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RE duction of Equations. If the Quantities of the Equation be fractional, make them on each side pure Fraćtions, then multiply them cross-wise and the Produćt may be reduced as before.

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which is by completing the Square, and is performed by the following Rule

Add the Square of half the Co-efficient of the unknowu

quantity to both sides of the Equation, and the Square will be complete,

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mainder.

Of PROBLEMS.

1. Problems are Questions to be solved.

2. The Solution of a Problem is, the Answer to a Question, or the Determination of the quantity sought.

3. The Problem has oftentimes various Answers, and therefore it is necessary to know when it is truly limited; which may be known by the following * *

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Rule. . .

When the Number of Quantities sought are equal the number of Equations (not depending on each other) the Question is truly limited.

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Ax. 1. If from the Sum of any two Quantities either Quantity be taken, the remainder is the other Quantity. Ax. 2. The difference of any two Quantities being added to the less, the Sum is the greater. Ax. 3. The Product of any two Quantities being divided by either Quantity, the Quotient is the other. Ax. 4. The Quotient of any two Quantities being multiplied by the less, the Product is the greater. Ax. 5. The Rećtangle of the Sum, and Difference of any two Quantities, is equal to the difference of their Squares. Ax. 6. The Difference of the Squares of the Sum, and difference of any two Quantities, is equal four times their Rećtangle. Ax. 7. The Sum of the Squares of the Sum, and dif

ference of any two Quantities, is equal twice the Sum of —their Squares. * - x+y

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Prob. 2. The Sum of two Numbers=za and their Product

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Prob. 4. The Sum of any two Numbers being 14, and the Sum of their Squares-106, required the Numbers? "

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Prob. 7. The Difference of two Numbers being 4, and their quotient 1, 8 required the Numbers ?

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Prob. 8. The IYifference of two Numbers 4, and the Sum of their Squares I off, required the Numbers 2 x

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