The Equals Sign and the Man Who Hated Repeating Himself

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Welcome to Origins. In this program we explore the technological and scientific roots of the modern world. Let's begin.

In 1557, a London printer put a new schoolbook on sale at the west door of Saint Paul's. It was called The Whetstone of Witte, and it is generally counted as the first algebra printed in English. Its author was a Welsh physician named Robert Recorde. Deep in the book, about to work through a list of equations, he stopped to complain about a phrase.

"To avoid the tedious repetition of these words, is equal to," he wrote, "I will set, as I do often in work use, a pair of parallels, or gemowe lines of one length." Then he drew them: two long horizontal strokes, one above the other. And he gave his reason. "Because no two things can be more equal."

Gemowe is an old word for twins, a cousin of Gemini. The first line he wrote with his twin strokes says, in modern terms, fourteen x plus fifteen equals seventy-one.

For most of history there was no such mark at all.

Algebra began as prose. In ninth-century Baghdad, al-Khwarizmi set out his famous problem as a sentence: a square and ten roots are equal to thirty-nine. The solution was a paragraph too. Halve the roots, multiply the half by itself, add it to the thirty-nine, and so on, like a recipe. For centuries, that was what algebra looked like. A reader had to hold the whole problem in mind while the words went by.

Symbols crept in one at a time, and mostly from commerce. The plus and minus signs turn up in German manuscripts of the 1480s, and reach print in 1489 in a Leipzig handbook of merchants' arithmetic. The unknown quantity was simply called "the thing," cosa in Italian, so algebra went by the name of the cossic art. That gives Recorde's title its joke. Cos is Latin for whetstone. This was the stone to sharpen your wits on.

Recorde was born around 1510 in Tenby, a harbor town in southwest Wales. He studied at Oxford, took a medical degree at Cambridge, and practiced in London. On the side he wrote something scarce in England: mathematics in plain English, for people with no Latin. There was an arithmetic, then a geometry, then an astronomy, each written as a conversation between a patient Master and an eager Scholar. The Whetstone was dedicated to the merchants of the new Muscovy Company, people with ships to navigate and accounts to keep.

He also worked for the Crown, and that went badly. He was controller of the mint at Bristol, then surveyor of silver mines in Ireland that lost money from the start. Along the way he made an enemy of William Herbert, soon to be Earl of Pembroke. In 1556 Recorde accused Pembroke of misconduct. Pembroke sued for libel and won. In February of 1557, Recorde was ordered to pay him a thousand pounds, a ruinous sum.

The Whetstone came out that same year, and its last page is strange. The Master is in the middle of a lesson on roots when he breaks off. "But hark," he says, "what meaneth that hasty knocking at the door?" The Scholar answers, "It is a messenger." The Master hears the message in his ear and says there is no remedy. He must neglect all studies and teaching. "My fortune is not so good, to have quiet time to teach." He promises to finish if he ever gets some peace, and the book ends with both of them saying amen.

Nobody can say how literally to take that scene. But the following summer Robert Recorde died a prisoner in the King's Bench prison in Southwark. His will left small sums to his nine children.

And his twin lines went nowhere. After 1557 the sign did not appear in print again for sixty-one years. It surfaced in 1618, in an anonymous appendix to an English book on logarithms, and in 1631 in three English works at once.

Across the Channel, meanwhile, it had rivals. A French monk used a square bracket. A German editor used two vertical lines. In 1637 René Descartes, in his Geometry, used a small looped figure that may have begun as the first letters of aequalis, the Latin for equal. Dutch and French mathematicians followed him for decades. Worse, on the Continent Recorde's two strokes were already taken. Different writers used them to mean a difference, or plus-or-minus, or a decimal point. And Kepler, Galileo, Pascal and Fermat used no equality sign whatsoever. They wrote out the word.

What settled it was calculus. Isaac Newton used the English sign. Gottfried Leibniz tried other marks in his private papers, but in print he used Recorde's. The historian Florian Cajori, who traced these marks through hundreds of old books, judged that if Leibniz had gone with Descartes, the looped sign would probably have won. By the early 1700s the contest was over. Along the way printers shortened Recorde's long strokes to the stubby pair used now.

Why did a mark for three words matter so much? Because of what it does to a sentence. With two strokes in the middle, an equation works like a balance, and the eye can take in both pans at once. Whatever is done to one side can be done to the other, and the balance holds. Take fifteen from both sides. Divide both sides by fourteen. X is four. That is much harder to do with a paragraph.

The sign is still not as simple as it looks. In 2006, researchers in Wisconsin asked 177 middle school students what the symbol means. At every grade, fewer than half said something like "the same as." Many wrote "the total," or "what the problem's answer is." For them it was a signal to go and compute. And the students who read it as a balance were the ones more likely to solve equations correctly.

Machines read it the first way too. In 1957, four hundred years to the year after the Whetstone, IBM delivered the first Fortran compiler. There, X equals X plus one is a perfectly good line. As a statement it is false. As an order it means: take the number, add one, put it back.

So the mark has had three lives: an author's shorthand, the hinge of algebra, and a command to a computer. It started with a man who had run out of patience for three words, and who had his reason ready. No two things can be more equal.

Thanks for listening to Origins.