THE ALGEBRAIST's COMPANION. Explanation of the Signs made Use of in this Algebra. + Signifies Addition, and is termed Plus, or more. less. · Signifies Division, or 24, or хху. N QSignifies Involution. yu Signifies Evolution, or the Root to be extrac ted. :: Signifies continued Proportion. Signifies Equality, or equal to Signifies the Root to be extracted. coSignifies the Square must be compleated. AXIOMS. 4 Χ Ι ο Μ M S. 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. A L G EB RA S a specious Arithmetic, or an Arithmetic in Let ters; it consists of Addition, Subtraction, Multiplication, Division, Involution, and Evolution, &c. or it is the Art of abstract Reasoning upon Quantity, by general and indefinite Representations, in order to resolve Problems, invent Theorems, and to demonstrate both. A D DI I TI O N. Rule ist. 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 17*, N. B. 1+2 in the Margin, opposite the third Step, shew you, that the first and second 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 Lettér after it, and the Sign of the greater Quantity before it. E X A M P L E S. 4* I 2Xy_76c+16 -3* 70 16xy+762—30 -120 2 i+23 To explain why an Affirmative Quantity added to a Negative, is made so much less, as the Negative is in Value, let us imagine a Person to have 40% Pounds; this Quantity represents the Value of his ready Money, Lands, Plate, Debts, &c. in his Porsession, or owing to him ; this 40x would be the real Sum the Person is worth, if there were no Debts due from him to others. But let us suppose he owes 17%, which, tho' it be something real, is yet of a direct contrary Nature to what is due to him, and must be expressed with a direct contrary Sign, and in the Nature of Debts, must destroy so much of the Person's Eltate, as the Debt is in Value. Hence, if the Debts equal the Estate, they destroy each other, and the Person is worth nothing. If the Debts exceed the Estate, the Person is then worse than nothing ; but if the Estate exceeds the Debts, it is an Affirmative, and the Person's real Worth is the Difference. Rule 3d. If the Quantities to be added be of different kinds, and such 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 several 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. 1+2 i3lx+y a-6 7m+12x+4dc+44—20 3x4+4abc-b6+20 2bb-3XX--2cbcm 15 3 2dd—2bb+abc+7 1+2+3/4/ zabc-b6+12+2dd Subtraction Subtraétion of Algebra I cular 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 fupposing them changed) for since it has been already observed, the subtracting any one Quantity from another, is the same in Effect as adding its contrary ; and since changing the Sign of every Member to be subtracted, 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. |