. A New general dictionary of arts and sciences, or, Compleat system of universal knowledge. common difference,be the< number of terms,laft term,fum of all the terms; And the feveral cafes are reducible into ten pro-pofitions, which are all folved by the two followinglemmata. Lemma I. In any arithmetic progreflion, it is,. Therefore s -\- s z=z a -\- i x na -\-lX • Confequently, i : — 2 J = •.■.a + /:s. From this the following corollaries naturally fol-low. ,Cor. T ^ n 1 2s — nl I •/ = = 2i—n/x- II. n =. -r-,=2X—r-,= 2 s Xa -j-/ a -\- i III. + / l_l± ^__ —naX— Cor. IV. s=-Xa + l
. A New general dictionary of arts and sciences, or, Compleat system of universal knowledge. common difference,be the< number of terms,laft term,fum of all the terms; And the feveral cafes are reducible into ten pro-pofitions, which are all folved by the two followinglemmata. Lemma I. In any arithmetic progreflion, it is,. Therefore s -\- s z=z a -\- i x na -\-lX • Confequently, i : — 2 J = •.■.a + /:s. From this the following corollaries naturally fol-low. ,Cor. T ^ n 1 2s — nl I •/ = = 2i—n/x- II. n =. -r-,=2X—r-,= 2 s Xa -j-/ a -\- i III. + / l_l± ^__ —naX— Cor. IV. s=-Xa + l =« X—I—• then any three of thefe termsbeing given, the other twoare eafily found. n X Lemma 11. In any arithmetical progreffion it isi:«—i::d:l—a. For, a, a-\-d, a + id, a+^d, a-\-d—iXd=:L That is, >!—ixd = i— a, by tranf-ppfition. Therefore, i:n — i::d:l — a. From this likewife the four following corollariesare decuced. Cor. I. « = /—n—\—nd-{■ d. Cor. II. „ = _-+i=:_-:r-. Cor. III. ^ = l- ^or. IV. /=:a-f « =.1—ax . n—I -^nd — d. Geometric Progression, or continued geometricproportion, is when the terms do increaf-i or de-creafe by uqual ratios : thus. », ar, arr. arrr, &c. increafing t—, &c. decreafing ( from a continual ( mult
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