An area of science concerned with the application of mathematical concepts to the physical sciences and the development of mathematical ideas in response to the needs of physics. Historically, the concept of mathematical physics was synonymous with that of theoretical physics. In present-day terminology, however, a distinction is made between the two. Whereas most of theoretical physics uses a large amount of mathematics as a tool and as a language, Mathematical physics places greater emphasis on mathematical rigor, and devotes attention to the development of areas of mathematics that are, or show promise to be, useful to physics. Henri Poincaré likened physics to a large library in which the books are furnished by the experimenters, and the index, without which the collection would be inaccessible and useless, by the mathematical physicists (in the older sense of the word). If that simile is accepted, there remains the remarkable fact that most of the organization and structure of the library, and even the existence of many of the books, is determined by the nature of the index, since both the language and the content of mathematics continue to have a powerful influence on ideas of the structure of nature. The results obtained by pure mathematicians, with no thought to applications, are almost always found to be both useful and effective in formulating physical theories.