Tuesday, September 24, 2019

Mathematical Beehives and the Peacock Fountain



Listed as once in the library of  Thomas Browne (1605-82) Beehives of Universal Mathematical Philosophy by the Italian mathematician and astronomer Mario Bettini (1582-1657) is a compendium of mathematics, physics and optics. Each chapter of Apiaria Universae Philosophiae Mathematicae (its Latin title) is a self-contained 'Beehive' in which a proposition or topic of early modern science is discoursed. Euclidean geometry, optics, acoustics, the camera obscura, mathematical discussion of the flight of projectiles, the art of navigation and the measurement of time are among the many studies and experiments discussed by Bettini in his Aparia, some of which are considered to be innovative contributions to the early scientific revolution. [1]

Bettini's Aparia went through a number of editions from its first publication in 1642. Browne's edition is dated 1656, just two years before the publication of his discourse The Garden of Cyrus. If Browne acquired an edition of Bettini's 'Beehives' in 1656, then it  potentially influenced either consciously or unconsciously, Browne's penning the discourse, The Garden of Cyrus. Either way Bettini's Aparia and Browne's The Garden of Cyrus are thematically united, jointly supplying evidence to their reader of how the principles of geometry pervade the world. In Browne's case this involves countless examples of the 'mathematics of nature' via the geometry of the quincunx pattern. 

Although the bulk of Browne's scientific writings are  to be found in his encyclopaedia Pseudodoxia Epidemica,  many of the topics covered by Bettini in Aparia also feature in Browne's discourse The Garden of Cyrus. For example, in the second proposition or 'Beehive' in Bettini's 'Beehives of Universal Mathematical Philosophy' the Jesuit scientist examines the mathematics of the spider-web - 


The spider and its web-making ability feature twice in Browne's Garden of Cyrus, firstly in his observing - 'that the woof of the neat Retiarie Spider, which seems to weave without transversion, and by the union of right lines to make out a continued surface.' and secondly  - 'And no mean Observations hereof there is in the Mathematics of the neatest Retiary Spider, which concluding in forty four Circles, from five Semi-diameters beginneth that elegant texture'. [2]

Bees

Browne also shared with Bettini an interest in bees. From the time of the ancient Greek philosopher Pappus of Alexandria to the Renaissance-era various mathematicians and philosophers believed bees to be Heaven-instructed mathematicians which were capable of 'geometrical forethought' and in possession of knowledge transcendent to humanity. 

Bee's important contributions to civilization consist of honey, a rare source of sweetness and wax, useful for many aspects of human commerce including candles for light. Honey and wax were both valuable contributions to the advancement of civilization until the advent of gas and electric lighting and the discovery of other sources of sugar. Evidence of human beekeeping, known as apiculture, can be found in Hindu, Hittite, Greek and ancient Egyptian civilizations and as such bees have fascinated poet, philosopher and scientist alike.  From the Roman poet Virgil's verse on apiculture in his fourth Georgic to Bernard Mandeville's inverted theory of the relationship between morality and economics in The Fable of the Bees (1719) to the mysticism of Maurice Maeterlinck's  Life of the Bee (1900) bees are frequently associated with activity, diligence, an industrious work-ethic and order. The collective nature of the beehive has been used as evidence supporting both communal and monarchical forms of government.

Thomas Browne makes a beeline towards advocating the wisdom of the 'curious mathematics' of bees in Religio Medici when proposing -

'Indeed what reason may not go to School to the wisdom of Bees, Ants, and Spiders ? What wise hand teacheth them to do what reason cannot teach us?..... in these narrow Engines there is more curious Mathematics, and the civility of these little Citizens, more neatly set forth the wisdom of their Maker;  [3]

Centuries before the Czech author Franz Kafka (1883-1924) described the horror of  Gregor Samsa's transformation into a giant beetle in his short story Die Verwandlung (1915) Thomas Browne in Religio Medici (1643) imagined himself  as a bee in flight -


'when homeward I shall drive

Rich with the spoils of nature to my hive,
There will I sit, like that industrious fly,
Buzzing thy praises'.....[4]



Browne's mystical awe in contemplation of the 'curious mathematics' of the bee in Religio Medici transforms into  a sharp-eyed 'ocular observation' of nature in The Garden of Cyrus in which the geometry of the beehive is closely examined-

