With great powers comes great responsibility

Katie Steckles

hose with an interest in the history of mathematical notation may have noted that in last month’s post I mentioned the symbol for equality, =, is often credited to Welsh physician and mathematician Robert Recorde, who used it as early as 1557 to denote equality. In his manuscript, The Whetstone of Witte, Recorde got bored with writing out in words, ’is equalle to’, and instead drew a pair of parallel lines of the same length, noting that ‘no two thynges can be more equalle’.

While this may not actually be the first recorded use of the equality symbol – Italian manuscripts dating from 1465 have been found to use a similar symbol – Recorde’s popularisation of the glyph has cemented his place for many as ‘the inventor of the equals sign’. But the book in which he first used it included many other mathematical notation and terminology conventions, some of which caught on – it was actually the first book in English to use the plus and minus signs – and others which… didn’t.

Power play

One particular topic from the book I found interesting was an alternative way of writing out powers of numbers. In modern notation, known as Cartesian notation (named after Descartes), we use indices to denote repeated multiplication:

When learning about such notation at school, we often learn that different powers of the same base value can be combined – but that doing so requires caution. For example, if we were to multiply together 𝑥³ and 𝑥²,  this would be three 𝑥s multiplied together, times two 𝑥s multiplied together. The result would be the product of 5 𝑥s, or 𝑥⁵. So, if we multiply powers of 𝑥 together, the powers are combined by addition: 𝑥³ × 𝑥² = 𝑥³⁺² = 𝑥⁵.

Similarly, if we raise a power of 𝑥 to another power – say, (𝑥²)³, we can consider this to be three copies of 𝑥² multiplied together, resulting in the product of 6 𝑥s, or 𝑥⁶. This correspondence between addition, multiplication and exponentiation echoes the same pattern found in logarithms: log a + log b = log(a × b), and  log aⁿ = n log a.

Hip to be squared

In Recorde’s time, this notation was all completely unknown. Numbers could be squared and cubed, and multiplied together, but algebraic notation was in its infancy and proper notation to describe higher powers hadn’t yet been introduced. Recorde noticed that if squaring and cubing were accepted notions, they could be combined using the rule for raising powers to other powers, and in this way we can obtain higher powers.

For example, a fourth power could be considered to be a square, squared, and a sixth power would be a square, cubed (as in the example above). Continued squaring of the square could be used to obtain all powers which are themselves a power of two, and cubing cubes would get the powers of three. Further powers could be obtained by squaring cubes and cubing squares.

This leaves a few gaps – for example, there’s no way to obtain a fifth power using only squaring and cubing. Recorde realised that in order to obtain other powers, we’d also need to be able to raise numbers to every prime power beyond two and three. Such powers were named sursolids, with 𝑥⁵ being termed the ‘first sursolid’ of 𝑥, then each prime power beyond that taking the next named sursolid – the second sursolid being 𝑥⁷, third sursolid 𝑥¹¹ and so on.

In this way, systematic terminology could be developed. Using the term ‘zenzic’ to denote squaring, ‘cubic’ for cubing, and sursolids as described above, Recorde could concisely combine these to form terms for other powers. For example, e.g. the square of squares, or ‘zenzizenzic’, would be 𝑥⁴, and the square of cubes, or ‘zenzicubic’ would be 𝑥⁶. This could be extended to such joys as the term for a sixteenth power, described as a ‘square of squares, squaredly squared’, pleasingly known as the zenzizenzizenzizenzic.

While this method of writing powers remarkably didn’t catch on, it’s an interesting alternative way of thinking about powers, and might be more than just an amusing diversion for someone currently learning about powers and trying to understand how numbers fit together more generally. There are several other areas of maths in which we have analogous notions to that of prime numbers, and in this case prime numbers themselves pop up again in these ‘prime exponents’, which can be combined to make any power.

The book, whose full title is The whetstone of witte, whiche is the seconde parte of Arithmetike: containyng the extraction of Rootes: The Coßike practise, with the rule of Equation: and the woorkes of Surde Nombers, covers a range of other interesting topics around powers and surds, including finding roots of numbers and many of the ideas behind modern algebra.

It’s well worth a look for the delicious script handwriting and illuminated initials on chapter heads (a sadly dying art), as well as Recorde’s hand-drawn diagrams of squares and cubes. Next time you write a plus, minus or equals, spare a thought for the physician from Wales who brought them to fame – and when you raise one number to the power of another, be thankful his system didn’t catch on.

The post With great powers comes great responsibility originally appeared on the HLFF SciLogs blog.