The Elves had quite a lot of time to do what interested them. They liked the stars, which is (among mortals) a good starting point for geometry. It is also said about the Noldor, that they always searched more suitable names for things they encountered or imagined, so I can imagine them playing with abstract concepts.

Is there any evidence that the Eldar developed mathematics? (Other than simple integer counting.)

Edit: The answers so far covered the everyday math, they had to have to accomplish their other works, quite well. So my second question is:

Is it possible, that they studied more advanced mathematics not for purpose, but for challenge and intellectual exercise, and for its beauty?

(It is said in the Silmarillion, that: "(the Noldor) delighted in building of tall towers." They didn't do it for watchtower, fortification, display of power, lighthouse, or any other purpose, for which towers are normally built on Middle-earth, but they enjoyed their beauty, and loved to challenge their powers, and to create things, which weren't before. Just like they cut gems, and scattered them on the shores.)

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    If the men have 9 rings, and the dwarves had 7..... how many rings did they have together. – Matrim Cauthon Jan 17 '17 at 15:52
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    @MatrimCauthon - The Hobbits invented second breakfast. I think we can all agree that they're the smart ones. – Valorum Jan 17 '17 at 16:00
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    Considering they built buildings, which implies architecture, one would think there would be some form of math to determine stress and other engineering factors to keep said buildings from collapsing. This is assuming there wasn't just some magic involved to keep even the most outlandish structures from falling down. – VBartilucci Jan 17 '17 at 16:06
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    @MatrimCauthon One Ring? – enderland Jan 17 '17 at 16:12
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    @VBartilucci Possible, but in the medieval times (although there were mathematics) the great cathedrals were built with instinct and traditional knowledge, since the physical and mathematical concepts needed for static analysis, (like pressure, stress or calculus) were missing. The builders were in fact far less mathematically learned, than the chief mathematicians of the time, (but of course, used some geometry to make harmonic and good-looking shapes.) – b.Lorenz Jan 17 '17 at 16:12


There's evidence for some elementary algebra in devising the Elvish calendar; in particular, they were aware that they couldn't quite fit the right number of days into a year, and devised leap days to compensate, and then compensated again for the deficiencies in that system:

Between yávië and quellë1 were inserted three enderi or 'middle-days'. This provided a year of 365 days which was supplemented by doubling the enderi (adding 3 days) in every twelfth year.

How any resulting inaccuracy was dealt with is uncertain. If the year was then of the same length as now, the yén2 would have been more than a day too long. That there was an inaccuracy is shown by a note in the Calendars of the Red Book to the effect that in the 'Reckoning of Rivendell' the last year of every third yén was shortened by three days: the doubling of the three enderi due in that year was omitted; 'but that has not happened in our time'.

Return of the King Appendix D: "The Calendars"

There's also some (limited) evidence for abstract mathematical thought, in that the Elves prefer a duodecimal number system; Tolkien remarks in Letter 344 that this is based on a simple mathematical observation:

The English use duodecimals and have special words for them, namely dozen and gross. The Babylonians used duodecimals. This is due to the elementary mathematical discovery, as soon as people stop counting on their fingers and toes, that 12 is a much more convenient number than 10.

The Letters of J.R.R. Tolkien 344: From a letter to Edmund Meskys. November 1972

As VBartilucci points out in a comment on the question, the advanced engineering seen among the Elves may hint at some practical mathematics, though I wouldn't want to come down on this definitively; based on Tolkien's intended themes, I would suggest that the Elves tend to prefer a more intuitionist approach to their craft, and of course one can't discount the presence of magic.

In any case, there's no evidence given of theoretical mathematics among the Elves.

