ARITHMETICAL INSTRUMENTS. 31 



of the Peaucellier movement to the extraction of the roots of 

 numbers. 



With regard to all these arrangements it must be observed that 

 the solutions which they afford are only approximate, and that the 

 degree of the approximation cannot be carried beyond a certain 

 point. This arises, not from any imperfection in the theory of the 

 instruments, but from the circumstance that the solution of the 

 problem is given by measurement ; and that all measurements are 

 necessarily approximate, and subject to errors which cannot be 

 reduced beyond a certain point. 



In this respect the analytical solution of a problem possesses a 

 great theoretical advantage above a solution obtained by geome- 

 trical or mechanical means. The analytical solution is indeed in 

 general approximate, no less than the geometrical or mechanical 

 one ; but the degree of the approximation is no longer limited : 

 for if we are dissatisfied with the degree of approximation we 

 have obtained, we can go back and repeat the process over again, 

 retaining small terms which we before omitted, until we arrive at 

 a result as near to the truth as we please. Of course this theo- 

 retical advantage ceases to have any practical importance when- 

 ever the degree of approximation attainable by the mechanical 

 appliance is sufficient for the purpose in view. 



At an earlier stage of the development of analytical science, gra- 

 phical methods for the solution of analytical problems were of more 

 importance than they are at present. When Descartes showed that 

 the solution of a biquadratic equation could be made to depend on 

 the determination of the intersection of a parabola by a circle, it is 

 possible that, at least in certain cases, the very best method of find- 

 ing the roots of a proposed biquadratic equation, which the resources 

 of mathematics could then supply, was to describe the parabola 

 and the circle, and actually to measure the ordinates of the points 

 common to the two. But the continual progress of improvement 

 in analytical processes, coupled with the greatly increased facility 

 in calculating the arithmetical values of analytical expressions, 



