ALGEBRAIC FORMS - HISTORICAL NOTE Prior to the year i800 the study of algebraic forms was con fined to questions of factors, multiples, powers, differentiation and elimination. Resultants, discriminants and in particular determinants were chief objects of interest. Early in the i9th century, however, the invention or formulation of projective geometry and the study of groups in the theory of equations drew attention to the operational character of mathematics as distinguished from its static aspect. It cannot be regarded as purely accidental that algebra, with its abstract groups of trans formations, followed after and eventually outran geometry with its groups of projectivities and quadric inversions. Projective geometry had been systematized and largely created between 1813 and 1840 by Jean Victor Poncelet, Plucker and M. Chasles. In 1841 the germ of a new movement in algebra appears when George Boole noticed that the discriminant of q+XQ, a homo geneous form with one parameter, must be a multiple of the discriminant of the same after transformation q'+XQ'; from this follows that the discriminant is an invariant. In the same year appeared an essay of C. G. J. Jacobi on functional determinants, emphasizing their invariance under linear substitutions. In 1842 Boole announced the invariance of polars. Three years later Cayley subsumed what was already known, and opened a new field of discovery, in the publication of his calculus of hyper determinants. This was in his view simply an extension, to sym bols, of the theorem on multiplication of determinants of like order whose constituents are actual numbers. This was in 1845; but already in 1844 had been published Friedrich G. M. Eisen stein's development of the invariants of a binary cubic and guar tic, and Hesse's first paper on the covariant now universally known as the Hessian. Aronhold (1849) found the invariants of the ternary cubic, and their combination in the discriminant. Sylvester's 4o years' labour in this field opened in 1851, and within four years he had invented most of the technical terminology and tapped most of the rich veins of the mines of the theory of invariants. He had even then shown the invariance of resultants under substitutions of higher order, and had given examples of combinants, and had named the catalecticant.
Charles Hermite's contributions began in 1851 with his dis covery of evectants, followed not long after by his law of rec iprocity for invariants of binary forms, and by explicit formulae for the transformations of a quadric form into itself. Several writers realized the value of the differential equations of inva riance and covariance (at present perhaps the most emphasized point of departure for the entire theory); among these were Francesco Brioschi, Sylvester and Cayley. The reference of invariant forms to sources is found first in the work of M. Roberts (1861), who proved, for binary forms, that the source of a product of covariants is the product of their several sources. Non Euclidean geometry was attached to the algebraic train by Cayley Meantime the new algebra had become accessible to wider circles by Salmon's publication of his Higher Plane Curves (1852) and Modern Higher Algebra (1859). French and Italian trans lations were widely circulated. In Germany it was Siegfried Aronhold who founded, in publications in 1858 and 1863, the symbolic notation which has been proved most effective for the explicit working out of invariants and their relations for particular forms and systems. It is, however, more to Clebsch, from 186 i on, that we owe the completion of this fundamental work. By its use Gordan (1868) proved the finiteness of the total system of covariants for any given binary forms, and so gave a climax to the earlier period of this new branch of algebra. In vention and discoveries have been less notable since that date, except for the climactic work of Hilbert, already described, and the systematizing publications of Study, Deruyts and the school of Sylvester. Important applications have been many, but the apparent tendency is to embody this theory as a typical and well developed part in the comprehensive theory of transformation groups, of Sophus Lie. The most nearly complete history of this theory is Franz Meyer's Bericht.
BIBLIOGRAPHY.-A. Clebsch, Theorie die biniiren algebraischen ForBibliography.-A. Clebsch, Theorie die biniiren algebraischen For- men (Leipzig, 1872) ; A. Clebsch and F. Lindemann, Vorlesung caber Geometrie (Leipzig, 1875) ; F. Meyer, Apolaritat and rationale Curven (Tubingen, 1883) , and "Bericht caber den gegenwartigen Stand der Invarianten-Theorie" in Jahresbericht der deutschen Mathematiker Vereinigung, vol. i. (1892) ; G. Salmon, Lessons Introductory to the Modern Higher Algebra, 14th ed. (1885) ; P. Gordan, V orlesung caber Invariantentheorie, herausgegeben von Kerschensteiner (Leipzig, 1887) ; E. Study, Methoden zur theorie der ternaren Formen (Leip zig, 1889) ; J. Deruyts, Essai d'une theorie generale des formes algebriques (Brussels, 1891) ; E. B. Elliott, An Introduction to the Algebra of Quantics (1895) ; H. Andoyer, Lesons sur la Theorie des Formes et la Geometrie analytique Superieure (1900) ; U. H. Grace and A. Young, The Algebra of Invariants (1903) ; E. Pascal, Reper torium der hoheren Mathematik, vol. i. pt. i., 2nd ed. (Leipzig, 191o) ; L. E. Dickson, Algebraic Invariants (1914) ; O. E. Glenn, Treatise on the Theory of Invariants (Boston, 1915) . (H. S. W.)