Normally, a sliderule at distance x has the value of log(x) written on it, which allows doing multiplications by moving along the sliderule, since log(ab) = log(a) + log(b).
Now imagine a sliderule onto which values of x^2/2 are written. This also allows you to multiply two numbers, because ab = (a+b)^2/2 - (a^2/2 + b^2/2).
a * b = f(a + b) - (f(a) + f(b))