The first instance of the concept of a symbolic mathematical equation is usually attributed to Robert Recorde, a Welsh mathematician, in a 1557 book with the rather verbose title ‘*The Whetstone of Witte, whiche is the seconde parte of Arithmeteke: containing the extraction of rootes; the cossike practise, with the rule of equation; and the workes of Surde** Nombers’*.

That is to say, Recorde invented the equals sign, which is still in use today. However, over the centuries it has undergone significant length contraction. As Piers Bursill-Hall remarked (you may enjoy reading his History of Maths lecture notes or, even better, attending the aforementioned lectures), the symbol for equality was a pair of ridiculously elongated parallel lines, like so:

Even though he invented one of the most useful mathematical notations in existence, Recorde had very bizarre reasons for doing so. In *The Whetsone of Witte*, he explained his reasoning for choosing the symbol to represent equality, namely that ‘no two things can be more equal than a pair of parallel lines’ (!).

Actually, the hyperbolic Fermat equation shown above is not entirely in Recorde’s notation, since the Cartesian notation for powers had not been adopted. Recorde had a brilliantly clumsy method of expressing powers of quantities. We still had the Hindu-Arabic numerals at this point thanks to Fibonacci’s text, *Liber Abaci* (which Bursill-Hall accidentally spoonerised, resulting in him saying ‘[…] the great mathematician, Liberace’). Hence, Recorde would have expressed the quantities in this equation as:

30042907 square

So far, so good. And you can probably guess the next one:

96222 cube

Again, nothing out of the ordinary. But once we get to higher powers, the notation appears slightly comical:

43 zenzizenzizenzic

What?** Zenzizenzizenzic**???

I have to admit that this is definitely my favourite obsolete mathematical notation. It is based on a Germanised spelling of the Italian word *censo* (meaning ‘squared’), and exists in the Oxford English Dictionary, noted for possessing more copies of the letter ‘z’ than any other word. Some books feature 16th powers, described as ** zenzizenzizenzizenzic** (starting to get ridiculous now), but this term unfortunately did not make it into the OED.

The fourth power, of course, was merely denoted *zenzizenzic*. The fifth power was called the *first sursolid*, and subsequent prime exponents were described similarly. There’s a sub-exponential-time algorithm for converting the *N*th power into Recorde’s notation:

- Factorise the number
*N*into a product of primes (with possible multiplicity). This can be done in sub-exponential time using the General Number Field Sieve. - Sort the prime factors into ascending order. This can be accomplished in O(log
*N*log log*N*) time with von Neumann’s merge sort. - Replace every instance of `2′ and `3′ with ‘square’ and ‘cube’, respectively. This takes linear time.
- Replace any remaining primes
*p*with ‘(π(*p*) − 2)th sursolid’, where π is the prime-counting function. This can be accomplished in time O(*p*^(½ + ε)) with the Lagarias-Odlyzyko algorithm.

Contrast this with linear time for expressing a power in Descartes’ notation, and it is clear that moving away from Recorde’s notation was probably a good idea.

How does one acquire a copy of the History of Maths lecture notes? I feel these could be appropriate entertainment for an early morning ferry trip tomorrow.

Nevermind, I found the link. 😛

Then again, I can’t access the pdf…

Oh, sorry, I think you have to be inside the .cam.ac.uk network. You’ll have to remind me on or after the 6th October, when I regain absolute power. 😛

If you can’t wait that long, you could always pester me on the 5th instead.

Alternatively, I could pester Ben Green… Oh, wait, he’s not at Cambridge. ;P

Being picky here, but these aren’t even lines. These are line segments 😛

I am enjoying thinking of Robert Recorde as reading his equations out loud in the style of legendary boxing announcer Michael Buffer.

“One pluus one equaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaals TWO!”