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.
Classical elements of a quaternion
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.
Scalar
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.
Vector
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.
Unit vector
A unit vector is a vector of length one. Examples of unit vectors include i, j and k.
Tensor
Note: The use of the word tensor by Hamilton does not coincide with modern terminology. Hamilton's tensor is actually the absolute value on the quaternion algebra, which makes it a normed vector space. Hamilton defined tensor as a positive numerical quantity, or, more properly, signless number. A tensor can be thought of as a positive scalar. The "tensor" can be thought of as representing a "stretching factor."
Hamilton introduced the term tensor in his first book, Lectures on Quaternions, based on lectures he gave shortly after his invention of the quaternions: it seems convenient to enlarge by definition the signification of the new word tensor, so as to render it capable of including also those other cases in which we operate on a line by diminishing instead of increasing its length ; and generally by altering that length in any definite ratio. We shall thus (as was hinted at the end of the article in question) have fractional and even incommensurable tensors, which will simply be numerical multipliers, and will all be positive or (to speak more properly) SignLess Numbers, that is, unclothed with the algebraic signs of positive and negative ; because, in the operation here considered, we abstract from the directions (as well as from the situations) of the lines which are compared or operated on.
Each quaternion has a tensor, which is a measure of its magnitude (in the same way as the length of a vector is a measure of a vectors' magnitude). When a quaternion is defined as the quotient of two vectors, its tensor is the ratio of the lengths of these vectors.
Versor
A versor is a quaternion with a tensor of 1. Alternatively, a versor can be defined as the quotient of two equal-length vectors.
In general a versor defines all of the following: a directional axis; the plane normal to that axis; and an angle of rotation.
When a versor and a vector which lies in the plane of the versor are multiplied, the result is a new vector of the same length but turned by the angle of the versor.