'The sexangular Cells in the Honeycombs of Bees, are disposeth after this order, much there is not of wonder in the confused Houses of Pismires, though much in their busy life and actions, more in the edificial Palaces of Bees and Monarchical spirits; who make their combs six-corner’d, declining a circle, whereof many stand not close together, and completely fill the area of the place; But rather affecting a six-sided figure, whereby every cell affords a common side unto six more, and also a fit receptacle for the Bee it self, which gathering into a Cylindrical Figure, aptly enters its sexangular house, more nearly approaching a circular Figure, then either doth the Square or Triangle. And the Combs themselves so regularly contrived, that their mutual intersections make three Lozenges at the bottom of every Cell; which severally regarded make three Rows of neat Rhomboidal Figures, connected at the angles, and so continue three several chains throughout the whole comb'. [5]

Its difficult to imagine the sheer profusion of natural life which existed in Browne's day. Bird and insect populations were much denser than today. Scientific evidence indicates there's been a 33% decline among the 130 plus species of pollinating insects in just the past 13 years. This decline is closely related to world food security and even, potentially, to the extinction of present-day civilization. 

A superb example of Browne's sharp sighted  'ocular observation' occurs in The Garden of Cyrus in the learned doctor's declaration - 

'He that would exactly discern the shape of a Bees mouth, need observing eyes, and good augmenting glasses; wherein is discoverable one of the neatest pieces in nature, and must have a more piercing eye then mine'. [6]

In modern times the Russian mathematician and esotericist, P.D. Ouspensky (1878-1947) speculated of bees-

'Having begun to alter their being, their life and their form, bees and ants, taken as individuals, severed their connection with the laws of  Nature, ceased to express these laws individually and began to express them only collectively. And then Nature raised her magic wand, and they became small insects, incapable of doing Nature any harm'.

'Ants and bees alike both call for our admiration by the wonderful completeness of their organisation, and at the same time repel and frighten us, and provoke a feeling of undefinable aversion by the invariably cold reasoning which dominates their life and by the absolute impossibility for an individual to escape from the wheel of life of the ant-hill or beehive. We are terrified at the thought we might resemble them'. [7]

Optics

Of the many facets of optics such as reflection, refraction, magnification and perspective, it seems as if  the study and understanding of the workings of the camera obscura was the 'holy grail' of the 17th century European scientific revolution. Mario Bettini describes the workings of the camera obscura in his Apariaand a rough description of its workings also occurs in The Garden of Cyrus. 

'wherein the pictures from objects are represented, answerable to the paper, or wall in the dark chamber; after the decussation of the rays at the hole of the hornycoat, and their refraction upon the Christalline humour, answering the foramen of the window, and the convex or burning-glasses, which refract the rays that enter it'.

In Bettini's Aparia the optical illusion of replicating the image of just one foot-soldier into a total of twelve foot-soldier's, advantageous strategy in military affairs, is demonstrated. (below).



In addition to Bettini's describing the camera obscura many other authors associated with its development are recorded in Browne's library [8].

Browne's strong interest in optics is celebrated in French artists Anne and Patrick Poirier's 'geometric garden' of twenty interconnecting sculptures in granite and two large-scale marble pieces, one of a brain, the other an eye, which were installed in 2007 near the physician's 17th century home at Hay Hill, Norwich. The Italian marble block, approximately 1.5 metre square has on its obverse an eye and the word 'Memorabilia' on its reverse.



The subject of acoustics is explored in the third volume of Bettini's Aparia ; a topic also included in The Garden of Cyrus -

'A like rule is observed in the reflection of the vocal and sonorous line in Echoes, which cannot therefore be heard in all stations. But happening in woody plantations, by waters, and able to return some words; if reached by a pleasant and well-dividing voice, there may be heard the softest notes in nature'. [9]

An authoritative Brunonian scholar perceptively notes of the geometric and mathematical content of The Garden of Cyrus -

'In long stretches of chapters 3 and 4 of Browne's discourse The Garden of Cyrus, the job of preserving the ubiquity of decussation (X) in nature is mathematical, the tapering cylindricality of trees, Archimedes on conic shapes, squaring the circle, and pyramids of light through the aperture of the eye. If The Garden of Cyrus is an almost mathematical work, suffused in the Euclidean pleasures of number and form, Browne also dwells in the near tactility and texture of his geometrical vocabulary, 'helicall or spirall roundles, volutas, conicall sections, circular Pyramids, and fustrums of Archimedes'. [10]

It was during the early scientific revolution (generally considered to begin with Nicolaus Copernicus's theological-challenging heliocentric universe, 'On the Revolutions of the Heavenly Spheres' in 1543 and culminating in the abstract mathematics and physics of Isaac Newton's Principia in 1687) that the study of optics, along with astronomy and botany among other subjects became accessible to educated and leisured enquirers, in particular from the ranks of priest and physician, Mario Bettini and Thomas Browne's respective professions. 

