THE

ALGEBRAIST'S COMPANION.

Explanation of the Signs made Ufe of in this Algebra.

+Signifies Addition, and is termed Plus, or

more.

Signifies Subtraction, and is called Minus, or lefs.

x Signifies Multiplication.

Signifies Divifion, or 24, or *xy.

Signifies Involution.

n

yu Signifies Evolution, or the Root to be extracted.

:: Signifies Proportion.

Signifies continued Proportion.

Signifies Equality, or equal to

Signifies the Root to be extracted.

CSignifies the Square must be compleated.

AXIOMS.

First, If to equal Things are added equal Things, their Sums are equal.

Second, If from equal Things, equal Things are taken away, their Remainders will be equal.

Third, If equal Things be multiplied by equal Things, their Products will be equal.

Fourth, If equal Things be divided by equal Things, their Quotients will be equal.

Fifth, Things equal to one and the fame Thing, are equal to one another.

ALGEBRA

'S a fpecious Arithmetic, or an Arithmetic in Let

ters; it confifts of Addition, Subtraction, Multiplication, Divifion, Involution, and Evolution, &c. or it is the Art of abftract Reafoning upon Quantity, by general and indefinite Reprefentations, in order to refolve Problems, invent Theorems, and to demonftrate both.

ADDITIO N.

Rule ift. If Quantities are alike, and have the fame Sign before them, to be added together, put down the Sum of the Coefficients with the common Sign before them, and the Common Letter after them.

EXAMPLES.

N. B. 1+2 in the Margin, oppofite the third Step, fhew you, that the first and fecond Steps are added together, and the Sum is placed against the third Step.

Rule 2d. If the Quantities to be added have dif ferent Signs before them, then put down the Difference of the two Coefficients, with the common Letter after it, and the Sign of the greater Quantity be

To explain why an Affirmative Quantity added to a Negative, is made fo much lefs, as the Negative is in Value, let us imagine a Perfon to have 40x Pounds; this Quantity reprefents the Value of his ready Money, Lands, Plate, Debts, &c. in his Poffeffion, or owing to him; this 40x would be the real Sum the Perfon is worth, if there were no Debts due from him to others. But let us fuppofe he owes

17%, which, tho' it be fomething real, is yet of a direct contrary Nature to what is due to him, and must be expreffed with a direct contrary Sign, and in the Nature of Debts, muft deftroy fo much of the Perfon's Estate, as the Debt is in Value. Hence, if the Debts equal the Eftate, they deftroy each other, and the Perfon is worth nothing. If the Debts exceed the Eftate, the Perfon is then worse than nothing; but if the Eftate exceeds the Debts, it is an Affirmative, and the Perfon's real Worth is the Difference.

Rule 3d. If the Quantities to be added be of different Kinds, and fuch as will not incorporate, put them down in any Order, one after another, with their proper Signs before them; and this is all the Addition they are capable of. The Addition of compound Algebraick Quantities, is performed by collecting the feveral Members of every particular Species into as many Sums as there are Species; and then putting down the Sums in any Order, with their proper Sign before them.

Subtraction of Algebra

S performed by changing the Sign of every particular Member of the Quantities to be subtracted, i. e. making every negative Member affirmative, and every affirmative Member negative, and then adding it to the other, (or fuppofing them changed) for fince it has been already obferved, the fubtracting any one Quantity from another, is the fame in Effect as adding its contrary; and fince changing the Sign of every Member to be fubtracted, renders that Quantity just the contrary to what it was before, it is evident, that after this Change it may be added, and that the Sum of this Addition will be the true Difference.