Aztec Fractions & Divisions of the Day

One of my pet peeves is when folks—even scholars who should know better—act like the absence of proof is proof of absence. This is especially true of pre-Spanish Invasion Indigenous Mexican cultures like the Nahua peoples whom I study.

A perfect example would be fractions. I have seen entirely too many people claim the “Aztecs” had no fractions. That’s absolute nonsense.

In the first place, we literally know their word for fraction: centlacotontli (“a bit broken from a whole”). And we know their words for “one half,” the main one being centlahcol. We know their word for “divide in half” (tlahcoxeloa) and “divided in half” (tlahcoxeliuhtoc). But that’s not all.

We know they could talk about thirds, as when, in discussing the leaves of a particular plant, a 16th-century informant said, “excan xeliuhticah” or “they are divided into three places [parts].” That -can construction shows up with other numbers. If I ask “Quezquican xelihui?” (How many places / parts is it divided into?), you could answer …

• “ca nauhcan” (it’s [divided into] four parts)
• “ca macuilcan” (it’s [divided into] five parts)
• “ca chicuaceccan” (it’s [divided into] six parts)
• “ca chicoccan” (it’s [divided into] seven parts)
• “ca chicuexcan” (it’s [divided into] eight parts)
• “ca chiucnauhcan” (it’s [divided into] nine parts)
• “ca mahtlaccan” (it’s [divided into] ten parts)

… and so forth.

I promise you that the feats of engineering that pre-Spanish Invasion Nahuas and other Mesoamerican civilizations accomplished didn’t happen without complex math with fractions. Just because the Spanish didn’t interview engineers to document their methods and terminology doesn’t mean that those didn’t exist.

And by the time Bernardino de Sahagún and others were trying to preserve some of the older knowledge, decades after the fall of the Triple Alliance, European methods for counting and time-keeping had displaced the systems previously in use.

Ah, time-keeping. Another subject that makes my eyes twitch.

A cursory search of the Internet might lead you to believe that, while Nahua and Maya people had complex calendar systems, a day was just a day and had no real, regular, functional divisions. And it’s true that they didn’t use hours / minutes / seconds.

BUT THEY DIVIDED THEIR DAYS. Of course they did.

There’s ample evidence of four major divisions:

• Hualmomana Tonatiuh (“sun presents itself here”)
• Nepantlah Tonatiuh (“sun at middle/ zenith”)
• Oncalaqui Tonatiuh (“sun enters yonder”)
• Yohualnepantlah (“divide of darkness/night”)

Subdivisions crop up:

• tlahuizcalpan (dawn) …
• tenihzalizpan (breakfast)
• tlacualizpan (lunch)
• tlahcah (mid-afternoon)
• teotlac (near sunset)
• tlapoyahua (dusk)
• netequilizpan (bedtime)
• tlahtalpitzalizpan (trumpet-sounding)
• ticatla (midnight)
• nezohua (blood-letting)
• tlathuinahuac (near-dawn)

Sure, we don’t know the full structure. But it’s clear that a regular system of … let’s call them “watches” … that to me looks a lot like the Roman dies civilis (civil day).

And there’s ample reference to priests burning incense during these watches. Why? Perhaps not just religiously. Perhaps for time-keeping. Sticks of incense made the same way from the same materials burn at the same rate. If you’re trying to keep these “watches” roughly the same from day to day, it seems to me that burning incense is a solid way to calculate that in lieu of clocks.

Think about the size of the Aztec Empire. Do we really think that a massive civic infrastructure and vibrant, complex economy could exist without a way to say, “Meet me a half-watch after lunch so we can do X”? The evidence is right there that these people had a system in place. Only cultural bigotry keeps us from seeing it.

Ditto the issue of leap days.

For centuries, experts have insisted that there was no addition of leap days in pre-Spanish Invasion Mexico … mainly because people who kept using the old dual ceremonial-solar calendar after the Invasion didn’t insert any days beyond the 365.

HELLO! DUH! What was missing?

The Indigenous astrologers and their temple / observatories!

Let’s back up. Each of the 18 20-day months has a name and a ritual that ties it to … a specific season, like Atemoztli (“the falling of water”) or Toxcatl (“dryness”). Without adjustments (like our leap year), these months no longer make sense after a few decades.

Now, outside of the 18 months of 20 days was a period of five days that didn’t belong to any month. In Nahuatl, this non-month is known as nemontemi, meaning “superfluous, but fills up [the calendar]” because of course 360 +5 = 365.

Recent scholarship has reached the (frankly obvious) conclusion that the nemontemi was occasionally extended to SIX extra days if the vernal equinox fell later than normal (by astronomer observation).

Here’s a great article by Dr. Magnus Pharao Hansen explaining the mechanism by which the leap day was added.

Again, the assumption that Mesoamerican science wasn’t advanced enough to recognize that the solar year is 1/4 a day shorter than the planet’s actual trek around the sun is just sheer cultural bigotry.

Oh, I forgot. They didn’t have fractions. 🤪🫠💀

An Aztec incense ladle.