Daniel Dockery

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Home of published musician, recording artist, mathematician, programmer, translator, artist, classicist, and general polymath.

Polygorials

June 21st, 2003

A little while ago I ran across and couldn’t resist the urge to acquire a copy of the last published critical edition of Nicomachus of Gerasa‘s nearly 2000-year-old Introduction to Arithmetic (Αριθμητικη εισαγωγη), the Hoche edition published by Teubner in Leipzig in 1866.

Cover of 1866 Introduction to Arithmetic

The provenance of the volume is interesting in itself. A small imprint on the inside front cover, “Paul Koehler—Buchhändler und Antiquar—Leipzig ’05“, shows where, thirty-nine years after its publication, it went through the bookshop of the famed German antiquarian and publisher Paul Koehler; in the rear, on the blank bottom of the last page of the index, is a stamp in Hebrew, stating that it was at one point part of the library collection of the University of Jerusalem; then on the FEP, the handwritten signature of the late, renowned philologist, linguist and classical scholar, Benedict Einarson (long of U. Chicago’s classics dept.). When I found the volume, it had already made its way back to a small bookshop in the UK.

In a recent period of recovery, I kept myself occupied by reading the little volume, and while so engaged found myself entertaining myriad ideas about polygonal numbers, a dormant fascination reawakened by Nicomachus. Skipping a good bit of middle ground and jumping to the point: I generalized the idea of the factorial function, which traditionally produces only the product of the first n natural or counting numbers, to a function which returns the product of the first n polygonal numbers for k-sided polygons. For simplicity of reference, I have called these “polygorials”, merging the two words polygonal factorials. In exploring these I discovered that a few have already appeared in the literature as other sequences in the Encyclopedia of Integer Sequences, though most were novel to this research, which also yielded other surprises, such as the fact that the ratio of the nth hexagorial to the nth trigorial yields the Catalan numbers.

For those interested, a small, introductory paper on the Polygorials details the work.


The pentalpha

February 2nd, 2002

Some notes on the historical role of the figure in response to someone’s assertion that it “symbolizes both mystical and diabolic aspects of the number 5”.

While the pentagram has a distinct and quite old history, the “diabolic” bit is a relatively modern connotation and typically a misunderstanding. None of the ancient sources consider it so. The earliest source I have seen to remark on any “diabolic” connection of any sort to the symbol of the five-pointed star is Éliphas Lévi and his remarks were strictly confined to the symbol with one point downward. Still, he offers no sources for his claim and there are no known precedents for it, so, like most of his writings, one is left to assume it’s purely an invention of his own mind, not a traditional symbol or concept.

When conceived of as five interlocked majuscule Greek Alpha’s, the pentalpha, the symbol actually played a significant role for the Pythagoreans, and is related as well to the golden ratio. The simplest form of the symbol can be traced as far back as Uruk IV (3500 b.c.e. or so) in Mesopotamia where it seemed to merely indicate any heavenly body. During the cuneiform period (2600 b.c.e. & after), the symbol—then called UB—represented a “region” or “direction”, some particular “heavenly quarter”. It is found frequently on potsherds in Uruk, and more frequently still on Jemdet Nasr and proto-Elamite tables (3000–2500 b.c.e.), though examples elsewhere during those periods are rare. There are also examples of the symbol turning up later (early 5th C. b.c.e) in Attic cups and shards. A series of coins bearing the word PENSU (Etruscan for 5) and the symbol have turned up in Greek, Aryan and Etruscan regions. During the Republic in Rome, the symbol was the sign for the building trades.

In the early Christian period it was frequently employed as a Christian symbol said to signify the five wounds of Christ. In the mediæval period it first began to turn up frequently in magical texts, the so-called Solomonic grimoires in particular, where it was usually called a pentacle/pantacle or pentangle, but most sources seem to agree that the specific terms “pentacle” and “pentangle” do not derive from the Greek “penta” but from an old French term meaning “to hang”, usually indicating a “talisman” of one kind or another for wearing about the neck. This seems to be supported by the fact that the Solomonic texts applied the same word equally across the board to both circles inscribed with the five-pointed figure as to those inscribed with a six-pointed hexagram. Both symbols have been called the “Seal of Solomon” through Jewish, Arabic, and theurgic sources.

The first known usage of the symbol in an English source is apparently in Sir Gawain and the Green Knight, stanzas 27–28 (c. 1380 c.e.) were Gawain carries a shield with

shining gules,
With the Pentangle in pure gold depicted thereon.

[…]

It is a symbol which Solomon conceived once
To betoken holy truth, by its intrinsic right,
For it is a figure which has five points,
And each line overlaps and is locked with another;
And it is endless everywhere, and the English call it,
In all the land, I hear, the Endless Knot.

In Tycho Brahe’s Calendarium Naturale Magicum Perpetuum (1582) the symbol is illustrated with a human body drawn within and the letters Y, H, S, V, H drawn at the points (the Hebrew letters יהשוה which were thought by the early Christian kabbalists—Ficino, Mirandola, Reuchlin, etc.—to signify “Jesus” by adding the letter ש in the center of the tetragram יהוה of the Old Testament texts). A contemporary of Brahe, Agrippa, pupil of the Abbot Trithemius, in his de Occulta Philosophia libri tres (available as vol. 48 in the Studies in the History of Christian Thought series from Brill Academic Publishers) presented a very similar illustration but showing the traditional “planetary” glyphs. Robert Fludd produced a similar symbol at the time, as did da Vinci following the symbolism of the Roman architect Vitruvius. Throughout the period the magical texts used it not as a symbol for “evil” but rather as a symbol intended to keep it away. That same idea crops up in Goethe’s Faust (1808) where the symbol prevents Mephistopheles from entering:

Meph.: Let me own up! I cannot go away;
A little hindrance bids me stay.
The witch’s foot upon your still I see.

Faust: The pentagram? That’s in your way?
You son of Hell explain to me,
If that stays you, how came you in today?
And how was such a spirit so betrayed?

Meph.: Observe it closely! It is not well made;
One angle, on the outer side of it,
is just a little open, as you see.

Biedermann, in his Dictionary of Symbolism, indicates that the pentagram was a Christian symbol all the way up through this period. The Roman Emperor, Constantine I, after defeating Maxentius and issuing the Edict of Milan in 312 c.e., credited his conversion to Christianity for his success and from that point on incorporated the pentagram, one point downward, into his personal seal. And while it was a lesser used Christian symbol than, say, the cross or the stylized fish, and may be arguably “rare” in appearance, there are no appearances of the symbol as representing any kind of “devil” or “fiend” at any known point preceding Lévi’s 19th-century writing. The symbol was still being used in a Christian context as late as 1824 (perhaps later) since it is indented on the gateposts of the churchyard of St. Peter’s in Walworth, England which was built in that year.

It did not become a widely recognized symbol for anything “diabolic” until the late ’60s when LaVey founded his “Church of Satan” and used the one-point-down form as the primary symbol of the same.

The Caduceus

October 8th, 2001

The following mailing list remark prompted some thoughts I thought I’d share:

If anyone wonders about the origin of the Caduceus, it actually goes all the way back to the book of Genesis. Moses had to make a copper serpent for the Israelites to look at when bitten by poisonous snakes (during a plague that God had sent) in order to be cured. More info here: http://www.publicsafe.net/wand.htm

Actually, caducei (or caduceum [a.k.a. caduceus] in the singular; κηρύκειον in the Greek) were simply herald staves, signs of peace which, much like the later “white flag”, allowed the messenger to travel unmolested. In their original form they were nothing more than olive sticks with stemmata (wreaths, garlands, etc.) attached to or twined about them; only in later times did the stemmata develop into a serpent motif in artistic representations. The caduceus was not associated with the Greek Hermes or Roman Mercury until some time later, and then only insofar as Hermes/Mercury was considered to be the messenger of the gods. As near as I can tell, caducei did not become symbols of the medical profession until after WWI, due to the exposure it received then by being the symbol the U.S. Army had adopted (around 1902) to indicate its medical department (which use was itself, as I understand it, based on a mistaken assumption). What mistake would that be? In the early 19th century a medical publisher adopted the symbol as part of his imprint—not because he published medical literature, but because he viewed himself as a deliverer of information: a herald, in other words, such as originally carried a caduceus. The usage in the medical profession can be seen as a mixture of two things: 1) the proliferation and perpetuation of a popular error (i.e., that described above) where others adopt the sign because they have seen it used elsewhere in the same context without realizing that the previous usage was itself incorrect; and 2) a confusion of the caduceus with the staff of the healer/medicine god Asclepius of ancient Greece, which was usually represented as a T-shaped staff with a single snake (the snake related mythologically to Asclepius) wrapped around it, which symbol was used to represent “doctors” even in ancient times. The snake and staff combination can be seen in many of the sculptures of Asclepius; e.g., see http://bellarmine.lmu.edu/faculty/fjust/Asclepius.htm. It can also be seen in the symbol of the Royal Society of Medicine. For more information, you might be interested in having a look at the W.J. Friedlander (Prof. Emeritus of Medical Humanities and Neurology at the U. of Nebraska’s College of Medicine) book The Golden Wand of Medicine which can be found on Amazon.

As to Moses, I take it they are referring to the scene depicted in the Book of Numbers, chapter 21, verse 8. It might, or it might not, interest you to know that the word “serpent”, “snake” or any related term appears nowhere in that verse. According to the Hebrew:

ויאמר יהוה אל־משה עשה לך שרף ושים אתו על־נס והיה כל־הנשוך וראה אתו וחי׃

YHWH said to Moses: make unto you a seraph and set it on a pole so that all who are bitten who look at it shall live.

This term seraph and its plural seraphim appear in seven places in the Tanakh:

YHWH sent seraph serpents against the people. They bit, and many Israelites died.

[The Hebrew נחש, meaning “serpent”, does appear in this verse, though not in 21:8.]

Numbers 21:6

YHWH said to Moses: make unto you a seraph and set it on a pole so that all who are bitten who look at it shall live.

Numbers 21:8

who led you through the great and terrible wilderness with its seraph serpent and scorpions, a parched land with no water in it, who brought forth water for you from the flinty rock;

[The Hebrew for serpent appears again here, but is curiously in the singular.]

Deuteronomy 8:15

Seraphs stood in attendance on Him. Each of them had six wings: with two he covered his face, with two he covered his legs, and with two he would fly.

Isaiah 6:2

Then one of the seraphs flew over to me with a live coal, which he had taken from the altar with a pair of tongs.

Isaiah 6:6

Rejoice not, all Philistia,
Because the staff of him that beat you is broken.
From the stock of a snake there sprouts an asp,
A flying seraph branches out from it.

Isaiah 14:29

The ‘Beasts of the Negeb’ pronouncement:
Through a land of distress and hardship,
Of lion and roaring king-beast,
Of viper and flying seraph,
They convey their wealth on the backs of asses,
Their treasures on camels’ humps,
To a people of no avail.

Isaiah 30:6

At least two of the Isaiah passages, those of chapter six, indicate that seraphim are angels; more particularly, that they are angels concerned with praising YHWH and “cleansing” his servants. E.g., Isaiah 6:3-7, in reference to the seraphim earlier described:

And one would call to the other,
“Holy, holy, holy!
YHWH of Hosts!
His presence fills all the earth!”

The doorposts would shake at the sound of the one who called, and the House kept filling with smoke. I cried,

“Woe is me; I am lost!
For I am a man of unclean lips
And I live among a people
Of unclean lips;
Yet my own eyes have beheld
The King YHWH of Hosts.”

Then one of the seraphs flew over to me with a live coal, which he had taken from the altar with a pair of tongs. He touched it to my lips and declared,

“Now that this has touched your lips,
Your guilt shall depart
And your sin be purged away.”

So from the Isaiah 6 descriptions, we can ascertain that a seraph: has arms, a distinct face, six wings (perhaps paired, but not necessarily so), can sing or chant, can speak to a man, can employ fire or some other burning thing such as a coal (or perhaps a venomous serpent? but see below for more on that) to “cleanse” “sin” (it should probably be noted that seraph itself seems to derive from the Hebrew verb meaning “to burn” [e.g., Isaiah 44:19, saraphti, meaning “I have burned”, etc.], so it seems logical that they would have some connection to fire); we might also suspect that they have hands or arms of some kind since how else would they use “tongs” to retrieve a piece of coal from the altar?

If we assume that Revelations 4:8 is a valid commentary on them, then we may also consider that they are apparently covered with eyes.

In Jewish lore and legend, and also in various derivative Christian sources later, seraphim were considered to be the highest “order” of angels, always surrounding the Throne and chanting the trishagion (“Holy, Holy, Holy”), or kedushah for the Hebrew term. According to the third book of Enoch, there are said to be only four of them, corresponding to the four “parts” of the world. In rabbinic sources, they are often equated with the chayyot—the “living creatures” described in the Merkavah literature and the visions of Ezekiel. According to the second book of Enoch, in addition to the six wings mentioned in Isaiah, the seraphim are said to have four faces (like the chayyot). The pseudepigraphal Revelation of Moses includes the reference: “there came one of the six-winged seraphim, and hurried Adam to the Acherusian lake, and washed him in presence of God” which, again, seems to indicate the sin-purging function (although, curiously, this time with water).

How do they relate to the serpents mentioned? Hard to say, really, and there seems no clear answer. Numbers 21:6 implies that the seraph serpents were sent by YHWH, so one could perhaps consider the angelic appellation to merely be signifying the role of the serpents as a “messenger” of YHWH (since the term “angel” only means “messenger” [which, I suppose, could tie us back into the caduceus topic]); the use of the particular “kind” of angel, the seraphim, could reflect possibly two things: 1) considering the root verb “to burn”, it may simply reflect the venomous or burning character of that particular kind of snake-bite, or 2) it could reflect the role of those snakes as similar to the “sin cleansing” role of the seraphim since the snakes were apparently sent as a punishment for their “sin” of speaking up against their predicament. It wouldn’t be the only case of YHWH sending one or more of his “angels” against people. Considering the Isaiah 14:29 and 30:6 references, they also seem to attribute “flying” to the seraph serpents (it is assumed that serpent is meant in the latter two references since the desert setting mentioned in context doesn’t seem to match the location given for the seraph angels—unless YHWH took his Throne to the desert for a bit of a vacation). I’m afraid I’m not at all knowledgeable about this particular subject, so I’ll ask: is there a venomous species of snake native to such an environment, which leaps or is capable of extreme leaps? (i.e., one that could, in time, through retellings, gain—in the literary context—the mythological capability for “flight”?)

On a tangent, it seems that the seraph Moses was instructed to make in Numbers 21.8 gathered a cult and received a formal name. In II Kings, 18:4, we find:

He abolished the shrines and smashed the pillars and cut down the Asherah. He also broke into pieces the bronze serpent that Moses had made, for until that time the Israelites had been offering sacrifices to it; it was called Nehushtan.

[Nehushtan is simply a concatenation of the two Hebrew words נחש, meaning serpent, and תן, meaning dragon or monster (as used in Ezekiel 32:2, “O great beast among the nations, you are doomed! You are like the dragon in the seas,” etc.), so if we assume the “bronze serpent” to have been real at all, it’s probably a safe assumption that it was of quite a large size. Curiously, “bronze serpent” seems to be a bit of a play on words in the Hebrew, since the adjective translated as “bronze” is written identically like the noun for “serpent” except with an additional final letter: הנחש הנחשת, ha-neHesh ha-neHeshat]

Bewer’s Literature of the Old Testament indicates that the Numbers 21:8 “explanation”, however, was merely a mythopoeic creation of the Elohist author—a bit of popular propaganda, in other words—intended to give an origin to the ancient Nehushtan idol that was at the time being worshipped in the temple. According to Bewer, it was thought that by explaining that the idol didn’t represent any sort of god or the like, but instead had simply been a temporary item created by Moses for a specific purpose—namely the healing of snake-bites—for a specific instance, that people would be dissuaded from further worship; Bewer continues, though, stating that the story ultimately had the opposite effect, causing people to increase sacrifices to it, only then considering it to be capable of healing them, in addition to whatever else they’d attributed to it previously.

Still, though, my point was: there’s no historical connection between the supposed seraph of Moses and the caduceus of Rome. Similarity of symbols doesn’t mean equality of symbols, nor does it imply that one was borrowed or derived from the other. It’s not at all uncommon to find wholly unrelated—though visually similar—symbols spread out across all kinds of disparate locales and cultures.

Daniel Dockery

animî nostrî dêbent interdum âlûcinâri

Home of published musician, recording artist, mathematician, programmer, translator, artist, classicist, and general polymath.

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