<<Has [scientific notation] been used in this way at all before the
invention of logarithms?
In astronomy brightness of stars is measured in 'magnitudes' that
are logarithmic, not sure when that started?>>
1. Probably not, because decimal fractions were introduced not all that
long before Napier introduced logarithms.
In developing logs, Napier was thinking of the relationship between
arithmetic and geometric progressions, and scientific notation is simply
one way of writing decimal numbers as a member of a geometric series:
mantissa x 10^n
Unfortunately my most useful source on the history of logarithms has
disappeared into the library's bindery, which appears to be the closest
Terrestrial equivalent to a black hole.
2. I don't have references handy, but the story of magnitudes goes
something like this:
Some Greek astronomer made a catalog of stars, in which he divided all
stars into 6 (I think) categories based on their apparent brightness.
When astronomers eventually developed techniques for actually measuring
brightness, they found that this catalog was quite accurate. They also
found that there was a geometric (or logarithmic) relationship between
the categories, as each one was approximately 2 and a half times as
bright as the next dimmer one. (The reason for this is that the eye
responds logarithmically rather than linearly to light).
The stars in the first category became "first magnitude", those in the
second category became "second magnitude", etc. For convenience the
ratio of brightness between integer magnitudes was set at the fifth root
of 100, which is 2.512, and magnitude of 1 was assigned a particular value,
which means that on the logarithmic scale you could have extremely bright
stars with negative magnitudes (Sirius, the brightest night star, has a
magnitude of somewhere between -1 and -2).
Since as telescopes got bigger, fainter and fainter stars could be seen,
it would eventually have been necessary to set up a logarithmic scale
(or perhaps give brightnesses in scientific notation!) However, the
logarithmic response of the human eye caused art to imitate life.
James A Landau