837] 
305 
837. 
ON THE SO-CALLED D’ALEMBERT CARNOT GEOMETRICAL 
PARADOX. 
[From the Messenger of Mathematics, vol. xiv. (1885), pp. 113, 114.] 
The present note has reference to Prof. Sylvester’s paper on this subject \l.c., 
pp. 92—96]. I cannot admit that D’Alembert and Carnot raised a well-founded 
objection “to the then and even now too prevalent interpretation of the meaning of 
the geometrical positive and negative ” : it appears to me that the objection was not a 
well-founded one. 
Consider through the origin K an indefinite line t'Kt, and measure off from K 
in the sense Kt a distance equal to the positive quantity a, and let m be the extremity 
of the distance thus measured off. There is not in the ordinary theory any reason 
why the distance Km should be = + a rather than = — a; it is = + a, if Kt be the 
positive sense of the line through K, and it is = — a if Kt' be the positive sense 
of the line through K; if it be undetermined which of the two is the positive sense, 
then the distance Km is = ± a, the sign being essentially indeterminate. 
The problem is from a point K outside a given circle to draw a line Kmm such 
that the intercepted portion mm' within the circle has a given value c. 
Supposing that the line from K to the centre meets the circle in the points 
A, B at the distances KA = a, KB — b ; then if Km = r, we have ab = r (c + r), or 
r = - \c ± V(i c2 + ab); viz. we have for r, not simultaneously but alternatively, the 
positive value — |c + VCic 2 + ab), and the negative value -\c — V(iC 2 -I- ab), the latter 
of these being the greatest in absolute magnitude; say the values are -f p x and — p 2 . 
We may with either of these values construct the point m; viz. we obtain m as one 
of the intersections of the given circle with the circle centre K and radius p u or 
else with the circle centre K and radius — p 2 (that is, radius p. 2 ); and attending to 
the intersections on the same side of the line from K to the centre, it happens that 
c. xii. 39