How The Infinitely Wide Spectrum Of Symmetry Makes It A Universal Phenomenon

Here's my third and concluding essay on the awesome concept of Symmetry.  I thought I had said the last word on the subject, but I was wrong.  Just read on!

++++++++++++++

Glossary & annotations
(in same order as in text)

Nataraja  --  Alternate name of Lord Siva. the Hindu God, depicted as a vigorous cosmic dancer striking a standard pose in all statues and pictures. 

Raaga/Taala  --  In Indian classical music, established patterns of melody/rhythm.

Aarohana/Avarohana  -  Established order of ascent/descent of notes in a raaga.

Ballet  --  Classical dance of Europe.

Bharatha-naatyam/Odissi/Kathak   --  Classical dance systems of South/South-East/North India.

Scotch reel  --  A traditional dance of Scotland, set to bagpipes music.

Flamenco  --  Dance tradition of Spain, with a bullfight orientation and gypsy colors.

Waltz -- A graceful, swirling type of Western music, ideal for ballroom dancing. 
       (Look for above dance forms in YouTube, for a rich experience)

+++++++++++++++

THE HINDU
Sunday Magazine
29 December 1991

Articulations
The panorama of symmetry

     Turning our view from the visual arts to the performing arts, we find that symmetry still figures as a fundamental criterion, which can be given up only at the cost of excellence.  This truth is most clearly visible in the case of dance, where the impact is mainly visual.  There is a strong resemblance between a well-performed dance and the steady flame of an oil-lamp (especially one exposed to a gentle breeze) or the lively flames of a fuel-wood fire.  

Naturally, in a good dance we observe the same quality of symmetry which we find in flames.  Good choreography obviously calls for a symmetrical vision.  The ancient temple sculpture of India was, among other things, a celebration of the symmetry of dance :  perhaps the most magnificent example of this is the classic pose of Nataraja. 

Music & symmetry

      Symmetry in music concerns the logical flow of its sound.  One may look for it separately in melody, harmony or tempo (incorporating rhythm), or in the balance achieved when integrating these elements.  In Indian classical music there is remarkable symmetry in the 'aarohana' and 'avarohana' of the ragas, and in the meticulous tempo set up by the various taalas.  Given these established patterns of melody and rhythm, the symmetry of Indian classical music is to a large extent pre-determined.  In the West, where no such compulsory patterns restrain the process of composition except in the case of certain forms of dance music, the structure of any arrangement for instrumental or vocal music is normally governed by a far more flexible regime of symmetry.

     A vital source of symmetry in Indian classical music is the 'sruthi' -- a continuous shimmering backdrop which not only anchors the basic tonic at the proper level but also provides a reassuring sound which is part of the music in its own right.  Western music does not normally adopt such an organic device for regulating the pitch or reinforcing the sound ;  but the persistent anchoring efforts in pop music and in instrumental jazz do give rise to a kind of sruthi, whether this is based on any theoretical requirement or not.  There is, by the way, a constant drone similar to sruthi in certain types of Swedish music. 

Dancing to music

     Except perhaps in very unorthodox displays of virtuosity, dance cannot be performed without appropriate parallel music (usually with an accent on rhythm).  We can easily recognize the symmetry between a system of dance and the music which is meant for it -- whether we are thinking of ballet or Bharatha-naatyam, Scotch reel or Odissi, Kathak or Flamenco.  The steady pulse of a beautiful waltz contributes  to the graceful swirl of ballroom dancing, and also reflects it.  Folk music and dance everywhere are more closely bound together than art music and dance, therefore this consonance between sound and vision is more prominent in their case.

     There are, of course, exceptions to this rule.  For instance, the showy gyrations and gymnastics of disco dancers or pop musicians (especially in video productions) often degenerate into wildly discordant movements and images, while the parallel music usually maintains a stable sonority.

Drama and cinema

     Is there symmetry in drama?  We do not have to search hard for an answer :  it would be enough merely to consider the composite nature of the dramatic art, which incorporates the elements of literature and the visual arts,, and often the other performing arts too.  In all shades of drama, ranging from simple rustic drama to sophisticated urban theater, there has to be a fine balance between artistic disciplines of different kinds.  It all boils down to a question of symmetry, of course.  In this regard, what is true of drama is also true of motion pictures, with appropriate modifications.

