of the Problem of Shortest Twilight. 183 



We thus see that the factor sin 2 § which arose out of the 

 process of finding (5) is really a. foreign factor introduced, as 

 those factors so generally are, by elimination : and we see too 

 how it arose; viz. by Bernoulli and D'Alembert having can- 

 celled the factor cosec - g from equation (4) merely because it 

 was a factor of the whole equation. Had this view occurred to 

 them, they would have depressed their biquadratic to a qua- 

 dratic at once, and have escaped the embarrassment created 

 by attempting to prove that certain roots contained in it did 

 not fulfill the conditions of the problem, and were therefore 

 inadmissible. But an important question here arose — what 

 is the signification of this result, and how did it make its ap- 

 pearance intermingled with the proper solution of the problem ? 

 It is not enough to take those parts of a result which we can 

 readily interpret, and reject all the others as useless or un- 

 meaning: though such is and always has been so greatly the 

 fashion, that scarcely a single author has insisted upon an 

 unflinching determination to consider every solution incomplete 

 which stops short of this consummation. It will, I have no 

 doubt, ere long, become a fundamental principle in the philo- 

 sophy of mathematics — that every part of an algebraical result 

 admits of complete interpretation, either by reference to the con- 

 ditions which were expressed in the fundamental equations, or 

 else in the hypotheses, tacitly made {in order to apply our trans- 

 forming operations,) in the various subsequent stages of the so- 

 lution. It is, however, so much the custom to discard, with- 

 out consideration or remark, all such results as do not admit of 

 a ready and obvious application to the immediate objects be- 

 fore us, that facility of interpretation is the rarest of all the fa- 

 culties of the geometer that we find in any considerable de- 

 gree developed. No factor ought to be rejected for which a 

 satisfactory reason cannot be given, nor ought it to be consi- 

 dered foreign till the step at which it was introduced, is distinctly 

 ascertained: and it bespeaks both a bad taste and unpardon- 

 ably negligent habits of research, to leave any part of the final 

 equation unexplained, even in the most trivial inquiry which 

 may be undertaken *. 



* The valuable discussion and illustrations of this principle, given by 

 Mr. Babbage in vol. ii. of the Cambridge Philosophical Transactions, cannot 

 be too earnestly recommended to every young mathematician who would 

 form a proper taste in the modern analysis, nor too diligently studied by 

 every one who aspires to a higher character than that of a mere algebraical 

 machine. 



With regard, however, to the metaphysical views entertained by that 

 distinguished philosopher, on the relations between the mind and certain 

 objects of mental action, as expressed in that paper, I offer no opinion 

 here, as I shall have occasion to moot that subject in another place. 



