Sliderules: The What and How.
A slide rule recently rescued from the tip as reported at a recent meeting begs the questions – what are slide rules, and how do they work?
Mankind on the Moon: Slide rules were used for space program calculations, indeed some were actually taken on the missions – in case calculations needed to be made (all of the astronauts were slide rule literate), in the photo, the astronaut is wearing a slide rule watch. Lord Merchiston (John Napier 1550  1617) was concerned with the time it took to
perform calculations – they had to be done using pen and paper, long division
and multiplication – a very time consuming business. Extracting square and cube
roots were a real trial. It was the 1600s, and calculations were essential for
the emerging fields of engineering and navigation, as well as the increasing
interest in science and astronomy. Merchiston, in a brilliant stroke, invented
logarithms. With these, any numbers
could be multiplied by merely adding the logarithms of the numbers, and then
looking up the antilog of the result to get the answer. (so, the log of 100 is
2, and the log of 1000 is 3, so to calculate 100 times 1000 , add 2 + 3 = 5,
the antilog of 5 is 100000, which is the answer). Division is achieved by
subtracting the logarithms. This was an astonishing advance in calculation,
indeed, and extracting square and cube roots is achieved by diving the
logarithm by 2 and 3 respectively. (Merchiston also invented the decimal
point). Tables of logarithms were published and advanced calculation
immensely. Of course, the results were
limited to 4 figure accuracy, using 4 place logs.
Within a few decades, William Oughtred, an Anglican vicar, arranged two logarithmic scales face to face, which could be used directly for calculation, without having to use logarithmic tables – Oughtred had invented the slide rule. (Usual rules have linear scales – equal distance between the numbers, however in logarithmic scales the numbers are squashed closer at the large end, as in the photo). By the mid 1800s, a young French artillery officer had developed the layout of the slide rule as we know it now – it was found to be so useful for artillery calculations it was adopted by the Military, then spread to the world of science and engineering.
The slide rule meant that complex calculations could be done immediately, and they became icons in movies and photos. It was usual to have photos of people such as Franck Whittle (inventor of the British jet engine) and Edward Teller (inventor of the Hydrogen bomb) with slide rules to show they were at work. The use of logarithmic tables for multiplication and division was taught in schools, and those students studying physics and chemistry also learned to use slide rules. All university engineering courses included a section on the use of slide rules. Slide rules for specific purposes (bomb aiming, navigation, space travel, pressure pipe calculations etc) were developed.
The beginning of the end for slide rule use occurred in the mid 60s, when an engineer developed the first electronic calculator – indeed, he had performed all the calculations necessary for the invention using a slide rule! It was the introduction of the HP calculator in 1972 that tolled the end of slide rule use, and early electronic calculators were called “electronic slide rules”. Slide rule production ceased in 1976, however there are still large numbers of slide rules extant, and batches of 'new old stock' are unearthed periodically. One Japanese manufacturer still has for sale (on their web site) their circular slide rule as sold in the 1970s.
Enthusiasts still collect and use slide rules, there are still annual slide rule competitions. The benefits of using a slide rule are real – use combines physical and mental skills. For example, from the photo, it can be seen that the decimal point is not shown. It is the same process to multiply each of the following: 218 x 325 or 0.218 x 0.0325. For each, the slide rule will yield the same answer, that is, 710 as in the photo. In order to get the right decimal point in the answer, it is necessary for the person to mentally estimate the answer and insert the decimal appropriately., thus requiring a level of mental arithmetic.
Of course, a slide rule does not rely on batteries! In the mid 1980s I recall that a person teaching navigation for solo yachtsmen insisted on each student learning to use both calculator and slide rule use in the course, arguing that noone knew when they may be midocean with dead batteries....... I cant imagine that such courses insist on slide rule use now, however I was recently interested to hear a Master Mariner, whose job it is to test and certify magnetic compasses on commercial ships in Australia waters (it is an annual requirement) say that the compass is required to be accurate, being the fall back navigational aide should the GPS system for some reason become unavailable. However, he also said he expects that it wont be many more years before the requirement will fall away, if for no other reason than sailors themselves will no longer be skilled enough to adequately use it.
