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Slide rules are based on the principle of logarithms which were developed by John Napier (British, 1550 - 1617). (Napier's Bones were an ingenious way of performing multiplication but not directly related to modern slide rules).
The next major step was by Edmund Gunter (British, 1581 - 1626), who developed the concept of a rule with logarithmic values engraved. This rule was used with compasses or dividers.
The true inventor of the slide rule was William Oughtred (British, 1574-1660) who was the first to use sliding scales.
A major step in the development was the invention of the sliding cursor by Victor Mayer Amédée Mannheim (French, 1831 - 1906).