But when Lines are compared to Superficies, or Lines are compared to Solids, fuch Comparisons are diftinguished from the former, by the Names of Duplicate, and Triplicate, (&c.) Proportions; fo that Simple, Duplicate, and Triplicate, &c. Proportions are to be understood in a different Senfe from Simple, Double, Treble, &c. Proportions, which are only as 1, 2, 3, &c. to 1; but thofe of Simple, Duplicate, Triplicate, &c. Proportions are thofe of a.aa.aaa., &c. to I. Or if the Simple Proportions

a

b

be that of a to b, whofe Ratio or Exponent is or

And if there are three, four, or more Quantities in, as 1.a.aa.aaa. a. as, &c. (as in the first Series, Sect. 2. of the laft Chapter.) Then, that of the firft to the third, fourth, and fifth, &c. (viz. I to aa.aaa. a. a) is Duplicate, Triplicate, Quadruplicate, &c. of the firft to the fecond (viz. of I to a ;) and by Inverfion, that of the third, fourth, fifth, is Duplicate, Triplicate, &c. of that of the fecond to the first (a to 1) per Def. 10. Eucl. 5. But the Name of thefe Proportions will appear more evident, and be eafier understood when they are applied to Practice, and illuftrated by Geometrical Figures, further

on.

Sect. 3. How to turn Equations into Analogies.

FROM the firft Section of this Chapter, it will be eafy 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 thofe four (or more) Quantities Proportional. By the 16 Eucl. 6. That is, if a bed, then is a :c::d: b, or c:a :: bd, &c. From whence there arifes this general Rule for turning Equations into Analogies.

RULE.

Divide either Side of the given Equation (if it can be done) into two fuch Parts, or Factors, as being multiplied together will produce that Side again; and make those two Parts the two Extreams. Then divide the other Side of the Equation (if it can be done) in the fame Manner as the first was, and let those two Parts or Factors be the two Means.

:

For Inftance, Suppofe a bad bd. Then a b:: d: b+d, or bab+d: d, &c. Or taking a d from both Sides of the Equation, and it will be a bb dad; then a :d :: b. Dr, bd :: b

a: a, &c.

a:b, Again, fuppofe a a+2ae2by+yy. Here a and a +2e are the two Factors of the firft Side in this Equation; for a + 2e xa=aa+2 a e.

Again, y and 26+y are the two Factors of the other Side; therefore, a:y:2b+ya+2e, or 26+ya+2e::a:y, &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 Inftance, Suppofe b c + b d = da+g, then b √da+g::√da+g: c+d, or √da+g:6::c+d: √da +8,&c;1

CHAP. VIII.

of Subftitution, and the Solution of Quadratick Equations.

W

Sect. 1. Of Substitution.

WHEN new Quantities not concerned in the firft Stating of any Question, are put inftead of fome that are engaged in it, that is called Subftitution. For Inftance, If inftead of be-de you put z, or any other Letter; that is, make z Nbc-dc. Or fuppofe aa+ba—ca+da=dc, instead of b-c +d put s, or any other Letter not engaged with the Question, viz.s=b-i+d, then a asa=de. That is, if c be greater

than bd, it is a a-sa-dc; but if b+d be greater than c then it is a a+sa=dc.

And this way of fubftituting or putting of new Quantities inftead of others, may be found very ufeful upon feveral Occafions; viz. in Order to make fome following Operations in the Queftion more easy, and perhaps much fhorter than they would be without it, as you may obferve in fome Queftions hereafter propofed in this Tract.

And when thofe Operations, in which the fubftituted Quantities were affifting or ufeful, are performed according as the Nture of the Queftion required, you may then (if there be Occafion) bring the original or firft Quantities into the Equation, in the Place (or Places) of thofe fubftituted Quantities, which is called Reftitution, as you may fee further on.

Sect. 2. The Solution of Duadzatick Equations.

WHEN the Quantity fought is brought to an Equality with

thofe that are known, and is on one Side of the Equation, in no more than two different Powers whofe Indices are double one to another, thofe Equations are called Quadratick Equations Adfected; and do fall under the Confideration of three Forms or Cafes.

When there happens to be more Terms in one of thefe Kind of Equations than two, and the higheft Power of the unknown Quantity is multiplied into fome known Co-ëfficients; you muft reduce them by Divifion; as in Sect. 4, of Chap. 5. and for the Fractional Quantities that may arife by thofe Divifions, substitute another Quantity doubled.

For Inftance, let baa+caa-ca-da-dc+cb, then aaca-da de + c b 2x, and if you please,

===

d

Make

for

de + cb
btc

put z. Then will a a— 2x az be the new Equation, equal to the other, being now fitted for a Solution.

Now any of thefe three Forms of Equations being thus prepared for a Solution, may be reduced to fimple Powers by cafting off the fecond or loweft Term of the unknown Quantity; which is done by Subftitution; thus, always take half the known Coefficient, and add it to (Cafe 1.) or fubtract it from (Cafe 2.) it's fellow Factor; and for their Sum, or Difference, Substitute another Letter; as in these.

2 and 67 a+b=vbb+de, per Axiom 5. 7b8a= √ bb + dc: — b

In Cafe 3. From Half the known Co efficient fubftract it's fellow Factor.

And this Method holds good in those other Equations, wherein the higheft Powers are a*, a, a3, &c. As, for instance,

Let 1a6+263 dc Cafe 1.
Put 2 a+b=e
22 396 +2ba3+bbee
3-14bbee-dc
4+cd5eebb + de
5 ww2 6e = √ b b + d c
2 and 67a3+b= √ b b + dc
7b8a3 = √bb+dc:—b

The fame may be done with all the reft, Care being taken to add, or fubtract, according as the Cafe requires.

But all Quadratick Equations may be more eafily refolved by compleating the Square, which is grounded upon the Confideration of raifing a Square from any Binomial, or Refidual Root. (See Sect. 5. Chap. 1.) Viz, if a + b be involved to a Square, it will be a a+2ba+bb; and if ab be fo involved, it will be a a2ba+bb. Whence it is easy to obferve, that a a+ 2 ba=dc (Cafe 1.), and aa-2bade (Cafe 2.), are imperfect Squares, wanting only bb to make them compleat. And therefore it is, that if half the known Co-efficient be involved to the fecond 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 aa+2ba=dc 2 b is b, which being squared,

But 2

bb bb

is bb.

I+23 aa+2 ba+bb=dc+bb Cafe 1.

3 w2 | 4a+b=dc+bb, as before.

2

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