ba a =V gbt 4 lat CC 6+ 964 + CC 961 + CC baa 5 + 4 964, bat 4 bat 7+ 4 4 4 4646 9ccb3 46 - 46 a V as was required. 46-46 For 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. 1, 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 Sett. 6, &c. But when the Powers of the unknown Quantities are either mixed with their Root, as a atbardd, &c; or do consist of different Powers, as a aatbaardd, &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 fhall be thewed further on. c H A P. CH A P. VI. Of Poportional Duantities; both arithmetical, Geometrical, and ulical. WH HAT 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 Arithmetical Progression. THOSE Quantities are said to be in the most simple or na tural Progreffion, that begin their Series of increase or decrease with a Cypher : Thus { fo:a: 2 a : 3a : 42:52: 6a : &c. increasing. 0:-a: -22:39:-42:-5a: -6a:&c. decreasing. Or Universally, putting a the first Term in the Progression, and 6 the common Excess or Difference. Şa:a te: a +22:a +32:2 +44:2 +52:2 +61:&c. -22:6- 34:4-46:2-54:a-6e:&c. In the first of these Series it is evident, that if there be but three Terms; the Sum of the Extreams will be double to the Mean. As in these, o:a: 2a: or, a:20:3a: or, 22:3a:44, &c. viz. 2a:to=ata:or, a +39=2a + 2a, &c. Also, in the second Series, either increasing or decreasing, it is evident, that if the Terms be a:ate:a+21, &c. increasing ; then ata +26, viz. 22 +2e the sum of the Extreams, is double to a te the Mean, or if they be a:a- e:Q — 22, &c. decreasing; then ata-2e: viz. 2 a 2e the Sum of the Extreamns, is double to are the Mean. And so it will be in any other of the three Terms. Secondly, if there are four Terms; then the sum of the two Extreams, will be equal to the Sum of the two Means; as in these, e:ate:a+2e: a +36, in the Series increasing; here a tat3e=ateta +24. Also in these, 2::& 3 e in the Series decreasing; here ata- 31=a-.-6+- 2 1, &c. in any other four Terms. Consequently, If there are never so many Terms in the Series, the Sum of the two Extreams will always be equal to the Sum of of any two Means, that are equally distant from those Extreams. As in these, a:ate: +22:+36:9 +42:a +5e: &c. Here atat se sa tiita +41 = a + 2 éta +3C, &e. 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 1. 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, the Product will be double the Sum of all the Series. Now for the easier resolving such Questions as depend upon these Progreffional Quantities. a = the first Term, as before. y = the last Term. N-the Number of all the Terms. 2 Then will a+yxN=2 S, by the precedent Consectary: Na + Ny that is, Na +Ny=2 S. Consequently =S, the Sum of all the Series, be the Terms never so many. Thirdly, In these Series it is easy to perceive, that the common Difference (R) is so often added to the last Term of the Series ; -as are the Number of Terms, except the first; that is, the first Term (a) hath Do 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 (ë) multiplied into the Number of all the Terms less Unity or t. That is, N - ixe=y- ,, the Difference betwixt the two Extreams, viz. Néme=y- a. CONSECTAR Y 2. 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 Series. Now by the Help of these two Consectaries, if any three of the aforesaid five Parts (viz. a. y... N.S.) be given ; the other two may be easily found. Na+Ny =S Thus, 2 as before. And 2 ly N-I 2 x NI 3y-a= Nece 3tel 4 y- a te=N. ly-ate =N, the Number of Terms. 7 Ny=2S--Na 2SNa 7-N 8 N =y, the last Term. -= a, the first Term. 2S a ty per Axiom 5. aty ++y=25, the Sum of all the Series. 16 y y -aa tye 2 Se ae - a bye. =e, the common Difference. 2 S-ay 3+ a 19 Nemetary, the latt Term. 19+ 20 Neta=yte 20 - Ne 211y+ Ne=a, the first Term. &c. 5, and i1!12 yy 13 214 2 2 ae 17 • 118 In like Manner you may proceed to find out any of the five Quantities (a.c.y. N. S.) otherwise, viz. by varying or comparing those Equations one with another, you may produce new Equations I Equations with other Data in them; the which I shall here omit pursuing, and leave them for the Learner's Practice. Sect. 2. Of Duantities in Geometrical Proportion. G EOMETRICAL Proportion continued has been already defined in Sect. 2. Chap. 6. Part 1. And what is there faid concerning Numbers in - may easily be applied to any Sort of Homogeneal Quantities that are in ::. The most natural and simple Series of Geometrical Propora tionals, is when it begins with Unity or 1. as As I.2. аа.ааа аааа. a', &c. in &c. are Terms in :: bb bb bbb bbb 64 64 85 &c. 25:223 Ora.b. aa ats That is, when all the middle Terms betwixt the two Extreams are both Consequents and Antecedents, that Series is in Geome. trical Proportion continued. Therefore in every Series of Quantities in all the Terms except the last are Antecedents; and all the Terms except the first are Consequents. But universally putting a the first Term in the Series, and the Ratio, viz. the common Multiplier, or Divisor; then it will be a.de.alb.acce.accce.ae.22. &c. in :: Osa . . &c. are in decr. a I. In any of these Series it is evident, that if three Quantities are in *:, the Rectangle of the two Extrems will be equal to the Square of the Mean; as in these, a:at.ar, here a xare =%6Xafg=aacb. &c, |