1 Two of the seasons as reckoned in Rivendell; the words respectively translate to "autumn" and "fading", and between them basically correspond to what we would call autumn (possibly with some overlap with early winter)

2 "Long year", equivalent to 52,596 solar days

  • You've got your citations off on yavië and quellë (would edit but isn't a big enough change) – Edlothiad Jan 17 '17 at 16:52
  • @Edlothiad They look fine to me; what's off about them? – Jason Baker Jan 17 '17 at 16:55
  • You've put a 2 instead of a 1 behind quellë haven't you? – Edlothiad Jan 17 '17 at 16:56
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    @Edlothiad Oh, yes; thanks for that. I was confused because what you call "citations" I think of as "footnotes" – Jason Baker Jan 17 '17 at 16:57
  • Oops, sorry. I guess citations was a bit vague (as it could've meant your quotes). – Edlothiad Jan 17 '17 at 16:58

There's no geometry proofs in the text of Lord of the Rings, but there are lots of activities that the Elves participated in that require mathematics of greater complexity than integer counting.

The elves we see in the movies and books live high in the trees in beautiful constructions. Now, Ewoks also lived in trees, but there's a huge difference in construction style: everything Ewok is over-engineered and over built, because they are NOT concerned with their city looking beautiful, and they aren't doing any computation during construction. It's possible to build a relatively safe construction intuitively using ropes and the like in trees, or big thick walls surrounded by buttresses using instinct alone... but it's hard to do so beautifully. Elven cities are beautiful constructions with graceful arches, flying buttresses, and artistic flourishes. This would require at least basic geometry to calculate dimensions and the ability to calculate load. It's easy to slam a thick buttress into the ground; to have a graceful load bearing pillar requires either a TON of wasteful experimentation or some form of calculation.

Elves understood metalworking. Celebrimbor and Gondolin crafted the Rings as well as Orcrist and Glamdring. Basic smithing can be done on an ad-hoc basis; a few general "recipes" can be used to combine metals. Swords and armor for regular troops would be nothing special; good enough quality to fight, but not pieces of art. However, advanced alloys require mathematics to calculate time, temperature, and ingredients to produce specific properties in the materials being forged. In the books, armor is "Early Middle" mail and scale shirts and the like; basic simple repeating shapes or thick slabs of metal. In the movies, plate armor appears in the "High and Middle Late" period style. Plate armor requires basic geometry to fabricate, as well as computation to build armor properly for the wearer. Hand-forging being slow, you wouldn't want to "seat of the pants" design a complete suit of plate armor, especially if you need to repeat that pattern over an entire group of soldiers.

Finally, Elves understood the concept of sailing. Any sailing beyond sight of shore requires navigation, and even dead reckoning needs mathematics to compute speed and time. Celestial navigation brings in angles as well. Basic geometry is required to navigate by map using the data collected to compute heading and current location.

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    I don't think that metalworking under a certain (and quite high) line needs to much mathematics. The Vikings for example, didn't done too much in mathematics, when compared to the ancient Greeks, but were able to forge far better steel. Also, as you said, in the books they preferred scale and chain mails. – b.Lorenz Jan 17 '17 at 16:57
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    The basis of their crafts and arts the elves learned from the Valar, and were able to use it trough lore and memory. – b.Lorenz Jan 17 '17 at 17:00
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    @b.Lorenz learning tradecraft from the Valar doesn't mean the Valar didn't teach them mathematics to go with it. Base metal forging is pretty simple, and so is basic steel (just add some carbon to it). Proper alloying is much more difficult. Making alloys require using weights and doing at least basic mathematics (i.e. Multiplication to adjust the quantity of alloy created by a recipe); once somebody has created an alloy, they need to be able to share the recipe which is itself somewhat mathematical. – Zoey Boles Jan 17 '17 at 17:12
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    Multiplication or recipes like 'take 5 parts from that strong iron found in the North Esoriach and 1 part of cooper, mix it in smeltery burning yellow or white.. ' are not so advanced mathematics. – b.Lorenz Jan 17 '17 at 17:18
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    They also used 'magic' to accomplish their works. Minas Tirith (The First Age one) had a spell bounding stone to stone, and it collapsed after Lúthien had broken it. – b.Lorenz Jan 17 '17 at 17:20

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