14 
the circles become the normals to the curve, and the given 
relation is supposed to exist between the radii and the 
abscissae which correspond to the centres of the circles. 
Taking the relation to be this, viz., that the square of the 
normal is proportional to the abscissa of the centre, he found 
the curve of intersection and solved the problem (Lagrange, 
Legons 1806, pp. 263, 264). But Leibnitz did not remark 
that his solution does not admit of an arbitrary constant, 
while it is evident that the problem leads naturally to a 
differential equation, whereof the integral cannot be com- 
pleted without the introduction of an arbitrary constant 
(ib. 265). Nor did John Bernoulli, discussing the same 
problem in his own way, remark that it belongs essentially 
to the inverse method of tangents, and that, consequently, 
the general solution depends on an integration which ought 
necessarily to introduce an arbitrary constant (ib. 272). 
Neither Leibnitz or John Bernoulli seem to have noticed 
the sort of contradiction which their solutions, arrived at by 
different methods, offered to the very principles of the dif- 
ferential calculus (ib. 274, 275). 
2. Taylor (1716) is perhaps the first who deduced a sin- 
gular solution directly from the derived equation, and who 
recognised the singularity. He obtained it by differentia- 
ting the differential equation (ib. 276). Clairaut (1739) 
was led to a differential equation whereof he obtained two 
different solutions by means of differentiation (ib. 278). He 
noticed the singularity, and states generally that there are 
differential equations which have two different solutions 
whereof one may be obtained without using the integral 
calculus (ib. 279). Lagrange showed that, in general, every 
derived equation is susceptible of a form such that the 
derivee of this form has two factors, whereof one corre- 
sponds to the complete primitive, and the other gives at once 
the singular solution, if there be one (ib. 211, 214, 216, et 
seq., and compare 180.) 
