The slide rule - a forgotten tool
The slide rule was developed and improved over a period of more than 300 years, and science and engineering design and construction would have been very difficult without its help. Nevertheless few, if any, technological objects have seen the sudden fall of the slide rule, following its meteoric rise. It disappeared virtually overnight in the early 1970s, due to the sudden availability of cheap electronic pocket calculators.
In ancient times the lack of development of numerical notations, as well as the poor understanding of mathematics, mitigated against the mechanisation of calculation. Calculations were generally limited to counting, using pebbles, sticks, etc. In ancient Rome merchants used counting boards and in China they used the abacus. At some stage, however, astronomy, navigation, artillery, and land surveying started to require more sophisticated calculation methods.
The forerunner of the slide rule was the sector, the invention of which is generally attributed to Galileo Galilei (ca. 1597). It consists of a hinged pair of graduated rulers, and in its early version it used the theory of similar triangles to calculate natural numbers, squares, cubes, reciprocals, chords, tangents, etc.
It took many steps by many people to develop the slide rule. The crucial foundations were laid by four Englishmen early in the 17th century:
The seminal invention for slide rule technology was that of the natural logarithm by Lord John Napier (1550-1617), a Scottish landowner with an interest inter alia in mathematics. He published a book in 1614, in which he described the theory of logarithms and showed a table of logarithms, which made it possible to reduce multiplication and division to addition and subtraction. The 'Napier's Bones' calculator came in the shape of a number of rods marked with numbers, which also allowed multiplication to be done by the addition of numbers (1617).
Ca. 1616 Henry Briggs (1561-1630), Professor of Geometry at Gresham College near London and a friend of Napier, explained the advantages of the base 10 logarithm, relative to the natural logarithm.
Edmund Gunter (1581-1626), Professor of Astronomy at Gresham, converted base 10 logarithms into distances along a straight edge, introducing the so-called 'Gunter Scale'. In 1618 he introduced a quadrant device with logarithmic scales for adding or subtracting distances with the aid of a hinged pair of rulers. Improved Gunter Scales were used by sea captains and ships' pilots to determine the hours of the day and the azimuth till the end of the 19th century, when the modern technical slide rule had established its ascendancy. Gunter also introduced a mechanical interpretation of a logarithmic scale in circular form. Initially the linear scales were held against each other by hand, while on the circular rules dividers were pivoted at the centre of the discs.
The Reverend William Oughtred, mathematics teacher, is generally recognised as the inventor of the slide rule. In 1631 he published 'The Circles of Proportion and the Horizontal Instrument', describing his circular slide rule. The original design had two discs with concentric logarithmic scales which rotated against each other. This design was implemented by Elias Allen, an instrument maker in the Strand in London, who reduced Oughtred's two discs to a single disc. Oughtred did not publish his invention, but a student of his, Richard Delamain, later claimed to have invented the circular rule, starting an argument which continued till Delamain's death.
Among others, Edmund Wingate has been credited with the introduction of a slide to a fixed body in 1626 to produce his so-called 'Rule of Proportion' by drawing log scales on two rules which were capable of being placed in juxtaposition.
In 1654 Robert Bissaker introduced a slide between fixed stocks, thus a producing a form which was very similar to the modern slide rule.
The use of logarithms for the slide rule is fundamental. By using log scales, i.e., representing distances on the scales by the logarithms of the figures marked on the scales, it becomes possible to carry out multiplication by adding figures on the scale, and division by subtracting figures:
Log (A x B) = log A + log B, and
Log (A / B) = log A - log B.
Types of slide rules
By the middle of the 17th century the slide rule had established itself, following which it continued to be gradually developed. Its advantages of speed and sufficient precision (2-3 places) were quickly recognised. The common forms are the rectilinear, the circular, and the tubular ones. Despite being the shape of Oughtred's original design, the circular slide rule has never been very popular. Its main advantage is that it can provide good accuracy, relative to its size and that it can therefore be made to fit into a pocket (for example, a waistcoat pocket Fowler rule had a diameter of ca. 7 cm). There may, however, be parallax problems in reading the circular scale, although circular slide rules could be surprisingly accurate.
The costly tubular rule achieved a very long scale by making it helical (for example, the Fuller rule, costing 5 Pounds Sterling in 1938, was capable of 5 digit precision, using a 30 cm long, 7.5 cm diameter body, providing an effective scale of 12.6 m).
