The major collector of Marcel Duchamp's art, his American friend and benefactor, Walter Arensberg, displayed an astuteness of intellect cerebral and clever enough to qualify him--in certain areas anyway--as a true colleague of the artist. A Harvard graduate, he was a sometime poet, and gentleman scholar. One of Arensberg's favorite hobbies, as we have seen, was crypanalysis (or cryptography) the composing and deciphering of secret messages. This observation about Arensberg lends further pith to the tantalizing encouragement of Walter Hopps: that the puzzling nature of the object making the hidden noise "could be figured out." Figure it out? But how? Through the agengy of an oracle? Logically? Analytically? We should be able to deduce, from this hint alone, that the secret was not intended to be utterly inscrutable; instead, let us regard the piece in the spirit of a challenging game, posing a riddle at least theoretically possible to solve. That makes it worthwhile to search for other clues, perhaps some kind of secret "signature," such as that for which Arensberg was searching through Dante's terza rime, possibly even to discover some distinguishing mark left by the amateur cryptanalyst himself, embodied in the conundrum.

Another of the lifelong, literary, and intellectual passions of Walter Arensberg centered on problems of Shakespearian authorship. His fascination with codes and cyphers led him to champion the cause of those who attribute the Bard's works to Francis Bacon. Without wishing to wander through that thicket of literary thorns, the present occasion nevertheless prompts us to mention one of Shakespeare's fantasy plays, A Midsummer-Night's Dream, and a memorable set of lines containing five "dies" in a single line. (There is, earlier in the speech, another die, and one more in the line that follows, in all making seven; but the five that come together, we think, must be reckoned quite extraordinary. For that matter, it must be admitted, there are four "thusses.") In the play within the play, the clown Bottom, in his role as Pyramus, thinking his beloved Thisbe dead, and moved by passionate remorse, commits a melodramatic suicide:

PYRAMUS (stabbing himself): Thus die I, thus, thus, thus. Now am I dead, Now am I fled; My soul is in the sky: Tongue, lose thy light; Moon, take thy flight: Now die, die, die, die, die.

DEMETRIUS: No die, but an ace, for him; for he is but one.

LYSANDER: Less than an ace, man; for he is dead; he is nothing.

[William Shakespeare (or Francis Bacon?), A Midsummer-Night's Dream, Act V, Scene 1.]

A few miles south of Pasadena--where Walter Hopps had arranged Duchamp's 1963 retrospective exhibition--stands the Huntington Library in San Marino, California. This charming institution houses the archival material which prompted publication by Macmillan (in April, 1988) of what they called "the literary event of the decade." A part of the academic world remains skeptical, while another faction is more forthrightly combative. The following account by Howell Raines, written for the New York Times, crisply chronicles one critical quest to define authenticity, while offering an object lesson on research methodology. The publication materialized as a book by Peter Levi, professor of poetry at Oxford, called A Private Commission: New Verses by Shakespeare, in which Levi claims to have discovered a new poem of fourteen verses written by Shakespeare in 1606.

But during his news conference at the Barbican Center, Levi acknowledged that he had not been aware until Sunday that the poem was first published in 1835 in New Facts Regarding the Life of Shakespeare, by John Payne Collier. The book by Collier is still available in the public library here [dateline London], and Collier's reputation as a forger seemed to cast doubt on Levi's argument for the authenticity of the poem.

In that regard, Levi admitted yesterday that he had not examined the original manuscript of the poem, which is in the Huntington Library in San Marino, California, and had been working from a photocopy.

This was an important point, because Levi's key piece of evidence that the poem is Shakespeare's is a handwriting expert's opinion that the initials at the end of the poem--variously read as "WSh" or "WSk" or "WSr"--were inscribed by John Marston. Most scholars agree that Marston, a colleague of Shakespeare's, could be trusted to identify Shakespeare's work.

But was the signature the work of Marston or of the forger John Collier, who wanted to give the impression that he had discovered unknown verses. "As for the signature," Levi said, "I have only the word of the Huntington Library, which, I'm sure, has experts of some integrity."

Yesterday ["...when our troubles seemed so far away..."], in a telephone interview, the chief librarian at the Huntington, Daniel Woodward, said the library never gave Levi an assurance about the signature and, indeed, regarded such verifications as the duty of the scholar, not the library....