Artists and audience

     In the case of the visual arts, architecture, audio and video recordings and cinema, the interaction of the viewers is not with the creators of art, but with their creations.  But in the case of live music, dance and drama, there is a direct communication between the artists and the audience.   The success of a performance depends to a large extent on the symmetry which exists in this sensitive relationship. 

Language & literature

     Let us now take a look at language and literature.  In an earlier essay we had noted that there is considerable logic in the grammar and composition of language.  It is a fact that wherever there is logic there is symmetry.  We had also seen that there are certain illogical elements in language, especially in the case of script and spelling in relation to sound ;  to that extent, symmetry is missing in language.  Of course, it is visibly present in the maintenance of alphabetical order, the compilation of dictionaries, and in the techniques of short-hand transcription.

     There is obvious symmetry in the rhythm of poetry in any language, and in its rhyming and alliteration.  John Ruskin, the English art critic of the 19th century, talked of the "symmetrical clause of Pope's logical metre."  While poetry can express profound and beautiful thoughts, it may also be majestic and lovely in its mere form, like music.

     Elegant prose too has a distinct rhythm, but this depends more on meaning.  Herbert Reed, the distinguished literary critic, says in his book 'English Prose Style' :  "Rhythm is born not with the words but with the thought . . .  The paragraph is a plastic mass. and it takes shape from the thought it has to express.  The words are like clay on a potter's wheel . . . "   But obviously thought is the seminal source of all literature, not merely of prose.  The ultimate symmetry of literature, whether it takes the form of a poem, essay, novel, short story or play, lies in the delicate equilibrium between thought and articulation -- in other words, in the balance achieved between perception and language.  It is equally true that elegance and beauty of style basically depend on the symmetrical arrangement of words, not only as they look on paper, but also as they sound when spoken.

Universal law of symmetry

     This survey will not be complete unless we take stock of certain other manifestations of symmetry -- which are illustrative and not exhaustive, and reinforce the impression that there seems to be a universal law of symmetry. 

     Let us consider sports, for instance.  There is symmetry in the whole ambiance of a hockey or football ground, tennis or badminton court, billiards able or carom board, ski slope or skating rink, athletic track or velodrome -- not only in the visual impact of the events, but in the competition between the sportspersons and (where relevant) the cohesion of team-work.  A high degree of symmetry exists in the infinite permutations of chess and in the intricate equations of bridge.

     There are many aspects of management where symmetry is a major factor.  Institutional success depends a great deal on the kind of balance which exists in the relationship between employers and employees, between management and ownership, and between the organization and its clients, customers and associates.  There must be order in the maintenance of accounts and the contents of balance sheets.  There must be compatibility in the planning of projects, and consistency in their implementation.

     Symmetry is of paramount importance in the context of rational argument, whether it concerns legal issues. audit scrutiny, political and social commentary, scientific inquiry or the pursuit of pure wisdom.  The force of impartial argument is derived from the balance implicit in objectivity.  One naturally looks for perfect logic in judicial pronouncements, the Auditor-General's reports to the legislature, editorials of responsible newspapers, scientific papers and philosophic treatises.

     As mentioned earlier, wherever there is logic, there is symmetry.  This is true whether we relate rationality to thought processes or to the scheme of things which exist in nature or in a man-made environment.  But symmetry and logic are not synonymous terms, and we must carefully note the distinction between them.  An important fact that needs to be underlined is that while there may be logic in something which is not symmetrical, there csn be no symmetry in anything which is not logical.

+++++++++++++++  

PostScript, 2018

Scientific sequel

When I started exploring the concept of symmetry in my monthly column 'Articulations' in THE HINDU, I knew it would call for several essays of the usual length permitted by the Editor.  But just when I had finished writing the third one -- which I imagined would successfully conclude the extremely intricate exercise -- the first two attracted a strong rejoinder from an eminent Indian scientist. 

It took me some time to respond, but after a few weeks I turned the argument into a question of science vs. language, proving my point.  That, of course, will be the topic of the next post! 

