XLent LInguistics

As was correctly answered in the comments to the previous post, I am now 40 years old (XL) and the next time that my age in Roman numerals will be the same length as my age in Western (Hindu-Arabic) numerals will be when I am 51 (LI).    49 is not a correct answer in this case because the Romans did not habitually use subtraction in this way; irregular formations like IL (49) and XM (1900) do sometimes occur irregularly, but normally one cannot ‘skip’ a power.  I can only be subtracted from V and X; X can only be subtracted from L and C; and C can only be subtracted from D and M.

As any schoolchild can tell you, one of the purported disadvantages of Roman numerals is that their numeral-phrases are long and cumbersome (e.g., 37 vs. XXXVII).  And of course, for many numbers that is true.  But for many other numbers (e.g., 2000 vs. MM) the Roman numeral is  equal in length or shorter than its Western numeral counterpart.     Note, in particular, that round numbers tend to be those that are shorter in Roman numerals; this is because, since the Roman numerals don’t have a 0, that numerals that in Western notation would have a 0 have nothing in their Roman counterpart.

Among numbers whose Roman numeral and Western numeral notations are exactly the same length, there doesn’t initially appear to be much of a pattern:

1, 5, 11, 15, 20, 40, 51, 55, 60, 90, 102, 104, 106, 109 …

but then we see a new sequence emerge –

111, 115, 120, 140, 151, 155, 160, 190 …

which are just the numbers in the sequence from 11-90 with an added C on the front, which makes sense, since you’re just adding a 1 in front of them, similarly.

You could be forgiven in thinking that these numerals come up frequently, since we’re in the midst of a giant cluster of years with this property –

… 2002, 2004, 2006, 2009, 2011, 2015, 2020 …

but then after that, it’s another 20 years before 2040.

Unsurprisingly, numerals containing 3 rarely have equal-length Roman equivalents and numerals containing 8 never do.     So once you get past 3000, these numbers become extremely rare.    Roman numerals don’t have a standard additive representation for 4000 and higher; you can write 4000 as MMMM, but normally one would expect a subtractive expression with M (1000) subtracted from 5000.  There are Roman numerals for 5000, 10000, 50000, 100000, etc., but they are extraordinarily rare, and the Romans during the Empire instead tended to place a bar (or vinculum) above an ordinary Roman numeral to indicate multiplication by 1000; thus,  IV=4000.  The addition of this feature creates a real conundrum: does the vinculum count as a sign or not?    If it does, then IVI=4001 has four signs; if not, then IVII=4002 does.

I’ll leave this aside and stop here to save all of our brains.   Thanks for playing!



The Case of the Missing Pi Day 4s

Yesterday was Pi Day, 3/14 (those who prefer days before months can have Pi Approximation Day, 22/7) and in celebration of this momentous annual event, I invited several of my American colleagues (who have learned to tolerate my numerical eccentricities) over to my house in Canada for an International Pi Day Pie Party, which was a great success.  And, of course, as befitting this event, we had Pie, complete with Pi (to two decimal places) on top:

It's blueberry!So far, so good.  (And for the record, it was very good).  There was only one problem: the local dollar store I went into had a very odd distribution of candle numerals: it had tons and tons of 0, 1, 2, and 9, some 3s, but no 4s, 5s, 6s, 7s, or 8s.  As a professional numbers guy, and also as a guy who needed a 4 for his pi(e), this was deeply disconcerting.

After a moment, I figured out why. Ordinarily, when stores buy products that come in different varieties from wholesalers, the default is to order the same amount of each variety.   In this case, the store had obviously ordered an equal amount of each numeral, but they were being purchased by consumers at different rates.    Now, there is nothing about the properties of the natural numbers that would lead to this observed distribution (if it were Benford’s Law in action, it would be 1 and 2 that would be in short supply). Rather, the explanation is a social one:   Many parents do not buy birthday candles for their child’s first, second, or third birthday, because, while, as my (thankfully childless) brother noted, “Babies love fire!”, parents of toddlers do not.    At the other end, by the time your kid is about 9, and certainly by the double digits, they’ve probably outgrown the ‘giant novelty numeral candle’ phase of their lives.  Ages 4-8 are the sweet spot, and thus these sell out much more quickly.

I also note that, for adults, decadal birthdays like 20 and 30 tend not to attract much numerological attention, whereas 40, 50, and 60 certainly do (not so sure about 70 and 80), and by 90 most of the clientele is deceased.    This doesn’t explain why there were so many 0s available – perhaps purchasers are aware of this phenomenon and order extra zeroes, but don’t take account of differential demand for the tens digits.

