The book that founded differential algebra: in Differential Equations from the Algebraic Standpoint, Joseph Fels Ritt turned polynomial algebra loose on whole systems at once. Joseph Fels Ritt (1893-1951) took his doctorate at Columbia University in 1917, joined its faculty in 1921 and stayed for the rest of his career, becoming Davies Professor of Mathematics in 1945 and being elected to the National Academy of Sciences in 1933. His book appeared in 1932 as volume fourteen of the American Mathematical Society's Colloquium Publications. It treats ordinary and partial algebraic differential equations as objects to be manipulated rather than solved, and builds an elimination theory that reduces the existence problem for a finite or infinite system to an application of the implicit function theorem. Across 10 chapters Ritt decomposes systems into irreducible ones, develops general solutions and resolvents, sets out constructive methods, and proves analogues of the Hilbert-Netto theorem and of Lüroth's theorem for form quotients before closing with Riquier's existence theorem and the partial case. The subject he opened here was carried on by his student Ellis Kolchin, and the ideas first assembled in these pages now underlie much of modern computer algebra. This edition features:The complete 1932 text, issued as volume fourteen of the American Mathematical Society's Colloquium Publications 10 chapters treating both ordinary and partial algebraic differential systems in a single unified framework Ritt's elimination theory, his resolvents and his constructive methods set out step by step in his own words The founding statement of a field that has since become central to computer algebra and elimination methods