A Treatise of Algebra. 71

Call’d Arithmetical, but according to their Ratios, Geo-metrical. If in a Rank of Quantities, as, a, b, c, d, (4,6 ,12, 14) the Difference of the first and second (b — a)is the same as the Difference of the third and fourth(d — c), those Quantities are said to be in an arithmeticalProportion ; but if the Ratio of the first and second bethe same as that of the third and fourth, that is, if thefirst be the same Part or Parts of the second as the thirdis of the fourth, then are they said to be in a geometricalProportion : Thus 2, 6 , to, and 30 are in a geometricalProportion ; because *• — (— 7). Moreover, when the

Difference, or the Ratio of every two adjacent Term (aswell of the first and second, and second and third, as ofthe first and second and third and fourth, istc.) is the same,then the Proportion is said to be continued : Thus 2, 4, '6, 8, 10, fcfr. is a continued arithmetical Proportion, and2, 4, 8, 16, 32 a continued geometrical one; for 4 — 2— 6 —— 4—8 — 6 — 10 — 8 and + =i,~T — (2(= i). These kind of Proportions are also called Pro-gressions, being carry’d on according to the same Lawthroughout.

Arithmetical Proportions.

T H E O R EM I. '

Jf four Quantities, as, a, b, c, d, or 6 , 9, 12, 15, be inProportion, the Sum of the two Means will be equal tothat of the two Extremes.

For since, by Supposition, b — a is —d — c, there-fore is b + c — d + a, by (J'ranspofition.

THEOREM II.

In any continued arithmetical Proportion ( 5,7, 9, II, 13,15) the Sum of the two Extremes , and that of every othertwo Terms equally di/iant from them, are equal.

For since, by the Nature of Proportionals, the secondTerm exceeds the first by just as much as its correspond-ing Term, the last but one, wants of the last, it ismanifest, that when these corresponding Terms are addedF 4 together,