17th Century Computation

As I wrote earlier, I have been reading Gamma: Exploring Euler’s Constant, by Julian Havil, which begins with a chapter about Napier and the invention of logarithms.

The historical motivation for logarithms is quite interesting. We think of addition, subtraction, multiplication, and division as elementary, while exponents and trigonometry are more advanced. In fact, the multiplication and division involved in the practical mathematics of c.1600, e.g. navigation and compound interest, was very time consuming. People were desperate to find a faster way. In fact, since the trigonometric functions were known, and tables of them were available, one serious suggestion was to use the identity

sin(A)cos(B) = (1/2)sin(A + B) + (1/2)sin(AB)

So to multiply X by Y you would start with

X = sin(A), Y = cos(B) or
A =arcsin(X), B = arccos(Y)


XY = sin(A)cos(B) = (1/2)sin(A + B) + (1/2)sin(AB)

XY = (1/2)sin(arcsin(X) + arccos(Y)) + (1/2)sin(arcsin(X) -arccos(Y))

The idea was that with this ugly mess you could use additions and subtractions, which were much faster than multiplication. See Logarithms:
History and Use
for a worked example.

With logarithms this is simpler:

log (XY) = log X + log Y, or

XY = 10(log10 X + log10 Y)

and you also can easily work with exponents. As Napier wrote

Seeing there is nothing that is so troublesome to mathematical practice, nor that doth more molest and hinder calculators, than the multiplications, divisions, square and cubical extractions of great numbers … I began therefore to consider in my mind by what certain and ready art I might remove those hindrances.

The first table of logarithms was published by Henry Briggs in 1617. Longer and more accurate tables followed quickly, and were soon established as an indispensable tool for computation.

In 1620 William Gunther simplified the use of logarithms by drawing a number line with the
positions of the numbers being proportional to their logs. Calculations could be done with a set of dividers without having to look up the the logarithms involved. Sometime around 1630 William Oughtred improved the process process by taking two of Gunther’s lines and sliding one along the other. And so the slide rule was invented. It just had two scales (C and D in standard terminology), but the underlying concept was established.

I wonder if I could make such a such a device and bring it to colgaffneyis camp as the latest innovation in computing technology :-)>

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