PREFACE.

V

Tract ix is a method of summing the series « + 6x+cx 24-dx 3 -)-e'.r 4 +&c, in the case when it converges very slowly }namely, when x is nearly equal to 1, and the coefficients a ,b, c, d, &c, decrease very slowly; the signs of all the termsbeing plus or positive :—a method which has been considereda great desideratum in infinite series.

Tract x contains the investigation of certain easy andgeneral rules, for extracting any root out of a given number ;exhibiting a general and very easy formula, to serve for allroots whatever.

Tract xi is a new method of finding, in general andfinite terms, near values of the roots of equations of this form,

x n — px n ~ l +qx n ~ — &c = 0 ; namely, having the termsalternately plus and minus : being one method more to beadded to the many we are already possessed of, tor deter-mining the roots of the higher orders of equations.

Tract xii treats of the binomial theorem ; exhibiting ademonstration of the truth of it in the general case-of frac-tional exponents. The demonstration is of this nature, thatit proves the law of the whole series in a formula of onesingle term only : thus, p, a, R, denoting any three succes-sive terms of the series, expanded from the given binomial

(1 + x) n , and if -f p a, then is = r, which denotes

the general law of the series, being a new mode of proving thelaw of the coefficients of this celebrated theorem. But, be-sides this law of the coefficients, the very form of the seriesis, for the first time, here demonstrated, viz, that the form

of the series for the developement of the binomial (1 + x ) n ,with respect to the exponents, will be 1 + ax + bx 2 + cx %+dx 4 + &c, a form which has heretofore been assumedwithout proof.

Tract xiii treats on the common sections of the sphereand cone: with the demonstration of some other new pro-perties of the sphere, which are similar to certain knownproperties of the circle. The few propositions whicli form