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The Quantification of Beauty

Laurence Shafe, University of Bristol

What do we mean by beauty? Can it be quantified—that is, reduced to basic elements of form and color combined according to certain rules? I would like to show how a number of Victorian artists answered these questions.
David Ramsay Hay (1798-1866) was a Scottish interior designer who became very influential both as a designer and a theorizer of beauty. He wrote a number of books on the rules of form and color culminating in his final work, The Science of Beauty (1856). In this he used mathematics to discover a scientific explanation for why we find certain images beautiful. His approach to the quantification of beauty was part of a classical revival that hoped to recover the secrets of ancient sculpture and architecture. Hay was aiming to rediscover those secrets using mathematics that were loosely based on the ideas of Pythagoras, particularly regarding the relationship between the beauty of music and the mathematics of musical scales, but his aim was to discover independently a logical system rather than to decode ancient texts.
Any attempt to quantify beauty immediately raises issues. For example, if beauty can be described by rules, then does this mean anyone can construct a beautiful object simply by following the rules? If beauty is determined by a single set of rules, then is it universal across all races? Hay’s answer to the first issue was that he saw his system as a training aid that would enable a student to become proficient more quickly by pointing out relationships that would otherwise take years to acquire through intuition and practice. On the second issue Hay saw his rules not as prescribing beauty but as describing an underlying framework common to all forms of beauty. The analogy I am reminded of is that of language—all languages differ but our common descent suggests we possess brain structures that impose certain constraints on the grammar of all languages. They do not determine the form of any one language but limit the range of possible languages. In an analogous way beauty may take many forms but must activate certain evolved brain structures.
I start by describing Hay’s method for quantifying beauty before putting it in the context of the art and science of the period. I then discuss one artist, Albert Moore (1841-1893), whose training and, I believe, whose methodology was influenced by Hay’s ideas, and finally I describe how ideas of absolute beauty competed with ideas of relative beauty partly as a result of Charles Darwin’s work.
Although Hay is not a well known name today he was described at the time of his death in 1866 as ‘the true Pythagoras’. In The Science of Beauty he took a Pythagorean and semi-mystical approach. He wrote, ‘Start with the ‘indivisible monad'—that is, the number one.@ He then gave procedures for generating the integers using addition, and he pointed out various facts he regarded as significant, such as the first four integers adding up to the number of digits on a typical human’s two hands.
Hay went on to provide a method for creating a table of ratios by sub-dividing between the monad and the duad and forming the reciprocal, one to a half. Further lines were created by dividing by the duad, giving a half to a quarter and so on. He then related this table to the musical scale where each fraction represented a note.
Imagine a musical instrument with a single vibrating string, the so-called ‘Pythagorean monochord’, and then hold down the string at the fractional position indicated; it will give the specified note. For example, if we stop a vibrating string half way along we obtain the octave of the fundamental note, and if we stop it two-thirds of the way along we get a perfect fifth. Then, through a further complex procedure he generated a palette of colors which he related back to the musical scale. In this way he claimed to have rediscovered the lost canon of beauty for forms and colors that he believed was known and used by the ancient Greeks. He justified this claim through his use of only simple rules and procedures, through its relationship with the musical scale used by the Greeks and by what I shall describe next—an empirical procedure for measuring the ratios he found in ancient buildings and sculpture and for showing how they conformed to his ratios.
To apply the table to a work of art or a building he converted the fractions into angular degrees by regarding each entry in the table as a fraction of the right angle, ninety degrees. He then showed how to measure the angular sub-divisions of various figures. For example, on the human face you can see the pivot point is the top of the head, and the angles are measured to the intersection of each major feature with the outline. The human body was measured using a similar scheme of drawing straight lines that intersect a horizontal line joining a major feature with the outline. Buildings were analyzed in a similar way. Hay pointed out that one advantage of his method was that it could be used to explain why both Aphrodite and Hercules are beautiful, as the angles stay constant as the body scales up. Hay extended his procedure by inscribing circles and ellipses within the figures in order to show how the more complex curves found in everything from figures to vases could be incorporated within his system.
