Maths history: Robert Recorde’s Whetstone of Witte
It’s close to the end of term — why not use part of a lesson to take your students for a tour through a document that introduced one of the most important symbols in mathematics? The document is Robert Recorde’s ‘Whetstone of Witte’, first published in 1557, and the symbol is the equals sign (=).
The passage where equals is introduced is quite well known. See for example this interesting article on 360, which puts the Whetstone into some historical context.
Transcribed:
Howbeit, for easie alteratiô of equations. I will propounde a fewe exâples, bicause the extraction of their rootes, maie the more aptly bee wroughte. And to auoide the tediouse repetition of these woordes: is equalle to: I will sette as I doe often in woorke vse, a paire of paralleles, or Gemowe lines of one lengthe, thus: ====, bicause noe .2. thynges, can be moare equalle. And now marke these nombers.
Transcription note: the first word of this is often transcribed ‘nowbeit’, but the very next page of this has a very clear example of a capital ‘N’ and a capital ‘H’ — it’s definitely an ‘H’!
There is much more to the book than this paragraph — for a start, it was the first book in English to use the + and – signs. Even just looking at the page where this paragraph appears is a very useful thing to do. Here is a scan of the double page spread where the equals sign is first used:
(Source: whetstone-of-witte-equals-sign.pdf )
It’s a very useful exercise, particularly if you’ve never tried to read a book published in the 16th century before, to spend five minutes or so trying to decipher this text.
On the bottom of the left-hand page we have six examples of things that look a lot like equations — but what are the odd looking symbols?
With a bit of squinting, you might be able to convince yourself that these are symbols that evolved into modern-day ‘x’, ‘y’ and ‘z’, so equation 5 for example would say ‘18x+24y=8z+2x’. That’s a good guess, but it’s not quite right. These were in fact symbols for powers of ‘the unknown’, including a symbol for ‘no unknown involved’ (in other words, a constant term). Earlier in the book Recorde gives (in quite painful detail) tables of symbols to use for various powers of the unknown, using quite a nifty powers-of-primes-style notation system. More on that another day!
Translated into modern notation, the six equations above are:
If you are giving this to a class, then you might find it useful, after making them stare at the original for a while, to give them the following transcription:
(Source: robert-recorde-equals-sign.doc)
This preserves all the idiosyncratic spelling and typography of the original. If you’re feeling particularly nice to your classes, you can always give them a version with modernised spelling (and also some slight grammatical editing).
You can then discuss how Recorde explains what to do with the first equation. In modernised form:
In the first there appears 2 numbers, that is
equal to one number, which is
. But if you mark them well, you may see one denomination on both sides of the equation, which never ought to stand. Wherefore abating the lesser, that is
out of both the numbers, there will remain
. That is, by reduction,
.
In the rest of this chapter, Recorde goes into great detail about how to rearrange these equations into the correct form (for him, the highest power of the unknown should be on its own on the left, with everything else on the right), and then gives quite a few examples of word problems, with worked solutions. These are very interesting in their own right — look out for some 16th century word problems in the next week or so!
Looking into the future, it is now only 48 years until the 500th anniversary of the equals sign. If any of us are still around in 2057 (fingers crossed: I’ll be in my late 70s), I hope to see the contribution of Robert Recorde properly recognised — I think a week or so of national celebration would be fitting!
[...] Maths history: Robert Recorde’s Whetstone of Witte [...]
Lessons taught; Lessons learnt » Blog Archive » The Whetstone of Witte online
28 Mar 09 at 4:13 am
Wow, thanks for that article. It was really interesting. I was reading about the equals sign in some maths book and was intrigued by that excerpt of Whetstone of Witte, I had to search the internet about it. But then again, those old works always seem interesting to me, not matter what they write about. I’m always surprised when I realise that there were actually people doing maths centuries ago, seems bizarre somehow.
Ben
10 Oct 09 at 11:20 pm