the fame Power (or Height) with the Index of the Surd, and take away the radical Sign; as in these EXAMPLES. 6 Suppofea = b + c Let 102 a = Suppofe 113aa-ba=d Let 132 aaba ddd10s Vaa = ? ¿RÓT. Sect. 6. Of Reduction by Evolution. WHEN any fingle Power of the unknown Quantity is on one Side of an Equation; evolve both Sides of the Equation, according as the Index of that Power denotes, and their Roots will be equal; as in these 36 EXAMPLES. Suppofelija a Suppofeta a = b b — dd a = bbdddder aaa = b+-+ I Or if any compound Power of the unknown Quantity be on one Side of the Equation (that hath a true Root of it's kind) evolve both Sides of the Equation, and it will be depreffed into lower Terms; as in these EXAMPLES. Suppofelijaa + 2 b a + b b = d dja a — za+2ba+bb 2 b a + b b = d d ce |amb=de I w2 |2| a+b= d Here follow a few Examples of clearing Equations, wherein all the foregoing Reductions are promifcuously ufed, as Occafion requires. By Help of thefe 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 anfwered: as in the first Example of Sect. 1, and 2. Or if any fingle 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 Roots, as a a+ba = dd, &c; or do confift of different Powers, as aaa+baadd, &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 Duantities; both Arithmetical, Granetrical, and Mufical. W HAT hath been faid of Numbers in Arithmetical Progreffon, 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 Decreafe with a Cypher: Thus { 0:0:22:30:4a5a: 6a : &c. increasing. 10:0:-2a: -3a: -4a:5a: -6 a: &c. decreafing. Or Univerfally, putting a the first Term in the Progression, and the common Excefs or Difference. {a:a+e:a+2e:a+3e:a+4e:a+5e:a+6e&c. Then L'a: a -e:a -- ·20:a 30:a - 4 c: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, o: a: 2 a: or, a : 2 a: 3a: or, 2 a: 3 a: 4 a, &c. viz. 20:40= =a+a: or, a+za: = 2a + 2 a, &c. a Alfo, in the fecond Series, either increafing or decreasing, it is evident, that if the Terms be a: a + c:a + 2 e, &c. increasing; then a + a + 2 c, viz. 2a + 2e the Sum of the Extreams, is double to ate the Mean, or if they be a -e: a -2e, &c. decreasing; then a ↑ a — 2 e viz. 2 a 2 e the Sum of the Extreams, is double to a ―e 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 + 3e, the Series increafing; here a + a + 3 c = a + e + a + 2e. Alfo in thefe, a: a — e: a →→→ 2e: a 3 e in the Series decreafing; here a + a ze=a-e-fa-- 2 e, &c. in any other four Terms. - in Confequently, If there are ever 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 distant from those Extreams. As in thefe, a: a+e: a +26:a+3e: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 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, 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. 2 =S, the Then will ay x N = 2 S, by the precedent Confectary : Na+Ny that is, Na+Ny 2 S. Confequently Sum of the propofed Series. Thirdly, In this Series it is easy 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 (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 1. That is, NI xeya, the Difference betwixt the two Extreams, viz. Ne-e — 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. H To wit, y = 4. вь Now |