The point about accuracy also indicates the fundamental limitation of the slide rule – the usual 10 inch rule will yield 3 figure accuracy. There were special slide rules (40 inch long) that would yield 4 figures, and cylindrical slide rules (the scales being spirally etched on the cylinder) that had scales equivalent to 80 feet would yield 5 figure accuracy. The first electronic calculator mentioned above itself had 12 figures, undreamed of on an analogue system. The use of 3 figure accuracy in engineering meant that engineers needed to be conservative in their designs – panels were made of thicker materials than actually necessary, etc, to avoid failure. Since the introduction of calculation to 12 figure accuracy, devices are more precisely designed – lighter and more reliable.
The slide rule was indeed a great step forward for mankind when it arrived, it is very pleasing to use (if you like puzzles, crosswords etc – I periodically take my slide rule, retained from my school days (!), dust it off and work through some textbook problems – just to keep my hand in). (Mind you, I still also use the RPN logic calculator I had as a uni student – although it is a virtual one, the original device alas ceased to function a decade ago).
There is a large amount of material on slide rules, how to use them, example problems to solve etc on the internet – just search for the Oughtred Society, whose web site has a wealth of material, including virtual slide rules that can be downloaded onto your laptop or tablet – work just like the real thing! Slide rules are plentifully available for purchase on ebay type platforms, and there are slide rule societies in existence, too.
My regret: There was a teaching slide rule (about 6 foot long, large scale, to be visible from the back of the class) at one of the educational institutions I worked at a decade or two ago – of course it hadn't been used since the 1970s, and languished in a store room. The lab assistant, noting my interest, asked if I wanted to have it. I declined (it wasn't, after all, mine). Alas, some years later, it was taken to the tip............ The Pickett 160 slide rule.Slide rules have ABCD scales, where C and D are the usual scales used, giving the interval 1 to 10 (and often called “X”). A and B are indeed merely C and D squared – that is, they range from 1 to 100 (ie, X^{2}). There is often a K scale, which is the cube (so ranges from 1 to 1000, that is, X^{3}). The inverse of the C scale is usually given in the centre scale , called CI, (ie, 1/X). Also, usually given at the bottom is a linear scale (called L), which yields the logarithm of the number (so, in the photo, the cursor shows that the logarithm of 710 is 850).
To multiply, say 218 by 325, as in the photo, set the 1 of scale C to the first number (218) on scale D, then using the cursor, find the second number (325) on C (so that is actually adding the second number to the first one), The answer (710) appears on the D scale, under the cursor. Note that you have to estimate the position of the “8” in 218, to be somewhere between the marks for 215 and 220. Likewise, close inspection will show that the cursor is just a shade below 710. The actual answer is of course 7085, which would round up to 709 to three figures. Given the thickness of the scales' lines (and the Pickett 160 was in fact a 6” slide rule, not the usual 10” model), the answer of 710 is close. To get the decimal point, one has to mentally do an approximate calculation, so 218 is close to 200, the estimate might be, 100 x 325 is 32 thousand, so twice that is close to 70 thousand, so the decimal in the answer is 71 000, not 710 as given by the three figures. Older engineers, brought up using slide rules and so having to mentally estimate the size of the answer to get the decimal point right would say that this mental exercise gives the person a feel for the size of the answer – ie, an idea of the actual physical entity being described by the number – an invaluable insight that only use of a such an analogue device can give. The Tachymeter scale on an analogue watch, solves 1/T (where T is time in seconds) to give the speed. The scale is on a rotatable bezel. Place the zero on the scale on the second hand when passing a milestone, then when passing the next milestone, the bezel will give the speed (in Km per hour, if using Km pegs on the road). This isn't a slide rule, really, as it doesn't involve logarithms, it is really a nomogram solving the expression 1/T. For example, if it took 30s between Km pegs, then the bezel would yield 120 km/hr. Although these are still readily available now, they are of limited use as most roads do not have Km pegs any more. Credits: Images come from the Oughtred website, the Tachymeter image from the web, and the slide rule example from the virtual Picket 160 slide rule.

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