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Most slide rules consist of three linear strips of the same length, aligned in parallel and interlocked so that the central strip can be moved lengthways relative to the other two. The outer two strips are fixed so that their relative positions do not change. Some slide rules have scales on both sides of the rule and slide strip, others on one side of the outer stips and both sides of the slide strip, still others on one side only. A sliding cursor with one or more vertical alignment lines can record an intermediate result on any of the scales.
In general, mathematical calculations are performed by aligning marks on the sliding central strip with marks on either of the fixed strips and then observing the relative positions of other marks on the strips. The marks engraved or printed on the strips are carefully placed to allow the handler to perform a number of important mathematical operations. The geometry of the markings determines which operations may be performed.
The rule has logarithmic scales. That is, a number is printed on each rule at a distance proportional to from the 'index', which is marked with the number 1. A logarithm transforms an operation of multiplication or division to one of addition or subtraction thanks to the rules and .
Since addition and subtraction are easily carried out using a number line, the slide rule effectively implements a number line with a sliding scale. By the use of the logarithmic transforms, the operations of multiplication and division can be carried out.
To multiply by , one aligns the index (the number 1) on the sliding scale with the number on the fixed scale, whereupon the number on the sliding scale becomes aligned with the number on the fixed scale.
The illustration below shows the multiplication of 2 with any other number. The index (1) on the upper scale is aligned with the 2 on the lower scale. The numbers on the upper scale (multipliers) correspond with the multiplication on the lower scale. Example: the 3.5 on the upper scale is aligned with the product 7 on the lower scale, the 4 with the 8 etc.
Where operations go 'off the scale' e.g. the user has to slide the upper scale to the left and use the index 10 or 100 instead of 1, and remember to adjust the result by this factor.
Slide rules calibrated on one side are called "simplex." Slide rules calibrated on both sides are called "duplex."
Typically two significant figures of precision are possible, with three being obtainable by expert users who can estimate the fraction between gradations. Some high-end slide rules have magnifying cursors that effectively double the accuracy, permitting a 10-inch slide rule to serve as well as a 20-inch.
Slide rules often have other mathematical functions encoded on other auxiliary scales. When they were in widespread use, the most popular were trigonometric, usually sine and tangent, logarithm of logarithm (base 10)The common logarithm is the logarithm with base 10. Before the early 1970s, hand-held electronic calculators were not yet in widespread use. Because of their utility in saving work in laborious calculations by hand on paper, tables of base-10 logarithms w (for taking the log of a value on a multiplier scale), natural logarithm and exponentialThe term exponential may refer to any of several topics in mathematics: Exponential distribution Exponential function Exponential growth, exponential decay Exponential time Matrix exponential Exponential map (in differential geometry) All relate in some f scales. Some rules included a PythagoreanPythagoras ( 582 BC 496 BC, Greek: Πυθαγρας) was an Ionian mathematician and philosopher, known best for formulating the Pythagorean theorem. Pythagoras, known as "the father of numbers", made influential cont scale, to figure sides of triangles, and a scale to figure circles. Others featured scales for calculating hyperbolic functions.
Specialised slide rules were invented for various forms of engineering, business and banking. These often had common calculations directly expressed as special scales, for example loan calculations, optimal purchase quantities, or particular engineering equations.
A number of tricks were used to get more convenience. Trigonometric scales were sometimes dual-labelled, in black and red, with complementary angles, the so-called "Darmstadt" style. Duplex slide rules often duplicated basic scales on the back. Scales were often "split" to get higher accuracy.
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