So how was multiplication done in the 18th Century.  Sure, you could and people did do long-hand multiplication just like you were taught in elementary school.  For example, we can multiply 384×56. The number with more digits is usually selected as the multiplicand:

LongMultiplication1

The long multiplication algorithm starts with multiplying the multiplicand by the least significant digit of the multiplier to produce a partial product, then continuing this process for all higher order digits in the multiplier. Each partial product is right-aligned with the corresponding digit in the multiplier. The partial products are then summed:

LongMultiplication2

Similarly, division can also be done long-hand. 

LongDivision

Of course, this is time consuming and tedious.  If you need to do arithmetic quickly or in a hostile environment, say as a ship’s navigator, an artillery captain, or as a surveyor.  For this you need a calculator.  So, lets quip out our 18th Century calculator, the Gunter’s Rule.

Edmund Gunter was an English clergyman, mathematician, geometer and astronomer best remembered for his mathematical contributions which include the invention of the Gunter’s chain, the Gunter’s quadrant, and the Gunter’s scale. The Gunter’s rule, is a large plane scale engraved with various scales, or lines. On one side are placed the natural lines (as the line of chords, the line of sines, tangents, rhumbs, etc.), and on the other side the corresponding artificial or logarithmic ones. By means of this instrument questions in navigation, trigonometry, etc., are solved with the aid of a pair of compasses.  It is a predecessor of the slide rule, a calculating aid used from the 17th century until the 1970’s.

So, how does it work.  Well to understand the Gunter’s rule, we should review some basic mathematical concepts.  I won’t bother you with the proofs but understand that

Using this simple relationship, we can multiply any two numbers by ADDING their respective logarithms.  So, what Gunter did was place a logarithm scale on a long board so that with the use of calipers, one can add logarithms rapidly.

Doing the problem from above:

384 X 56 = 21504

and

123546/17 = 7262

It was, of course, only a matter of time before someone got the idea of placing two Gunter’s rules side by side and the modern slide rule is born.  In 1622, William Oughtred, an Anglican Minister, placed two Gunter’s rules slide by side so that the scales slide by one another to perform direct multiplication and division.

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