. Encyclopaedia : or, A dictionary of arts, sciences, and miscellaneous literature; constructed on a plan, by which the different sciences and arts are digested into the form of distinct treatises of systems ... . ft or leaft be a, the difference betweenthe 1 ft and 2d b, and c, d, e, &c. the refpectivefirft term of their 2d, 3d, 4th, &c. differences ; thenfhall the feveral quantities themfeives be as in the an-nexed fcheme ; where each term is compofed of thefirft term together with as many of the differences asit is diftant from the firft term, and to thofe differencesjoining, for coefficien
. Encyclopaedia : or, A dictionary of arts, sciences, and miscellaneous literature; constructed on a plan, by which the different sciences and arts are digested into the form of distinct treatises of systems ... . ft or leaft be a, the difference betweenthe 1 ft and 2d b, and c, d, e, &c. the refpectivefirft term of their 2d, 3d, 4th, &c. differences ; thenfhall the feveral quantities themfeives be as in the an-nexed fcheme ; where each term is compofed of thefirft term together with as many of the differences asit is diftant from the firft term, and to thofe differencesjoining, for coefficients, the numbers in the Hoping oroblique lines contained in the annexed table of figu-rate numbers ; in the fame manner, he obferves, as thefame figurate numbers complete the powers raifed froma binomial root, as had long before been taught byothers. He alfo remarks, that this rule not only givesany one term, but alfo the fum of any number of fuc-ceffive terms from the beginning, making the 2d coef-ficient the ift, the 3d, the 2d, and fo on ; thus, thefum of the firft 5 terms is Sa+ iob+ iOc+ 5d+e. ift term ? • a 2d «+ b 3d - a-J-2^-f- c 4th a+3t>-\-3c+ d 5th - a+qb+bc + qd+e &c. & Vol. X. a+b a+ib In the 4th prop, it is fhown, that if the terms de-creafe, proceeding from the grearer to the lefs, thefame theorems hold good, by only changing the fignof every other term, as below. ift term - a 2d - - a— b 3d - - a—2b+ c 4th - - a—3b+3c— J 5fh - - a—4b\.6c — qd+e Sec. &c. Prop. 6th and 7th, treat of the approximate multi-plication and divifion of ratios, or, which is the famething, the finding nearly any powers or any roots of agiven fraction, in an eafy manner. The theorem for S railing r38 L O G A R Conftruc- raifmg any power, when reduced to a Ampler form, isticm of Lo- a a~~\m . f——-mj garithms. ibis, the m power of —, or—I is = _J= the m power of -5-, or — b o I ly, where s \& — a •+- I, and d — a ^ I, the fum and differe
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