’s promise to man never to unleash another
flood, transcends a supernatural significance into a natural
physical one.
Isaac Newton covered the topic in his Opticks [39]
(Fig. 4b), duly crediting de Dominis with being the first to
autonomously explain the phenomenon. However, Newton
was the first to give a scientific proof of the composition of
light spectrum, by claiming that white light consists of all
the colours of the rainbow. According to the English sci-
entist, colours could be separated into their spectrum by
allowing a light ray to pass through a glass prism. Newton
clearly showed that the iridescent figures generated by
prisms are caused by the decomposition of light into its
chromatic components, in opposition to the general idea
that colours were generated ‘‘within’’ the prism, which
would be endowed with this peculiar property, just as water
droplets were responsible for rainbows.
He also showed that red light is refracted less than blue,
thus providing the first scientific explanation of the main
features of the rainbow. However, the corpuscular theory of
light as formulated by Newton failed to explain the phe-
nomenon of the ‘‘supernumerary’’ rainbow or arc; indeed,
such an explanation would only come with the studies by
Edme Mariotte (1620–1684), who first described the ‘‘su-
pernumerary arcs’’ (1679) [35], and especially Thomas
Young, who managed to show how, in certain conditions,
light behaves as a wave and can interfere with itself.
In the decades following the publication of Newton’s
Opticks, the observation of the first phenomena of inter-
ference and diffraction, and the consequent studies on light,
led to discovering the wave theory of light. A crucial role
was played by the experiment performed by Young [46]in
1801 on the superposition of light rays emitted by two
point-like sources, which allowed him to be the first to
measure the wavelengths of different light colours
(Fig. 5). However, the wave-like behaviour of light was
completely accepted only some years later, when the
French physicist Augustin-Jean Fresnel (1788–1827),
managed, through studying polarisation, to interpret the
results of Young’s experiment by assuming, as already
suggested by Young himself in 1816, transverse rather than
longitudinal elastic waves.
The Newtonian approach was
also abandoned due to the experiments aiming to determine
the speed of light performed by Armand Hippolyte Louis
Fizeau (1819–1896) and Jean Bernard Le
´on Foucault
(1819–1868) in 1850, which showed how speed decreases
when the density of the medium increases, contrary to
Newton’s predictions.
Young’s work was later developed and improved in the
1830s by the English astronomer George Biddell Airy
(1801–1892) [5], who worked on the wave theory by
Christiaan Huygens (1629–1695) and Augustin-Jean Fres-
nel, while the phenomenon’s mathematical treatment was
partly carried out by the chemist Richard Potter
(1799–1886) in 1835, and published in 1838 in the pro-
ceedings of the Cambridge Philosophical Society [42].
Potter explained how the intensity of the rainbow’s colours
also depended on the size of the water drops. The meeting
of several light rays within a drop forms a curve called a
‘caustic’’, which is simply the envelope of a system of rays
corresponding to a maximum value of light.
Modern physical descriptions are based on the complete
and mathematically rigorous solution of the problem of the
optical scattering of an electromagnetic wave on a sphere
or a cylinder. This phenomenon is called ‘Mie scattering’
and follows from the studies published by the German
physicist Gustav Mie (1869–1957) and by Peter J.W.
Debye (1884–1966) in 1908. It is based on the equation for
electromagnetic waves formulated by the Scottish mathe-
matician and physicist James Clerk Maxwell (1831–1879),
who gave a precise mathematical description of the optical
aspects of the rainbow. The uninterrupted progress in both
theory and scientific computation throughout the 20th
century have led to an ever more complete understanding
of the phenomenon, taking into account the wave-like
properties involving interference, diffraction and polarisa-
tion, as well as the particle-like ones related to the
momentum carried by a light ray; in this regard an inter-
esting contemporary interpretation is due to the Brasilian
physicist Herch Moyse
´s Nussenzveig (1933-) [41] and, in
Fig. 5 Fig. 442 from Table XXX of Lectures by Thomas Young
(Vol. 1, 1807), which comprise the set of works presented to Royal
Society of London in 1802. The image shows the two-slit experiment
underlying the wave theory of light. A single ray of monochromatic
light passes through two tiny holes and consequently form light and
dark alternate fringes. T. Young, A Course of Lectures on Natural
Philosophy and the Mechanical Arts, 2 vols. London: Joseph Johnson,
See Brand [10], pp. 30–32.
See Crew [15] and Kipnis [29].
