BARYCENTRIC CALCULUS, a system of geometric analysis developed by August Ferdinand Mobius (q.v.) and founded on a generalization of the notion of centre of mass (or centre of gravity) of a system of mass particles. This explains the significance of the term "barycentric," from Gr. 0apvs, barys', heavy, and e 'rpoll, ken'tron, centre. Mobius was in possession of his method in 1823, and he published a general development of it in 1827.
If two particles of weights a and 0 are attached to the ends A and B, respectively, of a weightless rod AB, then there is a point P on AB such that the rod may be balanced, under the force of gravity, on a support at P. It will then exert on the support at P a pressure equal to that produced by a weight of a+0. It may be shown that the point P is situated so that its distances from A and B, respectively, are in the ratio of 0 to a; that is, so that AP : PB = (3 :a. This point P is called the centre of gravity (or centre of mass) of the particles at A and B. Similarly, there is a centre of gravity or balancing point P for particles of weights a, (3, y, placed at the respective vertices A, B, C of a triangle ABC; and the notion is extensible to any number of masses situated either in a plane or in space. The point P may be called the barycentre for the given masses with the given positions; and we may attach to it a weight equal to the sum of the weights of the given masses, since the system, when supported at the bary centre, or balancing point P, exerts at P a pressure equal to that due to the sum of the weights of the particles.
In the generalization of the foregoing notions, according to the method of Mobius, there may be associated with a given point A any real (positive, negative, or zero) numerical quantity a, called its weight, the weighted point being denoted by aA. The sum of two weighted points aA and (3B is defined to be the weighted point (a+(3)G, where G is a point on the directed line AB such that AG : GB =(3: a ; we then write aA+0B = (a+(3) G. By means of geometry it is proved that (aA+(3B)+yC= aA+(0B+yC)= (a+0+y)P, where P is the (generalized) centre of mass of particles of weights a, 0, y placed at A, B, C respectively. More generally, we have aA+0B+yC+ +XL= (a+(3+y+ +A) X, where in general X is a determinate point; this point is called the barycentre of the weighted points ceA, (3B, . . . XL. If a+3+y+ +X is zero, the point X lies at infinity in a determinate direction except when aA is the barycentre of 0B, yC, , XL, in which case aA+0B+ +XL van ishes and X itself is indeterminate.
If ABCD is a given tetrahedron of reference and if P is any given point in space, then weights a, 0, y, 6 may be assigned to A, B, C, D, respectively, so that aA+f3B +yC+6D = . It is evident that the quantities a, 0, 6 may be replaced by ka, kn3, k'Y, k6, without affecting the validity of this equation, if k is any real numerical quantity different from zero. Hence the numbers ce 3, y, 6 may serve as a set of homo geneous co-ordinates of P. Thus the barycentric calculus affords a remarkable system of co-ordinates for representing the points of space. In the case of a plane, three fixed points of reference in the plane are sufficient ; and in the case of a line, only two points on the line are needed. Since it furnishes such a set of co-ordinates, the barycentric calculus has been one of the im portant means leading to the development of the powerful methods of modern projective geometry. In another aspect it has also been the forerunner of important work in modern algebra. See A. F. Mobius, Der barycentrische Calcul (Leipzig, 1827), re printed in his Gesammelte Werke, vol. i., pp. 1-388 ; and Encyklopddie der mathematischen Wissenschaften (Leipzig), vol. iii., pp. 1289-1300. The former contains the original development of the barycentric cal culus by Mobius; the latter gives a brief summary and a history, with adequate references to the literature. (R. D. Ca.)