4. The only known type of algebra which does not contain arithmetical elements is substantially due to George Boole. Although originally suggested by formal logic, it is most simply interpreted as an algebra of regions in space. Let i denote a definite region of space; andNon-numerical algebra. let a, b, &c., stand for definite parts of i. Let a+b denote the region made up of a and b together (the common part, if any, being reckoned only once), and let a × b or ab mean the region common to a and b. Then a+a = aa = a; hence numerical coefficients and indices are not required. The inverse symbols −, ÷ are ambiguous, and in fact are rarely used. Each symbol a is associated with its supplement ā which satisfies the equivalences a+ā = i, aā = 0, the latter of which means that a and ā have no region in common. Finally, there is a law of absorption expressed by a+ab = a. From every proposition in this algebra a reciprocal one may be deduced by interchanging + and ×, and also the symbols 0 and i. For instance, x+y = x+xy and xy = x(x+y) are reciprocal. The operations + and × obey all the ordinary laws a, c, d (§ 3).
5. A point A in space may be associated with a (real, positive, or negative) numerical quantity a, called its weight, and denoted by the symbol αA. The sum of two weighted points αA, βB is, by definition, the point (α+β)G, where G divides AB so that AG: GB = β:α. Möbius’s barycentric calculus.It can be proved by geometry that
where P is in fact the centroid of masses α, β, γ placed at A, B, C respectively. So, in general, if we put
αA+βB+γC+...+λL = (α+β+γ+...+λ)X.
X is, in general, a determinate point, the barycentre of αA, βB, &c. (or of A, B, &c. for the weights α, β, &c.). If (α+β+...+λ) happens to be zero, X lies at infinity in a determinate direction; unless −αA is the barycentre of βB, γC,...λL, in which case αA+βB+...+λL vanishes identically, and X is indeterminate. If ABCD is a tetrahedron of reference, any point P in space is determined by an equation of the form
(α+β+γ+δ)P = αA + βB + γC + δD:
α, β, γ, δ are, in fact, equivalent to a set of homogeneous coordinates of P. For constructions in a fixed plane three points of reference are sufficient. It is remarkable that Möbius employs the symbols AB, ABC, ABCD in their ordinary geometrical sense as lengths, areas and volumes, except that he distinguishes their sign; thus AB = −BA, ABC = −ACB, and so on. If he had happened to think of them as “products,” he might have anticipated Grassmann's discovery of the extensive calculus. From a merely formal point of view, we have in the barycentric calculus a set of “special symbols of quantity” or “extraordinaries” A, B, C, &c., which combine with each other by means of operations + and − which obey the ordinary rules, and with ordinary algebraic quantities by operations × and ÷, also according to the ordinary rules, except that division by an extraordinary is not used.
6. A quaternion is best defined as a symbol of the type
q = Σαses = α0e0 + α1e1 = α2e2 + α3e3,
where e0, . . . e3 are independentHamilton’s quaternions. extraordinaries and α0, . . . α3 ordinary algebraic quantities, which may be called the co-ordinates of q. The sum and product of two quaternions are defined by the formulae
| Σαses + Σβses = Σ(αs+βs)es |
| Σαrer × Σβses = Σαr+βseres, |
where the products eres, are further reduced according to the following multiplication table, in which, for example, the
| e0 | e1 | e2 | e3 | |
| e0 | e0 | e1 | e2 | e3 |
| e1 | e1 | −e0 | e3 | −e2 |
| e2 | e2 | −e3 | −e0 | e1 |
| e3 | e3 | e2 | −e1 | −e0 |
second line is to be read e1e0 = e1, e12 = −e0, e1e2 = e3, e1e3 = −e2. The effect of these definitions is that the sum and the product of two quaternions are also quaternions, that addition is associative and commutative; and that multiplication is associative and distributive, but not commutative. Thus e1e2 = −e2e1, and if q, q′ are any two quaternions, qq′ is generally different from q′q. The symbol e0 behaves exactly like 1 in ordinary algebra; Hamilton writes 1, i, j, k instead of e0, e1, e2, e3, and in this notation all the special rules of operation may be summed up by the equalities
i 2 = j 2=k2 = ijk = −1.
Putting q = α+βi+γj+δk, Hamilton calls α the scalar part of q, and denotes it by Sq; he also writes Vq for βi+γj+δk, which is called the vector part of q. Thus every quaternion may be written in the form q=Sq+Vq, where either Sq or Vq may separately vanish; so that ordinary algebraic quantities (or scalars, as we shall call them) and pure vectors may each be regarded as special cases of quaternions.
The equations q′+x = q and y+q′ = q are satisfied by the same quaternion, which is denoted by q−q′, On the other hand, the equations q′x = q and yq′ = q have, in general, different solutions. It is the value of y which is generally denoted by q÷q′; a special symbol for x is desirable, but has not been established. If we put q′0 = Sq′−Vq′, then q′0 is called the conjugate of q′, and the scalar q′q′0=q′0q′ is called the norm of q′ and written Nq′. With this notation the values of x and y may be expressed in the forms
x = q′0q/Nq′, y = qq0′/Nq′,
which are free from ambiguity, since scalars are commutative with quaternions. The values of x and y are different, unless V(qq′0)=0.
In the applications of the calculus the co-ordinates of a quaternion are usually assumed to be numerical; when they are complex, the quaternion is further distinguished by Hamilton as a biquaternion. Clifford’s biquaternions are quantities ξq+ηr, where q, r are quaternions, and ξ, η are symbols (commutative with quaternions) obeying the laws ξη=ηξ=0 (cf. Quaternions).
7. In the extensive calculus of the nth category, we have, first of all, n independent “units,” e1, e2, . . . en. From these are derived symbols of the type
A1 = α1e1+α2e2+...+αnen=∑αe,
which we shall call extensive quantities of the first species (and, when necessary, of theGrassmann’s extensive calculus. nth category). The coordinates α1, . . . αn are scalars, and in particular applications may be restricted to real or complex numerical values.
If B1=∑βe, there is a law of addition expressed by
A1 + B1 = ∑(αi + βi)ei = B1 + A1.
this law of addition is associative as well as commutative. The inverse operation is free from ambiguity, and, in fact,
A1 − B1 = ∑(αi − βi)ei.
To multiply A1 by a scalar, we apply the rule
ξA1 = A1ξ = ∑(ξαi )ei.
and similarly for division by a scalar.
All this is analogous to the corresponding formulae in the barycentric calculus and in quaternions, it remains to consider the multiplication of two or more extensive quantities The binary products of the units ei are taken to satisfy the equalities
ei2 = 0,eiej = −ejei ;
this reduces them to 12n(n−1) distinct values, exclusive of zero. These values are assumed to be independent, so we have 12n(n−1) derived units of the second species or order. Associated with these new units there is a system of extensive quantities of the second species, represented by symbols of the type
A2 = ∑αiEi(2)[i = 1, 2, ... 12n(n−1)],
where E1(2), E2(2), &c., are the derived units of the second species. If A1 = ∑αiei, B1 = ∑βeiei, the distributive law of multiplication is preserved by assuming
A1B1 = ∑(αiβj)eiej ;
it follows that A1B1= −B1A1, and that A12=0.
By assuming the truth of the associative law of multiplication, and taking account of the reducing formulae for binary products,