Levi yesterday defended his failure to go to California as a matter of academic economics. "I can't afford to jump on a plane to California," he said. "I'm a professor of poetry."

"...I don't terribly mind what happens to my reputation," said Levi. "It would be a terrible act of cowardice not to have printed as a poem of Shakespeare what other people were too idle or too silly to notice."...The poem consists of tributes to be spoken by fourteen identified ladies presenting gifts to Anne Chandos at her betrothal party in 1607.

Levi cites this verse to prove the poem was worthy of England's greatest poet: "Witty, pretty, virtuous and fair, Compounded all of fire and air, Sweet, measure not my thought and me, By golden fruit from fruitless tree."

"That's beautiful. It could be Shakespeare, and I think the whole thing is by him," Professor Levi said of the poem.

[Howell Raines, "Much Ado Over Shakespeare Book," New York Times, reprinted in "Datebook," San Francisco Chronicle (April 26, 1988).]


The figure five, Duchamp, and Mallarmé come together in an intriguing document now lodged in the Arensberg Archives at the Francis Bacon Foundation in Claremont, California. This institution, which is also within a couple miles of both the Huntington Library in San Marino and the Pasadena City Art Museum, serves as a principal repository for the papers of Walter Arensberg, whose collection of artworks, we recall, included most of the Duchamp treasures now at the Philadelphia Museum of Art. In a folder of documents--not all of which, at this writing, have been fully catalogued by the Foundation--are some pages by Marcel Duchamp. Of particular interest is a page full of pentimenti, scribbles and doodles drawn over a text, the words of which have been made up from the typographic small-capitals in Mallarmé's poem, Un Coup de s. The whole page appears to be in Duchamp's hand. The present author has only seen a photocopy; but despite living in California (since "I am only a professor of art") regrettably has not yet had the opportunity to jump into a Maserati, say, and drive down to Claremont to inspect the original document. In any event, the section of text quoted on the page appears to read:

Quand bien même lancé dans des
circonstances éternelles du fond
d'un naufrage soit (Abime) le
Esprit] [Nombre] [Fiançailles]
matre existât-il commençât-il
et cessât-il se chiffrât-il, illu-
minât-il rien n'aura eu
lieu que lieu except
[Septentrion] [Nord]
peut-être une constellation.
[Toute] [Pensée] [Coup] [Dés]

The brackets here indicate words around which Duchamp has drawn boxes on his page. These are words distinguished in Mallarmé's text by being displayed in a lower-case font, not printed in small caps (as is the rest of the text which Duchamp quotes). The four words Duchamp has written at the bottom of the page with a box drawn around each word, Toute Pensée Coup Dés, form an abbreviation of Mallarmé's key line:

Toute Pensée émet un Coup de Dés.

[For a photographic reproduction of the final definitive proofs of the edition Mallarmé was working on when he died (the Lahure edition which, as such, never appeared), see Robert G. Cohn, Mallarmé's Masterwork: New Findings, Mouton, The Hague-Paris (1966).]

In the middle of the sheet appear a good many scrawls. Among these, however, there may be distinguished three square shapes. The superior square is marked with five dots, disposed like the pips on a die; just as another of the squares also seems to be marked, although it is partly obscured by the other markings on the page. These dice reiterate the theme of chance for Duchamp, who assumed the vanguard among modern artists consciously articulating aleatory functions--the operations of chance--carefully contriving to incorporate randomness of a statistically definable order into the creative act. The roots of this notion explicitly recall the idea expressed by Mallarmé, that "Un coup de Dés jamais n'abolira le Hasard," which literally translated: is either "A..." or "One throw of the dice will never abolish chance."

To the casual eye this seems to be merely a commonplace of gnomic wisdom, even of the homespun type. Looking closer we are led, not to a rejection, but a modification and intensification of this first impression. The word hasard stems from the Arabic az-zahr, dice or a kind of dice-game, so that a second, tautological, meaning is set up alongside the first one; this time it reads:"A throw of the dice will never abolish the dice throw," or approximately; (Mallarmé, like Joyce [and Duchamp], deals constantly in this kind of punning).

[Robert Greer Cohn, Mallarmé's Un Coup de Dés: an exegesis {a Yale French Studies publication, originally presented as the author's thesis (1946 and 1949)}, AMS Press, New York (1980), p. 9.]

