So recently professor Bowley showed us
log tables; how in the good old days he
used to use books like this to do big
calculations.
I'd suggest having a look at the video
before watching this one if you need to
get up to speed, but at the end of the
last one we promised we talked about how
these tables were created.
Now the log tables we were looking at
were in base 10, all numbers were sort
of converted to a power of 10, but that's
not how the first tables were created;
that's what we're going to explain now.
It's some pretty hardcore number
crunching at times but it is amazing to
think all those years ago people were
doing that kind of work to pave the way
for the scientists and mathematicians in
the years to come.
Now you're going to ask the question:
where do these numbers come from? How did
I make up tables, anti logarithms and
logarithms? And this was done by a man
called John Napier in the period between
1594 and 1614. This is before calculus,
before Newton's works, before Leibniz,
before Gauss - all these great
mathematicians, he was the one who did
the first bit which was to work out this
new function, the logarithm. And although
I've told you about logarithms to the
base 10 you could do logarithms to
whatever number, it doesn't have to be 10
to the power; it could be 3 to the power,
or 5 to the power - whatever you
want - or 0.3 to the power. So I'm going to
make up a number and show you the
method used by John Napier so you can
follow the logic; but I'm not going to
ask you to take 20 years which is what
he took in order to create the table, I'm
gonna make a very short abbreviated
table. The number I am gonna use, which
I'll call a, is a very strange number.
It's 1 minus 0.1 to the power of 10.
(Brady: That's our base is it?) - That's going to be the
base. And it makes the calculations as
easy as possible and I'll try to explain
why. So now I want to work out this
number to the power of
0.1. Now 0.1 is
different from 1 and 2 and 3 because
it's got a decimal point. And once you've
understood that you can get decimal
points out of this you can see how you
can build up a longer table. And this is
1 minus 0.1 to the 10 to the power of 0.1
which is 1 minus 0.1; 10 times
0.1 is 1 so this is 0.9.
0.9 is this number, this
exponent to the power of 0.1. If I
were to do this again and put in
0.2 to the power of 10 times
0.2, which is 1 minus 0.1 to the
power of 2. And now I have a choice here,
I'm going to use this result, 0.9
into 1 minus 0.1 - of course 0.9
minus 0.09 equals 0.81. So
I've got the next one; and you see I'm
doing it all the time by subtracting
this from that. - (This is tedious!)
Oh yeah! He took 20 years over this and he used a
much larger number than this. I'll just do
one more: 0.81 minus 0.081
equals 0.729. And it goes on and
on and on. This is working out things
which I can put in a table. So this is
just the same as I was explaining when
doing the log tables, this is just to a
different base. Now Napier used the
number 1 minus 1 over 10 to the 7 to the
power of 10 to the 7. - (That was his base?)
That was a one he used over a long period
to get his accurately as possible. So he
was using this process but with a more
complicated number. - (Those tables you were)
(using as a boy in school were base 10,)
(what's going on here?) - Oh there was a man
called Briggs who worked out that if you
calculated it to the base e, or to this
base, you could convert it to base 10. And
that sold the product much more easily
because people are used to dealing in base
10. And there's a simple transformation,
once you've got one function, the
exponential function rising up, or any
number to some power up rising up, you
can convert to base 10; and that's what
Briggs did and that helped sell the
product. So Napier did all the donkey work;
Briggs came in and got all the glory at
the end. If you go through this algebra
and you get to really long numbers you
have problems; because first of all if I
get eight decimal places and have to
subtract off another eight I'm bound to
make mistakes. And people did make
mistakes. And also if you get up to 10 to
the 7 decimal places you really ought
to be truncating at some point because
you can't do the sums. And so they were
errors introduced because people were
doing subtraction all the time. And so
one of the great problems was to
move all the errors from the tables
because if you're an astronomer and you
were relying on all these things to get
accurate measurements and accurate
calculations what you should see it
might disagree with the experiment - not
because of the experiment and the theory
not working but because the calculations
are wrong based on these tables. And so
somebody called Babbage tried to invent a
difference engine to try and remove all
the errors in all the mathematical
tables and couldn't do it.