Al-Khwarizmi solved equations without writing a single symbol. Eight hundred years later, Descartes wrote $xx+10x$ almost as we do. This chapter follows that crossing: the numbers that learned to calculate on paper, the figure that goes with the proof, the lost names of trigonometry, the letters that turned recipes into formulas, the small signs of the operations, Napier’s logarithms, the great dispute between Newton’s dots and Leibniz’s $d$, the passage from $\Sigma$ to $\int$, Euler’s functions, Cauchy’s limit, Galois’s table, the war between quaternions and vectors, differential equations and the signs of logic.
The thread running through it is a simple question: what does this way of writing let us do? Some notations, which we call active, carry the relations of the object in their very shape and suggest the next step. Others, which we call sterile, record what is already known but lead nowhere further.
Texts in this chapter
- Radix quadrati: mathematics has a script too
- Zero and place value: numbers that calculate
- Euclid: the figure and the proof
- Sine, leg and hypotenuse: trigonometry and its lost names
- From words to letters: algebra learns to write
- +, −, × and =: a short history of the signs
- Napier and the logarithm: the marvellous abbreviation
- Leibniz against the dots: active and sterile notation
- From exhaustion to the integral: Σ, ∫ and the number π
- Euler and functions: giving a rule a name
- Cauchy and the limit: how to write the infinitely small
- Galois and Cayley: an algebra for things that are not numbers
- Hamilton, Gibbs and Heaviside: the war of the arrows
- Differential equations: the law in an anagram, ∂ and ∇
- Boole, Frege and Peano: writing down reasoning
- References and credits