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computational terms, to Vijay Khare [28], who in 1975
obtained results similar to Mie’s.
3 The colours of the rainbow
The spectrum of the colours of the rainbow is continuous
and consists of a set of approximate ranges for each col-
; this is due to the structure of the human eye and to
the way the brain processes the data about the coloured
image from the photoreceptors, which differ from one
person to the next. What one actually gets by using a glass
prism and a point-like light source is a continuous spectrum
of wavelengths with no separate bands and hence no pre-
cise distinction of single colours. Isaac Newton, in his
treatise on optics, discerns just five primary colours: red,
yellow, green, blue and violet; only after more precise
investigations he added orange and indigo (Fig. 6a), thus
creating a seven-colour scale by analogy with the notes of
the musical scale,
based on the suggestions of Greek
sophists, who saw a relation between colours and musical
notes (Fig. 7).
Nevertheless, Newton himself admitted to a difficulty in
recognising the colours forming the rainbow: ‘‘My own
eyes are not very critical in distinguishing colours’’.
Between the end of the 18th and the beginning of the 19th
century, the structure of the visible spectrum was com-
pletely revealed. Later studies also clarified the phenomena
of the light outside the visible range: infrared was dis-
covered and analysed by William Herschel (1738–1822),
ultraviolet by Johann Wilhelm Ritter (1776–1810) and
Thomas Johann Seebeck (1770–1831). Seebeck also
described how light acts on silver chloride, an important
step towards colour photography.
Finally, research has concluded that indigo is not one of
the colours of the rainbow, but just a variation of wave-
length in the transition from blue to violet. According to
Isaac Asimov (1920–1992):
It is customary to list indigo as a color lying between
blue and violet, but it has never seemed to me that
indigo is worth the dignity of being considered a
separate color. To my eyes it seems merely deep blue
Indeed, defining the colours of the rainbow is different
from defining the spectrum itself, since the colours of the
rainbow are less saturated; for each particular wavelength,
there is a range of exit angles, rather than a single, invariant
angle, and the number of colour bands in a rainbow can be
different from the number in a spectrum, especially if the
suspended droplets are significantly large or small. Hence,
the number of colours in a rainbow is variable. If, on the
other hand, the word ‘‘rainbow’’ is inaccurately used to
denote the spectrum of colours, then the colours of the
rainbow do correspond to the main colours of the spectrum.
The light of a rainbow is almost completely polarised.
This phenomenon is due to the angle of refraction in the
drop being very close to Brewster’s angle, discovered by
the Scottish physicist David Brewster (1781–1868) in
Hence, most of the p-polarised light disappears
during the first reflection (and refraction) within the drop.
The circular shape of the rainbow is strictly related to a
problem of minima: the angle at which the sunlight
reflected by the water drops has a maximum of intensity is
Fig. 6 a Above The colours of the rainbow according to Newton: first
and second hypothesis. bBelow: the spectrum of colours; the image
shows an approximate representation of the colour associated to each
wavelength in the visible region. Under 400 nm (nm) and over
750 nm, colours fade to black, because the human eye is not able to
detect light out of this boundaries
Fig. 7 The description of the spectrum of colours, correlated,
according to Newton, to the musical scale, as given by David
Brewster (1781–1868) in his Memoirs of the Life, Writings and
Discoveries of Sir Isaac Newton, Edinburgh: Thomas Constable, 1855
See Berlin-Kay [4].
‘Ex quo clarissime apparet, lumina variorum colorum varia esset
refrangibilitate: idque eo ordine, ut color ruber omnium minime
refrangibilis sit, reliqui autem colores, aureus, flavus, viridis,
cæruleus, indicus, violaceus, gradatim and ex ordine magis magisque
refrangibiles’’ , Newton [40], Propositio II, Experimentum VII.
See Gage [22], p. 140.
‘When a ray of light is polarised by reflexion, the reflected ray
forms a right angle with the refracted ray’’, Brewster [11], p. 132.
Lett Mat Int
Author's personal copy
about 40–42with respect to the observer; this angle is
independent of the size of each suspended droplet, but
depends instead on their refractive index (a dimensionless
quantity that measures how much the speed of propagation
of electromagnetic radiation decreases while crossing
4 Conclusions
Drawing some conclusions from this short history of the
rainbow is not at all easy. There is a large bibliography
regarding this phenomenon, from scientific, historic,
artistic viewpoints and more. The history of the rainbow
certainly belongs to the history of science, and in particular
to that of physics and optics, for which it is an important,
epistemologically wide-ranging subject. However, when
studying this optical phenomenon, a scientific approach has
not always been favourably received. In literary or artistic
fields, after the scientific revolution of Enlightenment,
continuous attempts have been made to assign to this
physical phenomenon one or another meaning from sub-
jects such as philosophy, religion, mysticism, esotericism
or art. There have been, indeed, scholars, writers and artists
claiming that a physico-mathematical analysis of natural
phenomena lessens their allure. This is why we wish to
close this essay with some brief, more literary musings.