Jesuits such as Bettini made many contributions to the development of science and have been described as "the single most important contributor to experimental physics in the seventeenth century." By the eighteenth century the Jesuits had "contributed to the development of pendulum clocks, pantographs, barometers, reflecting telescopes and microscopes and to scientific fields as varied as magnetism, optics and electricity. They observed, often before anyone else, the coloured bands of Jupiter, the Andromeda nebula and Saturn’s rings. They theorized about the circulation of the blood, the theoretical possibility of flight, the way the moon effected the tides, and the wave-like nature of light.

Above all other Jesuit scientists however it was books by the polymath Athanasius Kircher (1601-80) which were avidly collected by Browne. A near exact contemporary to Browne, Kircher has been described as 'the supreme representative of Hermeticism in post-Reformation Europe'  and was a favourite read of the physician-philosopher, as the contents of his library reveals. Browne often wrote with his most recent reading in mind; its hardly coincidental therefore that the antiquarian artefact known as the Bembine Tablet of Isis is mentioned not once, but twice, in The Garden of Cyrus for Browne had  recently acquired Kircher's vast work of comparative religion Oedipus Aegypticus (Rome 1650-1655) in which the Bembine Tablet, the Rosetta Stone of its age, is reproduced and 'interpreted' by Kircher. Although frequently misapprehending the true meaning of the antiquities, Egyptian hieroglyphs and world religion myth he encountered, nevertheless Kircher paved the way for future study in comparative religion.  [11] 

Although Browne often purchased books swiftly upon their publication there's no easy way of ascertaining whether or not he acquired an edition of Bettini’s Aparia in the year of 1656 and even though Browne's The Garden of Cyrus (1658) shares subject-matter with Bettini's Aparia, it also ranges into topics as diverse as - Architecture, Biblical scholarship, Egyptology, comparative religion, mythology, gardening and plantations in antiquity, geometry, the Archimedean solids, sculpture, numismatics, games and sports including backgammon, knuckle-stones, chess, archery and skittles as well as paving-stones, battle-formations, optics, the camera obscura, perspective, acoustics, music therapy, zoology, ornithology, the kabbalah, astrology, astronomy and not least, botany, including speculations upon the related topics of  germination, generation, longevity and heredity. All these topics are used by Browne in order to supply his reader with evidence of the archetypal quincunx pattern's  eternal existence.

In essence the subjects of mathematics and geometry were viewed  in tandem during the seventeenth century, from both a practical, utilitarian perspective as well as from an esoteric view-point. Discoveries of mathematical laws and geometrical principles, 'the higher geometry of nature' were interpreted by early scientific enquirers, all of whom were religious-minded, as evidence of the wisdom of God, 'the supreme geometrician' in Browne's personal, mystical vision in The Garden of Cyrus whilst Bettini's Aparia is in essence a Counter-Reformation attempt to harness the rapid development  of  science to Church teaching and authority.

Bettini's Aparia is related not only in  its subject-matter but also in its frontispiece art-work to Browne's discourse. New study of the frontispiece to Bettini's Aparia by the Bolognese artist Francesco Curti entitled The Garden of Mathematical Sciences reveals it to exhibit the self-same fusion of scientific enquiry and esoteric symbolism as encountered in Browne's Garden of Cyrus. Curti's early colour engraving as such may be considered a worthy  'alternative' candidate to the frontispiece of Browne's The Garden of Cyrus. This relationship between Browne's textual discourse to Curti's visual artwork is rewarding to explore in depth. 