Significant Aspects Of Symmetry In Art & Artistic Settings, Architecture & Artifacts

Here is the second of my three comprehensive essays on Symmetry, which have survived -- and will surely continue to survive -- the severe tests of time :- 

+++++++++++++++

THE HINDU
Sunday Magazine

11 December 1991 

Symmetry in art
   
     In the first part of this essay we had noted the widespread presence of symmetry in nature and science.  Transferring our attention to the field of art, we find the role of symmetry to be no less significant.  Like science, art also has its deepest roots in nature.  The earliest forms of art in human experience were apparently an endeavor to represent nature in rudimentary drawings or etchings on rocks, or in carvings of wood and stone -- picturing the prominent landmarks of nature, which include the physical forms of life. With such a spontaneous beginning, it was inevitable that the symmetry inherent in nature would be permanently imprinted on the artistic instinct and perspective of mankind.

     It is not therefore surprising that even after superior techniques of sculpture and painting  were developed with the progress of civilization, preoccupation with nature has continued to be a compelling element of art for many centuries.  The highly symmetrical form of the human body was captured in the superb marble sculpture of ancient Greece in an idealized manner, and artists of Imperial Rome had looked mainly to Greece for inspiration.  So exquisite is the symmetry of the statue of Venus de Milo (second or third century BC) that even an ordinary viewer with a little bit of imagination can visualize, in his or her own way, the lovely shape of the lady's missing arms which have been lost to the ravages of time.

     In terms of reposeful grace, this masterpiece of Greek art has a striking parallel in the nude figure of David sculptured by Michelangelo more than 1500 years later in Italy.  Indeed, the whole art of the Renaissance in the 15th and 16th centuries in Europe  -- whether in sculpture or painting -- reflected an intense nostalgia for the grace, beauty and symmetry of the classical art of ancient Greece and Rome. 

Conflicting trends

     Subsequent artistic trends in the West veering away from the instinctive affinity with nature were presumably the result of accumulated and growing intellectual ennui in the realm of art, particularly since the last quarter of the 19th century.  But even where artists have revolted against merely copying nature, they have not always sacrificed symmetry in the works they have created.

     It is true that there have been several artistic movements in which arbitrariness has been the predominant element -- such as Expressionism (Munch), Fauvism (Matisse), Cubism (Picasso, Braque),  Dada (Duchamp, Ernst), Surrealism (Ernst, Dali), and Abstract Expressionism (Pollock).  But there have also been parallel trends where even artists with unconventional attitudes did not give up their concern for harmony -- such as those belonging to movements like Realism (Courbet), Impressionism (Manet, Pissaro, Renoir), and Post-Impressionism (Cezanne, Gauguin, Van Gogh).  There is fascinating symmetry in the severe straight lines and austere rectangles which figure in the abstract art of Mondrian.

     Moreover, the asymmetrical and anti-symmetrical elements of modern art have also tended to create their own brand of ennui in the artistic spirit :  and as the dawn of a new century draws closer, there seems to be a growing inclination among artists in the West to revert to less arbitrary styles.

     There is great symmetry to be found in the classical forms of Oriental art, whether Byzantine (Turkish-Roman), Greek Orthodox, Persian, Indian or Chinese.  In India, modern art styles have tended to copy bizarre and asymmetrical European models quite indiscriminately ; and by and large they have failed to make a forceful impact on the people, whose collective vision of the visual arts has always been rooted deeply in traditional concepts.  No wonder the National Gallery of Modern Art in New Delhi, which is supported by the Government at great cost, has only marginal relevance to the cultural life of the people.

Architecture and designs
     While symmetry can be devalued to some extent in the case of painting and sculpture, it is almost indispensable in architecture.  Buildings, bridges and other permanent constructions need to have a stable framework for retaining their forms ;  and stability in physical terms is invariably enhanced by symmetry in structure.  This is obviously the reason why, from the most ancient times to the present day, architects all over the world have generally tended to adopt symmetrical designs.  Architecture is concerned with engineering as well as aesthetics.  While science provides the foundation and the basic features of the framework, art prescribes the appearance which creates the visual impact and appeal of a structure :  and on both counts, symmetry happens to be a crucial factor. 

     Another area in which science and art have a common concern is that of industrial design.  Considerations of utility and engineering have an important bearing on the shape of innumerable products of modern industry ;  but the attractiveness of the package -- or of the shell enclosing mechanical, electric or electronic systems -- depends largely on its artistic value.  The design of a product may either be influenced by the preference of propsective consumers, or may be meant to influence their tastes ;  in either event, the designer usually faces questions of symmetry and visual appeal.  And in important functional matters like sreamlining an automobile body or devising a well-balanced internal mechanism, the engineer is vitally concerned with the connection between symmetry and efficiency.

