The oldest known tables of squares and cubes

0:00Somewhere around seventeen hundred years before the common era, somebody sat down and wrote out the squares. Not a rule for squaring. A list. The oldest such lists anyone has found were pressed into wet clay with a wedge, and then the clay dried, which is the only reason there is anything left to find. Squares and cubes, row after row. That is about three thousand seven hundred years ago.

0:30And the interesting question is not that somebody could square a number. It is why they bothered to write them all down. Here is what a list buys you. Suppose you write out the squares from one up to sixty. Sixty rows, and nothing else. Read a row from the left and you have squared something: forty-seven gives two thousand two hundred and nine. Now read the very same row from the right.

0:59You have a field, you know its area is two thousand two hundred and nine, and you want to know how long its side is. You do not calculate anything at all. You run a finger down the right-hand column until you find the number, and the answer is sitting next to it. Forty-seven. One list, sixty rows, and it answers a hundred and twenty questions — sixty one way and sixty the other.

1:29That second reading is not free, and it is worth seeing why it works. Running a list backwards only makes sense if no two different sides give the same area. If two rows carried the same number on the right, you would find it and still not know which row you were meant to be looking at. They do not. Sixty rows, sixty different areas, each one bigger than the last.

1:56And that is because a side is a length, and lengths are positive. Squaring does press two numbers into one — six and minus six both give thirty-six — but minus six is not the side of any field, so the folding never reaches the list. Now the cost, because a list always has one. It stops. Sixty squared is three thousand six hundred, and that is the last row on the tablet.

2:27The next square up is three thousand seven hundred and twenty-one, and it is simply not there. A rule does not stop. A rule handles a number it has never met. But if nobody has a rule yet, and somebody needs the answer this afternoon, you write the list. That is what a table is — the thing you build instead of a method. Every row you add costs one more line and buys you two more answers.

2:56And it is a thin net: of every number from one up to three thousand six hundred, exactly sixty of them are on that list. So who needed it? People measuring land, and people putting up buildings. A field has an area long before anybody has an algorithm. If you are dividing ground between two families, or laying out a foundation, then the question — what side gives me this area — is an ordinary day's work.

3:30It turns up centuries before anyone writes down a way to answer it. That is the setting these ideas came out of. And here is the part worth staying for: the words still show it. In Sanskrit, the word for a square is varga. A four-sided figure with equal sides — something you can draw. And varga is also the word for the second power. One word, doing both jobs.

3:59That is not carelessness. It is a fossil. The second power was met as an area first, as the amount of ground inside a drawn square, so when it came time to name the operation, it did not get a name of its own. It kept the shape's. The solid gets exactly the same treatment. Ghana is the cube — the block, the thing with three equal edges. And ghana is the third power.

4:30One word again, for the same reason. A number taken three times as a factor turned up as a volume before it turned up as an operation. So two of the powers are named after objects you could pick up and hold. Then there is the fourth power, and its name is varga-varga. Square-square. The square of the square. And that is not a label anybody had to choose. It is arithmetic.

5:01Take a number, square it, then square the answer. Two squared is four, and four squared is sixteen — and sixteen is two to the fourth. Three squared is nine, and nine squared is eighty-one, which is three to the fourth. Squaring twice is the fourth power, every time. So the name is not a name at all. It is a set of instructions. Which raises a fair question about the cube.

5:30If the fourth power can be built out of squares, why did the third power need a word of its own? Because you cannot get there. Squaring composes, and every step doubles: two, then four, then sixteen, then two hundred and fifty-six. Three is never on that list, and it never could be. Four works because four is two times two. Six would work, because six is two times three.

5:58But three does not split into two whole factors bigger than one at all. So the cube had to be given its own word. Not because anybody preferred it that way — arithmetic left no choice. Around the year four hundred and ninety-nine, Aryabhata wrote something down about that first word. He recorded that a four-sided figure with equal sides carries the name varga; that the number giving its area carries it; and that a quantity multiplied by itself carries it as well.

6:34Read that as testimony rather than as a definition. He was not handing out a new meaning. He was writing down that one word already covered all three, and that this is how it was used. Which is the evidence that the power came out of the figure, and not the other way round. Now the operation that runs the other way — the one that hands you back the side.

7:00The Sanskrit word for it is mula, and mula is the root of a plant. It also means basis, and cause, and origin. Think about what that is saying. If the area is what the square grows into, then the side is what it grew from. So a square root is varga-mula, and a cube root is ghana-mula. And the dates line up the way you would guess. The words for the powers are in use at least a hundred years — possibly three hundred — before the word for the root is.

7:35You name a thing before you name the business of undoing it. And then the word travelled. Arabic took the same idea and used jidhr, which is the Arabic word for a plant's root. Latin took it and used radix, which is the Latin word for a plant's root. That is the word sitting behind the sign you write over a number today. A hundred and twenty-nine years after Aryabhata, Brahmagupta uses another word for the same operation.

8:08Pada, which means a foot — and also a basis, a cause, an origin. He says the pada of a square is the quantity that it is the square of. Foot. Root. Origin. Three separate words for the same operation, and every one of them is a word about where something came from. Which is the whole point. Those lists were pressed into clay because somebody was out measuring ground, and the vocabulary never lost the fingerprints.

8:42You cannot compute a fact like that. You can only read it off the names.