These devices were standard tools of scientists in nearly every discipline, until they were superseded by electronic pocket calculators, as soon as these became comparatively affordable. To understand the principle, let’s look first at a device to do additions and subtractions. You can either draw out, or use ready made, two scales graduated in centimetres and millimetres. If possible, one scale should have the markings along the top edge, and the other along the bottom edge – this is often the case if you can find an older ruler, which has inches marked along the top edge. The two scales are placed against each other as shown in Fig. 1 (showing just the centimetre marks for clarity).
Fig. 1. Two simple linear scales with their values aligned
Now suppose we move the upper scale two divisions to the right, as in Fig. 2.
Fig. 2. Arrangement for adding (or subtracting) 2
We can see that every number on the lower scale is now two units larger than the number immediately above it on the upper scale. What we are actually doing is adding two lengths together by putting them end-to-end. For example, we can read off the results 2 + 4 = 6, 2 + 5 = 7, and so on. Of course, it should not be necessary to use any sort of calculating device to do sums as easy as this! If we include the millimetre marks, then we can use exactly the same method to do two-figure calculations like 56 + 27 = 83, which are slightly harder. The really clever bit is if we can estimate fractions of a millimetre, and gain another decimal place in accuracy. In practice, you could estimate a position like 23.5 mm fairly accurately, but it is more difficult to decide whether a value is actually 41.3 or 41.4 (a lot depends on the thickness of the graduations), so although you can do three-figure sums this way, sometimes the last figure could be out by one unit. This is a general fact about this type of calculation – you can aim for three-figure accuracy, and you should usually be correct. Four-figure accuracy can sometimes be achieved even on a device with a length up to about 25 cm while, if you consistently fail to get three-figure accuracy, then you are not doing it very well!
Subtraction can be done in exactly the same way. If you want to calculate 5 – 3, then place the 3 on the upper scale opposite the 5 on the lower scale (Fig. 2 already shows this set-up), and read the answer on the lower scale under the 0 on the upper scale – the answer is 2.
How do we make a slide rule to do multiplication? Suppose that we have such a device, and we are using it to multiply by 2. Then we would place the 1 on the upper scale immediately above the 2 on the lower scale, as shown in Fig. 3.
Fig. 3. Constructing a simple multiplying slide-rule
So 1 × 2 = 2, which we know already. By analogy with Fig. 2 (in which every number on the top scale is increased by +2 on the bottom scale), we would now expect that every number on the top scale is increased by multiplying by 2. If we ask what number on the lower scale should come under 2 on the upper scale (the number temporarily shown as “a”), then clearly this should be 2 × 2 = 4. The number “b” on the lower scale coming under the “a” on the upper scale will be a × 2 = 4 × 2 = 8. In the same way we can replace the symbols c, d, e, by the values 16, 32, 64, and so on. This would lead to a slide rule that, in principle, could do multiplications, but… It is clear that there are big gaps in the scale between the values 1, 2, 4, 8, etc., and it would be rather difficult to estimate where other numbers such as 3, 5, 27, etc should go.
If the marks on the scale are at 1 cm intervals, then the mark at N cm from the left of the scale will carry the number x = 2N. What we want to do is to find where on the scale a given number x should be placed. Suppose it is at a position y cm. Then we know that the formula x = 2y applies for any integer y. We assume that it continues to hold when y is not an integer. The “inverse” of this formula is y = log2x, (meaning logarithms to base 2) so the answer to the problem is that numbers on the scale have a “logarithmic” spacing. In practice, logarithms to any base will work, and the convention is to consider a slide rule to be based on logarithms to base 10.
If we had started off with 1.01 as the multiplier, instead of 2, then we would have a scale with many more graduations, and it would be possible to estimate the positions of the 1.1, 1.2, 1.3, … marks quite accurately. This is the sort of thing that John Napier (1550–1617) did when he invented logarithms – but using a multiplier like 1.0000001 to get even higher accuracy. [Actually he used the multiplier 0.9999999 and worked backwards! but the principle is the same.] The multiplying slide-rule was invented by William Oughtred in 1622.
So, if we know about logarithms, then the construction of a slide-rule for multiplication follows a simple recipe: the mark corresponding to the number x (e.g. 1.1, 1.2, 1.3, etc) should be placed at a distance proportional to log(x) from the left-hand end of the scale. Even if we didn’t know anything about logarithms, we could still in principle construct a multiplying slide rule using the method just outlined.