The Garden of Mathematical Sciences


The colour engraving and frontispiece to Bettini's Aparia entitled The Garden of Mathematical Sciences (above) by the Bolognese artist Francesco Curti (1603-1670) conjures a garden in which mathematics is associated with nature. In what is a highly symmetrical and artificial composition combining art with nature, Curti's engraving depicts a Villa courtyard with an extensive background landscape. In its foreground stand ten antique vases, each of which has optical phenomena etched upon it,  a scientific instrument  growing from it as if a flower, and a stem with a geometric shape attached to it. Curti's ornate vases represent the vigorous growth of  mathematical science during the early scientific revolution in which understanding of geometry and mathematics advanced understanding in subjects as  diverse as architecture, navigation, art-perspective and optics. [12] 

Centre-stage in Curti's Garden of Mathematical Sciences there is a sculptured stone basin supported by two entwined water-nymphs or Naiads, female spirits once believed to preside over fountains, wells, streams and freshwater. A peacock alights upon the water basin's sculptured ornamentation with one foot upon a sphere its other mysteriously grasping a staff with a single eye at its tip. Water streams from its fanned feathers, creating a perpetual fountain. Two hedged gardens, rough pasture, bees in flight, a geometrical spider-web, two mystical statua and the figure of Mercurius holding an armillary sphere while standing upon a pyramid of six  beehives can also be seen.

A comparative study of Curti's engraving to Browne's discourse is assisted by the fact that The Garden of Cyrus  is itself a highly visual work in its abundance of  visual imagery; both 'Garden' art-works may loosely be defined as possessing characteristics associated with Mannerist art. 

The art-historian John Shearman noted that characteristics of  Mannerist art included - Hidden classical references, refinements, interlacing of forms and unexpected and departures from common usage. The Hungarian art-historian Arnold Hauser noted that Mannerist art delighted in symbols and hidden meanings and that it catered for an essentially international cultured class, was a refined and exclusive style, with an intellectual and even surrealistic outlook. He also noted that Mannerist art was inclined towards esoteric concepts in its symbolism. In words easily applicable to either 'Garden' art-work Hauser defined the qualities and excesses of Mannerist art thus -

'At one time it is the deepening and spiritualizing of religious experience and a vision of a new spiritual content in life; at another, an exaggerated intellectualism, consciously and deliberately deforming reality, with a tinge of the bizarre and the abstruse.' [13]

Thus, although differing in medium, both 'Garden' art-works with their utilization of multiplicity and variety, juxtaposition of art and nature, along with their fusion of scientific enquiry to esoteric symbolism, easily conform to the artistic style and objectives of Mannerist art. However, such is the stylistic contrast between Browne's two philosophical discourses that while the stoicism of Urn-Burial with its survey of human grief, passion and bereavement, couched in oratorical prose is utterly Baroque in theme and style; its diptych  companion, The Garden of Cyrus with its procession of examples from art and nature involving great variety and multiplicity and many esoteric allusions is exemplary of Mannerist artistic traits.

In Curti's Garden of Mathematical Sciences the superimposed symbols of fountain and peacock are worthwhile looking at closely.   


Victorian-era, Gothic-style fountain, Plantation Gardens, Norwich.

Fountains feature prominently in gardens from the Renaissance era onwards. The functional aspect of the fountain, to provide drinking-water, was superseded as a purely decorative and entertainment feature in gardens. In addition to creating health-inducing negative ions, fountains also camouflage conversation from prying ears in public, urban spaces. Many of Rome's famous fountains were constructed during the seventeenth century including Bernini's fountain of the Four Rivers, the  Trevi Fountain and the so-called Bee Fountain. 

Contemporary to the construction of such large-scale public fountains Jacob Dobrzenski (1623-97) a Professor of mathematics and medicine of Nigro Ponte, Ferrara, published a book in 1657 with the intriguing title of, 'New and More Pleasing Philosophy on the Wonderful Spirit of Fountains' (Nova et amenior de admirando fontium genio philosophia).

    
    15th c. illustration from De Sphera Modeni,Italy. 


The alchemical symbolism of the fountain was developed through Bernard of Treviso's story of a King who is rejuvenated after bathing in a fountain. Trevsio's story was included in the 17th century anthology known as the Theatrum Chemicum. A Fountain of Love is also mentioned on several occasions by the philosophical alchemist Gerard Dorn in his Speculativa Philosophia  in the first volume of the Theatrum Chemicum a copy of which was once in Thomas Browne's library. [14]

'Approach the fountain here, Body, so that you may drink your fill with your Mind and not thirst any more for Vanities. O admirable efficacy of the fountain, which makes one from the two and brings peace between enemies ! The fountain of Love can make Mind from Spirit and Feeling Soul, but here it makes one man from Mind and Body. [15]

Alchemical literature and iconography frequently alludes to a fountain of Youth in which the magical powers of its waters restore and rejuvenate; like the philosophical bath the mercurial character of the fons mercuralis in which mercury is transformed  means it is dualistic, being poisonous as well as healing,  apt symbolism of the underlying unity of the trickster god of alchemy.

In his late work Mysterium Coniunctionis - An inquiry into the synthesis and separation of psychic opposites (1963) C.G.Jung likens the everlasting fountain to psychic processes, thus -

The ever-flowing fountain expresses a continual flow of interest towards the unconscious, a kind of constant attention or "religio" which might also be called devotion.....If attention is directed towards the unconscious, the unconscious will yield up its contents, and these in turn will fructify the conscious like a fountain of living water.   [16] 


The myth of how the peacock got its many 'eyes' and how it became a bird sacred to the goddess Juno is recounted in Ovid's Metamorphoses, a source-book of inspiration to Renaissance painter, poet and sculptor alike. The Roman poet relates how the hundred eyes in the head of Argus took their rest two at a time while the others kept watch on guard. Wherever Argus stood he was looking at Io, and had Io in front of him even when his back was turned. Zeus ordered Hermes to assassinate Argus. The goddess Juno had the hundred eyes of Argus preserved forever, into a peacock's tail. [17]

The subject of Juno and the hundred eyes of Argus became a popular theme during the seventeenth century.  European artists including Rubens, Velasquez and many others  were inspired by the Greek myth. [18]


Avian symbolism is frequently encountered in alchemical iconography in which the raven, swan, pelican, dove, owl and peacock also feature. Several symbolic attributes are associated with the peacock, these include it being, like the phoenix, a solar bird from its wheel-like fanned display of feathers, as a symbol of rebirth and immortality from its supposed incorruptible flesh, as a symbol of multiplication from the many 'eyes' upon its fanned feathers, while the optical effect of iridescence produced by its feathers is likened to the numinous experience of the alchemist engaged in experiment.

Symbols can endure paradox. Whilst the peacock, like the phoenix is a solar symbol from the way in which it spreads its tail in the shape of a wheel,  the many 'eyes' upon its fanned feathers are analogous to the starry night sky.

C.G. Jung notes - 'The peacock is an old emblem of rebirth and resurrection, quite frequently found on Christian sarcophagi' [19] a fact which Thomas Browne noted  in Urn-Burial when writing of early Christian funeral iconography depicting,  'the mystical figures of peacocks, doves and cocks'. 

Jung also states-


'The caudo pavonis announces the end of the work, just as Iris, its synonym, is the messenger of God. The exquisite display of colour's in the peacock's fan heralds the imminent synthesis of all qualities and elements, which are united in the  "rotundity" of the philosophical stone'. [20] 

Jung  likened the iridescence of peacock's feathers to alchemical experimentation stating - 'The chemical causes of the cauda pavonis are probably the iridescent skin on molten metals and the vivid colours of certain compounds of mercury'. [21]  

The optical effect of iridescence on silk may have been known  to Thomas Browne when very young for his father was a wealthy silk merchant. In  Pseudodoxia Epidemica he notes

'And from such salary irradiations may those wondrous varieties arise, which are observable in Animals, as Mallards heads, and Peacocks feathers, receiving intention or alteration according as they are presented unto the light'.[22]


The 19th century mythologist De Gubernatis stated-

'The serene and starry sky and the sun are peacocks. The deep-blue firmament shining with a thousand brilliant eyes, and the sun rich with the colours of the rainbow, present the appearance of a peacock in all the splendour of its eye-spangled feathers. .....It is commonly said of the peacock that it has an angel's feathers, a devil's voice, and a thief's walk'. [23]

On a mundane level the many eyes of the peacock's tail may be interpreted as symbolizing  the watchfulness of the observer during the alchemical  opus while at a higher level poly optics symbolizes the alchemical stage of Multiplication. Crucially, in Jung’s view the motif of the all-seeing 'eyes' of the peacock - polyophtalmia (many eyes) - is associated with ‘multiple consciousness’ that is, with the various quasi-conscious states  which exist in the unconscious. Multiple eyes symbolize what Jung calls 'multiple luminosities' of the unconscious. Particularly, polyophthalmia ‘indicates the observing consciousness is the observing agent of the psyche. Polyopthalmia can also symbolically illuminate the concept of foreknowledge, that is, not about knowing something in advance (‘fore’) but rather instead about being able to observe what is already in existence through a simultaneous multiplicity of perspectives. Thus, the many eyes of the displayed tail feathers of the peacock can be said to symbolize a non-linear multiplicity of perspectives. [24 ] 