The profound influence on Duchamp of Mallarmé's play with chance was attested by the artist himself. Calvin Tomkins in his sparkling survey of Duchamp's work, also comments on the important theme of chance, referring to the often-quoted statement by the artist:

The whole idea of chance, which was later to become the indispensable tool of a number of artists who saw it as a means to make their work conform more closely to the conditions of life, interested Duchamp in a unique way. He believes that chance is the expression of the subconscious personality. "Your chance is not the same as my chance," he has explained, "just as your throw of the dice will rarely be the same as mine."

[Calvin Tomkins, The Bride and the Bachelors: Five Masters of the Avant-Garde, Viking Compass Edition, New York (1968), p. 33.]

A sublime mystique has surrounded the difficult poem of Mallarmé, but despite the oblique revelations and the extreme modesty of the author, one tradition of criticism and interpretation, following Claudel, Gide, Régnier, Dujardin, Valéry and more recently Robert G. Cohn, has come to regard Un Coup de Dés as Mallarmé's masterpiece, no less than a brilliant attempt at THE poem of humanity.

Un Coup de Dés jamais n'abolira le Hasard is an "orphic explanation" of the universe, telling tautly, poetically, the story of the rise and fall of all...for Un Coup de Dés, I submit, resembles Finnegans Wake more than anything we have in all literature, not excluding the Divine Comedy....Mallarmé is a metaphysician of the very highest order, and his Poem is put together as rigidly, in one sense, as a mathematical theorem.

[Cohn, Mallarmé, pp. 9, 5.]

The complex, dense, paradoxical, synoptic language of Mallarmé can be unravelled to reveal structures within structures, including a four-fold cycle much like that James Joyce drew from Giambattista Vico, corresponding to the four times of day, seasons of the year, the "four-polar symmetry expressing return," and so forth. These same Viconian ideas were fairly well-known in the France of Mallarmé's time through Michelet and Comte. These internal cyclic aspects of the Poem so closely related to Finnegans Wake also receive expression in the etymological pun on ds and hasard.

Vico shared Mallarmé's belief in universal language and worked a good deal with (dubious) etymologies.

[Cohn, Mallarmé, p. 12 f., notes 22, 23, 24. Cohn relates these schemata to Joyce's Four Old Men in Finnegans Wake, p. 18 f., n. 35.]

The old French hasard comes from the Moorish Spanish azar (an unlucky throw of the dice or an accident) from the Arabic triliteral root Y-S-R (Ya-Sin-Ra). Most of the common verbs associated with this root convey the notion of gentleness, easiness and tractability, whether of man or beast. Several of the word forms indicate left-handedness, again with connotations of gentleness in contrast to the force and power of the right hand. But the verb in the form Ya-Sa-Ra means to draw lots with arrows; and the derived form Yasur indicates the winner at the game in arrows; and maysir is the game of casting arrows, the game of "hazard" itself. It is natural here to recall the very similar connection we noted above between dice and arrows (and yarrow stalks) in ancient China. And finally, the Ya-Sin-Ra root in Arabic yields one of the names that may be used in reference to God, Al-Muyassir, meaning the Bestower, the Disposer of all things. This is how one may call the name of the Arrow-Divining, Dice-Throwing God.


In his Charles Eliot Norton Lectures for 1971-1972 at Harvard University, Octavio Paz, a critic and a poet in the Spanish language and subsequent recipient of the Nobel Prize for Literature, developed his interpretation of twentieth-century poetics based upon the interacting themes of analogy and irony. For him, Marcel Duchamp and James Joyce epitomize the aesthetic tradition of self-awareness, with a quality of meta-irony also found in the poetry of Ezra Pound and T. S. Eliot, but perhaps above all in that exacting Poem of Stephane Mallarmé, Un Coup de Dés, as we might say, "A Toss of the Dice."

With Mallarmé was born a form which belongs neither to the nineteenth century nor to the first half of the twentieth century, but to the present. [Ezra] Pound's poetic system consists of presenting images as clusters of signs upon the page: ideograms, not static but moving, like a landscape seen from a ship or, rather, like constellations moving toward or away from each other on the surface of the sky. The word "constellation" calls forth the idea of "music," with its innumerable associations from the erotic harmony of bodies to political harmony among men, conjures up the name of Mallarmé. Here is the heart of analogy. Pound is not a disciple of Mallarmé, but the best part of his work belongs in the tradition begun by Un Coup de Dés. In the Cantos as in The Waste Land analogy is continuously torn apart by criticism, by ironic consciousness. Mallarmé and Duchamp: analogy ceases to be a vision and turns into a system of permutations. Like the erotic-industrial, mythic-ironic duality of the Large Glass, all the characters in The Waste Land are real and mythic. Reversability of signs and significances: in Duchamp, Artemis is a "pin-up," and in Eliot the image of heaven and its rotating constellations turns into a deck of cards spread out on a table by a fortune teller. The image of cards leads to that of dice, and the latter once again to the image of the constellations.

