In this week’s Suite101 article, Michael Martinez notes that Tolkien researchers haven’t overlooked the fundamentals of Tolkien’s numerals, but the fundamentals of mathematics in Middle-earth may be easier to discern than previously believed.
Here is an excerpt:
It requires a lot of nerve to criticize one of the most respected Tolkien linguists in the world, and to suggest he may be wrong, especially when you are no linguist yourself. But I’m going to suggest that Helge Fauskanger, and other linguists who have followed his lead or confirmed or accepted his work, may be wrong. About Elvish numbers. Numerals, that is, not the populations of Elves.
A great deal of work has been done on Elvish numerals, and Helge has only built upon the work of others. Indeed, until I began looking into the possibility of constructing a numeral system (I was sure others had attempted to do this), I hadn’t really given the matter much thought. Every time someone asked, “Are there tengwar for numerals?” I would quickly say, “No, not in any published text I have seen.” Well, that’s what you get for not looking at the published texts before answering questions. It occurred to me that there must, indeed, be at least one candidate for an example of Elvish numerals: Aragorn’s letter to Sam. And when I checked Sauron Defeated I was pleased to read two examples of Elvish numbers. After all, Aragorn mentions a specific date in the letter.
Now, these two tengwarin numbers are not unknown examples. Helge and others have been aware of them for years. Since I never explored the minutiae of Elven writing systems in that kind of detail the examples were quite obscure to me. So I’m hardly revealing anything new here. Except that most people, I am sure, do not know about these numbers. Are there other examples of tengwar numerals? Yes. And no. Yes, some people have tried to construct tengwar numeral systems. No, not so far as I know, are there any other examples by the hand of J.R.R. Tolkien.
David Doughan and Julian Bradfield published a tengwar numeral system in Quettar Special Publication no 1 in 1987. Daniel Smith has used the Doughan and Bradfield work as part of the Tengwar resource for the Electronic Tolkien Encyclopedia Project. Doughan and Bradfield’s work, which would at most be compatible with The Lost Road and Other Writings but not later books, has apparently been the foundation of all tengwar research.
The basic system of tengwar numeral construction that Doughan and Bradfield documented is very simple. It is also designed for a base-ten (decimal) number system. However, Helge and other linguists argue that the Elves may have used a base-twelve (duo-decimal) number system. Why? Because Tolkien said that “the Eldar preferred to reckon in sixes and twelves as far as possible”. However, this statement comes in Appendix D to The Lord of the Rings, which dicusses calendars, not numbering and counting systems. Now, there is another source for Elvish reckoning, but let’s first look at the most well-known information provided in the published texts.
Please click on the link below to read the entire article.