But even when I figure out all the substitutions, deciphering the entire notebook will be a long project.
This branch of algebra he notably enriched, and to him is also due the notion of a group of substitutions (see Equation: Theory of Equations; also Theory of groups).
The walls of the nave are adorned with mosaics of the 6th century; the scenes from the New Testament above the windows date from the time of Theodoric, while the somewhat stiff processions below, of virgins on one side and of saints on the other, are substitutions of the latter half of the 6th century for representations which probably contained some allusion to Arianism or episodes in the life of Theodoric (so Ricci).
Dumas gave especial attention to such substitutions, named metalepsy (µeraXntks, exchange); and framed the following empirical laws to explain the reactions: - (1) a body containing hydrogen when substituted by a halogen loses one atom of hydrogen for every atom of halogen introduced; (2) the same holds if oxygen be present, except that when the oxygen is present as water the latter first loses its hydrogen without replacement, and then substitution according to (1) ensues.
Under the general heading "Algebra and Theory of Numbers" occur the subheadings "Elements of Algebra," with the topics rational polynomials, permutations, &c., partitions, probabilities; "Linear Substitutions," with the topics determinants, &c., linear substitutions, general theory of quantics; "Theory of Algebraic Equations," with the topics existence of roots, separation of and approximation to, theory of Galois, &c. "Theory of Numbers," with the topics congruences, quadratic residues, prime numbers, particular irrational and transcendental numbers.
How would you define substitutions? Add your definition here.