In the richly coloured and detailed engraving for Salomon Trismosin's Splendor Solis by Jörg Breu the Elder (1480-1537) a peacock is depicted encased within an alchemical vessel (above).

The peacock's  fanned feather display exhibits the short-lived nature of all manifestation, since its forms appear and vanish as swiftly as the peacock displays and furls its tail. Indeed, to the present-day the sudden appearance of a rainbow (the peacock's close symbolic relation) caused by the optical effect of light refracted through water, retains a fragment of a once potent numinosity to those seeing it occur in nature. 

Although Juno is named in The Garden of Cyrus  the bird sacred to her, the peacock is not, however, geese, ducks, cormorant, bittern, owls, swallows along with butterflies, bees, beavers, rattlesnakes, lambs and carp as well as elephants and whales are mentioned in the discourse.

Browne was in fact a keen bird-fancier, keeping at one time or another a cormorant, owl, bittern, golden eagle and even an ostrich so he may well have approved of a peacock on a frontispiece for his discourse,  stating in the dedicatory epistle of The Garden of Cyrus‘noble spirits contented not themselves with Trees, but by the attendance of Aviaries, Fish Ponds, and all variety of Animals’.

In many ways the symbols of peacock and fountain in Curti's engraving are near-identical in their symbolic meaning, that of a numinous and revivifying phenomena accompanying the alchemist and/or early scientist in their quest. The appearance of the cauda pavonis of the peacock is considered to be a dramatic indicator of success in the opus while the fountain is similarly associated with flourishing and growth in the alchemical opus.

In essence Curti's Garden of  Mathematical Sciences captures the moment of revelation. As such it depicts a 'Light-bulb' moment as experienced by the alchemist/scientist whilst engaged in experiment in the laboratory. The light-bulb did not of course exist during the 17th century, and a more natural, if at first, seemingly paradoxical imagery is employed by Curti to express the  short-lived psychic experience of revelation.  

In modern times the 'Light-bulb moment' can be traced in origin to a character in Max Fleischer's early Betty Boop cartoons (1935-1937). Grampy is an eccentric inventor who entertains his guests by building self-playing musical instruments out of household gadgets. Whenever presented with an unexpected new problem, Grampy puts on his thinking cap, a mortarboard with a light-bulb on top. When the light-bulb lights up Grampy is able to solve his problem and build a new gadget to solve the problem.



The two mid-seventeenth century 'Garden' art-works text and image are related to each other not only in title,  chronology and subject-matter,  but also, crucially, in their self-same fusion of scientific enquiry with esoteric symbolism. Juxtaposed to its depiction of scientific instruments in Curti's Garden of  Mathematical Sciences allusions to Pythagorean  number symbolism can be seen; the self-same fusion of nascent scientific enquiry to esoteric symbolism permeates Browne's mystical vision of the inter-connection of art and nature in The Garden of Cyrus. 


The Renaissance was an era in which  the 'Re-birth' or 'rediscovery' of various forms of knowledge occurred. Its useful to realise that this included the 'rediscovery' of esoteric writings such as the Corpus Hermeticum by so-called Gnostic authors, as well as 're-discovered' texts, foremost of which was the discourse known as the Timaeus by the ancient Greek philosopher Plato.

Second only to the many myths included in the Judaeo-Christian Bible, Plato's discourse the Timaeus was the most frequently consulted hand-book which influenced and inspired hermetic philosopher and alchemist alike during the Renaissance.  In  what is his most Pythagorean work, Plato's  Timaeus recounts how the demiurge created the world in the geometric form of a globe. The round figure is proposed to be the most perfect one, because it comprehends  all other figures and  is therefore the most omnimorphic of all figures, each point on its surface being equidistant from its centre. The sphere is featured above all other shapes in the frontispiece engraving The Garden of Mathematical Sciences with no less than ten spheres in total around each of the two enclosed gardens of Curti's Neoplatonic landscape view from a courtyard villa. 