Hay’s work was of interest to the design reform movement, as it provided a system for analyzing and assisting with the design of simple, geometric forms. The process of finding simple forms underlying the complexity of art and nature inspired designers such as Richard Redgrave and Christopher Dresser. Redgrave incorporated many of Hay’s ideas in his 23-stage course, which was adopted by the Government Design Schools as part of their training program.@
The Schools of Design were Government funded institutions intended to train designers for industry and help produce goods that could compete with superior designs from France and Germany. Despite, or perhaps because of, Government special committee reports and frequent changes in the directors and management, the objectives of the Schools remained unclear. This is best seen in the provision of figure drawing classes. These classes were introduced and then withdrawn at regular intervals, reflecting disagreements concerning the skills required by artists compared with designers. Many were concerned to maintain a clear distinction between them and maintain the Royal Academy’s exclusive role in authenticating artists. This distinction between the training of designers and fine artists created a clear division between decorative and fine art, causing a problem for students who wished to combine the skills of both disciplines and become rounded artists. Some students, such as Albert Moore, whom I will discuss next, attended external life classes and sought training at the Royal Academy after their training at a School of Design. Moore was therefore able to combine the mathematical analysis which was part of decorative design with the classical figure of fine art.
Albert Moore is little known today but in the 1870s he was a leading artist of the Aesthetic Movement—a movement based on the idea of representing beauty for its own sake—that is, without any narrative or moral lesson. In 1886 Cosmo Monkhouse wrote that Moore had a ‘claim to stand in the 1st rank of living English artists’.@
Moore was unusual in that he was trained at the School of Design in York but later attended the Royal Academy in London. The critic Sidney Colvin recognized Moore’s analysis of the sources of beauty when he wrote, ‘Mr. Moore stands almost alone in the power of finding out for himself and in common nature the sources of this ideal loveliness’.
His work is remarkably consistent from about 1865 to the end of his life. He creates compressed, often intensely decorated space that resists narrative interpretation and he often draws attention to the paint surface, although his classical figures date his work to the mid to late Victorian period. His style was a unique combination of decorative design applied to working from the nude model. No lectures, books or letters by Moore remain; all we have is a biography by his student Alfred Lys Baldry, press reviews, a few letters to his friends and his working drawings.
The grid you can see on this figure study was part of Moore’s method, and its structure was used to anchor key parts of the work. The grid or grids (as there was often, as here, a curvilinear and a rectangular grid superimposed) and even the shape of the vase hint at Hay’s theories; many of his drawings also have mathematical calculations in the margins.@ Robyn Asleson in her monograph on Moore suggests the grids imply an underlying set of construction lines that further tie the composition together.
His intensely patterned surfaces and formal arrangements led critics to complain that his paintings were merely decorative. For example, after his death a review of Baldry’s biography in The Times said it ‘does not follow he was in any sense a great man, … or his art other than decorative’.@ Many Victorian critics thought an emphasis on the decorative meant the work was deficient both morally and intellectually although others believed that he had managed to bridge the gap between the geometric disciplines of decorative art and fine art.
One way Moore bridged the gap was through his use of the classical figure. This referenced the intellectually acceptable implication that the artist had a classical education and indirectly suggested the Platonic ideal of beauty, which was also the inspiration for Hay’s Science of Beauty. Figure painting and particularly the classical figure were important signifiers of Moore’s identity as a leading artist. You can see from his sketches that he used a rigid system of constraining grids to impose a formal arrangement that blurred the distinction between the figures and the flatness of the decorative patterns on the curtains, tiles and fabric. Hay was trying to discover the rules that determine the eternal, fixed and absolute Platonic ideal of beauty, and Moore also used a formal process to try to quantify beauty by reducing the figure to its essential decorative and formal elements of line, shape and color.
Hay’s ideas implied there are absolute rules of form in art and nature, and this implication is compatible with the then current belief in divinely-created, fixed species. The paleontologist Richard Owen, who coined the word ‘dinosaur’, had noted the striking similarities in many animal structures such as a person’s hand, a mole’s claw, a dolphin’s flipper and a bat’s wing. These homologies, as they are known, were explained by Owen as variations on an ideal pattern, a sort of schema in the mind of God that he used as a mould to guide his creation of each species.
