The Slide Rule
Logarithms you can slide: a plain Mannheim rule, and the line it grew from.
A slide rule is a table of logarithms cut into wood. The distance from 1 to any number on its scales is the logarithm of that number, so laying one length after another adds logarithms, and adding logarithms multiplies numbers. Set the slide, move the hairline, and read three figures of the answer without a pencil.
Plate I. The C and D scales, the cursor, and the reading glass. The brackets above are the lengths being added.
Drag the slide or the cursor. Or focus either with Tab and use the arrow keys: each press goes to the next graduation of D, and Shift goes to the next figured one.
Plate II. Nine scales and one hairline. Every scale is read against D.
Drag the slide or the cursor, or use the arrow keys on either.
Plate III. A line of numbers and a pair of dividers, after Gunter; then a second line laid against it, after Oughtred.
Drag either point to open the dividers, or the head to carry them. With the keyboard, each arrow press moves to the next graduation.
¶Commentary
Logarithms. John Napier published his logarithms in 1614, in the Mirifici logarithmorum canonis descriptio, to turn the multiplications of astronomy into additions. Henry Briggs, after visiting Napier, recast them to base 10, so that multiplying by 10 adds exactly 1. He printed the logarithms of the first thousand numbers in 1617 and the great Arithmetica logarithmica in 1624. Base 10 is what lets a single cycle of the scale serve for every decade: 2, 20, and 0.002 share a mantissa and so share a mark.
Gunter's line. Edmund Gunter, professor of astronomy at Gresham College, had the idea around 1620 of drawing the logarithms instead of printing them: a line on a rule on which the distance from 1 to n is log n. With a pair of dividers the line multiplies, divides, and works proportions, as on Plate III. For two centuries Gunter's scale, with its lines of numbers, sines, and tangents, was a navigator's instrument.
Oughtred's slide. William Oughtred saw, around 1622, that two such lines held side by side and slid along each other do the work of the dividers, and do it for every number on the line at once. He described both a straight form and a circular one; his account was printed in 1632, in The Circles of Proportion, translated from his Latin by William Forster. Richard Delamain had published a circular rule in 1630, and the two men quarrelled bitterly over who was first.
Mannheim's rule. In the 1850s Amédée Mannheim, a French artillery officer, settled the arrangement most later rules kept: squares on A and B above, C and D below, all read together through a sliding cursor with a hairline. That is the rule of Plates I and II. Later patterns added cubes, reciprocals, logarithms, and trigonometric scales on the face and back, and the slide rule stayed in every engineer's pocket until the electronic calculator of the 1970s.
Why the scale is self-generating. A rule multiplies because the length from 1 to ab is the length from 1 to a plus the length from 1 to b; for a scale that increases steadily, this forces the length to be a fixed multiple of log. So the scale can be made from itself. The step from 1 to 2 also goes from 2 to 4 and from 4 to 8; the step from 1 to 3 goes from 3 to 9. Mark enough such steps and the rest fall between them. The stepping dividers of Plate III do exactly this.
Accuracy. On a rule 25 cm long, one division is about a millimetre or less, and a careful eye splits it into fifths. That gives about one part in a thousand: three significant figures, four near the left end where the scale is open. It is enough for engineering. The reckon beside each plate shows the exact value of what you set, so you can see the rule's error, which is the reader's error, since the scale itself is exact.
The decimal point. The rule knows only figures. The product 240 × 0.035 is set exactly as 2.4 × 3.5 and reads 840; a rough estimate, 200 × 0.04 = 8, puts the point: 8.40. Slide rule users learned to carry this estimate in their heads, and it is a habit worth having still.
Further. Florian Cajori, A History of the Logarithmic Slide Rule and Allied Instruments (1909), still the standard history of the early rules; Peter M. Hopp, Slide Rules: Their History, Models and Makers (1999), for the instruments themselves.