Symmetry in settings

     When discussing painting and sculpture, we have so far referred to symmetry only in the sense of an artistic work displaying a symmetrical pattern, whether as a reflection of the symmetry which exists in nature, or in a different form altogether.  We must also observe the fact that when a faithful or even impressionistic image of a real object or person is created in a painting or sculpture, there is a symmetrical relationship between the subject and the work of art.  This aspect is more conspicuous in the case of photography, where the picture seeks to represent a subject or scene faithfully unless any deliberate distortions are introduced by manipulating the normal process itself.  And what is true of photography in this regard is also true of cinematography, TV transmission and video recording.

     In the case of cinema, there is a symmetrical link between the scene which is shot and the film, and between the film and the immensely magnified image on the screen.  Similarly, in a video recording there is a symmetrical link between the original scene  and the magnetic tape, and between the tape and the monitor screen.  In the case of live television, there is a direct symmetrical link between the image telecast and the one received, just as there is a symmetrical bond between inputs and outputs in the case of the telephone, telegraphy, gramaphone, sound-recording tape or compact disc. The continuing efforts of inventors to develop more and more sophisticated techniques -- with the ultimate objective of reproducing sound and vision with the highest fidelity -- are in fact a constant search for finding effective methods for achieving greater symmetry in the whole system.  

    There is, however, a serious paradox in the evolving situation.  The state of technology in cinema and television has already reached a very high level today ;  and the possibility of duplicating reality with still greater accuracy by deveoping perfect three-dimensional imagery creates the legitimate fear that it may only lead to a reduction in the artistic value of the viewer's experince.  This would mean that a delicate equilibrium which is conducive to artistic excellence may be lost, spoiling the symmetry of the whole equation. 

Decorations and artifacts

     Symmetry would seem to have been a basic element of decoration from the earliest times, whether in relation to objects of art or articles of utility ;  and by and large it continues to be so today.  In architecture, symmetry is found not only in the harmonious shape of a structure and its constituent parts, but in interior and exterior xevoration as well.  Thus, the symmetry visible in the arabesques, friezes, mosaics, frescos or stained glass of olden days is matched by the symmetry one sees in the carpets. mantels and chandeliers of more recent times, and in today's linoleum, wall-paper or even functional tiles.

     We have noted the relevance of symmetry in modern industrial products.  This is reminiscent of the symmetry which has generally charaterized the artifacts of human workmanship from pre-historic periods to the present times, through successive civilizations.  There is a compulsive urge in human nature to look for beauty in addition to utility even in things which are intended to serve ordinary purposes.  This calls for elegance and symmetry in curtains and carpets, cutlery and chinaware, cooling and washimg machines, and so on endlessly.  To what extent this need is actually satisfied by contemporary industrial culture geared to mass production is an issue which has to be studied separately.

                                                                    (to be continued)

+++++++++++++++

PostScript, 2018

Blending beauty and utility     

As I was writing the last paragraph of this essay in 1991,  I could hear loud echoes of the following thought expressed by Mr. P.N. Haksar, the distinguished Indian diplomat-turned-cultural-philosopher, a couple of years earlier :-

"It must be ensured that the cultural life of the individual, no less than that of the community, should seek to relate aesthetic fulfilment to the everyday activities of life.  Encouragement should be held out to creative activity which locates both beauty and utility in the artifacts of cultural as well as material production." 

A significant sequence of his profound reflections can be seen in my blog :  When & How Chairman Haksar Wrote His Monumental Essay On Art & Culture. 

The Intricate And Intriguing Dimensions And Manifestations Of Symmetry

My contributions to THE HINDU, one of India's most prestigious English-language newspapers, were spread over half a century in successive spells and columns which had several different characters and colors.  The initial spell in the 1960s featured articles cast in the mould of classical English essays ;  this was followed by reviews and sketches on the Carnatic and Western music scenarios in India's capital city New Delhi in the late '80s and early '90s. 

I also wrote a column titled 'Articulations' in 1990-93, recording the reflections of an articulate layman on the intricate manifestations of art and culture, as well as some psychological or philosophic aspects of life.  In the following recent blog I had explained how this column happened to be launched :   When & How Chairman Haksar Wrote His Monumental Essay OnArt & Culture.   

Earlier, in several successive blogs I had also recalled half a dozen essays on friendship, starting with the following one :  Frontiers Of Friendship : How, Beyond Natural Boundaries, Close Friendship Becomes Troublesome Bondage! 
            
