Scientia Gratiii Scientiae It is now thirteen years since the first book that discusses transfinite graphs and elec trical networks appeared 50]. This was followed by two more books 51] and 54] which compiled results from an ongoing research effort on that subject. Why then is a fourth book, this one, being offered? Simply because still more has been achieved beyond that appearing in those prior books. An exposition of these more recent re sults is the purpose of this book. The idea of transfiniteness for graphs and networks appeared as virgin research territory about seventeen years ago....