 In his highly influential Oration on the Dignity of Man (De hominis dignitate) of 1486 the Renaissance humanist scholar Pico della Mirandola (1463-94) famously justified the importance of the human quest for knowledge within a Neoplatonic framework. Pico della Mirandola is also credited with re-introducing the 'mystical mathematics'  of Pythagoras to Renaissance Europe. The Greek philosopher Pythagoras was worshipped and venerated as a god for almost one thousand years before institutions teaching his ideas were closed down at the Fall of the Roman empire. Pythagoras taught that -

'By number, a way is had, to the searching out and understanding of everything able to be known'. 

Pythagoreans believed the number ten to be the number of totality and perfection containing within it all other numbers. It was depicted in Pythagorean teachings in the form of the tetractys a pyramid of dots (1+2+3+4) representing universal principles. 

Pythagorean numerology and Platonic shapes abound in Curti's illustration The Garden of Mathematical Sciences. The sphere is featured in repeated groupings of  ten as well as ten bees in quincunx formation and in ten vases in a 2 x 5 arrangement in its foreground.  The  number of chapters  in Browne's diptych discourses total ten and the figure X along with citations from Plato's Timaeus loom large throughout the pages of The Garden of Cyrus from its very opening  to its Platonic meditation upon the figure X as a symbol of the soul.


                       



Radiating from the heart of the tetkratys pattern the hexagon can be seen, believed by Bettini, among others, to be 'proof' of the transcendent mathematical ability of bees in their construction of hexagonal honeycomb cells. The quincunx pattern (four corner dots of a square with one at the centre as upon dice) celebrated for its ubiquity in art and nature in Browne's Garden of Cyrus can also be discerned at the centre of the tetkratys. 

Although the figure of quincunx  is mentioned in classical antiquity it was during the Renaissance  that the idea of it being a pattern which transcends the realm of the artificial originates. The idea can be found in book 4 of the Italian Renaissance scholar Giambattista Della Porta's agricultural encyclopedia Villa (1583-1592) in which Della Porta (1535-1615) asserts that the quincunx pattern in addition to featuring in gardens and plantations, 'is to be found in each and every single thing in nature'. An illustration of the quincunx pattern was 'lifted'  from Della Porta's agricultural encyclopaedia Villa by Thomas Browne for the frontispiece of his 'Garden' discourse  (below)




Magnification of Curti's frontispiece  reveals the very same  quincunx pattern occurs in the hedge panels surrounding the gardens of Curti's imaginary Villa, in the formation of bees in flight, as well as the double 2 + 1 + 2 arrangement  of the ornates vases in its courtyard foreground.

In conclusion, Curti's Garden of Mathematical Sciences features two quite different approaches and interpretations of number which  co-existed during the 17th century before going their separate ways. It alludes to Pythagorean numerology as well as promoting the new 'observational' sciences of optics and astronomy and is therefore a strong candidate as an alternative frontispiece to Browne’s 'Garden' discourse for these two quite different interpretations of number, that of Pythagorean number symbolism and a utilitarian approach to number during the early scientific revolution occur in both Curti's Garden of Mathematical Sciences (circa 1660) as well as in Browne's  1658 discourse The Garden of Cyrus .  

Notes

[1] Mario Bettini's book is listed in the 1711 Sales Catalogue  of Thomas Browne's library on p. 28 no. 16 under Folio by its half-title Fucaria & Auctaria ad Apiaria Philosophiae Mathematicae 1656. 
[2]  The Garden of Cyrus chapter 2
[3] Religio Medici Part 1:13
[4] Religio Medici Part 1:15
[5] The Garden of Cyrus
[6] The Garden of Cyrus
[7] A New Model of the Universe: Principles of the psychological method in its application to problems of Science, Religion an Art. by P.D. Ouspensky RKP 1931
[8] Optic books in Browne's library include - Alhazen  - Opticae Thesaurus Libri X, Basle 1572 Francois d'Aguillon - Opticorum Libri 6, Antwerp 1613 Johannes Kepler - Ad Vitellionem Paralipomena, Frankfurt 1604 Athanasius Kircher - Ars Magna Lucis & Umbrae, Rome 1646 Christoph Scheiner - Rosa Ursina sive Sol, Bracciano, 1630
[9] The Garden of Cyrus
[10]  Thomas Browne Selected Writings ed. Kevin Killeen OUP 2014 
[11] Oedipus Aegyptiacus  1711 Sales Catalogue page 8 no. 91
[12] Francesco Curti colour image courtesy of Getty Images, with thanks for fair usage. This image has been available online since December 31st 2016. The full size of Francesco's Curti's colour engraving is approximately 30 x 40 cm. There are in fact two different versions of the  frontispiece for The Garden of Mathematical Sciences. Early editions include a frontispiece by Matthiae Galasso/Matthias Galassus while later editions feature Francesco Curti's colour engraving.


