may be said to be logically complete. It is incomplete if the cycle of its operations be not a closed one. An algebra, for example, that admits evolution and the logarithmic process, but precludes the imaginary and the complex quantity is logically only the fraction of an algebra. The inability of the earlier algebraists to recognize this fact made it also impossible for them to carry out the algebraic processes of evolution and the taking of logarithms to any except real and positive numbers.

With this fundamental characteristic of the algebra of complex quantities the reader will find it interesting to compare the defining principles of Peirce's linear associative algebras, outlined in his memoir on that subject,* and Cayley's observations on multiple algebra and on the definitions of algebraic operations, contained in his British Association address at Southport in 1883, and subsequently amplified in a paper published in the Quarterly Journal of Mathematics for 1887.**

87. Scale of Equal Parts. Every magnitude, real, imaginary or complex, as represented by a straight line, can be measured by means of an arbitrary scale of equal parts, called units, and can be expressed in terms of the assumed unit by a rational number, real, imaginary or complex, with an error that is less than any assignable number. The method may be exemplified as follows:

* Benjamin Peirce: Linear Associative Algebra (Washington, 1870), or American Journal of Mathematics, Vol. IV (1881), p. 97.

**Cayley: Presidential Address in Report of the British Association for the Advancement of Science, for the year 1883, and on Multiple Algebra in the Quarterly Journal of Mathematics (1887), Vol. XXII, p. 270.

88. For Real Magnitudes. Suppose the scale of equal parts to be constructed, of which the arbitrary unit shall be

and suppose the outer extremity, P, of the given magnitude OP to fall between the mth and (m + 1)th points of division

of the scale. Then m is the integral number of full unitlengths contained in OP, and OM, or mXj, differs from OP by a magnitude that is at least less than MN, that is, less than I Xj, so that

OP=jXm+j × (<1),

where (< 1) means 'something less than 1.'

Divide the segment MN into r equal parts and suppose P to fall between the hth and (h+1)th points of division. Then h is the integral number of th parts of the unit j by which OP exceeds OM and the excess of OP over OM+j× (h/r) is less than the 7th part of j; that is

OP=j×(m + 1 ) + j× (< ;).
外

Divide the rth part of MN upon which P falls into equal parts, and suppose P to fall between the kth and (k+1)th points of division. Then, by the same reasoning as in the preceding case,

If, in the continuation of this process, P eventually falls upon one of the points of division last inserted, OP is exactly measured by a rational number of the form

But it may be that the process will never end. The successive rational numbers will then be more and more nearly the measures of OP, but never exactly. If the process be continued to the (q + 1)th term, it is obvious that the error committed in assuming the rational number thus obtained as the measure of OP will be less than 1/r; that is

h

k

OP = j × (m + + + + ... + 1 ) + ƒ× (< ;-) ·

Х

and by increasing q sufficiently this error 1/r may be made less than any number that can be assigned.

If the scale be decimal, r is 10 and this numerical measure of OP assumes the form

OP=jX (m. hkl...p)+jx (<0.000... I),

a decimal number, m being the integral part of the rational term, p and I occupying the 9th decimal place.

If the magnitude be negative, the number which is its approximate measure will be affected with the negative sign, but will differ in no other respect from the number that measures a positive magnitude of equal extent.

The common measure of the two magnitudes OP and j, here sought, may be described as their greatest common measure of the form 1/2. When the process continues ad infinitum, they have no exact common measure.

89. For Complex Magnitudes. If the magnitude be complex it is of the form

OA+iAP=x+ iy

in which x and y are real. By the process just described

x

Fig. 38.

y

-X

these two real constituents may be separately measured by rational numbers in terms of an arbitrary unit, with errors that, in each case, are less than an arbitrarily small number. Thus let m and n

be the rational numbers that measure x and y, with errors respectively less than and 7; then

x + iy

€

=j× (m+ in)+j×{(<©)+i (<n) }

n

=jX (m + in) +jxVe+r {( < vet) + (<vet)}

=j× (m + in)+j×{<Vê+n2 • cis & },

=

2

where cos = €/1/e2+n2, sin n! √ e2 + n2, and consequently tan =ŋ/e, or $=tan−1 ( ŋ /€).

Since € and ʼn are separately less than an arbitrarily small real number, so is Ve+ncis less than an arbitrarily small complex number. Hence the rational complex number m + in is the measure of xiy within any required range of approximation.

go. Definitions.* The general cyclic functions, of which the circular and hyperbolic ratios of Chapter II are special forms, are defined by the following identities, in which w and w' stand for complex quantities and

These are read: sine of w, cosine of w, etc., with respect to modulus K. They may be appropriately called modulocyclic, or modo-cyclic functions.

When ki, (a) and (b) assume the special forms

* Cf. American Journal of Mathematics, Vol. XIV (1892), p. 192, where tentative definitions of these functions, slightly different from the above, were given.