In this last section I would like to contrast Hay’s ideas for quantifying beauty and Owen’s ideas of divinely created species with ideas of contingent and relative beauty derived from evolutionary science and independently explored in art.
Charles Darwin’s theory of sexual selection, which can be interpreted as a theory of beauty, undermined the idea of absolute beauty by introducing a mechanism based on arbitrary variations. Walter Pater understood this profound shift in thought when in 1866 he wrote, ‘Modern thought is distinguished from ancient by its cultivation of the “relative” spirit in place of the “absolute.”’ He had been studying Classics at Oxford in 1859, when Darwin’s just-published Origin of Species was causing controversy and discussion among his fellow students. Pater's comment shows that Darwin’s ideas had an immediate impact not just on science but on the arts.
Charles Darwin argued that the reason for homologies is that we all share common ancestors. Darwin’s explanation had profound cultural consequences as it meant that each type of animal was not fixed but forever changing. As Pater said, ‘“type” itself … is not but is only always becoming’.@ So the form of an animal, which determines its beauty, is not fixed and absolute but forever changing.
So how did Darwin explain beauty, for example, the beauty of a peacock’s tail? His answer occupied two-thirds of his 1871 book The Descent of Man, and he called it sexual selection. In certain species the appreciation of beauty, typically by the female, evolves in parallel with the more and more extreme display of the characteristic regarded as beautiful. That is, the recognition of beauty and the beautiful trait co-evolve: both sets of genes are passed on to the children of both sexes. This results in a form of positive feedback and rapid evolutionary change. Beauty is therefore not concerned with fitness and the ability to survive; it is arbitrary and contingent and changes over time, like fashion. It is not absolute and timeless like the idea of classical art. Darwin’s replacement of the absolute by the relative was associated with similar profound changes in art.
James McNeill Whistler (1834-1903), a good friend of Albert Moore, made it clear in his Ten O’clock lecture of 1885 that beauty was not to be found directly in nature but in the works of an artistic genius. Whistler believed the artist deconstructs nature to reconstruct beauty from its component parts in the same way a musician uses notes to produce a beautiful melody. Although his reference to musical scales harks back to Hay, Whistler’s intention was very different. He was not seeking an eternal ideal but saw beauty in the ephemeral, formless and transient. Darwin found beauty in nature that was itself contingent. Whistler freed beauty from nature by introducing the intellect. He believed beauty could not be created by copying nature, and it was not concerned with detailed mimetic representation. He described the ‘foolish sunset’ admired by most people. ‘Nature’, he wrote, ‘contains the elements, in color and form, of all pictures, as the keyboard contains the notes of all music’. It was nature that was quantified, and it was the task of the artist to create beauty through artistic genius by reassembling elements found in nature. The idea of the artistic genius was a Romantic notion, but Whistler's denial of beauty in nature, his dissolution of form and his assembly of ambiguous forms from components parts aligned his ideas more with modernism and abstraction than with Romanticism.
During the Victorian period beauty was an important topic for both scientists and artists. Many theories were developed to explain why we find certain forms and colors beautiful, but the only one that survived to the present day was Darwin’s theory. Hay sought to recreate the lost ancient secrets of formal beauty through mathematics and careful measurement. Moore was an outsider who developed a precise methodology that balanced the repetition and symmetry of decorative forms with the sensuous beauty of the human figure, the two forms of beauty discussed by Darwin.
Whistler turned beauty on its head; rather than search for beauty in nature he found it within the mind of the artist who sees beauty in the ephemeral, the formless and the contingent. I end with a quote from Whistler's Ten O’clock lecture; we find beauty, he said,
‘when the evening mist clothes the riverside with poetry, as with a veil—and the poor buildings lose themselves in the dim sky—and the tall chimneys become campanile—and the warehouses are palaces in the night—and the whole city hangs in the heavens, and fairyland is before us.’@

Laurence Shafe is interested in the interaction between art and science in the nineteenth-century. To understand this cultural interchange he is currently studying the ways in which contemporary ideas about beauty, progress, nature and gender influenced both Charles Darwin and artists of the mid-nineteenth century. The research is for a doctorate at the University of Bristol supervised by Professor Elizabeth Prettejohn.