This little book is the outgrowth of a series of lectures given to a group of high school teachers and published as a mimeographed booklet by the School Mathematics Study Group. The aim of the booklet was to illustrate many ways in which mathematical methods have helped discovery in science.

The present edition has the same objective. However, we now aim at a group of readers who, we assume, are interested in mathematics beyond the level of high school mathematics. We have added material, and we occasionally use some calculus and more intricate arguments than before. We hope that we will appeal to college students and general readers with some background in mathematics. This has also led to a change in style of exposition and choice of material. If we succeed in giving an impression of the beauty and power of mathematical reasoning in science, the purpose of our work will have been achieved.

We thank Professor R. Richtmyer for his comments on our treatment of relativity, in particular for his illuminating remark that the Lorentz transformation in space gives us more physical insight than that in one dimension. This leads to a simplification in deriving the conservaton laws of mechanics, which was elegantly done and woven into Section 7.10 by Professor P. D. Lax, whom we thank also for many other remarks that helped us clarify the exposition.