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But when Lines are compared to Superficies, or Lines are compared to Solids, such Comparisons are distinguished from the former, by the Names of Duplicate, and Triplicate, (&c.) Propora tions ; lo that Simple, Duplicate, and Triplicate, &c. Proportions are to be understood in a different Sense from Simple, Double, Treble, &c. Proportions, which are only as 1, 2, 3, &c. to I; but those of Simple, Duplicate, Triplicate, &c. Proportions are chofe of a . aa. aaa . , &c. to 1. Or if the Simple Proportions
6 be that of a to b, whose Ratio or Exponent is
a in Then
is the Exponent of the Du
es is the Exponent of the b b b Triplicate Proportions, &c.
And if there are three, four, or more Quantities in :, as 1.a.aa.aaa .24.a', &c. (as in the first Series, Sect. 2. of the last Chapter.) Then, that of the first to the third, fourth, and fifth, &c. (viz. I to aa. aaa .at .a) is Duplicate, Triplicate, Quadruplicate, &c. of the first to the second (viz. of I. to a ;) and by Inversion, that of the third, fourth, fifth, is Duplicate, Triplicate, &c. of that of the second to the first (a to 1) per Def. 10. Eucl. 5. But the Name of these Proportions will appear more evident, and be easier understood when they are applied to Practice, and illustrated by Geometrical Figures, further on.
Sect. 3. How to turn Equations into analogies.
FROM the first Section of this Chapter, it will be easy to con
ceive how to turn or diffolve Equations into Analogies or Proportions. For if the Rectangle of two (or more) Quantities, be equal to the Rectangle of two (or more) Quantities; then are those four (or more) Quantities Proportional. By the 16 Eucl. 6. That is, if absid, then is a :c::d:b, or c:a::b:d, &c. From whence there arises this general Rule for turning Equations into Analogies.
RU L E.
Divide either Side of the given Equation (if it can be done) into two such Parts, or Factors, as being multiplied together will product that Side again; and make those two parts the two Extreams. Then divide the other Side of the Équation (if it can be done) in the same Manner as the first was, and let those two Parts or Factors be the two Means.
For Instance, Suppose ab tad=bd. Then a:b::d:6td, or b:a::b+d:d, &c. Or taking a d from both sides of the Equation, and it will be ab=bd-ad; then a :d::ba:b, or, b:d::b-a:a, &c.
Again, suppose a a + 2ae=269 +9y. Here a and a +2e are the two Factors of the first Side in this Equation ; for 2+26 xa=aa + 2a e. Again, y and 2b + y are the two Factors of the other Side,
y therefore, a:y::26+y:a +2e, or 26+y:a +2e::a:9, &c.
When one side of any Equation can be divided into two Factors, as before, and the other Side cannot be so divided, then make the Square Root of that Side either the two Extreams or the two Means. For Instance, Suppose bc+bd=da +8, then b: v dats::v da +8:0+d, or vida +8:6::ctid: vda +8,&c.?
CH A P. VIII.
Of Substitution, and the Solution of Duadratick
Sect. 1. Of Substitution.
WHEN new Quantities not concerned in the first Stating of
any Question, are put instead of fome that are engaged in it, that is called Substitution. For Instance, If instead of ✓be-dc you put z, or any other Letter ; that is, make z= Voc-da. Or suppose aa toba-ca tode=de, instead of bac + d put s, or any other Letter_not engaged with the Question, vit. s=bmitd, then a a Fsa=dc. That is, if c be greater than 6 td, it is a a-sa=dc; but if 6+d be greater than a then it is a a +sa=dc.
And this way of substituting or putting of new Quantities instead of others, may be found very useful upon several Occasions; viz, in order to make some following Operations in the Queftion more easy, and perhaps much shorter than they would be without it, as you may observe in some Questions hereafter proposed in this Tract.
And when those Operations, in which the substituted Quantities were affifting or useful, are performed according as the Niture of the Question required, you may then (if there be Occasion) bring the original or first Quantities into the Equation, in the Place (or Places) of those substituted Quantities, wbich is called Reftitution, as you may see further on.
Sect. 2, The Solution of Duadratick Equations.
HEN the Quantity sought is brought to an Equality with
those that are known, and is on one side of the Equation, in no more than two different Powers whose Indices are double one to another, those Equations are called Quadratick Equations Adfected ; and do fall under the Consideration of three Forms or Cases.
When there happens to be more Terms in one of these Kind of Equations than two, and the highest Power of the unknown Quantity is multiplied into some known Co-ëfficients; you must reduce them by Division; as in Sect. 4. of Chap. 5. and for the Fractional Quantities that may arise by those Divisions, subftitute another Quantity doubled.
For Instance, let baa trasaca-da=dctcb, then aacada dctcb
= 2*, and if you please, 675 btc
btc Сс 2
put z. Then will aa-2x0=z be the new Equabtc tion, equal to the other, being now fitted for a Solution.
Now any of these three Forms of Equations being thus prepared for a Solution, may be reduced to simple Powers by casting off the second or loweft Term of the unknown Quantity ; which is done by Substitution ; thus, always take half the known Coëfficient, and add it to (Case 1.) or subtract it from (Case 2.) it's fellow Factor; and for their Sum, or Difference, Substitute another Letter; as in these.
Letilaa + 2ba=dc Cafe 1. Put 2 a+b= e 2 Q23 a 2 + 2 ba +bb=i 3-14 bble-do 4+dc 5 ce=bbtdc
5 w? 6 e=Nbbt de 2 and 617 atb=vbb + dc, per Axiom 5. 7
bl8la=vbbtdc: - -b
"Letla 4 ---2bo=dc Case 2.
Put 2 a
bb teede 4 foi o s'eedctob 5 w
6em v dit bb 2 and 6a-bVdctub 7682= = b ta det bb
In Cafe 3. From Half the known Co ëfficient substract it's fellow Factor.
Thus, Let 12 ba-aardo
Put 216 2Q?1368
60-2batoarce I+31 416b=dctee 4-dc51bb-do
5 w26 le=vbb - dc
7tal8/6= atubb - do 8-V, &c.Igla=bvba
And this Method holds good in those other Equations, wherein the highest Powers are at, a', a®, &c. As, for inftance,
The same may be done with all the rest, Care being taken to add, or subtract, according as the Case requires.
But all Quadratick Equations may be more easily resolved by compleating the Square, which is grounded upon the Consideration of raising a Square from any Binomial, or Residual Root. (See. Sett. 5. Chap. 1.) Viz, if a + b be involved to a Square, it will be a a +2ba+bb; and if & -b be fo involved, it will be aa2ba+bb. Whence it is easy to observe, that a a + 2be=do (Cale 1.), and a a--2ba=dc (Case 2.), are imperfect Squares, wanting only 66 to make them compleat. And therefore it is, that if half the known Co-ëfficient be involved to the second Power, and the Square be added to both sides of the Equation, the unknown Side will become a compleat Square.
Here half the Co-efficient Thus Let Ilaa +2ba=dc 2b is b, which being squared, But| 2
bb=b8 Lis 66.