If, as Virginia Woolf (1882–1941) points out in her
novel To the Lighthouse, published in 1927, the rainbow
represents the transience of life and man’s mortality—‘‘it
was all ephemeral as a rainbow’’—the cold physico-
mathematical representation of an optical phenomenon that
is ‘‘ephemeral’’ and evanescent, unreachable and insub-
stantial, deprives the natural event of all the poetry of an
event that strikes human imagination and fancy, opens the
soul to hope and life and moves George Gordon Byron
(1788–1824) to write ([13], p. 45):
Or since that hope denied in worlds of strife,
Be thou the rainbow to the storms of life!
The evening beam that smiles the clouds away,
And tints to-morrow with prophetic ray!
Johann Wolfgang von Goethe (1749–1832), in his essay
about the theory of colours [25] wrote that the scientific
analysis performed by Newton would have ‘‘paralysed the
heart of nature’
:Einen Regenbogen, der eine Viertel-
stunde steht, sieht man nicht mehr an (A rainbow that lasts
a quarter of an hour is seen no more).
The same opinion
was shared by Charles Lamb (1775–1834) and, as we
already recalled, John Keats. In the two poets’ opinion,
Newton ‘‘had destroyed all the poetry of the rainbow, by
reducing it to the prismatic colours’’ and, while dining in
1817, they offered a toast to ‘‘Newton’s health, and con-
fusion to mathematics!’’.
Translated from the Italian by Daniele A. Gewurz.
1. Ainger, A.: Charles Lamb. Cambridge University, Cambridge
2. Albertus Magnus: Opera omnia. In: Borgnet, A. (ed.). Vive
Paris (1890)
3. Asimov, I.: Eyes on the Universe: a history of the telescope.
H. Mifflin, Boston (1975)
4. Berlin, B., Kay, P.: Basic color terms: their Universality and
evolution. University of California, Berkeley (1969)
5. Biddell Airy, G.: On the intensity of light in the neighbourhood of
a caustic. Trans. Camb. Philos. Soc. 6, 379–402 (1838)
6. Blay, M.(ed.): Les figures de l’arc-en-ciel. Carre
´, Paris (1995)
7. Boyer, C.B.: Robert grosseteste on the rainbow. Osiris 11,
247–258 (1954)
8. Boyer, C.B.: The rainbow: from myth to mathematics. Yoseloff,
New York (1959)
9. Boyer, C.B.: The rainbow, from myth to mathematics. Princeton
University, Princeton (1987)
10. Brand, J.: Lines of light: the sources of dispersive spectroscopy,
1800–1930. CRC Press, Boca Raton (1995)
11. Brewster, D.: On the laws which regulate the polarisation of light
by reflection from transparent bodies. Philos. Trans. R. Soc.
Lond. 105, 125–159 (1815)
12. Bridges, J.H.: The opus majus of Roger Bacon, vol. I & II.
Williams and Norgate, London (1900)
13. Byron, G.G.: The bride of Abydos: a Turkish tale. J. Murray,
London (1813)
14. Corradi, M.: La teoria dei colori di Johann Wolfgang von Goethe.
In: Rossi, M., Marchiafava, V. (eds.) Colore e colorimetria.
Contributi multidisciplinari, Vol. X A, pp. 401–712. Maggioli, S.
Arcangelo di Romagna (2014)
15. Crew, H.: The wave theory of light; memoirs of Huygens, Young
and Fresnel. American Book Company, New York (1900)
16. Crombie, A.C.: Robert grosseteste and the origins of experi-
mental science. Clarendon Press, Oxford (1983)
17. De Dominis, M.A.: De radiis visus et lucis in vitris perspectivis et
iride tractatus Marci Antonii de Dominis, Per Ioannem Bartolum
in lucem editus: in quo inter alia ostenditur ratio instrumenti
cuiusdam ad clare videndum, quae sunt valde remota excogitati.