In order to overcome the lack of addition and subtraction capabilities of the slide rule, an 'Addmaster' machine was produced in the 1980s in Germany, combining a small mechanical adding machine with a slide rule (at that late stage, however, electronic calculators had already won the battle with the slide rule).
The earliest standard for rectilinear slide rules was developed by a 19 years old French artillery officer, Amedee Mannheim (ca. 1855). It was the first standard rule to be supplied with a cursor. It has the following scales:
- Scale A: a two-cycle logarithmic scale placed on the bottom edge of the gap in the stock.
- Scale B: an identical scale on the top edge of the slide.
- Scale C: on the bottom edge of the slide there is a scale running from 1 to 10; this is the single-cycle logarithmic scale, which is used for most calculations.
- Scale D: an identical scale below Scale C on the top edge of the bottom of the gap.
Over the years the functionality was improved, for example the importance of the introduction of the log-log scales should be mentioned. It was first suggested by Peter Mark Roget (the author of the Thesaurus) to the Royal Society in London in 1815, and the increasing calculation requirements in the areas of thermodynamics, physics and electrical engineering led to its re-invention with improvements in 1881.
Until ca 1930, many slide rules had only the four basic scales. Since then this arrangement has been found to be limited in scope and over the years it has been supplemented with a multitude of additional scales, including square roots, squares, cubes, sines, tangents, reciprocals, and log-log.
Specific technology areas require their own specific mathematics and hence specific function slide rules, and many slide rules were implemented for general and specialist uses; in particular increased accuracy of the marking of the gradations led to great improvement in the accuracy of calculations.
Special slide rules have been manufactured for electrical, radio, nuclear, mechanical, chemical and mechanical engineers, navigators, surveyors, gunnery applications, commerce and financial applications and many others. During the Apollo missions Moon-bound astronauts carried slide rules as backup to electronic calculators.
General use of the slide rule
The complexities of learning to use the slide rule discouraged its use by the general population, but in spite of certain common limitations, it became the undisputed symbol of engineering. Typically the divisions on the slide rule marked a scale which could be read to a precision of two significant figures, while the user estimated the third figure. As it did not provide the place of the decimal point in the calculations, the user had to estimate the order of magnitude of the result. This promoted a feel for the calculation, for example that the result of the product of 0.128 and 0.9365 is ca 0.12, not 0.012, 1.2, or 12. (While direct multiplication shows an answer of 0.1198720, because one of the inputs is known to only 3 decimal points, the answer must not be reported with more than 3 decimal points; it would be prudent to report the answer as 0.120).
The availability of 3 significant places is sufficient for many calculations, as it corresponds well with many input data to engineering formulas, say for calculating how much concrete to use for a certain application. It would, however, not be satisfactory for navigation of a satellite trajectory.
In summary, the slide rule was instrumental in teaching the user two important things about engineering or science:
- magnitudes come from a feel for the problem and do not come automatically from calculating machines;
- results are approximations and should only be reported as accurately as the inputs are known.
The electronic calculator
For many years both mechanical and electronic calculators existed in parallel with the slide rule. Texas Instruments started to manufacture cheap and reliable electronic pocket calculators in 1972, and while these were still expensive in 1973, price breakthroughs came as early as 1974, ringing the death knell for the slide rule. By 1975, 50 million electronic calculators were made each year; soon just about every engineer and scientist had one, and slide rules were to be found only in museums.
The supremacy of the calculator did not last long. Within about 10 years the next development had arrived, and the powerful computer is now an indispensable partner in any design process. Calculations can now be done very quickly and with very high accuracy, but it is precisely because of its immense power that the computer can also be a source of over-confidence of the user. Many engineers and scientists have now been educated using electronic calculation, i.e. without ever using slide rules, and we are now seeing some bad effects of this. For example, it may be tempting for an engineer to take on design work outside his area of expertise simply because a software package is available. There may also be a tendency to design structures in which every part is of 'minimum adequate' strength and weight, as this undoubtedly produces the most economic structure. Such extreme optimisation, however, leaves little or no room for error, whether in terms of manufactured parts, execution of the design, or the computer calculations. In this regard there have been cases where structural failures have been attributed to such computer-optimisation.
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