XXVI APPENDIX THE FIFTH* 



observations is either the result which the strictest ap- 

 plication of sound principles would declare to be the 

 most probable truth, or else very near to it. 



It is, in the meanwhile, a most remarkable circumstance 

 that a method so simple, and so conformable to common 

 sense, as that of averaging, should first turn out to be 

 incorrect, except upon a supposition never contemplated 

 in thinking of the evidence of this rule, and should after- 

 wards prove to be always nearly correct, for large num- 

 bers of observations, on account of the tendency of all 

 admissible suppositions to confound themselves, as the 

 number of observations increases, with that one parti- 

 cular supposition, which makes the common notion 

 absolutely correct. My own impression, derived from 

 this and many other circumstances connected with the 

 analysis of probabilities, is, that mathematical results 

 have outrun their interpretation : and that some simple 

 explanation of the force and meaning of the celebrated 

 integral, whose values are tabulated at the end of this 

 work, will one day be found to connect the higher and 

 lower parts of the subject with a degree of simplicity 

 which will at once render useless (except to the his- 

 torian) all the works hitherto written. 



APPENDIX THE FIFTH. 



ON THE METHOD OF CALCULATING UNIFORMLY DE- 

 CREASING OR- INCREASING ANNUITIES. 



AN authority from which I rarely differ has spoken 

 thus, " A few writers on these subjects, of late years, 

 have employed the differential and integral calculus in 

 their investigations. We have not yet seen any fruits 