Thoma Baglionus, Venice (1611)
18. Descartes, R.: Discours de la me
´thode pour bien conduire sa
raison, et chercher la verite
´dans les sciences Plus la Dioptrique,
les Meteores, et la Geometrie qui sont des essais de cete Methode.
Ian Maire, Leyde (1637)
19. Descartes, R.: The World and Other Writings (English translation
by Gaukroger, S.). Cambridge University, Cambridge (2004)
See for instance Venturi [45], Mascart [36], Minnaert-Lynch-
Livingston [38], Nussenzveig [41], Greenler [26], Boyer [9], Blay [6],
Lynch-Livingston [33], Lee-Fraser [31], Maitte [34].
For further reading about the controversy Goethe-Newton about
colours, see [14].
From Goethe’s Maximen und Reflexionen.
For the last two quotations, see [1], p.86.
Lett Mat Int
Author's personal copy
20. El-Bizri, N.: A philosophical perspective on Alhazen’s optics.
Arab. Sci. Philos. 15, 189–218 (2005)
21. Fleischer, J.: De iridibus doctrina Aristotelis et Vitellionis.
Iohannes Crato, Wittenberg (1571)
22. Gage, J.: Color and meaning. art, science and symbolism.
University of California, Berkeley (1994)
23. GaziTopdemir, H.: Kamal Al-Din Al-Farisi’s explanation of the
rainbow. Humanit Soc Sci J 2, 75–85 (2007)
24. Gedselman, S.D.: Did Kepler’s Supplement to ‘Witelo’ Inspire
Descartes’ Theory on the Rainbow? Bulletin American Meteo-
rological Society 70, 750–751 (1989)
25. Goethe, J.W.: Zur Farbenlehre. J.G. Cotta’schen Buchhandlung,
¨bingen (1810)
26. Greenler, R.: Rainbows, Halos, and Glories. Cambridge Univer-
sity, Cambridge (1980)
27. Hackett, J.: Experientia, Experimentum and Perception of
Objects in Space: Roger Bacon. In: Aertsen, J.A., Speer, A. (eds.)
Raum und Raumvorstellungen im Mittelalter, pp. 101–120.
Walter de Gruyter, Berlin (1998)
28. Khare, V., Nussenzveig, H.M.: Theory of the rainbow. Phys. Rev.
Lett. 33, 976–980 (1974)
29. Kipnis, N.: History of the principle of interference of light. Bir-
¨user, Berlin (1990)
30. Krebs, E.: Meister Dietrich (Theodoricus Teutonicus de Vriberg)
sein Leben, seine Werke, seine Wissenschaft. In: Beitra
¨ge zur
Geschichte der Philosophie des Mittelalters, Texte u. Unter-
suchungen, part V, pp. 5–6. Aschendorff, Mu
¨nster (1906)
31. Lee, R.L., Fraser, A.B.: The rainbow bridge: rainbows in art,
myth, and science. Penn State Press, University Park (2001)
32. Lindberg, D.C.: Roger Bacon’s theory of the rainbow: progress or
regress? Isis 57, 235 (1966)
33. Lynch, D.K., Livingston, W.: Color and Light in Nature, 2nd edn.
Cambridge University, Cambridge (2001)
34. Maitte, B.: Histoire de l’arc-en-ciel. Seuil, Paris (2005)
35. Mariotte, E.: Essais de physique, ou Me
´moires pour servir a
science des choses naturelles. Michallet, Paris (1679–81)
36. Mascart, E
´.: Traite
´d’Optique, 3 voll. Gauthier-Villars, Paris
37. Maurolico, F.: Photismi de lumine et umbra ad perspectivam, and
radiorum incidentiam facientes. Diaphanorum partes, seu libri
tres. Editio princeps. Tarquinius Longus, Naples (1611)
38. Minnaert, M.G.J., Lynch, D.K., Livingston, W.: The nature of
light and color in the open air. Dover Publications, New York
39. Newton, I.: Opticks or, a Treatise of the Reflexions, Refractions,
Inflexions and Colours of Light: also Two Treatises of the
Species and Magnitude of Curvilinear Figures. Sam. Smith and
Benj. Walford, London (1704)
40. Newton, I.: Optice: Sive de Reflexionibus, Refractionibus,
Inflexionibus and Coloribus Lucis Libri Tres. Marci-Michaelis
Bousquet, Lausanne/Geneva (1740)
41. Nussenzveig, H.M.: The theory of the rainbow. Sci. Am. 236,
116–127 (1977)
42. Potter, R.: Mathematical considerations on the prohlem of the
rainbow, shewing it to belong to physical optics. Trans. Camb.
