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2xb13baa+bc-bd=48—4a a 3+4aa4baa+4aa+bc-bd=48 4+bd5baa+4aa+bc=48+ bd 5-bc6baa+4aa=48+ b d - b c

48+bd-bc

6÷6+417 aa= 6+4

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2+aa 4 4aa=125316
445 aa= 31329

2

5 w 6

a=√ 31329=177, the Value of a required.

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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 thofe 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 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 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.

CHAP. VI.

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

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

Sect. 1. Of Duantities in Arithmetical Progzelsion.

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

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

20:-a:24:3a-4a: -5a: -6 a: &c. decreafing. Or Univerfally, putting a the firft Term in the Progreffion, and e the common Excefs or Difference.

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a: a -e:a- 20:a- ·3e:a-4e:a—5e:a — 6e:&c. In the first of thefe 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, oa: 2a: or, a:2a:3a: or, 2a:3a:4 a, &c. viz. 2a:+0=a+a: or, a +3a=2a+za, &c.

Alfo, in the fecond Series, either increafing or decreasing, 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 ae the Mean, or if they be a: a-e: a — 2e, &c. decreasing; then aa -2e: 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 thefe, a: a +e: a + 2e: a +36, in the Series increafing; here a +a+30=a+e+a+2e.

Alfo in thefe, a: a —e: a — 28:α- 3 e in the Series de creafing; here a + a — 36=a-eta- 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 équally diftant from thofe Extreams. As in these, a: a+e: a + 2 é: à +3e: a + 4 é:a+5e: &c. Here a+a+5e=a+e+a+4e=a+2e+a+ze, &c. And if the Number of Terms be odd, the Sum of the two Extreams will be double to the middle Term, &t. às in Corol, 1. Chap. 6. before-mentioned.

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 eafter refolving fuch Questions as depend upon these Progreffional Quantities.

Let

a the firft Term, as before.

y= the laft Term.

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=2 S, by the precedent Confectary: Na+Nys, the

that is, Na+Ny=28. Confequently

2

Sum of all the Series, be the Terms never fo many. Thirdly, In thefe 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 no Difference added to it, but the laft Term hath fo many times (*) 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, NI xey➡d, the Difference betwixt the two Extreams, viz. Nee =)—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,

N.

Bb

1 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 easily found.

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Na+Ny

=S

2

as before.

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N, the Number of Terms.

e

28

=y, the laft Term.

N

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14 x 2 e 15 yy —aa+ae+ye = 2 Se 15ae16yy -aa+ye =2Se е-ав y e 17 y y -aa=28e-ae-ye

16

1718

yyaa

2 S -a-y

=e, the common Difference.

3+ 19 Ne-e+ay, the laft Term. 19 e20 Ne+a=y+e

20 Ne 21 yeNea, the firft Term.

&c.

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

'I

Equations

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