Classical Hamiltonian quaternions

William Rowan Hamilton invented quaternions, a mathematical entity in 1843. This article describes Hamilton's original treatment of quaternions, using his notation and terms. Hamilton's treatment is more geometric than the modern approach, which emphasizes quaternions' algebraic properties. Mathematically, quaternions discussed differ from the modern definition only by the terminology which is used....let space be now regarded as the field of progression which is to be studied, and POINTS as states of that progression. ...I am led to regard the word 'Minus,' or the mark −, in geometry, as the sign or characteristic of analysis of one geometric position (in space), as compared with another (such) position. The comparison of one mathematical point with another with a view to the determination of what may be called their ordinal relation, or their relative position in space... William Rowan Hamilton invented quaternions, a mathematical entity in 1843. This article describes Hamilton's original treatment of quaternions, using his notation and terms. Hamilton's treatment is more geometric than the modern approach, which emphasizes quaternions' algebraic properties. Mathematically, quaternions discussed differ from the modern definition only by the terminology which is used. Hamilton defined a quaternion as the quotient of two directed lines in tridimensional space; or, more generally, as the quotient of two vectors. A quaternion can be represented as the sum of a scalar and a vector. It can also be represented as the product of its tensor and its versor. Hamilton invented the term scalars for the real numbers, because they span the 'scale of progression from positive to negative infinity' or because they represent the 'comparison of positions upon one common scale'. Hamilton regarded ordinary scalar algebra as the science of pure time. Hamilton defined a vector as 'a right line ... having not only length but also direction'. Hamilton derived the word vector from the Latin vehere, to carry. Hamilton conceived a vector as the 'difference of its two extreme points.' For Hamilton, a vector was always a three-dimensional entity, having three co-ordinates relative to any given co-ordinate system, including but not limited to both polar and rectangular systems. He therefore referred to vectors as 'triplets'. Hamilton defined addition of vectors in geometric terms, by placing the origin of the second vector at the end of the first. He went on to define vector subtraction. By adding a vector to itself multiple times, he defined multiplication of a vector by an integer, then extended this to division by an integer, and multiplication (and division) of a vector by a rational number. Finally, by taking limits, he defined the result of multiplying a vector α by any scalar x as a vector β with the same direction as α if x is positive; the opposite direction to α if x is negative; and a length that is |x| times the length of α. The quotient of two parallel or anti-parallel vectors is therefore a scalar with absolute value equal to the ratio of the lengths of the two vectors; the scalar is positive if the vectors are parallel and negative if they are anti-parallel.