Now, if we lived in a perfect world where suppliers and store owners had full information about their stock and made perfectly rational decisions, purchasers would notice such discrepancies and perhaps order more of the missing numerals.  The local dollar store, however, does not occupy such a world.  Fortunately, this being Windsor, Ontario, there was another dollar store across the street, and while it also had a skewed distribution, lo and behold, it did have one lonesome 4 for purchase (seen above).   Thus my Friday Pi Day pi display supply foray was saved.  Yay! (Try saying that in Pig Latin.)

Actually, this is not the first time I had encountered this phenomenon.  Back in 2008, when American gas prices first regularly began to hit $4.00 a gallon, the New York Times reported on, of all things, a shortage of numeral 4s, because their number sets were purchased with an equal distribution across all ten digits (presumably with extra 2s and 3s purchased individually to deal with those dollar amounts).  Once that leading digit got to 4, there was a temporary shortage, leading to some store owners writing their own makeshift 4s until new ones could arrive.

Thus, while we think of linguistic and symbolic resources like numerals as being effectively infinite, in contexts like these, you can indeed have shortages and surpluses.   Thankfully, now that we’re on to the Ides of March and our Pi Day shortage is dealt with for another year, I can store these candles for future use, if I want.  The pie, on the other hand, has gone to a better place.  Because, while you may sometimes need to ration your fours, let’s hope we never live in a world where we have to ration pie.

What’s so improper about fractions?

Yesterday, as part of the Wayne State Humanities Center brownbag series, I gave a talk entitled, “What’s so improper about fractions? Mathematical prescriptivism at Math Corps”, based on my long-term ethnographic research in Detroit.   For those of you who might be interested, you can watch the video below (or on Youtube itself), and the powerpoint is available for download here.

Cistercian number magic of the Boy Scouts

“You know that in ancient times religion, astronomy, medicine, and magic were all mixed up so that it was difficult to tell the beginning of one and the ending of the other and to-day the Gypsies, hoboes, free masons, astronomers, scientists, almanacs, and physicians still use some of the old magical emblems.  So there is no reason why the boys of to-day should be debarred from using such of the signs as may suit their games or occupations and we will crib for them the table of numerals from old John Angleus, the astrologer.  He learned them from the learned Jew, Even Ezra, and Even Ezra learned them from the ancient Egyptian sorcerers, so the story goes; but the reader may learn them from this book.” (Beard 1918: 91)

So begins the chapter, “Numerals of the Magic: Ancient System of Secret Numbers”, by Daniel Carter Beard in his 1918 volume The American boys’ book of signs, signals, and symbols, which you can download from Google Books for free.  Beard was one of the founders of the Sons of Daniel Boone in the early 20th century, which merged with the Boy Scouts of America (of which Beard was a key founder) in 1910 when that famous group was formed.  Beard wrote a number of popular books intended for boys in the Scouting movement, including this one.   Scouting books today do not, as a rule, make reference to esoteric Egyptian sorcery or Freemasonry or ‘John Angleus’ (who is Johannes Engel (1453-1512)) or ‘the learned Jew, Even Ezra’ (Abraham ibn Ezra (1089-1164)), or, for that matter, have a chapter on number magic at all.   At least, I never heard about it, and I was in Scouts for over a decade.  But we are fortunate that this one did, because it has a couple of real treasures inside, not previously recognized as such.

Let’s take the second one first.  It appears on p. 92 immediately following the passage I just quoted:

(Beard 1918: 92)

For those of you familiar with my book Numerical Notation, these are the numerals used primarily by Cistercian monks from the 13th – 15th centuries, and thereafter described in early modern numerology and astrology for several centuries, though largely at that point as an intellectual curiosity rather than a practical notation.    David King’s wonderfully detailed Ciphers of the Monks (King 2001), which is one of the few books at that price point (somewhere around $150, if I recall) that may be worth it, lists every example the author could find of these numerals, from medieval astrolabes to Belgian wine barrels to 20th-entury German nationalist texts.    It’s extremely comprehensive.  However, it does not mention Beard’s book – and why should it? What a bizarre place to find such a numerical system!   It’s what I describe as a ciphered-additive system, which is to say that there is no zero because none is needed: there is a distinct sign for each of 1-9, 10-90, 100-900, and 1000-9000.   The Cistercian numerals are a little anomalous typologically; another interpretation of them would be that they are positional, but use rotational rather than linear position – the signs for 9, 90, 900, and 9000 (e.g.) are rotations or flips of one another, so we could consider them the same sign (9) in four different orientations.     Zero is superfluous (thus not present) because unlike linear texts, there is no ‘gap’ to be accounted for by an empty place-value.

I became curious and tried to figure out why Beard attributed these to ‘Angleus’ and to ‘Even Ezra’.    Engel’s Astrological Optics was translated into English (1655) but contains no Cistercian numerals, and King doesn’t note him as using or depicting the system.  Similarly, ibn Ezra was not a known user of the system.   And I haven’t even been able to find any other source that attributes the system to those individuals; rather, it’s almost always Agrippa of Nettelsheim or Regiomontanus who are invoked in the scholarship.  We know that Beard was a Freemason, so he may have had access to some Masonic texts that said as much, but I can’t find any such reference, and King doesn’t mention any likely sources either, although he does note that many Masons (especially in France) were familiar with the Cistercian system.    So it’s not entirely clear where Beard learned about the system (although see below), and he’s got a lot of things mixed up in the account.

The other numerical treasure in Beard’s book is even more fascinating, although it appears in the previous chapter on codes and ciphers and is less prominent, on p. 85, the ‘tit-tat-toe’ numerals:

(Beard 1918: 85)

So what we see here, again, is a ciphered-additive decimal system in which there is a ‘family resemblance’ between 9, 90, 900, and 9000 (and the other numbers so patterned), but no zero.  The signs are designed after their place in a hash / tic-tac-toe / octothorpe with the power indicated through ornamentation.  As a ciphered-additive system, it’s like the Cistercian numerals (although the signs are completely different) but instead of placing signs around a vertical staff, the signs are constructed into a box.  Note that the signs in each numeral-phrase are not strictly ordered, but are packed compactly in whatever way suits the resulting box aesthetically. This is one of the advantages of ciphered-additive systems that, if desired, for cryptographic purposes or for any other reason, the signs can be re-ordered without loss of numerical meaning.   But I know of no system quite like this, where numerals are arranged in a box-like shape, or where there is such a novel means of forming individual signs.

Beard is explicit that this system is newly designed: “The tit-tat-toe system of numerals here shown for the first time is entirely new and possesses the advantage of being susceptible of combinations up to four figures which suggests nothing to the uninitiated but a sort of Japanese form of decoration”  (Beard 1918: 84).   He claims that the alternate name ‘Cabala’ is just another name for the tit-tat-toe, which is a highly dubious claim, but he is clearly trying to invoke a connection between his newly-developed system and Jewish mysticism – in the hope that Boy Scouts will use it as a numerical code.  Ciphered-additive numerals are rare enough in the modern era – most of the systems are obsolescent at best.  So it’s fascinating to see a twentieth-century system right at the moment of its development.   It’s also fascinating to see how mystical, spiritual, and numerological knowledge from early-modern authors is incorporated into a manual for Boy Scouts and recommended for use in cryptography.

We’re not quite done, though.  Based on some of the (otherwise uncited) quotations in Beard’s book, I concluded that he was taking some of his ‘insight’ about the ‘Cabala’ from L.W. De Laurence’s Great Book of Magical Art (1915), which was a popular American book of spiritualism and Oriental mysticism at the time.  And, looking into de Laurence’s book, lo and behold, what did I find?

(De Laurence 1915: 174)
(De Laurence 1915: 174)

De Laurence, whose work is also not noted by King, gives a more standard attribution than does Beard for what we now know to be the Cistercian numerals: he attributes them to the ‘Chaldeans’, which is a very common descriptor for the system and is even found in the scholarly literature.  He doesn’t mention Angelus or Even Ezra or any other of the medieval and early modern authors who use the system, so it’s still a mystery how Beard made that attribution.  But, given that there really are not a lot of texts that discuss this system at all, I suggest that Beard encountered them through De Laurence and possibly confounded their origin with some other understandings he had picked up along the way, possibly through Masonic writings.

It’s not every day that I discover a new numerical notation system, and it’s great to do that, even when it’s one that  seems to have been developed once but never adopted more widely.   So it was neat to find the ‘tit-tat-toe’ system, even if it never appeared anywhere else.  But I also found it fascinating to track the transmission of the much more widespread (but still under-appreciated) Cistercian numerals through their roundabout path to a Scouting manual for boys.    As King’s book amply demonstrates, the system has a tendency to show up in the oddest places, so perhaps we should (ahem) ‘be prepared’ to find them anywhere.

Beard, Daniel Carter. 1918. The American boys’ book of signs, signals and symbols. Philadelphia: Lippincott.

De Laurence, L. W. 1915. The great book of magical art, Hindu magic and East Indian occultism. Chicago, Ill., U. S. A.: De Laurence Co.

King, David A. 2001. The ciphers of the monks: a forgotten number-notation of the Middle Ages. Stuttgart: F. Steiner.

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