There are other operations in the machine. There are two pairs of storage registers that can be _coupled_ together so that we can handle problems requiring numbers of 46 digits instead of 23. Registers 64 and 65 can be coupled, and registers 68 and 69 can be coupled. There is another storage counter, No. 71, that has an extra property. We can read out the number it holds times 1, or times 10¹², or times 10⁻¹², as may be called for. As a result of this counter, we can do problems requiring 144 registers storing numbers of 11 digits each, instead of 72 registers storing 23 digits each. Bigger statistical problems can be handled, for example. There are some minor sequences of operations, or _subroutines_, that can be called for by a single code. The subroutine may be a whole set of additions, subtractions, multiplications, divisions, and choices, having a single purpose: to compute some number by a _process of rapid approximation_ (see Supplement 2). There are built-in subroutines for some special mathematical functions: the _logarithm_ of a number to the base 10, the _exponential_ of a number to the base 10, and the _sine_ of a number. (See Supplement 2.) There are also 10 changeable subroutines, each of 22 coding lines, which can be called in, when wanted, by the main sequence-control tape or by each other. These subroutines constitute the Subsidiary Sequence Mechanism, and are extremely useful. They have _A_, _B_, and _C_ fields just like the main sequence-control, but they are given information by plugging with short lengths of wire instead of by feeding punched paper tape. RAPID APPROXIMATION FOR A LOGARITHM Up to this point in this chapter the author has tried to tell the facts about the Harvard machine in plain words. But for reading this section, a little knowledge of calculus is necessary. (See also Supplement 2.) If you wish, skip this section and go on to the next one. What is the process that the machine uses to compute any desired logarithm to 23 digits? Suppose that we take for an example the process by which the machine computes log_{10} 49.3724. We choose a 6-digit number for simplicity; the machine would handle a 23-digit number in the same way. The process uses 2 fundamental equations involving the logarithm: the sum relation log (_a_·_b_·_c_···) = log _a_ + log _b_ + log _c_··· and the series relation _h_² _h_³ _h_⁴ logₑ(1 + _h_) = _h_ - ———— + ———— - ———— + ···, │_h_│ < 1 2 3 4 The error in this series is less than the first neglected term. Now, the machine stores the base 10 logarithms (to 23 decimal places) of the following 36 numbers: 1 1.1 1.01 1.001 2 1.2 1.02 1.002 ... ... ... ... 9 1.9 1.09 1.009