[Octavio Paz, Children of the Mire: Modern Poetry from Romanticism to the Avant-Garde, Harvard Univ. Press, Cambridge (1974), pp. 66, 136.]

In her doctoral dissertation on the subject of chance, Harriett Watts compares Mallarmé's constellation of pluralistic readings emerging from the hasard of the poem with Duchamp's notion of "canned chance," the phrase from Duchamp's Green Box which relates to Three Standard Stoppages (1914) and to With Hidden Noise of two years later. In this context she also quotes Professor Paz on the extension of the Symbolist idea of the work of art as a double of the universe which then exists beyond the further influence of chance, as

the Book.

Mallarmé has left hundreds and hundreds of notes describing the physical characteristics of this loose-leaved book; the form in which its leaves will be distributed and combined at every reading so that each combination will produce a different version of the same text...but the Book does not exist; it was never written.

[Paz, Children of the Mire, p. 77; quoted by Harriett Ann Watts, Chance: A Perspective on Dada, U.M.I., (1975, 1980), p. 49.]

In a sense, of course, Mallarmé's Un Coup de Dés was a monumental attempt to construct an indication of what such a cosmic book would be like. The exquisite structure and syntax, or what Cohn calls the "armature" of the poem, its grouping in accordance with numerical schemata, the outrageous subtlety of polarities and dimensionalities, of patterns within patterns, Mallarmé worked out to a fanatical order of detail: not only involving the interlacing etymologies of each word, but also their typographical disposition on the double-page spreads--as in the imagery of the Big Dipper constellation formed by clusterings of words--down to the mystical attributes of individual vowels and the different fonts in which they appear printed.

[The idea] consists in seeing the universe as a language, a script. But it is a language in unending movement and change: each sentence breeds another sentence, each says something which is always different and yet says the same thing....The metaphor which consists in seeing the universe as a book is very ancient and appears also in the last canto of Dante's Paradise...

The pluralities of the world--leaves blown here and there--come to rest together in the sacred book; substance and accident in the end are joined. Everything is a reflection of that unity, not excluding the words of the poet who names it. In the next tercet, the union of substance and accident is presented as a knot, and this knot is the universal form enclosing all forms. This knot is the hieroglyph of divine love.

[Paz, Children, pp. 71, 75. This famous phrase we have referred to above: as Dante says in Canto XXXIII, 91: La forma universal di questo nodo...("The universal form of this knot..." or less precisely, "The form that knits the whole world....").]

For Dante it was possible to believe that the knot was tied from the three persons of the Holy Trinity and that the key with which to read the book of the universe was supplied by the Bible. But for Mallarmé and modern poets after him, the magical text is not so easily identified with any material tome; it lies, rather, in the process of the Great Work, in the face of accepting the reality of Nothingness.

In his youth, in the years of isolation in the provinces, [Mallarmé] had the vision of the Work, a work he compared to that of the alchemists, whom he called "our forefathers." In 1866 he confided to his friend Cazalis: "I have confronted two abysms: one is Nothingness, which I reached without knowing Buddhism ...the other is the Work." The work: poetry confronting nothingness....The universe resolves itself in a book: an impersonal poem which is not the work of the poet Mallarmé, vanished in the spiritual crisis of 1866, nor of anyone else. It is language which speaks through the poet, who is now only a transparency.

[Paz, Children, p. 76 f.]

But the language, since it is poetic--for Mallarmé, as for Beaudelaire--is also magical and enciphered. Thus the poet becomes not only a translator and a cryptanalyst, but also inevitably a cryptographer, whose work generates new ciphers for every new exegesis.

Each poem is a reading of reality; this reading is a translation; this translation is a writing, a new code for the reality which is being unravelled. The poem is the universe's double: a secret writing, a space covered with hieroglyphics. To write a poem is to decipher the universe only to create a new cipher.

