Aki points us to this fun 1990s-style webpage from Jeff Miller. Last year we featured his page on word oddities and other trivia. You might also enjoy his page, Earliest Uses of Various Mathematical Symbols.

Here’s an example:

The equal symbol (=) was first used by Robert Recorde (c. 1510-1558) in 1557 in The Whetstone of Witte. He wrote, “I will sette as I doe often in woorke use, a paire of parralles, or Gemowe lines of one lengthe, thus : ==, bicause noe 2, thynges, can be moare equalle.” Recorde used an elongated form of the present symbol. He proposed no other algebraic symbol (Cajori vol. 1, page 164).

Here is an image of the page of The Whetstone of Witte on which the equal sign is introduced.

The equal symbol did not appear in print again until 1618, when it appeared in an anonymous Appendix, very probably due to Oughtred, printed in Edward Wright’s English translation of Napier’s Descriptio. It reappeared 1631, when it was used by Thomas Harriot and William Oughtred (Cajori vol. 1, page 298).

Cajori states (vol. 1, page 126):

A manuscript, kept in the Library of the University of Bologna, contains data regarding the sign of equality (=). These data have been communicated to me by Professor E. Bortolotti and tend to show that (=) as a sign of equality was developed at Bologna independently of Robert Recorde and perhaps earlier.

Cajori elsewhere writes that the manuscript was probably written between 1550 and 1568.

Lots more great examples at the above link.

I’m a bit captivated by how readable that Recorde quote is. For some reason I had the impression that 500 year-old English would be nearly impossible for a modern normie to interpret.

That’s within a generation of Shakespeare. You may be thinking of Middle English, which had pretty much transitioned into early Modern English by the mid 1500s. Of course, it’s not like these language changes are discrete—it’s a continuum of changes continuing through to today.

I once read a statement of the quadratic formula from a very old text, written without any of the modern mathematical notation. It was a horror, a densely worded paragraph that was utterly unenlightening, unlike the quadratic formula in modern notation, which I can see in its entirety in my mind’s eye. I don’t know how anyone managed to do much of any mathematics at all prior to the invention of mathematical notation.

It is like trying to write code on a tablet or phone.

What amazes me is that it’s less than 50 years from 1618 to Newton.

We went from inventing = notation to calculus rather rapidly eh?

And would that mean that most of Galileo’s work is devoid of the = sign? Would never have realised!

I’m curious about what preceded many of these symbols. How were the concepts conveyed? Did people just use a hodgepodge of symbols to indicate equality before the equals sign? Or words?

Compare Euclid’s proof with a modern proof here: https://mathcs.clarku.edu/~djoyce/elements/bookIX/propIX3.html

Wow—that’s hilarious.

Note also, that while there were few equations, there were many diagrams. The Hobbes-Wallis debates over the use of equations in geometry went on for 25 years.

Hobbes: Your treatise of the Angle of Contact, I have before confuted in a very few leaves. And for that of your Conic Sections, it is so covered over with a scab of symbols, that I had not the patience to examine whether it be well or ill demonstrated.

Hobbes’s first complaint is that “symbols, though they shorten the writing, yet they do not make the reader understand it sooner than if it were written in words.” He remarks that there is then a double labor in understanding, first reducing the symbols to words, then to attend to the ideas thereby signified. But as we have seen, Hobbes was to realize that the significance of symbols, or “certain figures, as if a hen had been scratching there” was more than simply that they were a shorthand which he found tedious and inefficient. Rather, like the terminology of the school divines, these symbols were a cloak for more devious goings-on. Signifying at one moment a length and at another a volume, the use of these symbols appeared to license trains of reasoning, such as “spurious” confutations of Hobbes’s duplication, trains of reasoning, according to Hobbes, which were in fact false. (Source: https://seis.bristol.ac.uk/~plajb/research/papers/Squaring_the_Circle.html)

A 14th century mathematical proof from the pretty brilliant Nicholas Oresme. https://mathshistory.st-andrews.ac.uk/Bookpages/Oresme6.gif

A probability example, from an anonymous author around 1400 (with a few modern emendations in brackets) of the proposition that if x is wagered on the first to three chess wins, and they stop after two games, both won by A, then fair settlement in expected value would be 3/4 of the aggregate stake.

Two men play chess, and [each] bets one ducat on [being the first to win] three games. It happens that the first man wins 2 games from the second. I ask: if the game proceeds no further, how much of the second man’s ducat will the first have won? Suppose that the first man has won x from the second in the first game; you must see that he has won the same amount in the second game as in the first, by the same reasoning. Therefore he will have won another x, and so will have won 2x in two games. So the second man, the loser, will still have out of his ducat 1 ducat minus 2x. It is to be known that if the one who has lost two games were to win two further games from his companion, neither would have won anything from the other. Now let us suppose that the second man begins by winning one game from the first; I say that he wins in this game the 1 ducat minus 2x that the first had won. The reason is that if the one who had won the first two games had won the third game too, he would have won from the first [read: second] man the whole remaining part of his ducat, and so conversely the second wins from the first 1 ducat minus 2x, so he wins back 1 ducat minus 2x out of the stake that the first player had won from the second, that is, 2x. There will then remain to the first man 4x minus 1 ducat. So the second man before he wins his second game will have 2 ducats minus 4x. Now observe that for the first man (who has won two games), that if the second man has won the [next?] two games he must have won the third game; there would be nothing he [the second] had not won [back], in all reason, of what the first had in his [the second’s] ducat. If the first had won that third game he would have won 2 ducats minus 4x, so likewise that is what the second [actually] wins from the first. Now we are supposing the second man wins his second [that is, the fourth] game; therefore he has won from the first 2 ducats minus 4x, and also he must have won back everything the first had won, because they have won two games each. Now look at how much the second wins from the first in his second [the fourth] game: he wins 2 ducats minus 4x. Now [in the equation 4x − 1 = 2 − 4x] we should add one ducat to each side, and will have on one side 4x and on the other 3 − 4x; then add 4x to each side, and there will be 8x equal to 3 ducats. Then divide the number by [the number of] xs, which gives ⅜ for the value of x. This is what the first man wins in the first game, and in the second game he wins a further ⅜ ducats which makes , that is, ¾. This is how much the first man has won when they play no further than two games; and thus you proceed in similar situations.

Franklin, James. The Science of Conjecture (pp. 294-295). Johns Hopkins University Press. Kindle Edition.