addcdd-acc-c3 Now, if neither the Terms aa-cc and 2a-2c had 4ad-2cc not had a common Divifor, the propos'd Fraction would have been irreducible. And this is a general Method of finding common Divifors; but most commonly they are more expeditiously found by feeking all the prime Divisors of either of the Quantities; that is, fuch as cannot be divided by others, and then by trying if any of them will divide the other without a a3—aababb—13 Remainder. Thus, to reduce to the leaft Terms, you aa-ab muft find the Divifors of the Quantity aa-ab, viz. a and a-b; then you must try whether either a, or ab, will alfo divide a3—aababb—b3 without any Remainder. The End of the Third P AR T. ARITH ARITHMETICAL INSTITUTIONS. 425. PART IV. Of the Doctrine of EQUATIONS. A CHAP. I. Of Equations in General. DEFINITION I. N Equation is the Expreffion of the Equality between two or more DEFINITION II. 426. The Registring an Equation is the noting down in the Margin how it is formed from one or more preceding ones; as in the following Process: where note, that if any Number be inferted in the Regifter which is not the Number of fome foregoing Step, it is diftinguished by a Line drawn over its Head, as the Number 5 in the 9th Step following. And this Method of Regiftring Operations was first introduced by the ingenious Dr. John Pell. DEFINITION III. 427. Reduction of Equations is the bringing an unknown Quantity to one fide, that its Value may be discovered; and is performed fix feveral Ways, viz. by Addition, Subtraction, Multiplication, Divifion, Involution, and Evolution. DEFINITION IV. 428. Reduction by Addition is the tranfpofing or removing a Defective Quantity to the contrary fide of the given Equation, with the Sign + before it. Ex. gr. 429. Reduction by Subtraction, is the removing a Pofitive Quantity to the contrary fide of the Equation, with the Sign-before it. Ex. gr. 430. Hence it is plain that any Quantity may be transposed to the contrary fide by only changing its Sign. COROLLARY II. 431. And if the fame Quantity be affected with the fame Sign on both fides an Equation, or with different Signs on the fame fide, it deftroys it felf. Ex. gr. If b+2c=b+y, then by fubtracting b from both fides of the Equation, 2c=y: So if y-a-r-a by adding a to both fides, y=r. COROLLA RY III. 432. If all the Terms in an Equation be tranfpofed to the fame fide, they will equal o. Ex. gr. If a=x-x, then a-xzo. If aa=zba+bb then aa-2ba-bb-o (In. 64.) DEFINITION DEFINITION VI. 433. Reduction by Multiplication is the bringing an Equation out of Fractions, by multiplying their Denominators into every Term of the Equation. Ex. gr. 434. Reduction by Divifion is performed by dividing the highest Power of the unknown Quantity by every Factor into which it is multiplied, and every Term in the Equation by every Factor which can be found in them all. 435. Reduction by Involution is the bringing an Equation out of Surds by tranfpofing the Surd Quantity to one fide, and then involving each fide as the fractional Exponent fhall direct. Example And this Method of Registring Operations was first introduced by the ingenious Dr. John Pell. DEFINITION III. 427. Reduction of Equations is the bringing an unknown Quantity to one fide, that its Value may be discovered; and is performed fix feveral Ways, viz. by Addition, Subtraction, Multiplication, Division, Involution, and Evolution. DEFINITION IV. 428. Reduction by Addition is the tranfpofing or removing a Defective Quantity to the contrary fide of the given Equation, with the Sign + before it. Ex. gr. 429. Reduction by Subtraction, is the removing a Pofitive Quantity to the contrary fide of the Equation, with the Sign-before it. Ex. gr. 430. Hence it is plain that any Quantity may be transposed to the contrary fide by only changing its Sign. COROLLARY II. 431. And if the fame Quantity be affected with the fame Sign on both fides an Equation, or with different Signs on the fame fide, it deftroys it felf. Ex. gr. If b2c=b+y, then by fubtracting b from both fides of the Equation, 2c=y: So if y-a-r-a by adding a to both fides, y=r. COROLLARY III. 432. If all the Terms in an Equation be tranfpofed to the fame fide, they will equal o. Ex. gr. If a=x-z, then a-x+x=0. If aa=zba+bb then aa-2ba-bbo (In. 64.) DEFINITION |