CONNECTED WITH HUMAN MOETALITY. 
523 
would not only require great labour, but leave very small confidence of an accurate result. 
And it is owing to this circumstance that, in addition to having a Table of the expres- 
sions of the Napierian logarithm, for every age, of the chance of persons of any age 
living X years, I propose also, for every age, to have the Napierian logarithm of the 
quotient of the expression which gives the chance of a person of any age being dead 
divided by its first term, namely, x multiplied by the coefficient it has in the value of 
the chance, as in many cases such a Table will introduce great facility. 
There are much more intricate cases for calculation, which the law of mortality 
enables us to overcome ; I allude to annuities and assurances depending on conditions 
of survivorships among the party who are involved in the annuity. Now I observe, 
from having the Napierian logarithm of the chance of each individual surviving x years, 
or having the Napierian logarithm of the expression after it is divided by its first term, 
of the chance of his being dead, and adding the coefficients of all the first terms 
together, all the second terms together, &c., and finding the anti-Napierian loga- 
rithm of the result, and multiplying this by all the first terms, which were directed to 
divide the various chances, we have the natural value of the chances compounded out 
of them. 
Art. 10. Previously to proceeding, I venture to introduce a notation which I have 
found convenient with respect to \ital algorithm, as the theory, and the application of it 
to important objects, introduces very large numbers, and also extremely small fractions, 
of which, in both cases, there is only a necessity to attend to a very few of the significant 
figures; thus, suppose we had the number 897654321, and that it answered for suffi- 
cient accuracy only to consider the number as 898000000, I would write it 898 by 
the @ 1 mean the six noughts which are not written down ; and if we had the deci- 
mal fraction -00000000763, in which eight noughts occur before the significant figure, I 
would write it @763; and if these two quantities had to be multiplied together, I 
should write it 898 @x (8)763 ; and (@763 consists of eleven places of decimals, 
and 898x763 = 685174; and this, if four significant figures were sufficient, I would 
write 6852 (@); to which add (@, we have 6852 (@); and adding to the left (l^, as 
(@763 signifies *00000000763, the product will stand (l^6852(@, and would signify 
that three figm-es to the right are to be cut off as a decimal, and that the product is 
6-852, because eleven places of decimals, including the first significant figure being- 
multiplied by 10", leaves 11 — 8 three places of decimals; and if we had to add 68-52 
_ 68-52 1 
to (@)23, it would be _|_.Q 22 j = 68-543 ; and so of other cases. The great use of this 
notation will appear in the construction of the Collecting Table, and its application to 
the analytical anti-Napierian logarithms. The algorithm of vital statistics introducing 
the necessity of intricate entanglements of common and Napierian logarithms and anti- 
Napierian logarithms, and reverses of those operations in analytical expression, I think 
it expedient to enter more particularly on the nature of logarithms than has been 
