I am looking for concrete information on "induction" in the old sense of the word. Now it is used to refer to proof by "mathematical induction"; however, I am aware that it also means inductive reasoning, apparently a type of reasoning you can use to discover theorems.
My main concern is whether there is anything of substance in this. To illustrate what I mean, here is an example.
When I was reading an old geometry book, the author kept using words like "synthesis". At the time I thought he just fancied using this term to mean "combining the previous theorems, we can prove this", or perhaps it was just a word whose real meaning was lost in translation. The point is that I had no idea there was a deeper meaning to this word: The Greek method of synthesis. And after reading up on this in a 17th century textbook, I realized how powerful the method actually is, and I went on to use it to solve geometry yproblems that I literally could not solve before. (Proof, I missed one of them in a math competition I took last year, and still could not solve it on my own a year later, but I magically became able to solve it after reading about this method.)
In that context, my question is whether the old meaning of the word induction also has a deeper meaning to it than say the one Polya gives in his book "How to solve it". I am talking about the one that Gauss refers to when he says "we discovered this theorem by induction", referring to the Quadratic Reciprocity Law. What exactly is this "induction"? Is it a general method for discovering mathematical truths? Is it a long forgotten gem like the synthetic approach I talked about above?
Sorry if anything isn't clear; please ask and I'll try my best to clarify.
EDIT: William Whewell's words on the man who devoted his life to mastering the Greek method of synthesis (that is, Isaac Newton):
I fear there are similar things to say about say Euler and induction, if it weren't for the fact that so many people do not know what induction means, or for the fact that the word induction has now practically been over-written by mathematical induction.