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In the firft bargain, an annuity in reverfion for 12 years, to commence 9 years hence, was fold for 1000l. the annuity will therefore be found by Cor. 3. in which all

the quantities are given, but a=px.


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and by inserting numbers, viz. p=1000, t=12, n=9, r=.05, and R=1.05; and working by logarithms a=175.029=175 1. 78.

Next, having found a, the fecond renewal is made by finding the present worth of the annuity a in reverfion, to commence In the years. 13 years hence, and to laft 8 canon (Cor. 3.) infert for a, 175.029, and let t=8, n=13, and r=.05 as before, p= 599.93=599 l. 18 s. 6d. The fine requi


As thefe computations often become troublesome, and are of frequent use, all the common cafes are calculated in tables, from which the value of any annuity, for any time, at any intereft, may easily be found.

It is to be obferved alfo, that the preceding rules are computed on the fuppofition


of the annuities being paid yearly; and therefore, if they be fuppofed to be paid half yearly, or quarterly, the conclufions will be fomewhat different, but they may eafily be calculated on the preceding principles.

The calculations of life annuities depend partly upon the principles now explained, and partly on physical principles, from the probable duration of human life, as deduced from bills of mortality.






Of the General Properties and Refolution of EQUATIONS of all Orders..


Of the Origin and Compofition of Equations; and of the Signs and Coefficients of their Terms.

N order to refolve the higher orders of


equations, and to investigate their general affections, it is proper first to confider their origin from the combination of inferior equations.

As it would be impoffible to exhibit particular rules for the folution of every order of

of equations, their number being indefinite; there is a neceffity of deducing rules from their general properties, which may be equally applicable to all.

In the application of algebra to certain fubjects, and efpecially to geometry, there may be an oppofition in the quantities, analogous to that of addition and fubtraction, which may therefore be expreffed by the figns and. Hence these figns may be + understood, by abftraction, to denote contrariety in general; and therefore, in this method of treating of equations, negative roots are admitted, as well as pofitive. In many cafes the negative will have a proper and determinate meaning; and when the equation relates to magnitude only, where contrariety cannot be fuppofed to exist, these roots are neglected, as in the cafe of quadratic equations formerly explained. There is befides this advantage in admitting negative roots, that both the properties of equations from which their refolution is obtained, and alfo those which are useful in the many extenfive applications of algebra, become

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