(Greater than, The Difference between b and c, when it is not known which of them is the Greater. b Is to be Involved, or raised to fome Power b Is to be Evolved, or fome Root to be Extracted out of it. 2 √b, (or √ b), ↓ b, ↓ b, c. Signifie the Square-Root of b, the Cube-Root of b, the Biquadrat-Root of b, &c. respect ively, B 2 Befides Befides the foregoing Signs, which are commonly used, I make ufe of others which are not Common, and are as follow. Some Quantity indefinitely Lefs than the Term that next precedes it, is to be added. Some Quantity indefinitely Lefs than the Term which next precedes it, is to be fubtracted. Either ore, when it matters not which of them it is. There are other Signs which are to be used in the Geometrical Part of Algebra, and which will be explain'd in Book II. All the Quantities concern'd in any Question or Problem may ftand in any Order at Pleasure, Viz. The moft convenient for Operation: As a + bd, may ftand thus, bda, or thus, ad+b, or thus, d+a+b, &c. thefe ftill being the fame, tho' differently plac'd. That Quantity which hath no Sign before it (as generally the leading Quantity hath not) is always understood to have the Sign before it, as a isa, or b· d is b d, &c; for the Sign is an Affirmative Sign, and therefore all leading, or pofitive Quantities are understood to have it, as well as thofe that are to be added. But the Sign being a Negative Sign, or Sign of Defect, there is a Neceffity of prefixing it to that Quantity to which it belongs, wherever the Quantity ftands. When any Quantity is taken more than once, you must prefix its Number to it, as 34 ftands for three times a, and 76 ftands for feven times b, &c. All Numbers thus prefixt to any Quantities, are call'd Co-efficients, or Fellow-factors, because they Multiply the Quantity; and if any Quantity be without a Co-efficient, it is always fuppofed or understood to have an Unit prefixt to it; as a is 1 a, or be is 1 bc, &c. All Quantities that are exprefs'd in Numbers only (as in Vulgar Arithmetick) are called abfolute Numbers. Like Quantities are those which are exprefs'd by the fame Letters under the fame Power; as b and b, a a and a a, cd b and 6 cd b. &c. Unlike Quantities are fuch as are exprefs'd by different Letters, or by the fame Letters under different Powers ; as a and b, c df, and cd, b3 and b3, &c. Thofe Thofe Quantities that are reprefented by fingle Letters, as 4, b, c, d, &c. or by feveral Letters that are immediately join'd together, as a b, cd, or 7 b d, &c. are called Simple, or fingle Quantities. con But when different Letters, or unlike Quantities are nected together by the Signs or, as a t b, or a—b, or ab + dc, or aaa, &c. they are called Compound Quan tities. When a Compound of divers Quantities is jointly confidered as one, it is exprefs'd, either by putting them between two Colons, or by drawing a Line over them; As a a+ba: or aa+ba, fignifies the Square-Root of a aba, confider'd as one Quantity: And c d +e: × f, or c + d + exf, fignifies that the whole Compound Quantity c+d+e, is Multiplyed by f. Algebraick Integers are fuch Quantities as are not express'd Fraction-wife: As a, a + b, &c. But when Quantities are exprefs'd, or fet down like Vulgar at d bat de Fractions, thus, or a÷b, or b) a, òr b or > &c. They are called Fractional or broken Quantities. Sect. 2 c-p Of the Tracing the Steps ufed in bringing The Method of tracing the Steps ufed in bringing the Quantities concern'd in any Question to an Equation, is best perform'd by Regiftring the feveral Operations with Figures and Signs plac'd in the Margin of the Work, according as the feveral Operations require, being very useful in long and tedious Opera tions. For Inftance, If it be requir'd to fet down and Register the Sum of the two Quantities a and b, the Work will stand Thus, I4 b 1+23a+b and First fet down the propos'd Quantities b, over-against the Figures 1 and 2, in the fmall Column (which are called Steps) and against 3 (the third Step) fet down the Sum, viz. a+b; then against the third Step, fet down I 2, in the Margin, which denotes that the Quantities against the first and fecond Steps are added together, and that thofe in the third Step are their Sum. To illuftrate this in Numbers, Suppofe a=9, and b = 6, then it will be Thus Thus, 146 216 = 6 1+2/3│a+b=9+6=15 Again, if it were required to fet down the Difference of the fame two Quantities, it will be Or if it were required to fer down their Product, then it will be Thus, I a = 9 26 =6 Note, Letters fet, or join'd immediately together, (like a Word) fignifie the Rectangle, or Product of those Quantities they reprefent; as in the laft Example, wherein a b=54 is the Product of a=9, and b= 6. Arioms. 1. If equal Quantities be added to equal Quantities, the Sums of thofe Quantities will be equal. 2. If equal Quantities be taken from equal Quantities, the Quantities remaining will be equal. 3. If equal Quantities be Multiplyed by equal Quantities, the Products will be equal. 4. If equal Quantities be Divided by equal Quantities, the Quotients will be equal. 5. Thofe Quantities that are equal to one and the fame thing, are equal to one another. 6. Every Whole is equal to all its Parts taken together. PART 7 PART I. Of whole Quantities. CHA P. II. Addition of whole Quantities. ADdition in Algebra may be easily learn'd, by observing the following particular Rules or Cafes. Rule 1. When Simple and like Quantities having like Signs are to be added. Add the Co-efficients, or prefixt Numbers together, and to their Sum adjoin the Letters common to each, or in either of the faid Quantities. Laftly, to this Sum prefix the common Sign, and you'll have the Sum required. Ex. I. -+ b N.B. Ex. fignifies Example. Ex. 2. -a Ex. 3. 20b ca 34bc a Ex. 4. -52bc -81bc Ex. 6. . ༡2༢༡ 1 + 2 + 3 + 4 | 5 | 1 4 0 6 a b c d The Reafon of thefe Additions is evident from the Work of common Arithmetick; for suppose b to represent I Crown, to which if I add 1 Crown, the Sum will be 2 Crowns, or 2 b (or +2b) as in Ex. I. Or if we fuppofe Want, or Debt of I Grown, to which if an other Want or Debt -a to reprefent the of 1 Crown be added, the Sum must needs be the Want or Debt of 2 Crowns, or 2 a, as in Ex. 2. And fo for all the rest. Rule 2. When Simple and like Quantities having unlike Signs are to be added. Add all the Affirmative Ones into one Sum, and all the Nega tive Ones into another (by Rule 1.); then prefix the Difference of the |