Fig. 4. Set-up for multiplying 1.7 by numbers up to about 5.9
Fig. 4. shows what an actual slide-rule looks like. It has more than just the two scales needed for multiplication, but for the moment we show just two of the scales, which run from 1 to 10. These are found on the lower edge of the moving slide (the C scale) and the edge of the fixed section adjoining C (the D scale). In the illustration, the slide is situated at the position necessary for multiplying by 1.7, i.e. the 1 on the C scale is placed over the 1.7 on the D scale. Looking at the 2 on the C scale, we find it over 3.4 on the D scale, showing that 1.7 × 2 = 3.4. Looking at 2.5 on the C scale, we find it over 4.25 on the D scale, showing that 1.7 × 2.5 = 4.25, and so on. By finding a number on the D scale and reading off the value on the C scale, we can divide by 1.7, for example, 2/1.7 can be read off as 1.18, or 1.176 if you are good at this sort of thing (and not so limited by the screen resolution).
There is an obvious problem in that the scales only run from 1 to 10. What if you want to multiply 17 by 250? You use the reduced values 1.7 and 2.5 again, but the answer is obviously not 4.25. Here you have to learn the art of estimating the “order of magnitude”. 17 × 250 will be of a similar magnitude to 20 × 200 = 4000, so the answer, instead of 4.25, must be 4250. In other words, you have to do a quick, rough, calculation in order to estimate the order of magnitude of the result – i.e. whether it is in the hundreds, or the thousands, or the millions, … – and the slide-rule provides the actual digits. People who have used slide-rules a lot should therefore have a better “feeling” for the sizes of numbers. Slide-rules also teach you not to quote too many figures in the result of a calculation: if the numbers entered into the calculation are of 3-figure accuracy (quite common in experimental measurements), then it is nonsense to quote the result to 6-figure accuracy. Conversely, if you are doing an experiment that is really capable of giving better than three-figure accuracy, you would not choose a slide-rule for analysing the results.
Another problem is when the result goes “off the scale”. In Fig. 4, you can see that 1.7 × 5 = 8.5, but 1.7 × 6 is off the scale. To overcome this, all you need to do is to place the 10 mark on the C scale over the 1.7 mark on the D scale – a factor of 10 is of no importance, since you have to estimate the order of magnitude anyway. See Fig. 5. Reading under the 6 on the C scale, you find the result 1.02 on the D scale. This can’t be correct, because 1.7 × 6 must lie between 6 and 12, so you know that the answer must be 10.2. Similarly we can see that 1.7 × 7 = 11.9. Another way of avoiding results going off the scale is to use a circular design, in which the markers for 1 and 10 are in the same place. Circular slide rules are much less common than linear ones, but have been incorporated into watch dials (often sold as a motorist’s watch or a pilot’s watch, which can calculate average speed, fuel consumption, etc.)
Fig. 5. Set-up for multiplying 1.7 by numbers greater than 5.9
In addition to the C and D scales, a “serious” slide-rule can also include:
A and B scales. These are similar to C and D, but go from 1 to 100 instead of 1 to 10. A value x on the D scale is aligned with x2 on the A scale (see Fig. 6). A cursor (a sliding transparent window with a fine vertical line on it) is necessary to carry out this process. It is not advisable to square a number this way, as all the subsequent work has to be done on the A and B scales, with a slight loss of accuracy (in other words, one should simply multiply x by x). The converse process is much more useful: setting a value x on the A scale means that Öx is used on the D scale. Similarly, the K scale gives the cubes of numbers on the D scale, but is more useful for finding cube roots.
S and T scales. These give trigonometric functions. The angle, x, is found on the S scale, and the aligned value on the C scale is sin(x). Similarly, the T scale is used for tangents.
Inverse scales. These are given names like CI or DI for example. The DI scale is the same as the D scale, but switched over left-to-right. Multiplying by a value on the DI scale is equivalent to dividing by that value on the D scale. The reason for including these is that a calculation involving alternate multiplications and divisions, like a / b × c / d × e / f… can be carried out very quickly, while a string of multiplications, like a × b × c × d × e × f… takes more movements of the slide and the cursor. By replacing half of the multiplications by divisions (using the DI scale), the calculation can be speeded up. When a scale has values that increase from right to left, its numerical values are often printed in red, to reduce the chance of misreading the scale.
Folded scales. Sometimes, the C and D scales are duplicated, but shifted by a factor p. This is because p comes near the halfway point on the scale (the exact halfway point is Ö10, but p occurs in calculations much more often than Ö10). Using these scales, CF and DF, a factor of p in a formula can be included without having to make an extra move of the slide – simply switch from the D scale to the displaced DF scale. These scales can also be used to avoid a result going off the scale. If you know the answer is going to be near 1 (or 10), but you don’t know which side of 1 it will be, then doing the calculation on the displaced scales is guaranteed to give a result that does not go off the scale.
Fig. 6. The front and back views of a good quality slide rule (left hand end)
Slide rules of a high quality, or very early versions, are now collectors’ pieces. They can, however, still be useful and never suffer from jammed mechanisms or damage from leaking batteries! An all-plastic version can be used outdoors in a tropical rainstorm. Try using a pocket calculator in that situation.