Setting of the values comes in tree slightly different varieties:
Using the slide rule notation these operations are written as:
Many of you know that if you have a number near 1, you can can get extra precision e.g. .963*23 = (1 - .037)*23 = 23 - .037*23 = 23 - .851 = 22.149. We can extend this to other products. There is a little pencil & paper work for addition and subtraction. An Addiator on the reverse of the slide rule is ideal. Suppose we have a product 35.76*829.4. The problem is then 3.576*8.294 and remember 10^3. It is a judgment call but lets divide 3.576 by 4 and 8.294 by 8. We subtract 3.576 from 4 and get .424. On the slide rule find .424/4 and .294/8. We have .106 and .0368. Multiply these and get .00390. Add these three last numbers and get .0731. Multiplying this by 32 we get 2.34. Subtract this from 32 and get 29.66. Applying the power of ten we have 29660. The answer from my calculator is 29659.344. We have four significant figures in our answer. This will work for quotients too.
In short notation this is written as
|->uU; |=>vV, or, with further simplification:
This operation allows to perform calculations in the following
If we match value u on scale U with formula d=f(u) against v on scale V with formula d=g(v) then, because d is the same for both formulas, u and v are in the following algebraic relationship:
For example, the value u on scale C with formula
If one of the U or V is on body and another is on slide, then these formulas depend on some constant (position of the slide), thus allowing to use slide rule as a function table for the whole family of functions.
This actions changes formulas of the scales located on the slide from d = f(u) to d = 1-f(u). In some cases this increases range of function that can be represented by appropriate choice of a pair of slide rule scale.
Reversing slide turns C into CI (C inverted) with formula d = 1 - log10x. The relation between these scales now is
Unlikely, the basic slide rule operation is ternary in
nature, that is, it has 3 operands.
This operation can be described
in the following way: move u on scale U to v on scale V and read the
result r from scale R under w on scale W. In short notation this is
however, you will often be required to use cursor to perform this:
If U, V, W and R have formulas (respectively) d=f(u), d=g(v), d=h(w),
d=t(r) then (see picture)
Note, that here R is scale on body, and W is scale on slide. This case is
most useful -- since the result obtained on a body scale can often be used
in subsequent calculations without reading. However, situation when the result
is read from some slide scale is also valid. Referring to the same
r is one of the input parameters and w is a result
Scales C and D are present on every slide rule, that is why this operation is so important.
Proof. Indeed, let U, V, W, R have formulas d=f(u), d=g(v), d=h(w), d=t(r) respectively. Then operation (2) yieds formula
Note however, that this is exactly (1) with u=1. This shows that multiplication is just a particular case of one of the basic ternary operations. Using other ternary operations (produced by different choice of scales) it is possible to construct additional multiplication algorithms.
For example, polyphase layout (K,A[B,CI,C]D) allows the following 11 ways to multiply 2 numbers u and v:
|[D,C]||1C->vD; uC=>D(r)||Standard multiplication|
|[D,CI]||uCI->vD; 1C=>D(r)||AKA multiplicaton by means of red digits|
|[A,B]||1C->vA; uB=>A(r)||Like standard, but using A/B scales|
|[A,B,D]||uB->uD; vB=>A(r)||May be useful|
This method requires 2 slide movements, but it uses only left and right indexes
of scale D, thus actually, using only one scale.
Dividing two numbers
Like multiplication, division, as described by slide rule manual
is the particular case of (1), with w=1.
Note that this is exactly the same basic operation that was used to produce
Again by using different choice of scales it is possible to come up with different u/v algorithms. Polyphase layout (K,A[B,CI,C]D) allows the following: