Can anyone suggest

Hi Surajit,

As soon as someone mentions the word "infinitesimal", the one book that first and foremost comes to the mind may not necessarily be the most appropriate source to begin on this topic; the book is: "Principia..." by Newton himself.

However, from what I (very) vaguely remember, there is supposed to be one edition of this book (or a book with some extensive excerpts of it or so) interspersed with some expert comments coming from "Chandra" S. Chandrasekhar (the 1983 physics Nobel laureate).

Certain books by Prof. Morris Kline do contain passages regarding the development of this concept. Also another book of title: "Mathematics: Its Contents, Methods and Meaning" or so, an English translation of an original Russian book (MIT Press, early 1960s). If I remember it right, none less than Prof. Kolmogorov himself was one of the main contributors. Please see my message here in recent times at iMechanica (I will insert the link later on) for the applicable "ideological" caveats re. "dilectical materialism" and all...

Most any book on history of mathematics would have passages on the infinitesimals, but I what I sometimes obnoxious about them is that they so easily slip into settling scores (usually *against* Newton) as to the priority of the concept. (The abovementioned Russian one happens to carry too classic a touch to do that, however---from what I now remember almost 1.5 decades later.)

As far as I know and recollect, Archimedes knew about the so called "method of exhausion" whereby, to find the area of the circle (the same task as estimating the value of PI), you first discretize the circle into an n-sided polygon and then sum up the areas of its constituent triangles. By increasing n, the process can be made to converge, and a both-sided convergence is possible. Archimedes knew about this all, and wrote about it too. But whether he originated it or not is not known for sure (though it is universally accepted that he had enough of genius to have originated it independently---a circumstance that has happened often enough in mathematics and even in science.) So, going through his writings/translations might be fun. 

There also is a recorded precedence by a certain Kerala-based mathematicians (I unfortunately forgot his name) who also had made some nice observations about the convergence of certain infinite series (written in Sanskrit, in the *verse* form!).

But notice that while the pre-Renaissance thinkers knew about convergence, I doubt if they had reached that abstract enough a level whereby you can actually isolate the idea of the infinitesimal from such behavior as is displayed by certain infinite series. As such, the credit for the invention of the infinitesimal properly seems to go to Newton and Leibniz, both of whom seem to have reached the same idea of the infinitesimal independently. In particular, however, I have the impression (and I could be wrong) that among the two, Newton was more emphatic to assert about the infinitesimal as a separate mathematical object in its own right. The reason is a bit roundabout: It was this emphasis of his which made it more difficult to understand what the idea was exactly about, and he couldn't explain the matter to his cotemporaries very well. It took at least one more century before people formulated it in the now familiar epsilon-delta terms.

Hope this helps. (And thanks for bringing up a topic of my liking!)