Xll PREFACE. 



it must be remembered that an annuity of, say 3 a year, 

 diminishing by l every year, is equivalent, by the 

 first part of the rule, to an annuity of which the suc- 

 cessive payments are as follows : 



3, 2, l, 0, (-l), (2), (3), &c 



That is, the first part of the rule, when the annuity is 

 extinguished during the tabular life of the party, gives 

 the value of his interest upon the supposition that he 

 is to begin to pay as soon as he ceases to receive. If 

 then, this is not to be the case, the value of his interest 

 must be increased accordingly. 



4. The method of the balance of annuities, or the 

 determination of complicated annuities by the addition 

 and substraction of simple ones. This has been done 

 before ; but it has not, to my knowledge, been carried to 

 the extent of making all the questions which commonly 

 occur deducible from the fundamental tables, without 

 the aid of any new series. It is desirable that the 

 beginner should be accustomed to deduction by reasoning, 

 without having recourse to the mechanism of algebra, 

 which, as a quaint editor of Euclid observed, " is the 

 paradise of the mind, where it may enjoy the fruits of 

 all its former labours, without the fatigue of thinking/' 

 Of no part of algebra is this more true, than of the 

 method by which complicated annuities are deduced 

 from simple ones, by the resolution of the series which 

 represent them into the simpler series of which they are 

 composed. The education of an actuary does not neces- 

 sarily imply the study of geometry ; and such processes, 

 for instance, as those by which are found the values 

 of a contingent insurance or a temporary insurance 

 (pp. 222. 226.), will serve, as far as they go, to ac- 



