CONGRUENCE (Lat. congruentia, from con gruare, to agree). In geometry, plane figures which (*In be superposed so as to coincide throughout_ are said to be congruent. This is the Euclidean definition of equality, and indi eates both quality of area and similarity form. The symbolfor congruence signifies these two properties. ln general it is not neces sary actually to superpose the figures. If the equality of certain parts is known. the equality of the other parts can be established—e.g. if two sides and the included angle of one triangle are equal to the corresponding parts of another, the triangles are congruent. since the remaining parts are also equal and similarly placed. Con gruence is related to axial and central sym inetry (q.v.), and constitutes an important the ory of geometry. Congruency, in modern geom etry, signifies a system of elements satisfying a twofold condition. Of all possible lines, those particular lines which satisfy a given condition are together called a complex, and those which satisfy two conditions are called a congruency e.g. all lines which intersect a given circle form a complex. and all which intersect two given circles form a congruency. The order of a con gruency is the number of its rays co-planar with a given plane; the class of a congruency is the number of its lines concurrent in a given point.
In the theory of numbers, two integers are said to be congruent with respect to a third. called the modulus, when their difference is exactly divisible by the modulus. Thus. 12 and 7, 27 and 12, are congruent with respect to 5 as a modulus, since (12-7) and (27-12) are divisible by 5. This relation is expressed
thus: 12=7 (mod 5), 27 ._=1 12 (mod 5), and, in general, cu —b (mod c). When two integers are congruent with respect to a third, either is called the residual of the other with respect to this modulus. A few fundamental theorems of congruences are: (1) if a',, a',, . . .
(to the same modulus), then a, + a, + .=-= -4- . . . a o(2) If a = a', then 7111E vd. (3) If a E- a', b', then ab (4) H a =a', then a' E a". (5) If a, E a„ a' . . , then G a, , . . 0 (a'„ a'„ . . . ). G designating any rational integral func tion of Tr,, a, .. .
In algebra, the congruence of functions is con sidered in addition to the congruence of num bers. When the elements considered are of the form ax b the congruence is called linear.
When the elements are of the form c, the congruence is called quadratic, and so on. To solve a congruence is to find the values of the unknown quantity which satisfy the con gruence. Thus. to solve the quadratic congru ence a 39 (mod 49) is to find the number whose square gives a remainder 39 when divided by 49. These numbers are 23. 26.
As to geometry. consult: Henrici. Geometry of Congruent Figures (London, 1888) ; Rennin and Smith, :Veto Plane and Solid Geometry (Bos ton, 1900) ; Plfieker, Nene Geometric des Ucarirs gegriindct auf die Betrachtung der Linic als Raumelement, edited by Clehsch (Leipzig. 1868) : and as to algebra. Salmon. Modern High er Algebra (Dublin, 1S76). and Pund. Algebra ?nit Einschlags der clementaren Zahlentheorie (Leipzig. 1899).