16-nnene 36|2s — nne + ne — 2na. :√:yy+ey + ee 2se: 41 + 42a=e±√:yy+ey+ee - 25e: 23e 40 w. 2.41a- e 2y+e nnn. e e e The Theorems by which any Queftion in are inferted in the 3, 4, 6, 8, 9, 10, 11, 13, 30, 32, 35, 37, 39, 42, 47 and 49th Steps. may be folv'd, 17, 20, 24, 28, * See Part X. СНАР. CHA P. II. Of Geometrical Propoztions. Sect. 1. Of Geometrical Pzopoztions continued. I Definition. WHen a Rank or Series of Numbers or Quantities, do either increase by one common Multiplicator, or decreafe by one common Divifor, thofe Numbers or Quantities are faid to be in Geometrical Proportion continued. As 2, 4, 8, 16, 32, 8. Here 2 is the common Multiplyer] 5 common 2729, 243, 81, 27, 9, 8c. Here 3 is the common Divifor. I r,rr,r3,r4, &c. Here is the common Multiplyer, I I I I Note, the common Multiplyer (or Divifor) is called the Ratio, and fhews the Habitude or Relation the Numbers or Quantities have to one another; viz. whether they are Double, Treble, Quadruple, &c. In order to folve fuch Questions as relate to Geometrical Pro portion continued, I will the leaft Term. y=the greatest Term. Suppofes Sum of all the Terms. in the Number of Terms. the common Ratio; but Note that r must be I Lemma. If from the Sum of any Series in Terms be fucceffively Subtracted; I fay the leaft and greatest the firft Remainder is equal to the Product of the fecond Remainder and common Ratio. Q, 2 Demon Demonftration. a being the leaft Term of any Series in, and r = the common Ratio; the Least, Leaft but one, Least but two, Least but three, &c. Terms will be equal to a, ar, ar, ar3, &c. refpectively: Alfo y being the greatest Term of the faid Series; the Greateft, Greatest but one, Greatest but two, Greatest but three, Er. Terms will be equal to y,,,, &c.refpectively : And the Means, or middle Terms we will denote by &c. by which Means our Series in be writ thus Increafing may Then a + ar+ ar2 + ar3 +8c.+1+2+4+;(= 9 + 2 + 2 + 3 + 8c. + ar3 + ar2 + ar + a) =s Wherefore, by Transposition, ar + ar2 + ar3 +&c.+2+2+2+y=s—a 3 ་་་༽ And a + ar + ar2 + ar: +8c + 2 + 2 + 2 = 5-y But the first Part of the laft Equation x is manifeftly equal to the first Part of the laft but one Equation; Consequently sy: xr = s-a. Scholium. It is manifeft, by viewing the foregoing Series in that aisy; for the Exponent of r in the leaft Term of that Series is c, in the leaf Term but one is I, in the leaft Term but two is 2, &c. and therefore univerfally the Exponent of r in the zz1" or n |