NOTES AND REFERENCES. 
605 
of the word, and if so does the notion of distance in the new sense ultimately 
depend on that of distance in the ordinary sense ? 
As to my memoir, the point of view was that I regarded “coordinates” not 
as distances or ratios of distances, but as an assumed fundamental notion, not 
requiring or admitting of explanation. It recently occurred to me that they might 
be regarded as mere numerical values, attached arbitrarily to the point, in such wise 
that for any given point the ratio x : y has a determinate numerical value, and that 
to any given numerical value of x : y there corresponds a single point. And I 
was led to interpret Klein’s formulae in like manner ; viz. considering A, B, P, Q 
as points arbitrarily connected with determinate numerical values a, b, p, q, then the 
logarithm of the formula would be that of (a — p) (6 — q) 4- (a — q) (b — p). But Prof. 
Klein called my attention to a reference (p. 132 of his second paper) to the theory 
developed in Staudt’s Geometrie der Lage, 1847 (more fully in the Beitràge zur 
Geometrie der Lage, Zweites Heft, 1857). The logarithm of the formula is 
log (A, B, P, Q), and, according to Staudt’s theory (A, B, P, Q), the anharmonic ratio 
of any four points, has independently of any notion of distance the fundamental proper 
ties of a numerical magnitude, viz. any two such ratios have a sum and also a product, 
such sum and product being each of them a like ratio of four points determinable by 
purely descriptive constructions. The proof is easiest for the product: say the ratios 
are (A, B, P, Q) and (.AB', P', Q'): then considering these as given points we 
can construct P, such that (.A', B', P', Q') = (A, B, Q, R): the two ratios are thus 
(A, B, P, Q) and {A, B, Q, R), and we say that their product is (A, B, P, R) 
{observe as to this that introducing the notion of distance, the two factors are 
AP.BQ an( j and thus their product = P, R), which 
AQ.BR 
is the foundation of the definition}. Next for the sum, we construct Q, such that 
(A', B\ P', Q') = (A, B, P, the sum then is (A, B, P, Q) + (A, B, P, Q); and if 
we then construct N such that (A, A), (Q, Q), (B, S) are an involution, we say that 
{A, B, P, Q) + (A, B, P, Q) = (A, B, P, S). {Observe as to this that again introducing 
, . . . . . AP.BQ AP.BQ, AP.BS j 
the notion of distance the last mentioned equation is j^q~p>p + AQ~BP = AN BP ’ 
is BQ q. which expresses that S is determined as above; in fact the equation 
AQ AQ, AN 
^ ~~ ^ + -—P — -— S is readily seen to be equivalent to 
a — q a — q, a — s 
1, b + s , bs 
1, 2 a , a 2 
1, q + qu qqi 
= 0}. 
It must however be admitted that, in applying this theory of Staudt’s to the theory 
of distance, there is at least the appearance of arguing in a circle, since the con 
struction for the product of the two ratios, is in effect the assumption of the relation, 
dist. PQ + disk QR = dist. PR.