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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.)

Rand al'Thor
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b.Lorenz
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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
  • Perhaps I'm equating geometry and mathematics incorrectly. The Golden ratio and the concept that arches were good structures certainly pre-date calculus and other advanced methods, but I presumed the basic math that existed / must have been used would qualify as a "yes" to the question. If we're discussing the kind of higher math that would be used to solve the "Three-body problem", then I'd guess no, if only cause they likely had no use for it. – VBartilucci Jan 17 '17 at 16:17
  • You are sort of right, since geometry is one of the most ancient branch of matematics. But the Golden ratio or the geometric construction of arches were required to make the buildings good-looking and naturally harmonic, not to prevent them from collapsing. The architectural usefullness of arches was based on empirism. – b.Lorenz Jan 17 '17 at 16:28
  • One of the points of my question is, that they might studied mathematics just for fun, not because it was useful. – b.Lorenz Jan 17 '17 at 16:30
  • Although the elves loved the stars, I do not recall any reference to other planets. And it is planetary motion which requires more sophisticated mathematics. – Verdan Jan 17 '17 at 19:17
  • The Star of Earendil, being indentical with Venus, was a planet, but you are probably right, since cosmology before the Fall of Númenor was more simple, (the stars all moved together or maybe stood still, and the Moon and the Sun orbited the flat Arda 'vertically') it didn't required so much math. The Trees of Valinor didn't move at all. – b.Lorenz Jan 17 '17 at 19:38
  • @Verdan See http://scifi.stackexchange.com/q/113632/31051 – Jason Baker Jan 17 '17 at 20:08
  • @Edlothiad This is why we need this question :) – Mithical Jan 18 '17 at 09:34
  • @Mithrandir, well if we're not using the answer from that there yet, then we should go to the tag wiki, which states: The Lord of the Rings is an epic fantasy novel written by J. R. R. Tolkien. The book has spawned a trilogy of billion-dollar-grossing films directed by Peter Jackson. This to me, means it is not relevant here and not simply "for the franchise". I'm just trying to help that question by not adding to the pile of incorrectly tag tolkien questions :) – Edlothiad Jan 18 '17 at 09:36

2 Answers2

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Little

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

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Jason Baker
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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.

Zoey Green
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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 Green 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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    "To make one sword requires 5 parts of North Esoriach iron and 1 part of copper" Now make me 150 swords, fast. Multiplication, @b.Lorenz. Sure, a single smith might just follow the recipe from scratch over and over again for 150 swords, but it's more time and fuel efficient to use a larger smelter and smelt larger masses of metal. Perhaps they just mindlessly threw in batches of metal and used integer counting, but SOMEBODY must have figured out it would be faster to do the batching in their head first, and then throw it all in. – Zoey Green Jan 17 '17 at 17:45
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    Also, anything can be hand-waved away with "magic did it." While it's possible that everything Elven was simply held together by magic, this isn't explicitly stated. It also leads to "fail-deadly" situations; if magic is required to keep something together, lack of magic makes it fall apart, and it'd be a shame if you were depending on it NOT falling apart. I expect elves used "magic" like Trekkies use "structural integrity fields," as a supplement to engineering, not as a replacement. – Zoey Green Jan 17 '17 at 17:47
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    I didn't said that they used magic everywhere, just that one of their buildings probably had magical strengthening. (Since Sauron took the Tower with force from Orodreth, it's doubtful that he could ask anybody for the magic word, so it's possible that the 'structural integrity spell' originated from Sauron, and not from the original builders.) But I again deny that Multiplication would mean very advanced math, and it don't looks like, that the eldar had to mass-produce anything in haste. Sauron for his Uruk-hai, yes, but not the elves. – b.Lorenz Jan 17 '17 at 21:04
  • Ancient Greece, Egypt, Inca, Maian are examples you do need a good understand of geometry to build something big with architecture. You do need at least a basic o geometry (triangles equivalence!) to build tall towers. To sail, count years, make astronomical observations you do need a lot of math. For sure elves can reconise and apreciate the beautiful and magic of math ao contraire of most users here. This is the only acceptable answer – jean Jan 18 '17 at 09:56
  • @ZoeyBoles You're very wrong that "basic metal forging is pretty simple". For starters you can't check the quality of the ore. When you don't have any way of measuring temperature other than "this colour looks about right", getting a decent result is a highly-skilled task. And within the limits of the medieval technology we see in Tolkein, a "larger smelter" simply isn't possible, because that smelter is the size of a stove and the fire is driven by an apprentice with a set of bellows. Perhaps you can cast a few sword blanks at once, but 150? Nope. – Graham Jan 18 '17 at 12:50
  • Re the plate armour, hand-forging is exactly how it was done in the past. There was never computation of shapes - it was a case of starting with a sheet of metal roughly the right size, and bashing and trimming it to fit. Remember that Elves have been around for thousands of years, and so far as we know they don't tend to change body shape. That suit of armour could be centuries old. Unless you're actually at war, there's no rush. And when they were at war and needed it quickly, they used scale armour which is quick to produce. – Graham Jan 18 '17 at 12:56
  • Re sailing, no medieval sailor would dream of going out of sight of land - every sea voyage simply involved following the coastline. Columbus took a huge risk, with ships which were not designed for ocean-going, because there was no such thing as an ocean-going ship back then. Back with Tolkein, the Elves did not need to know how to navigate because they had no idea where they were going - in fact that was the whole point of the voyage into the West being a one-way trip. – Graham Jan 18 '17 at 13:01
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    @Graham I agree with you about medieval navigation. But the elves (At least the Falathrim) could navigate in the open sea. In The Silmarillion, chapter 20, Nirnaeth Anoediad, it is stated, that the ship of Voronwe sunk in a great storm within sight of the coasts of Middle-earth, while returning in despair from an unsuccessful attempt to sail into Valinor. So they left line of sight with the land, and the very end of the voyage, being close to home, they got killed. – b.Lorenz Jan 18 '17 at 14:26