Let me now  share with you a set of three seminal essays on the concept of symmetry, which were followed by a lively exchange of clashing ideas between an eminent Indian scientist and myself :-            

__________

THE HINDU

Sunday Magazine

17 November 1991

Reflections on symmetry

     In my last column I had expressed the view that other things being equal, the universality of music in terms of space and time depends on the degree of its symmetry.  This raises a pertinent issue : what exactly constitutes symmetry in music?  For finding a convincing answer one must naturally start with a proper definition of symmetry.  However, the effort to expain tends to become an exploration, and one is astounded by the far-reaching significance and apparently unlimited manifestations of symmetry.  So then, let music wait!

     The true meaning of symmetry is not limited to parallelism -- namely that two things (or two sides of the same thing) look alike in an absolute or inverted sense -- although that is the most popular connotation of the word.   In a much larger frame of reference, symmetry implies good proportion, balance, harmony and equilibrium ;  concordance, congruity, consonance and consistency ;  and even orderliness, grace and beauty.  And if one took a careful look at things,  one would realize that symmetry so defined exists on an intriguingly wide scale in nature, as well as in science, art and other concerns of civilized life.

Symmetry in universe

     Nowhere in nature is symmetry present as a more awesome phenomenon than in the meticulous order of the universe, where countless celestial bodies of colossal size exist in the form of noble spheres  -- either in a solid form or as a mass of energy -- and are endlessly tracing elliptical paths in the cosmos without  collision, indicating a perfectly balanced equilibrium.  The gravitational forces which are responsible for this amazing consistency in the perpetual motion and recurring relative positions of stars and planets do not operate in an arbitrary manner, but obey certain eternal laws of nature.  That is why astronomical research and predictions are able to reach out to infinity in terms of time and space.

     There is great symmetry in the way light behaves . . . in the infinite directions in which it tends to get dispersed unless obstructed . . .  in the constant and inexceedable speed at which light beams travel, and in the absolutely straight line of their course which can stretch over millions of light-years till they encounter something which stops, retracts or reflects them.  There is symmetry in the refraction of light . . .  in the way a prism breaks it up into the basic colors of the spectrum, or a wet sky conjures up a magnificent rainbow.  There is symmetry in the constant speed of sound, and in the echo which results when it is reflected.  There is symmetry, too, in the velocity of electricity and radio waves, and in the radiation of heat and other forms of vital energy. 
 
Symmetry on earth

     There is symmetry in the shape of the earth, and in the composition of its atmosphere.  One can see perfect order in the latitudes and longitudes which are man-drawn lines on paper or on a globe.  There is symmetry in the directions -- north, south, east and west -- and in the compass which indicates them.  There is consistency in the altitudes at which different kinds of clouds float or cruise in the sky ;  there is symmetry in the waves of the seas and their alternating tides.  There is symmetry in one's view of the horizon, which resolves itself into a perfect arc or circle as one moves up in the sky regardless of the nature of the landscape, in the way lighthouse beams constantly sweep and scan the horizon, and in the way they are observed from any given spot as flashes which recur with unerring regularity. 

     There is great symmetry to be found in crystals and snow-flakes.  Trees may not always be symmetrical in shape unless they are coniferous or palm ;  but leaves, flowers and fruits are almost always symmetrical in form.  Even where trees and plants grow in irregular shapes, the configuration of forests has an inherent pattern.  There is symmetry in the atoms and molecules of the elements, whether they are in a solid, liquid or gaseous state, and in the temperatures at which those conditions are altered.  There is precision in the annual cycle of the seasons :  spring, summer, rain, autumn and winter.  There is symmetry in the way water invariably finds its mean level no matter how intricately it is connected -- particularly in the way the surfaces of all the oceans have a common, immutable level which forms the basis for all geographical calculations of altitude and depth.  There is symmetry in the flame of a candle or oil-lamp, and even in the volatile flames of a blazing log-fire. 

     There is a strong element of symmetry in the physical features of most living creatures . . .   in the limbs and bone structures of mammals ;  in the twin eyes and ears which make three-dimensional vision and stereophonic hearing possible ;  and in the almost invariable matching of the left and right sides of animate bodies.  There is remarkable balance in the shape of fish, which are streamlined by nature to overcome the resistance of water, and in the wings which sustain the airborne condition of birds.  There is symmetry in the traits which characterize any given species on this earth ;  and even between different species there is a striking similarity in the sexual division and functions.     

