on the Flexure of Thin Bars. 325 



the same arc 36 inches long, is put down .000240 ; by the gol- 

 den rule 60 : 36 : : 400 : 240. Admitting the truth of this 

 entirely new doctrine, it would be very easy to prove that a 

 crooked line is no longer than a straight one having the same 

 terminations. The question might here present itself to the 

 anthropologist, how it is possible that an intellect capable 

 of computing the reduction of any arc to its chord, could 

 have entertained for one single instant such an idea % But, 

 seriously, every tyro in trigonometry is acquainted with the 

 equation, sin x — x = \ x 3 — T ^ a; 5 + &c, from which it is pretty 

 clear that the reductions are as the cubes of the lengths of 

 the arcs at least, and that, therefore, the reduction .000240 

 ought to have been .000086. But even this method of reduc- 

 tion is inapplicable, for the curve is not circular, and the re- 

 duction on one part of it is not equal to that on another ; nei- 

 ther indeed is the reduction when the curvature is upwards 

 necessarily equal to the reduction when the curvature is down- 

 wards. 



From this specimen, it is quite clear that no dependence can 

 be placed in any results at which Captain Kater may have ar- 

 rived by computation, an error of this kind in a matter so 

 purely elementary leaving room for the expectation of other 

 blunders in simple matters. Perhaps, in the estimation of the 

 value of his micrometric divisions, similar mistakes may have 

 been committed. Nor, in looking more closely at the manage- 

 ment of the experiments, is this feeling of insecurity at all 

 lessened, for it is found that Captain Kater's idea of straight- 

 ness led him to adopt, as the best attainable straight rule, 

 a steel harpsichord wire stretched by a bow ; and that he 

 placed the whole weight of the micrometer apparatus upon 

 the bar whose flexure he was examining. 



It is almost needless for me to add, that, after pondering on 

 these matters, the idea of attempting to draw any safe result 

 from the recorded observations was abandoned. 



