What was Sir Francis Bacon's idea of 'induction' as a scientific method?

I would only add; 1) Bacon’s useful “eliminative
induction” and 2) his difference from the Aristotelian
inductionists.

Bacon’s eliminative induction; For instance,
in order to find the root cause of something (like rain), you would inductively examine
all the elements of many different situations, times and atmospheric conditions when it
is raining. Then you would also look for those same elements and conditions when it is
not raining. Thereby, you could eliminate some elements which do not contribute directly
to the cause of rain. This has been very useful in the development of the scientific
method.

 Bacon's version of induction differed from the
Aristotelian induction.

Think of a ladder to knowledge; you
(inductively) pick up particulars on your way up to the top where you form a general
theory about what you've been investigating and gathering data. Aristotelians say you
could then move down the ladder (deductively) to about the middle and make general
axioms there as well.

Where Bacon differed from the
Aristotelians is that he advocated endlessly moving up the ladder, collecting more data
and eliminating false propositions; as if he wanted to focus on the process itself; for
the sake of the process, rather than for the sake of reaching a general deduction with
which one could then move back down the ladder to apply to more particular things
(general deduction: all new cars are shiny - all new BMW's are shiny). Bacon would say
to take your time and keep moving up the ladder slowly because what if we find out that
not all new cars are shiny; then all the middle of the ladder axioms could be false. His
thinking is logical but not practical. Most scientists have accepted the idea that their
general deductions may someday prove to be wrong, but have accepted that if they are
currently applicable, to use them. For example, Newton’s theory of gravity predicted the
movement of the planets to a great degree and scientists were able to use his theory for
other particular (middle of the ladder) instances of gravity. Then Einstein came along
and improved upon the theory. While we had to rethink all those instances in which we
applied Newton’s gravity theory, it was still more beneficial for us that we did rely on
Newton’s effective theory. (It served us well enough for over 200
years.)