How Slide Rules Work
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INTRODUCTION
The survival of our species owes much to our brain, specifically, its ability to observe, analyse, and plan. Planting crops and storing grains for the winter were some of the earliest uses of these abilities. Measuring and calculating are foundational elements of observation, analysis, and planning. Computation, upon which our modern society depends, is but an extension of those ancient measurement and calculation techniques.
Calculations operate on operands obtained through measurements. Counting was the oldest form of measurement. In prehistory, humans counted by scratching marks on bones. Next to evolve was a ruler etched with markings. Thereafter, humans were marking, measuring, calculating, tracking, and predicting the movements of the Sun and the Moon using stone pillars, astronomically aligned burial mounds, and sun dials.
By around 3000 BC, Sumerians invented the sexagesimal (base-$60$) number system, and they were using the abacus by 2700 BC. The abacus was one of the earliest devices that mechanised calculations, and it is still in extensive use, throughout the world. A cuneiform clay tablet from 1800 BC shows that Babylonians already knew how to survey land boundaries with the aid of Pythagorean triples. Egyptians improved upon these techniques to survey property boundaries on the Nile flood planes and to erect the pyramids. By 220 BC, Persian astronomers were using the astrolabe to calculate the latitude, to measure the height of objects, and to triangulate positions. Greeks constructed truly advanced mechanical instruments that predicted solar and lunar eclipses. The sophistication and refinement exhibited by the Antikythera mechanism from around 200 BC continues to amaze modern engineers.
Ancient astronomy measured, tracked, and predicted the movements of heavenly objects. But when celestial navigation came to be used extensively in global trade across the oceans, we began charting the night sky in earnest, and thus was born modern astronomy. Astronomical calculations involved manually manipulating numbers. Those calculations were tedious and error prone.
In 1614, a brilliant Scottish mathematician John Napier discovered logarithms. Perhaps it would be more appropriate to say Napier invented logarithms, for his discovery was motivated by his desire to simplify multiplication and division. Arithmetically, multiplication can be expressed as repeated additions, and division as repeated subtractions. Logarithmically, multiplication of two numbers can be reduced to addition of their logarithms, and division to subtraction thereof. Hence, multiplication and division of very large numbers can be reduced to straightforward addition and subtraction, with the aid of prepared logarithm and inverse logarithm tables.
In 1620, Edmund Gunter, an English astronomer, used Napier’s logarithms to fashion a calculating device that came to be known as Gunter’s scale. The markings on this device were not linear like a simple ruler, but logarithmic. To multiply two numbers, the length representing the multiplicand is first marked out on the logarithmic scale using a divider and, from thence, the length representing the multiplier is similarly marked out, thereby obtaining the product, which is the sum of the two logarithmic lengths. Gunter’s scale mechanised the tedious task of looking up numbers on logarithm tables. This device was the forerunner of the slide rule.
The first practical slide rule was invented by William Oughtred, an English mathematician, in 1622. Oughtred used two bits of wood graduated with Gunter’s scale to perform multiplication and addition. Then, in 1630, Oughtred fashioned a brass circular slide rule with two integrated pointers. This device was a significant improvement over Gunter’s scale, in terms of practicality and usability. The photograph below shows a brass circular slide rule that is a contemporaneous clone of Oughtred’s.
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