## Monday Morning Math: Whose Triangle?

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Today’s snippet isn’t a who, or at least not a single who.  There is a triangle that many people learn about in school, since it pops up in some interesting places.  It starts off like this

with each number equal to the the sum of the two numbers above it.  It’s a pretty interesting triangle, but who first came up with it?

In the United States it is often referred to as Pascal’s triangle, after Blaise Pascal.  This isn’t so much because he invented it (he didn’t) but because he wrote so much about it, in a book entitled Treatise on the Arithmetical Triangle (but in French, so Traité du triangle arithmétique), written in 1654 and published 11 years later.  Here’s how he wrote it:

But the Triangle was known before that.  Here’s a picture from 100 years earlier, in a book by Niccolò Tartaglia in Italy:

Here’s one from 250 years before THAT, by Zhu Shijie in his book Si Yuan Yu Jian from 1303 in China:

There’s another version around this same time period by Omar Khayyam in Persia (modern day Iran) although I didn’t have a picture of that to include. But he wasn’t the earliest either: here’s one from 550 years before THAT (so roughly 900 years before Pascal)

This is from a manuscript in Raghunath Library, Jammu and Kashmir, in India from 755 (according to Wikipedia) where the figure was called the Staircase of Mount Meru (Meru-prastaara).  This, too, is unlikely to be the earliest: there are indications that the earliest manuscripts showing the arithmetic triangle are copied from even earlier ones.

So who first described this Arithmetic Triangle?  We don’t know, although we can say with assurance that it was well over a thousand years ago.