It is often thought that the sole contribution of the medieval Islamic civilization to modern mathematics and science was its preservation of the legacy of the Greeks during the Dark Ages in Europe until  Europeans could again appreciate its value.  Recent research in archives both in Europe and the Islamic world have begun to change this perception.  Not only did Islamic scientists and mathematicians preserve and protect the Greek heritage, but they developed it further in numerous areas and began the development of entirely new areas of mathematics.  Although not all of the Islamic material was transmitted to Europe during the late Medieval period, enough of it was, by Islamic and Jewish scientists as well as some of their Christian counterparts, to influence strongly the developments in mathematics and science in the Renaissance and beyond.
 
 

Although the earliest ideas of combinatorics -- the ideas of permutations and combinations -- come from ancient India, it was not until medieval times that Jewish and Islamic scientists developed these ideas in detail.  In working out the basic formulas, the mathematicians were led to consider various ideas of proof, including the idea of mathematical induction. We will look at the contributions of some of these mathematicians and see how their ideas finally spread to western Europe during the Renaissance.