Symmetry in shapes

     So far as shapes are concerned, symmetry invariably  concerns two dimensions, and sometimes three.  It will hardly be disputed that the circle and the sphere are the most perfectly symmetrical shapes which exist.  Symmetry can be axial, radial or bilateral, depending on whether it has a reference to a bisecting line, central point or dividing plane.  When we consider only two dimensions, we observe the symmetry of the circle, ellipse (or oval), square, rectangle, and equilateral triangle, pentagon, hexagon, and so on.  

     In three dimensions, we find symmetrical structures in the sphere, cone, pyramid (which you might call a 'square cone'), cylinder and cube.  The whole exercise of solving the intricate problem set up by the Rubik's Cube is a quest for restoring a symmetrical scheme of things from an extremely disturbed state.

Symmetry in science

The fundamental connection between nature and science is obvious, and has two distinct aspects.  Science is concerned with the discovery and interpretation of natural phenomena or laws ;  it also seeks to counter or harness the forces of nature for the purposes of mankind, whether these are constructive or destructive.  And since symmetry figures strongly in nature, so it does in science also.

Thus, the discipline of physics is concerned with deciphering the symmetry which governs the universe as well as the microscopic atoms of the elements.  Physics and astronomy together deal with complex issues concerning gravity, relativity, displacement in space and time, homogeneity of space, properties of heavenly bodies -- all of which raise vital questions of symmetry.  In chemistry, we come across equations and formulas which indicate the reactions which ensue when elements interact with one another, altering the symmetry of things.  Geology studies the symmetry of the earth's structure.  In biology scientists are deeply rooted in nature, studying the symmetry of plant and animal life ;  research in microbiology is essentially  concerned with the symmetry of living cells.

     No less significant is the status of symmetry in mathematics.  It has a conspicuous presence in geometry, which deals with magnitudes in space and the visual shapes of things ;  it is also present, though less perceptibly, in other branches of maths like algenra and calculus.  Of course, it is difficult to mark the borderline between mathematics and certain other sciences like physics and astronomy which are concered with the symmetry inherent in the laws of nature.  In statistics, there is always symmetry in the compilation and tabulation of data and their systematic analysis.

Symmetry in technology

     Symmetry can be found in the elementary tools devised by primitive men as well as in the sophisticated machines produced by high technology today.  There was symmetry in the blunt instruments of the stone age and in the original wheel whenever it was invented, as there is in the complicated gears or mysterious electronic systems of modern machinery.  There is symmetry in the way a sensitive gyroscope works, or in the way a simple bicycle in motion balances itself.

     The physical shapes which vehicles have assumed from early times to the present day provide sriking examples of symmetrical forms.  The streamlined structure of boats and ships, aeroplanes and rockets is the result of the necessity to overcome the resistance set up by natural forces.  The basic objective of hydrodynamics and aerodynamics is to discover the most effective symmetry of objects moving in water and air.  Of course, it took researchers condiderable time to realize that automobiles and rail-bound trains moving on the ground also need streamlining. 

Symmetry in standards

     Symmetry is inevitable in standardization, whether it concerns instruments, weights and measures, currency or the diverse products of modern industry.  A forceful illustration of this is provided by the identical specifications of gramophone records, photographic and cinematic film and magnetic recording tapes adopted all over the world.  We tend to take things for granted when we casually play a long-playing record or video cassette made in India or Bulgaria on a reproducing machine made in America or Japan.  The remarkable symmetry of the whole system which makes this possible may not be perceived unless one consciously looks for it, but it is there on a global scale all the time.   By contrast, the spread of the computer culture in the world today is perhaps retarded to some extent because of the inadequate symmetry of the existing software, although the computer itself represents a marvelous symmetry in science and technology. 

                                                               (to be continued)  
 _________

PostScript, 2018
Symmetry in articulations

Looking at this tightly-packed text written more than 25 years ago, I can't help feeling that it couldn't be so relevant and readable today -- as I am sure it's likely to be even after a hundred years from now -- if there hadn't been a substantial element of symmetry in the whole composition.  This impression is strongly reinforced by the two essays which followed, as you will see.  But  then, of course, that must be true of almost all my reflections as an articulate layman!