i6? A Treatise of Algebra.

proceeding as above, we shall have 2 -j- — -f- To)? 4*

i. 4 '

2 - , * S 3'A , o8^ + X 27/ _ * ... ,

-7T- +lS " + -^2i thatls ’ *♦”+

1.23# = 6.5; whence e = .03d and *=3.036; thelik*

of any other.

It may be observed that this Method, as all the Power*of t above the first are rejected, only doubles the Num*ber of Places at each Operation : But, from what istherein shewn, it is easy to fee how it may be extendedso as to triple (or even quadruple) the Number of Plac«»at every Operation ; but as the Trouble, in every Ope-ration would be increased in Proportion, it is manifest thatlittle or no Advantage would be reaped therefrom.

Hitherto we have treated of Equations which includeonly one unknown Quantity ; but if there be two Equa-tions given, and as many Quantities (xand j) to be deter-mined, one of those Quantities is first to be exterminated,and the two Equations reduced to ode, according towhat has been already shewn,‘inSect. 9. but if this can-not be conveniently done, (as in many Cafes it cannot)or the unknown Quantities be so entangled as to renderthat Way impracticable, the following Method may beof use.

Let the Values of x and y be estimated pretty nearthe Truth (which from the Nature of the Problem mayalways be done) and let the Values so estimated be de-noted by/and g, and what they want of Truth, by sand t respectively, that is, let f s — x and g -f- / —y :Substitute these Values of x and y, in both Equations,rejecting (by reason of their Smallness) all the Termswherein more than one single Dimension of the Quan-tities s and t are concerned : Let all the Terms in thefirst Equation, which are affected by s, be collected, un-der their proper Signs, and denoted by As ; in like Man-ner, let those affected by /, be denoted by B/ ; and thoseaffected neither by s nor /, by Qj Moreover let theTerms of the second Equation wherein s and t are con-

«m«d. be denoted by as and bt respectively, and let the

known