QUATER'NIONS, the name given by its inventor, sir W. R. Hamilton (q.v.), to one of the most remarkable of the mathematical methods or calculi, which have so enormously extended the range of analysis, while simplifying its application to the most formidable problems in geometry and physics.
It would be inconsistent with our plan to give even a complete though elementary analytical view of this calculus; hut it is possible, by means of elementary and algebra alone, to give the reader a notion of its nature and value.
For this purpose, it will be necessary to consider some very simple, but importaet,. ideas with reference to the relative position, of points in space. Suppose A and B to be any two stations, one, for instance, at the top of a mountain, the other at the bottom of a coal-pit. Upon how many distinct numbers does their relative position. depend? This can, be easily answered, thus: B is so many degrees of longitude to the e. or w. of A, so many degrees of latitude to the n. or s. of A, and so many ft. above or below the level
of A. THREE numbers suffice, according to this mode of viewing the question, to determine the position of B when that of A is given. Looking at it from another point of view, suppose A to be the earth, B a fixed star. To point a telescope at B, we require to know its altitude and azimuth, its latitude and longitude, or its right ascension and declination. Any of these pairs of numbers will give us the direction of the line AB, but to determine absolutely the position of B, we require a third number—viz., the length of AB. Hence, it appears that any given line AB, of definite length and direc tion, is completely determined by three numbers. Also, if the line ab be parallel and equal to AB, it evidently depends on the same three numbers. Hence, if we take the expression (AB) to denote (not, as in geometry, the length of AB merely, but) the length and direction of AB; we see that there will be no error introduced, it we use it in the _ following sense: