to the third, their Sum will be 46; and if the first be added to the third, their Sum will be 36. What are the Numbers? That is, a te=22. e+y=46. and a+y=36. Now the Queftion is perfectly limited, each fingle Quantity having but one fingle Value, to wit a = 6, e16, and y = 30. N. B. If the Number of the given Equations exceeds the Number of the Quantities fought; they not only limit the Queftion, but oftentimes render it impoffible, by being propofed inconfiftent one to another. Having truly stated the Queftion in it's fubftituted Letters, and found it limited to one Anfwer (or at least so bounded as to have a certain determinate Number of Answers) then let all thofe fubftituted Letters be fo ordered or compared together, either by adding, fubtracting, multiplying, or dividing them, &c. according as the Nature of the Queftion requires, until all the unknown Quantities except one, are caft off or vanifhed; but therein great Care must be taken to keep them to an exact Equality; and when that unknown Quantity, or fome Power of it (as Square, Cube, &c.) is found equal to thofe that are known; then the Question is faid to be brought to an Equation, and confequently to a Solution, viz. fitted for an Anfwer. But no particular Rules can be prefcribed for the cafting off, or getting away Quantities out of an Equation; that Part of the Art is only to be obtained by Care and Practice. And when that is done, it generally happens fo, that the unknown Quantity which is retained in the Equation, is fo mixed and entangled with thofe that are known; that it often requires fome Trouble and Skill to bring it (or it's Powers, &c.) to one Side of the Equation, and those that are known to the other Side; (ftill keeping them to a juft Equality) which the ingenious Mr Scooten in his Principia Mathefeos Univerfalis, calls Reduction of Equations. The Bufinefs of reducing Equations (as of most, if not all Algebraick Operations) is grounded and depends upon a right Application of the five Axioms proposed in Page 146, and therefore, if thofe Axioms be well understood, the Reafon of fuch Operations muft needs appear very plain, and the Work be eafily performed; as in the following Sections. R Sect. 1. Of Reduction by addition, EDUCTION by Addition is grounded upon Axiom 1. and is only the tranfpofing (viz. the removing) of any Negative Quantity from either Side of an Equation to the other Side, with the Sign before it; as in these EXAMPLES. bd Again, Suppofej labd1 For 31 Let I b = dj 1 + dj 2! aa- ・d=c-aa I + 34a=d+b2+aa|3|2aa=c+d Note, When any abfolute Number is Let|1|34-4=6—a regiftered in the Margin, you muft draw I + 4/2/3a=6+4—a 2+a34a=6+4=10 a Line over it, to diftinguish it from the other Numbers. As in the 2d Step of this Example. Let 1a adcb=dd — 2 ba I+b2aa-dc=dd- 2 ba+b 2 + dc 3 aa=dd 2 ba+ b + d c 3+26a4a a + 2 ba = d d + b + d c Suppofe 12 dad = cc-3 baa-aa a 1 + ada 3+d RE Sect. 2. Of Reduction by Subtraction. EDUCTION by Subtraction is grounded upon Axiom 2. and is performed by tranfpofing (or removing) any Affirmative Quantity from either Side of the Equation, to the other Side, with the Sign before it; as in thefe EXAMPLES. Suppofe1a6-dl Suppofe 1a a+de+b=dd + 2 ba 1-2 baj2jaa 2 ba + dc + b = d d 2de3aa2ba+b= dd-dc -4=2 Let Let 1 aaa+d=cc+3ba a + 2 da 1-3 baaz aaa- 3 bad + d = c c + 2 da 2-2 da 3-d 3aaa- 36 cc-d Sect. 3. Of Reduction by Multiplication. FRACTIONAL Quantities, in any Equation, are brought into whole Quantities by multiplying every Term in the Equation with the Denominators of the Fractions, per Axiom 3; as in these Ixa+b aaa ba -bb aa- .bb at b aaa= baaabba a − b b b a + b b b b a+b 3aaaa+baabaaa-bba a-bbba+bbb b Sect. 4. Of Reduction by Division. WHEN any Quantity (either known or unknown) is in every Term of an Equation, if the whole Equation be divided by that Quantity, it will be reduced into lower Terms, per Axiom 4, as in thefe following Examples. EXAMPLES. Suppofe | Ibaa+bca = b c " \ - ÷ Ial 21aa+ca=cd 12 19 1 aa=7@ Let 1 ffaa+ffcaa-ffa=ffda+ffdda 1÷ff 2 aa+caa- -ada+dd a 3a+ca-1=d+dd 2 a Or when the unknown Quantity is multiplied (viz. joined) with any that is known; let the whole Equation be divided by the known Quantity, that fo the unknown may be cleared; as in thefe I Suppofe 1 b baaa-2bba a = b d a + c ba 1÷ba 2 baa- 2 ba=d+s Let 49 da a +42aa7bca + 21c a 7daa6aa = bea+ 3 ca 23 1÷7 2013 7da + ba 3÷ 4 a= bc+36 74+6 Sect. 5. Of Reduction by Involution. WHEN there happens to be an Equation, between any ho mogeneal or like Surds, take away the radical Signs from the Quantities, and they will become rational; as in thefe EXAMPLES. Suppofe va=√d+c Let|1|3 Vaa= 3√db+bc & per Sect. 4. a=d+c1@3|2| da= db+bc Chap. 3. Or if one side of the Equation confifts of Surd Quantities, and the other Side be rational, then involve the rational Quantities to the the fame Power (or Height) with the Index of the Surd, and take away the radical Sign; as in these EXAMPLES. Let|1a6 |Suppofe | I |✔a = b + c 122 a=bb2bc+cc IQ 2 a Sect. 6. Of Reduction by Evolution. WHEN any fingle Powers 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 thefe 1|aa=bb=dd | Let | aaa=b3 + 3 bbc +3 bcc + c3 Suppofe|1|a I u 2 a = √ + c 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. Suppose 1 aa+2ba + b b = d d | a a − 2 b a + b b = d d c c I w2|2 a+b= d =dd| a -b=dc Here follow a few Examples of clearing Equations, wherein all the foregoing Reductions are promifcuoufly ufed, as Occafion requires. |