I. The great between CHAP. IX. Of Algebraical Addition, Subftra&tion, Multiplication, and Divifion. N (*) Algebraical Operation's Letters are made Use of to denote, either the Difference Numbers themselves, or the Things number'd. The great Difference between Algebraical and common Arithmetick lies in this, that Algebra admits of negative Quantities, which common Arithmetick does not, Algebra and common Arith metick, wherein it lies. 2. 'An affirmative and nega Where it is to be noted, that, as by an affirmative Quantity, is denoted fome real Thing it felf; so by a negative Quantive Quan tity, is denoted the Defect or Want of tity, what, fome real Thing. An affirmative Quantity, either actually has, or elfe is (†) fuppos'd to have the Sign + affixt to the Left-hand of it; and a negative Quantity, always has the Sign actually affixt to the Left on it. Infomuch, that any (*) Algebra denotes in the Arabick Language, as much as the Great or Excellent Art. (†) Namely, it is ufual, not actually to affix the Sign to a fingle affirmative Quantity, or to an affirmative Quantity, that ftands at the Beginning of any compound Quantity. Quantity, - Quantity, which has no Sign fo affixt to it, is to be esteem'd an Affirmative. Thus, 3A (or 3B, or 3C, &c.) may denote three Shillings; and then -3A (or -3B, or 3C, &c.) will denote the Want of three Shillings. Now, that the four primary Operations in Algebra, are agreeable to thofe in common Arithmetick, but where the Nature of negative Quantities caufe fome Variation, will appear as we go along the faid Operations. Algebraical Addition I. The Addi Quantities gathers into one Sum Quantities, of tion of the fame Denomination, i. e. Quantities of the fame which are expreft by the fame Letter or Denomina Letters, and by the fame Sign, Affirmative or Negative. For Inftance: EXAMPLE I tion. -A or -1. common Arithmetick. EXAMPLE III. bc A+ B— Cori+1—i -6A —6. 4A+2B—5C 4+2--5 7 a -7A -7. 5A+3B-6C 5+3-6 Namely, 2. Namely, as in Example I, 2 Shillings (denoted by 2A) added to 5 Shillings denoted by 5A) make together 7 Shillings (denoted by 7A), fo in Example II, the Want of 6 Shillings (denoted by -6A) added to the Want of one Shilling (denoted by -A) makes together the Want of 7 Shillings, denoted by →7A. If the Quantities be of feveral DenomiThe Addi- nations, then, there are two different tion of Quantities Ways of Working, according to the of feveral two different Cafes that may happen: Namely, Denomina tions. 3. Cafe the ift. Cafe ift, If the Species or Letters, by which the Quantities are expreft, be different, then they can't be properly added, i. e. can't be gather'd into one Sum; but can only be connected together by the Sign of Addition, (viz. + either expreft or understood. Thus, EXAMPLE IV. Where it is to be noted, that in this Cafe Algebra is agreeable to common Arithmetick. For therein likewife Numbers of feveral Denominations can't be properly added, while they continue fuch, but only connected by the Sign of Addition, which it is ufual to understand, not to express. Thus 2 Shillings and 3 Pence can't be added together, while they continue in the faid different Denominations, but by writing them together, S. d. d. thus, 2 2d. Cafe 2d, If the Letters be the fame, 4. but the Signs of the given Quantities be Cafe the different, then the faid Quantities destroy one the other in a like Proportion or Number. EXAMPLE VII. EXAMPLE VIII. EXAMP. IX. The Reason of thus Working will be evident, if the Nature of negative Quantities be confider'd. For a negative Quantíty denoting the Defect of a Thing, hence to add a negative Quantity, is in reality to fubftract. Hence, in Example |