Posted by: Wayne Robinson | August 10, 2010

Innovations of the Seventeenth Century – The Slide Rule

One of the earliest practical applications of logarithms was the development of the slide rule.  

Imagine, if you will, two rectangular strips of wood placed flat on the table with their long edges together. Starting from the left, mark the numbers 0 to 9 in even intervals along the edge where they touch in the manner of a rule. This pair of rules can be used for simple addition. Say you wanted to add 2 to 3. Slide the top rule along to the right so that the 0 lines up with the 2 on the lower rule. Go to the 3 on the upper rule and read the result, 5, at the adjacent point on the lower rule.  

Slide rule in position for 2 × n

Now if instead of addition, you wanted to do multiplication, you could treat the numbers on the scale as logarithms and instead of the numbers write the anti-lograithms, so 0 becomes 1, 1 becomes 10, 2 becomes 100, and so on. You can fill in the numbers in between proportioanlly if you like. Our equation above, rather than 2 + 3 = 5 becomes 100 x 1,000 = 100,000.  

Edmund Gunter invented the logarithmic scale in 1620 and in 1624 published The description and use of sector, the cross-staffe, and other instruments for such as are studious of mathematical practise. Gunter used a pair of dividers to measure the distance along his rule to find the result. In the example above, he would place one point on the 1 and open them so the other point was on 1,000. He then moved the first point along to 100 and read the result from the other point.  

A Gunter Rule
A Gunter Rule in the HP Museum 

In 1622, William Oughtred used two of Gunter’s rules placed side by side in the same manner as we just did in the paragraphs above, to create the first recognisable slide rule. Oughtred developed a circular slide rule in 1630, but thought it a distraction and could see no use for it. Neither of Oughtred’s innovations would have seen the light of day if it wasn’t for his student William Forster taking the credit and publishing them in 1632. Oughtred was also responsible for the × symbol to denote multiplication and the abbreviations sin and cos for the sine and cosine functions respectively.  

A Navigation Slide Rule c. 1800
English boxwood navigational slide rule c. 1800 

The cursor was first proposed by Isaac Newton in 1675 but took a while to catch on. The first modern style of two fixed outer rules with sliding middle section was released by Robert Bissaker in 1654. Specialist slide rules included Henry Coggeshall’s timber and carpenter’s rule of 1677, Thomas Everard’s gauging rule of 1683, used to determine the content of ale, wine and spirits barrels and to calculate the excise tax. With the addition of extra scales for squares, cubes, trigonometric functions and logarithms of logarithms (the log log scale was introduced by Peter Roget (yes, that one) in 1815), slide rules continued in scientific and engineering applications until about 1975 when the pocket calculators such as the Texas Instruments SR range (SR as in “Slide Rule”) became relatively affordable.

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