[Paz, Children, p. 72.]


James Keys, poet, polymath, and alter ego of the mathematician G. Spencer Brown, in a profound footnote, rehearses the process of this--or any other--general program of Creation. He counts with technical precision the steps from the Void; but to follow his count it is essential to distinguish between cardinal and ordinal numbers--and this awareness has become very muddled by popular misconstructions and by the inattention of educators. In one part of his extensive comments, Keys outlines a rectification of the conventional archetypal sequence while he associates the formal, mathematical states with certain historical and cultural symbolic representations of them, as in with Buddha-states of the Tibetan cosmogony, or the Persons of the Trinity in Christian tradition.

The story of creation can of course be protracted indefinitely. To cut a long story short, it turns out that there are five orders (or "levels") of eternity, four of which are existent (although not of course materially existent, this comes later) and one which is non-existent.The non-existent order is of course the inmost, the one the Greeks called the Empyrean. In the mathematics of the eternal structure the five orders are plainly distinguishable, and it is a fact of some interest that the early Greek explorers, who were not so well equipped mathematically as we are today, nevertheless confirmed, from observation alone, that the number of eternal regions or "heavens" stands at five.

At the next level, travelling outwards from within, an extraordinary thing happens. As we come into the sixth level (i.e. the fifth order [Order number Five], recollecting that the first level is of order zero) by crossing the fifth "veil"--mathematically speaking a "veil" is crossed when we devise an "outer" structure that embodies the "rules" of the structure next within--when we cross this fifth veil, a strange thing happens. We find that we cannot in fact cross it (i.e. it is mathematically impossible to do so) without creating time.

The time we create first, like the first space [given the cardinal number One], is much more primitive and less differentiated than what we know in physical existence. The time we set our watches by is actually the third time. The first time is much less sophisticated. Just as the regions of the first space have no size, so the intervals of the first time have no duration. This doesn't mean, as it might suggest in physical time, that the intervals are very short, so short that they vanish. It means simply that they are neither short nor long, because duration is not yet a quality that has been introduced into the system. For the same reason, all the heavenly states, although plainly distinguishable from one another, are in reality neither large nor small, neither close together nor far apart.

Everything reflects in everything else, and the peculiar and fundamental property of the fifth order of being reflects itself all over the universe, both at the physical and metaphysical levels. An interesting reflexion of it in mathematics is the fact that equations up to and including the fourth degree can be solved with algebraic formulae. Beyond this a runaway condition takes over making it impossible to produce a formula to solve equations of the fifth or higher degrees. A similar "runaway" condition applies, as we shall see in a moment, when we cross the fifth "veil" outwards into the first time.

It requires only a moment's consideration to see that what we call time is in fact a one-way blindness, the blind side being called "the future." Once we proceed into any time, no matter how primitive, we come out of heaven, i.e. out of eternity, out of the region where there is no blindness and where, therefore, in any part of it, we can still see the whole. And as we proceed further and further out into each successive and less primitive time and space, our blindness at each crossing is recompounded. It is thus easy to come out, hard to find one's way back in.

[James Keys, Only Two Can Play This Game, Julian Press, New York (1972), footnote No. 1, pp. 123 ff.]

Although the world of AI (artificial intelligence) and the theoretical branch of computer design in general have been slow to grasp it, this grand iconic image offers a potentially rewarding tool and perhaps a clue for solving some of the complexities of parallel programming. In new models, simultaneous (parallel) processing transcends lineal tree logic, yet in designs for new-genereation supercomputers the requirements of physical proximity are increasingly difficult to tolerate as constraints on the speed of information processing. The key lies in our understanding the architecture of heaven, or eternity. The necessary arrangement of the heavenly or eternal realms (with a paradigmatic five-steps-from-the-void) can indeed be seen, but not while retaining our conventional attachments to habitual vision of the sort we find so useful in the everyday world. Given the special meanings of formal language, we might say of this empyrean exercise:

...to experience the world clearly, we must abandon existence to truth, truth to indication, indication to form, and form to void.

If we distinguish anything at all, then "all this"--including in the end the physical universe--is how it must eventually appear. In short, what I prove is that all universes, whatever their contents, are constructed according to the same formal principles.

[G. Spencer Brown, Laws of Form, p. 101. Keys, Only Two Can Play This Game, p. 110.]