The biggest difference between the two versions is the various ensigns, banners and disembodied armoury in Galassus's version being replaced in Curti's engraving by the figure of Mercurius holding a banner with Papal ensigns. Both versions depict an armillary sphere, symbolic in Mathias Gallius's version to the world-wide influence of the missionary Jesuit Order. In Curtius's version it is Mercurius, the messenger of revelation and guiding 'deity' of alchemy who is featured in the frontispiece's symbolism.
[13] John Shearman Mannerism London, Penguin/Baltimore, MD, 1967 
and Arnold Hauser Mannerism. The Crisis of the Renaissance and the Origin of Modern Art
[14] Theatrum Chemicum Sales Catalogue page 24 no. 124
[15] Ibid.
[16] CW 14: 193
[17] Ovid Metamorphoses  Book 1 500-746 Penguin 1955
[18] Artists inspired by the Greek myth of Juno and the peacock include - Peter Paul Rubens (1577 – 1640) Juno and Argus, c. 1610, oil on canvas, 249 x 296 cm.  (Post illustration) Other seventeenth century paintings on the theme of Juno, Argus and the Peacock include- Claude Gellée ‘Mercury Lulling Argus to Sleep with the Sound of His Pipe’ (1662) - Cornelis Bisschop (1630-1674) Circle of Cornelius van Poelenburgh (circa 1650) - Govert Flinck (1615-60) circa 1635-45 - Jacob Jordaens circa 1620 - Carel Fabritius ( circa 1645 and circa 1647) Velázquez (1659) Hendrik Goltzius (1615) Antonio Balestra (1666-1740)
[19] C.W.  Vol. 9i: 686
[20]  C.W.  381 n. 2
[20]   C.W.  vol. 14 396
[21] CW 9i 581 n. 129
[22] Pseudodoxia Epidemica
[23] Angelo De Grubernatis Zoological Mythology II London 1872
[24] - Time and Timelessness: Temporality in the theory of Carl Jung By Angeliki Yiassemides

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The bee is considered to be the most important living creature on the planet

Recommended listening

Alchemical literature of the sixteenth and seventh centuries frequently alludes to  the transformative power of music, most notably in Michael Maier's Atalanta Fugiens (1617). The twentieth century musical genre of Jazz  - an art-form which thrives upon experiment and which has the meditative and melancholic music genre of the 'blues', almost equivalent to the Nigredo stage of the alchemical opus - is a worthy contender for representing certain prerequisites and templates of alchemy,  the musician in the studio or in performance expressing inner experience as much as the alchemist  in his laboratory engaged in the alchemical opus. 

A highly-stylized cry of the peacock can be heard in the legendary tenor saxophonist Stan Getz's interpretation of pianist/composer  Jimmy Rowles  The Peacocks (1975)    







John Coltrane (1926-67) and Stan Getz (1927-1991) were the t
wo tenor saxophonists who dominated 20th century JazzLike chalk and cheese to each other, each possessed a unique technique and interpretative skill, as their respective performances and recordings demonstrate. If Stan Getz's The Peacocks may be considered as expressive of the nigredo stage of alchemy, John Coltrane's rendering of The Night has a Thousand Eyes is an albedo fountain of musical notes.








The English composer William Alwyn (1905-1985) in his autobiography Winged Chariot states of his 5th symphony  Hydriotaphia (1973) 'Browne's wonderful prose sets the mood of each section and is an expression of my personal indebtedness to a great man whose writings have been a life-long source of solace and inspiration'.  Alwyn's Naiades (1971)  a Fantasy Sonata for flute and Harp aurally depicts the water-nymphs of antiquity, as seen supporting a water-basin in Curti's colour engraving.  




This blog essay for the artist Rinat Baibekov with thanks for encouragement.

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