Descartes’s Logarithm Machine
In the Geometry (1952), Descartes considered the problem of finding n mean proportionals (i.e. geometric means) between any two lengths a and b (with a < b). That is, the problem is to find a sequence of lengths, beginning with a and ending with b, such that the ratio of any two consecutive lengths is constant. Hence the terms of the sequence have a constant ratio of r, and form a geometric sequence beginning with a and ending with b.
In modern algebraic language, the problem is to find a sequence x 0,x1 ,…, xn+1 . k such that for some fixed ratio r, xk = a r , and xn+1 = b. That is, n mean proportionals between a and b are n numbers between a and b, not counting a and b themselves. For a modern algebraist, solving this problem involves finding a (n+1)st root: b = a ⋅ rn+1 b r = n+1 a A. Descartes’ linkage machine from the Geometry Book II of the Geometry shows the following illustration.