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.
Functions, operations, transformations, substitutions, correspondences, are but names for various types of relations.
When a homogeneous polynomial is transformed by general linear substitutions as hereafter explained, and is then expressed in the original form with new coefficients affecting the new variables, certain functions of the new coefficients and variables are numerical multiples of the same functions of the original coefficients and variables.
This arose from the study by Felix Klein and Sophus Lie of a new theory of groups of substitutions; it was shown that there exists an invariant theory connected with every group of linear substitutions.
In the theory of surfaces we transform from one set of three rectangular axes to another by the substitutions 'X=' by+ cz, Y = a'x + b'y + c'z, Z =a"x+b"y-l-c"z, where X 2+Y2+Z2 = x2+ y2+z2.
In general in space of n dimensions we have n substitutions similar to X l = a11x1 +a12x2 + ï¿½ ï¿½ ï¿½ + ainxn, and we have to express the n 2 coefficients in terms of Zn(n - I)i independent quantities; which must be possible, because X1+X2+..."IL Xn =xi+x2 +x3 +...+4.
In addition, and transform each pair to a new pair by substitutions, having the same coefficients a ll, a12, a 21, a 22 and arrive at functions of the original coefficients and variables (of one or more quantics) which possess the abovedefinied invariant property.
= (A11+A22)n by the substitutions 51 = A l, E1+ï¿½1 2, 52 = A2E1+ï¿½2E2, the umbrae Al, A2 are expressed in terms of the umbrae al, a 2 by the formulae A l = Alai +A2a2, A2 = ï¿½la1 +ï¿½2a2ï¿½ We gather that A1, A2 are transformed to a l, a 2 in such wise that the determinant of transformation reads by rows as the original determinant reads by columns, and that the modulus of the transformation is, as before, (A / .c).
If we solve the equations connecting the original and transformed unbrae we find (A ï¿½) (- a 2) =A i( - A 2) + ï¿½'1A1, (A ï¿½) a1 = A2(- A2)+ï¿½2A1, and we find that, except for the factor (A /), -a 2 and +ai are trans formed to -A 2 and +A i by the same substitutions as x i and x 2 are transformed to i and E2.
Are said to be contragredient when the linear substitutions for the first set are x =A1X+u1Y-}-v1Z-?--..., y = A2X+,u2Y +v2Z ï¿½..., Z = A 3 X +ï¿½3Y -1v 3 Z - -..., and these are associated with the following formulae appertaining to the second set, .`"?.
All the forms obtained are invariants in regard to linear transformations, in accordance with the same scheme of substitutions, of the several sets of variables.
Restricted Substitutions We may regard the factors of a binary n ip equated to zero as denoting n straight lines through the origin, the co-ordinates being Cartesian and the axes inclined at any angle.
- The cyclical meaning of the succession of zodiacal signs, though now obscured by interpolations and substitutions, was probably once clear and entire.
The motive of some of the substitutions was to avoid the confusion which must have ensued from the duplication of previously existing native asterisms; thus, the Egyptian and Greek Lions were composed of totally different stars.: Abstractions in other cases replaced concrete objects, with the general result of effacing the distinctive character of the Greek zodiac as a " circle of living things."
We have as Leibnitz's remaining legacy to later logicians the conception of Characteristica Universalis and Ars Combinatoria, a universal denoting by symbols and a calculus working by substitutions and the like.