Lectures on Quaternions

Abstract

A list of interesting quotations: On real and imaginary ''In the present Volume as has been already observed I have thought it expedient to present the quaternions under a geometrical aspect as one which it may be perhaps more easy and interesting to contemplate and more immediately adapted to the subsequent applications of geometrical and physical kinds And in the passage which I have made in the Seventh Lecture from quaternions considered as real or as geometrically interpreted to biquaternions considered as imaginary or as geometrically uninterpreted but as symbolically suggested by the gene ralization of quaternion formula it will be perceived by those who shall do me the honour to read this work with attention that I have employed a method of transition from theorems proved for the particular to expressions assumed for the general which bears a very close analogy to the methods of Ohm and Peacock although I have since thought of a way of geometrically interpreting the biquaternions also.'' (Preface, 16) On vectors ''In its synthetic aspect then I regard the symbol B -- A as denoting the STEP to B from A namely that step by making which from the given point A we should reach or arrive at the sought point B and so determine generate mark or CONSTRUCT that point This step which we shall always suppose to be a straight line may also in my opinion be properly called a VECTOR or more fully it may be called the vector of the point B from the point A because it may be considered as having for its office function work task or business to transport or CARRY in Latin vehere a moveable point from the given or initial position A to the sought or final position B.'' (15). ''To illustrate more fully the distinction which was just now briefly mentioned between the meanings of the Vector and the Radius Vector of a point we may remark that the RADIUS VECTOR in astronomy and indeed in geometry also is usually understood to have only length and therefore to be adequately expressed by a SINGLE NUMBER denoting the magnitude or length of the straight line which is referred to by this usual name radius vector as compared with the magnitude of some standard line which has been assumed as the unit of length.'' (16). ''But in the new mode of speaking which it is here proposed to introduce and which is guarded from confusion with the older mode by the omission of the word RADIUS the VECTOR of the sun HAS itself DIRECTION as well as length. It is therefore NOT sufficiently characterized by ANY SINGLE NUMBER such as 95 (were this even otherwise rigorous); but REQUIRES for its COMPLETE NUMERICAL EXPRESSION fl SYSTEM OF THREE NUMBERS such as the usual and well known rectangular or polar co ordinates of the Sun or other body or point whose place is to be examined AMONG which ONE MAY be what is called the radius vector but if so THAT RADIUS MUST (in general) be associated with TWO OTHER polar co ordinates or determining numbers of some kind before the VECTOR can be numerically expressed. A VECTOR is thus (as you will afterwards more clearly see) a sort of NATURAL TRIPLET suggested by Geometry and accordingly we shall find that QUATERNIONS offer an easy mode of symbolically representing every vector by a TRINOMIAL FORM ix jy hz which form brings the conception and expression of such a vector into the closest possible connexion with Cartesian and rectangular co ordinates.'' (17)