to pay, at the present low rate of interest, near 19 years' purchase for an annuity for life, whereas, if the common rate of interest were still at 5 per cent., he ought to pay full 15 years' purchase; and, as there were always more sellers than buyers, the common price was generally under this rate. OBJECTIONS TO DOCTRINES, ON THE PROBABILITY OF DURATION OF LIFE, ADVANCED BY SEVERAL WRITERS. The following ingenious objections were published some few years ago in a popular periodical, by Mr. Hawes : The probability that a life of any given age will continue in being to the end of any given term, being a fraction whose denominator is the number of persons living at the given age in any table of observations, and whose numerator is the number of persons living at an age older by the given term, than the given age, and in the case of joint lives, it being the product of the probabilities that each of the single lives shall continue in being to the end of the given term, is a doctrine that was suggested by Dr. Halley, adopted by M. De Moivre, adhered to by Mr. Simpson, confided in by Mr. Dodson, espoused by Dr. Price, embraced by Mr. Morgan, and assented to by a late writer, Mr. Bailey. The purport of these observations is to represent the fallacy of such a doctrine. The definition of a fraction is taken to be as follows:-The numerator denotes the number or quantity, and the denominator the distinguishing name of what is numbered. The subject of the present investigation being that of time, that is, its component and fractional parts, it follows that the measure of the probability of the duration of human life must be expressed by a fraction, whose denominator is a period of time composed of a specific number of years, and whose numerator is a portion of such period composed of a less number of years and a fractional part of a year. In the first example of Mr. Bailey's first practical question (chap. 12) he asserts, "The probability that a person, whose age is 20, shall attain to the age of 50, or live 30 years, is, according to the observations of M. De Parcieux, as given in Table 3, equal to And the probability that a person, whose age is 40, shall attain to the age of 70, or live 30 years, is, ac 310 657 581 814 cording to the same observations, equal to But the probability that both those persons shall live to the 581 310 end of 30 years, is equal to multiplied by that 180110 " is, equal to 534798 814' 657 By consulting nature, in preference to imagination, or to any received doctrine, it is found that the proba bility of a person, whose age is 20, attaining the age of 50, or live 30 years, is, according to the observations of M. De Parcieux, as given in Mr. Bailey's third table, equal to years, instead of per fraction 25.6689 581 21.3882 30.0000 years, as ; and the probability that a person, whose age is 40, shall attain to the age of 70, or live 30 years, is, according to the same observations, equal to 23.4056 30.0000 years, instead of 14.1552 30.0000 years, as per fraction 657 310 Thus every step in true knowledge, affording a glimpse of what lies next beyond it in the scale of nature, the same unerring law evinces the probability that both those persons shall live to the end of 30 years, is It appears that the most essential point of consideration attached to this subject has been wholly overlooked by every author whose name is here mentioned, namely, to keep within the verge of probability. Had this been attended to, that anomalous mode of procedure of multiplying causes without necessity, as evidenced by Dr. Halley's sixth and seventh uses of his Breslau Table, would never have been introduced into the science; nor the fallacy of supposing that a year (instead of being composed of certain portions of time) was made up of a continually fluctuating number of human beings, as taught by the same author in the second use of the same Table, and relied on, as well as amplified, by every celebrated author on the subject since. In his second example (page 356), Mr. Bailey affirms, "The probability that a man, aged 46, shall attain to the age of 56, or live 10 years, is, according to the observations made in Sweden, as given in Table 14th, equal to And the probability that a woman, aged 40, shall attain the age of 50, or live 10 3096 3991 years, is, according to the same observations, equal to 4027 But the probability, that both those persons 4733 3096 4027 shall live 10 years, is equal to multiplied by 4773' 12467592,, that is equal to 18889403 3991 Now the probability that a man, aged 46, shall attain to the age of 56, or live 10 years, (as in the aforesaid example), will be found equal to 7.7574 10.0000 8.9219 10.0000 instead of And the proba bility that a woman, aged 40, shall attain to the 9.2425 50, or live 10 years, will be found equal to 10.0000 In the third example he states "The probability that each of three lives, aged 20, 30, and 40, shall live 15 years, is, according to the observations made at Northampton, as given in Table 25th, equal to 5123 2448 and respectively. But the probability that 3248 4385' 3635 4010 all those lives shall continue so long, is equal to the product of the three fractions multiplied into each other whence such probability will be denoted by 31883927040 81801385700. Now the probability that each of three lives, aged 20, 30, and 40, shall live 15 years (according to the Northampton observations), will be found equal to 15,0000' 15.0000" 15.0000 years, respectively, in |