These principles can be illustrated by the formal steps that must be taken ("all-at-once") in the orders of creation. This structure

corresponds to the void, the form, the axioms which see the form...Then you get the arithmetic, which is seeing what becomes of the axioms. And then you be it to do it, and in being it and doing it you find that, being and doing, you see the generalities of it, and that is the algebra. And while you are seeing you notice you have got equations...and suddenly you decide: "Aha! Supposing what it equals goes back into what it comes from?" Now you have generated time and matter all at once. There can be no matter without time. Time and matter come simultaneously. But this is the first matter in which the orders are counted, and it's called the "crystalline heaven," but it is not, really, a heaven.

In the construction of matter, all that happens is that we create the temporal and the material together by imagining that the outside feeds back into the inside. We then have a succession of marked and unmarked states generated by an oscillator function...Once you are in time, everything is a vibration.

[Keys, AUM Conference Transcript, pp. 96, 104, 106, 108.]

In the context of some brief reviews, James Keys drew parallels between these formal states or relationships and various literary, religious and artistic expressions, including Dante, the Gospel accord-ing to Thomas, and the author of The Divine Names, Dionysius the Areopagite, the Early Christian mystic to whom (mistakenly) St. Denis, the first Gothic church in the Ile-de-France was dedicated in 1144.

The secret sayings of Jesus of Nazareth, many of them so much deeper and stronger than what we find in the canonical gospels as to make it a different order of book. For example, it says much more clearly (gives an exact recipe, in fact) what you actually have to do to enter eternity. [In The Divine Names] the parallel accounts of the emergence of time, i.e. the statements of what we have to do to construct an element that doesn't exist in any of the five orders of eternity. We attempt to recount, in other words, what are the essential magic spells for creating a temporal existence, just as books such as the Gospel of Thomas aim to give the essential magic whereby these spells may be reversed.

[Keys, Only Two Can Play This Game, pp. 104, 108.]

In this order of complexity, this space we enter following the fifth crossing from the void, we discover--we are for the first time able to imagine--those entities commonly called numbers. They exist in what has been called the crystalline heaven, which is Order number Five (counting from the void as "zero"); that order is:

with the first time...what is called the astral plane in magic. It is the last of the material existences. Its structure is transparent and crystalline. In the middle ages it was projected out and called the crystalline heaven, although it is not, technically, an eternal region. It is where the eternal regions are first plotted and counted, for there are no numbers in eternity itself. You cannot count without time. When we proceed from here into the heavens themselves, we lose all numbers in a blinding flash as we return through the fifth veil into the outer heaven. From here on, if we are to survey what we see mathematically, we have to use Boolean elements, which are not numerical.

[Numbers] nevertheless, do exist. But not in the physical universe...Common arithmetic for university purposes, which for a less vulgar name is called the Theory of Numbers, one of the most beautiful sciences in all of mathematics, is the science of the individuality of numbers. A number theorist knows each number in its individuality. He knows about the relationships it forms, and so on, as an individual, as a constant. An algebraist is not interested in the individuality of numbers; he is interested in the generality of numbers. He is more interested in the sociology of numbers...he is not interested in individuals at all.

[Keys, Only Two, p. 134 f.; AUM Conference transcript, pp. 43, 45.]

Previously we sought to provide a link to certain basic information about number with our reference to Warren Sturgis McCulloch's essay "What Is a Number, that a Man May Know It, and a Man, that He May Know a Number?" Here, we justify our methodological use of number by telling where we may find a number and how to count it, literally, digitally. In one of the easiest ways to demonstrate this count:

Hold the palm of one hand in front of your face.

With the index finger of the other hand, count off the states or orders of eternity, beginning with your thumb.

Call the thumb, "Order Zero" (though it is the FIRST counted!)

Count the gap (or "valley") between the thumb and the adjacent index finger stands for the first crossing.

Call the index finger "Order One," which stands for the Form.

Then count the next interdigital gap as the second crossing.

Call middle finger "Order Two," the Axioms.

Then count the next gap as the third crossing.Call the ring finger "Order Three," the Arithmetic.Then count the next gap as the fourth crossing.

Call the little finger "Order Four," the Primary Algebra.

THEN count the fifth crossing, which, you see, is different from all the others, and not a gap or a valley at all because you can go past the wrist, all the way around the palm of your hand and return to your thumb. In the next state after the fifth crossing, "Order Five," the Algebra may contain equations of the second degree.