QUATERNIONS (Lat. quaternio, group of four, from quaterni, four each). A branch of mathematics invented by Sir William Rowan Hamilton (q.v.) in the first half of the nine teenth century. It extehds the idea of complex numbers (see COMPLEX Numnr.k) to three-dimen sional space, and besides being interesting as a branch of pure inathematies. it finds numerous a pplieations in physics. The first concept pecu liar to the quaternion theory is that of rector. A line-segment AB has not only length, but ^' also direction, and two line-segments AB, A'11' are considered equal when they have the same absolute length and the same direction, e.g. in the parallelo gram ABB'A'. AB, A'B' are called rectors-1( Latin. vertarrs, carriers), be cause they are consid ered as `carrying' the points A. to the points B, respective ly. It is therefore evident that a vector may be transported parallel to its original position with out alteration in character. and hence
that it may be con sidered as a symbol of translation. The sum of two vectors, All and BC, is con sidered to be that vector which carries A to C, viz. AC. This does not mean that the absolute value of AB plus that of BC equals that of AC, but- that (direction being also considered) a force AB plus another BC, or AB + AC in the figure, equals the force All. It is there fore evident that, with this definition of addition, the sum of the sides of a tri angle or of any other closed polygon, eon sidered as vectors, is zero. Therefore, if we have three given rectangular vectors, OX, OZ. and any other vector. OP, can be resolved into three vec tors respectively, parallel to (hence equal to parts of) OX. OY, OZ. These are RQ, QP, Olt; for OR