Sect. 4. Reduction by Diviliou.

When you find out any Divifor of, or any Quantity that will exactly Divide the Equation;

Divide the Equation by that Divifor, and you'll reduce it in to lower Terms.

Note, a fimple Divifor may be easily discover'd, only by the Infpection of the Equation: But, in order to difcover a Compound Divifor, and then to apply it as above directed, you are to reduce all the Terms of the Equation to one Side, making them equal to 0,

Examples.

Suppofelibaaa+14bcaa= 49bcda 17ba2aa + 2ca7cd

Letila32caa-baazbca - bbc

=

·3bca + bbc2a3 +2caa-baa- 3bca+bbc=0 2:a― -6:13aa + 2ca bc=0

When the greatest Power of the unknown Quantity is Multiplyed by any that is known ;

Let the whole Equation be Divided by that known Quan tity, that fo the faid Power may be cleared. As in thefe

Suppofe

Sect. 5. Reduction by Involution.

When any of the Powers of the unknown Quantity is adfected, in one or more Terms of the Equation, with one or more Surd Indices, or Radical Signs, thefe Surds being in their moft Simple Terms;

Tranfpofe, if need be, fuch Terms of the Equation as are moft fit for your Purpose: Then Involve each Side of this Equa tion according to the Denominator (1 being the Numerator) of the leaft Index of the Surds. Repeat thefe Directions, if need be, and the Power, or Powers of the unknown Quantity will, in fome Cafes, become Rational: But for an univerfal Method you must have Recourfe to the Scholium in the latter End of this Part.

Suppofe vaid+c: 1 2 2a =d+c

Leta+c = √ ba 10324 + c = √ bbbaan 223/aa+2ac + cc = b3 a3

Let 1:3cba-c3: — c = √ Ea

I + c2:3cba — c3 := c + √ ba

2333cba — c3 = c3 + 3cc √ ba + zcba + ab √ ab Tranfp.42c3 = : 306 + ab : × √ ab

4C 2/5)+4c° = 9bst a + 6bbccaa + b3a3

Sect. 6. Reduction by Evolution.

When one Side of an Equation contains only known Quan-. tities, and the other Side, containing fome Power or Powers of the unknown Quantity, has a Rational Root;

Evolve each Side of the Equation, according to the Index denominating that Root; and you'll reduce the Power or Powers of the unknown Quantity to lower Dimensions.

Suppofe 1 aa 2ab+bbcc + 2df - fg

I us 22 a b = √ : cc + 2df-fg:

Here follows one Example of clearing Equations, wherein all the foregoing Reductions are promifcuously us'd, as occafion requires.

4

4

IX √4C 2:Can + 3bbc: — -N:caa 3bbc:

√4baa

22 3 caa + 3bbc - 21: caa+3bbc: × √: can—zbbc: + caa 3bbc4baa

-

2 √:ccat 9ccb4:=4baa

That is

42caa·

3bbc

For caa + 3bbc + caa

And

2caa

2√:caa+3bbc:×↓:caa_3bbc:=21/ :ccaa—9ccb4; √:cca4 9ccb4: 2baa

5+v: 6caa2baa + √:cca+ - 9ccb4: zban =v: ccat 9ccb4 :

6- 2ban 7 can

702 8ccat 4bcat + 4bba4 = ccat= 8 Tranfp. 99ccb4 = 4kcaa — 4bbar

9 = b 109ccb 3 = 4 car

10-40-4611

9ccb4

By

By the Help of thefe Reductions (properly applyed) the unknown Quantity (a) or its Powers are cleared and brought to one Side of an Equation; and if the unknown Quantity (a) is found thofe that are known, then the Question is anfwer'd, as in the ft. Examples of Sect. Ift, 4th, 5th, and oth.

Or, if any fingle Power of the unknown Quantity (a) is found equal to thofe that are known, then the refpective Root of the known Quantities is the Anfwer, as in the three last Examples.

But when different Powers of the unknown Quantity are contain'd in an Equation, as aa+ba = dd, or a3 da'cf, Bc. then it is an adfected Equation; the Method of refolving which, that is of finding the Values of a therein has been fhewn in Part IV. Chap. 3. And other Methods for the fame Purpose shall be fhewn further on,

CHA P. II.

How to reduce Equations, containing two or more unknown Quautities into a single Equation.

T hath been fhewn in the preceding Chap. how to reduce fingle Equations: But, in the Solution of Questions, these are frequently to be first deduc'd from others that contain two or more unknown Quantities. It will therefore be proper likewife to fhew how to Exterminate by two, three or four, &c. Equations (concern'd in any limitted Queftion, and not depending upon one another) one, two or three, &c. unknown Quantities, But fince upon the preceding Sect. these in this Chap. depend, it is proper that the foregoing thould (as they are) be first treated off.

Sect. 1. The Extermination of one unknown Quan tity by two equations.

Cafe I,

When the Quantity to be Exterminated is only of one Dimenfion in one of the Equations, its Value is to be fought in that Equation; and then this Value and its Powers to be Subftituted for that Quantity and its refpective Powers in the other Equation. Thus,

If 1.a+b=e, and 2. aa + ca=dce, that a may be Exterminated.

By ft. ae-b; then, by this Step and 2d. :e-b: x :e-b: +cx:eb: dce ; which Step, being reduc'd, by Chap. I. gives ee2be + 2ced - bb + cb.

Cafe 2.

When the Quantity to be taken away is of two, or more Dimensions in both Equations, the Value of its greatest Power must be fought in both; then, if thofe Powers are not the fame, the Equation that contains the lesser Power must be Multiplyed by the Quantity to be taken away, or by its Square, or Cube, &c. that it may become of the fame Power it has in the other Equation. Then the Values of thofe Powers are to be made equal, and there will come out a new Equation, where the greateft Power or Dimenfion of the Quantity to be taken away is diminith d.

And by repeating this Operation, the Quantity will, at length, be taken away; thus,

If 1. aee+be+c=0, and 2.fee + ge+b=0; and e is to be taken away.

Then, by the ift.ee: be + c:a, and, by the 2d. - ee g+bf. By the two laft Equations :be+c: ÷ a (= — ee) =: ge+b:÷f. This Equation Multiplyed by af gives fbe +fc=age + ah. And, by Tranfpofition, =age+ab. and Divifion *e =: ab - fc: fb-ag:

e, we have

Again, Multiplying the laft Equation by -ee:fce-abe :: fb - ag: From the 3d. and laft Equations be+c: ÷ a = : :fce abe::fbag: And, by Multiplication, Tranfpofition and Division, e : agc — bcf: aab afc agb+fbb:. By the laft Equation, and that mark'd with we have agc-fbc

*

aabafc-agb+fbb

ab fc fb -a ag

which is an Equation exclufive of the Quantity e, as was required. This Equation, when reduc'd, by the laft Chap. gives bbaa - 2cfha - Igha+rgga+bbfh — b·fg+ccff=0.

Note, Ey the first or fecond Equations and that mark'd with, you may more expeditiously Exterminate the Quanti ty e, by Cafe 1.

Note ello, the Quantity e may be Exterminated out of the preceding Example, by finding the Value of e in eithr of the firft to Equations, by Part X. and then proceeding by Cafe j. This in fome Caf ́s, happens to be a more conce. Methed than, but not so univerfal as, the foregoing one