How has Euclid influenced math?
(1) Euclid's "Elements" was used as the primary math textbook from Euclid's time (circa 300 b.c.e) to the 19th century. The 13 books comprised geometry, algebra, and number theory.
(2) Euclid's "Elements" is the earliest book that uses an axiomatic system. Euclid realized that you need some undefined terms, as well as some notions that should be accepted without proof (i.e. axioms or postulates.) Then he proved various assertions using only those undefined terms, postulates, and theorems already proven.
This axiomatic system has been copied in a number of fields. The United States' Declaration of Independence starts with "We hold these truths to be self-evident...", thus the concepts of people's inalienable rights to life, freedom, and pursuit of happiness are taken as postulates in the political system so devised.