In the first bargain, an annuity in rever- . fion for 12 years, to commence 9 years hence, was fold for 1000 l. the annuity will therefore be found by Cor. 3. in which all FR the quantities are given, but a=px R 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 l. I I 7 d. Next, having found a, the second renewal is made by finding the present worth of the annuity a in reversion, to commence 13 years hence, and to last 8 years. In the canon (Cor. 3.) insert for a, 175.029, and let t=8, n=13, and r=.05 as before, p= 599.93=599 1. 18 s. 6; d. The fine required. As these computations often become troublesome, and are of frequent use, all the common cases are calculated in tables, from which the value of any annuity, for any time, at any interest, may easily be found. It is to be observed alío, that the preceding rules are computed on the supposition of of the annuities being paid yearly; and therefore, if they be supposed to be paid half yearly, or quarterly, the conclusions will be somewhat different, but they may easily 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, 1 OF A L G E BRA PART II. Of the General Properties and Resolution of EQUATIONS of all Orders. CH A P. I. Of the Origin and Composition of Equations ; and of the Signs and Coefficients of their Terms. IN N order to resolve the higher orders of equations, and to investigate their general affections, it is proper first to consider their origin from the combination of inferior equations. As it would be impossible to exhibit párticular rules for the solution of every of every order of equations, their number being indefinite; there is a necessity of deducing rules from their general properties, which may be equally applicable to all. In the application of algebra to certain subjects, and especially to geometry, there may be an opposition in the quantities, analogous to that of addition and subtraction, which ́ may therefore be expressed by the signs + and — Hence these signs may be understood, by abstraction, to denote contrariety in general; and therefore, in this method of treating of equations, negative roots are admitted, as well as positive. In many cases the negative will have a proper and determinate meaning; and when the equation relates to magnitude only, where contrariety cannot be supposed to exist, these roots are neglected, as in the case of quadratic equations formerly explained. There is besides this advantage in admitting negative roots, that both the properties of equations from which their resolution is obtained, and also those which are useful in the many extensive applications of algebra, become |