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content Example of Secte) is found equal co antities is the
By Help of these Reductions (properly applied) the unknown Quantity (a) or it's Powers, are cleared and brought to one Side of an Equation; and if the unknown Quantity (a) chance to be equal to those that are known, the Question is answered: as in the first Example of Sect. I, and 2. Or if any single Power of the unknown Quantity (a) is found equal to those that are known, then the respective Root of the known Quantities is the Answer ; as in the first four Examples of Sect. 6, &c.
But when the Powers of the unknown Quantities are either mixed with their Root, as a atba=dd, &c; or do consist of different Powers, as a aa tbaa=dd, &c: Then they are called Affected, or Adfected Equations, which require other Methods to resolve them; viz, to find out the Value of (a) as shall be lhewed further on.
CH A P. VI. of Proportional Duantities ; both arithmetical,
Geometrical, and Pubcal. W H AT hath been said of Numbers in Arithmetical Progref
fron, Chap. 6. Part 1. may be easily applied to any Series of Homogeneal or like Quantities.
Sect. 1. Of Duantities in arithinetical Progression.
of any two Means, that are equally distant from those Extreams. As in these, a:dtera +2ė:8 +36:8 +44:8 +52: &c. Here a tia #seateta+ 40 =0 +2 état 31, &€. And if the Number of Terms be odd, the Sum of the two Extreams will be double to the middle Term, &c. as in Corol. 1. Chap. 6. before-inentioned.
CONSECTARY i. Whence it follows, (and is very easy to conceive) that if the Sum of the two Extreams be multiplied into the Number of all the Terms in the Series, thi Product will be double the Sum of all the Series.
Now for the eafier resolving such Questions as dépend upon these Progreffional Quantities.
ra = the firft Term, as before.
y = the last Term.
N= the Number of all the Terms.
Then will a+yx N=2 S, by the precedent Confectary:
Nat Ny that is, Nat Ny= 2 S. Consequently ~ Sum of all the Series, be the Terms never so many. Thirdly; In these Series it is easy to përceive, that the comimon Difference (6) is so often added to the laft Term of the Series ; as are the Number of Terms, except the first ; that is, the first Term (a) hath no Difference added to it, but the last Term hath so many times (e) added to it, as it is distant from the first.
Consequently, the Difference betwixt the two Extreams, is only the common Difference (e) multiplied into the Number of all the Terms less Unity or t. That is, N-Ixey - d, the Difference betwixt the two Extreams, viz. Né y a .
Whence it follows, that if the Difference betwixt the two Extreams be divided by the Number of Terms less 1, the Quotient will be the common Difference of the Séries.