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 Queftion is answered: as in the first Example of Sect. 1, and 2. Or if any fingle Power of the unknown Quantity (a) is found equal to thofe that are known, then the respective Root of the known Quantities is the Anfwer; 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 a+ba=dd, &c; or do confift of different Powers, as aaa+baa=dd, &c: Then they are called Affected, or Adfected Equations, which require other Methods to refolve them; viz. to find out the Value of (a) as fhall be fhewed further on.

CHAP.

CH A P. VI.

Of Proportional Quantities; both Arithmetical, Geometrical, and Musical.

WHAT hath been faid of Numbers in Arithmetical Progreffion, Chap. 6. Part 1. may be easily applied to any Series of Homogeneal or like Quantities.

Sect. 1. Of Duantities in Arithmetical Progzefsion.

THOSE Quantities are faid to be in the moft fimple or natural Progreffion, that begin their Series of increase or decrease with a Cypher:

So:a: 2a 3a 4a5a6a: &c. increafing.

Thus {

lo:-a:-2a: -3a: —4a: —5 a: -6a: &c. decreafing. Or Univerfally, putting a the firft Term in the Progression, and e the common Excefs or Difference.

Then{

a:a-e:a- 2e:a

a:a+e:a + 2e:a+3e:a+4e:a +5e:a+6e: &c. 3e:a. •4e:a—5e: a-be: &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 thefe, o:a: 2a: or, a: 2a: 3a: or, 2a: 3a: 4a, &c. viz. 2a:+0=a+a: or, a +3a=2a+2a, &c.

Alfo, in the fecond Series, either increafing or decreafing, it is evident, that if the Terms be a:a+e: a + 2e, &c. increafing; then a+a+2 e, viz. 2a + 2e the Sum of the Extreams, is double to ate the Mean, or if they be a: a-e: a—2e, &c. decreafing; then a+a2e: viz. 2 a — 2e the Sum of the Extreams, is double to ae the Mean. And fo 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, a:a+e:a+2e: a + 3e, in the Series increafing; here a+a+30=a+e+a+ze. Alfo in thefe, aa—e: a 2e: a 3 e in the Series decreafing; here a+a—ze=ae + a — 2 e, &c. in any

other four Terms.

Confequently, If there are never fo 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 diftant from thofe Extreams. As in thefe, a:ate: à +22:a +30:a+4e: a +5e: &c. Here a+a+5e=a+e+a+4e=a+2e+a+3e, &c. 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. I. Chap. 6. before-mentioned.

CONSECTARY 1.

Whence it follows, (and is very eafy 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 eafter refolving fuch Questions as depend upon these Progreffional Quantities.

a the firft Term, as before.

y

the last Term.

Let the common Excefs, &c. as before.

N= the Number of all the Terms.

S the Sum of all the Series, viz. of all the Terms.

Then will a+yx N=2S, by the precedent Confectary: Na+NYS, the

2

that is, Na+Ny=2 S. Confequently Sum of all the Series, be the Terms never fo many. Thirdly, In these Series it is eafy to perceive, that the common Difference (e) is so often added to the laft Term of the Series; as are the Number of Terms, except the firft; that is, the firft Term (a) hath Do Difference added to it, but the laft Term hath so many times (e) added to it, as it is diftant from the first.

Confequently, the Difference betwixt the two Extreams, is only the common Difference (e) multiplied into the Number of all the Terms lefs Unity or t. That is, Nixeya, the Difference betwixt the two Extreams, viz. Nee=y-a.

CONSECTARY 2.

Whence it follows, that if the Difference betwixt the two Extreams be divided by the Number of Terms lefs 1, the Quotient will be the common Difference of the Series.

To wit, Ni

Bb

Now

Now by the Help of these two Confectaries, if any three of the aforefaid five Parts (viz. a. y. e. N. S.) be given; the other two may be eafily found.

Na+Ny=s

Thus,

2

as before.

=N, the Number of Terms.

14 x 2 e 15 yy—aa+ae+ye=2Se 1.5 -ae 16yyaa+ye = 2 Se-ae 16-ye 17 yy—a a = 2 Se—ae — ye.

20

yy- -aa

1718

2 S

a-y

e, the common Difference.

3+a19 Ne-e+ay, the laft Term. 19 e20 Ne+ay+e

Ne21ye --- Nea, the firft Term.

&c.

In like Manner you may proceed to find out any of the five Quantities (a. e. y. N. S.) otherwife, viz. by varying or comparing thofe Equations one with another, you may produce new

I

Equations

Equations with other Data in them; the which I fhall here omit pursuing, and leave them for the Learner's Practice.

Sect. 2. Of Duantities in Geometrical Proportion.

GE

EOMETRICAL Proportion continued has been already defined in Sect. 2. Chap. 6. Part I. And what is there faid concerning Numbers in may eafily be applied to any Sort of Homogeneal Quantities that are in.

The most natural and fimple Series of Geometrical Propor tionals, is when it begins with Unity or I.

As I. a aa aaa aaaa. a', a", &c. in

For I a: a: a a : : a a ¦ à a a ¦ ¦ à a a ¦ à a a a, &c.

That is, when all the middle Terms betwixt the two Extreams are both Confequents and Antecedents, that Series is in Geometrical Proportion continued. Therefore in every Series of Quantities in all the Terms except the laft are Antecedents; and all the Terms except the firft are Confequents. But univerfally putting a the firft Term in the Series, and e the Ratio, viz. the common Multiplier, or Divifor; then it will be

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 thefe, a: a. aee, here a xace = aixaf = aaee. &c.