Philos. Soc. 6, 141–152 (1838)
43. Sivin, N.: Science in ancient China: researches and reflections.
Ashgate Publishing, Brookfield, Vermont (1995)
44. Theodoricus de Vriberg: Opera Omnia, voll. I-IV. In: Cavigioli,
J.-D., Imbach, R., Mojsisch, B., Pagnoni-Sturlese, M. R., Rehn,
R., Steffan, H., Sturlese, L., Wallace, W. A. (eds.) Corpus
Philosophorum Teutonicorum Medii Aevi II/1-4, Meiner, Ham-
burg (1994)
45. Venturi, G.B.: Commentari sopra la storia e le teorie dell’ottica.
I. Masi, Bologna (1814)
46. Young, T.: The Bakerian Lecture: on the theory of light and
colours. Philos. Trans. R. Soc. Lond. 92, 12–48 (1802)
47. Young, T.: The Bakerian Lecture: experiments and calculations
relative to physical optics. Philos. Trans. R. Soc. 94, 1–16 (1804)
48. Dictionary of scientific biography. American Council of Learned
Societies, Charles Scribner’s Sons, New York (1970–1990)
Massimo Corradi is an asso-
ciate professor of history of
science at the University of
Genoa, where he teaches several
courses in history of science,
technology, building tech-
niques, and landscaping. He has
been a visiting professor at the
´Catholique de Lou-
vain La Neuve, the Open
University of London, and the
Universidade de Brasilia, as
well as an invited lecturer in
several national and interna-
tional conferences on history of
science and building techniques and the relations between mechanics
and architecture. He is the author of some 120 papers about structural
engineering, mechanics applied to building, restoration and consoli-
dation of structures and history of science and technology.
Lett Mat Int
Author's personal copy
Full-text available
Web 2.0 is a technology that enables collaboration between citizens and the government to actualise the concept of open government. The adoption of technology in the public sector is known to have positive impacts in fostering public sector innovation, including in developing countries with various problems that require collaboration between citizens and the government. However, currently studies concerning open government technology adoption in developing countries are limited. Indonesia as an example of a developing and democratic country provides a suitable context to explore this phenomenon. Moreover, there are many local government institutions in Indonesia, especially on Java Island, that have implemented Web 2.0 to support open government initiatives. Therefore, this research aims to explore key factors influencing the adoption of the Web 2.0 platform by local government institutions in Indonesia. Multiple theories are combined to guide the exploration of the influencing factors related to technology, organisations, management, and the environment. Based on a multiple case study involving five local government institutions in Indonesia, we identified fifteen factors affecting the adoption of Web 2.0 technology. This study offers both research and practical implications that can be considered by government and other key stakeholders to optimise public participation and information access through the adoption of open government initiatives.
One of the prominent material properties of an optical material is the refractive index. It is the only property that governs various naturally occurring spectacular phenomena such as the formation of a rainbow, color dispersion during sunrise and sunset, and many more. The diameter of a rainbow is characterized by the refractive index of the droplets of water. The calculation of how often an illuminated ray refracts through one medium to the next (for example, from air to water) is determined by a refractive index. A high refractive index droplet helps create a smaller radius of a rainbow. For example, rainbows produced by seawater are less than plummeting rainbows, and saltwater has an improved refractive index than freshwater. Experiments in the classroom arouse new ideas, questions and contribute to developing clarity on Principles of Physics. In this paper, we explore the importance of the material property, refractive index demonstrating dispersion by creating rainbow spectrum, diffraction, and interference fringes in the classroom utilizing the tools available in a laboratory are presented. The experiment on Rainbow makes students observe a well-defined spectrum caused by white light undergoing dispersion through a prism. The experiment on Diffraction and double-slit Interference makes use of locally handmade slits exposed to Sodium lamps to produce fringes. Method to determine the wavelength of light is presented. When performed in the classroom, a variety of questions were posed by students leading to peer interaction and presentation of their ideas.
This work provides a perspective on the creation of a scientific discipline. The reader is led to meet the actual people who have contributed to this field and know their trials as well as breakthroughs. From 1800 to 1930, Brand preserves the thread of scientific thought and activity through six generations of working scientists.
Part I. The Four General Causes of Human Ignorance; Part II. The Connexion of Philosophy with Theology; Part III. The Study of Language; Part IV. Mathematical Science.