√b, (or ↓ b), √b, b Lefs c, or c Subftracted from b. b Multiplyed by c, or b Multiplying c. b Divided by c, or e Dividing b. b Equal to c. As a is to b, fo is b to c, fo is c to d. As a is to b, fo is c to d. a, b, c, d, have equal Differences. The Difference of a and b, is equal to that of c and d. b Is Greater than c. b Is Lefs than c. 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. sb Is to be Evolved, or fome Root to be Extracted out of it. the Cube Root of b, the b, &c. Signify the Square Root of Biquadrat-Root of b, c. refped bely B 2 Befides the foregoing Signs, (which are commonly us'd) I make use of others, which are not Common, and are as follows. be Signifies, Some Quantity indefinitely Lefs than b, is to {be added to b. Some Quantity indefinitely Lefs than b, is to be Subftracted from b. Either ore, when not material which of them it is. There are other Signs which are to be used in the Geometrical Part of Algebra, and which I will explain in Book II. All Quantities concerned in any Queftion or Problem may ftand in any Order at pleasure, viz. The most convenient for Operation; as a + bd, may ftand thus, b-da, or thus, a d+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 underftood to have the Sign before it, as a ista, or b―d is + bd, &c. For the Sign is an affirmative Sign, and therefore all' leading, or Pofitive Quantities are understood to have it, as well as they 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, whereever the Quantity ftands. When any Quantity is taken more than once, you must prefix its Number to it, as 3a ftands for three times a, and 7b ftands for feven times b, &c. All Numbers thus prefixt to any Quantities, are call'd Coefficients, or Fellow-Factors; because they Multiply the Quantity ; and if any Quantity be without a Coefficient, it is always fuppofed or understood to have an Unit prefixt to it; as a is 1 a, or bc, is ibè, &c. Áll Quantities that are exprefs'd in Numbers only, (as in Vulgar Arithmetick) are called abfolute Numbers. Those Quantities that are reprefented by fingle Letters, as a, b, c, d, &c. Or by feveral Letters that are immediately join'd together, as ab, cd, or 7bd, &c. are call'd Simple, or fingle whole Quantities. But when different Quantities reprefented by different or unlike Letters are connected together by the Signs or, as 4 + b, or a—b, or a b + dc, or a + aa. They are call'd Compound whole Quantities. And when Quantities are exprefs'd, or fet down like Vulgar ab+de, &c. They are cal a Fractions, thus or Ъ a+d, or b led Fractional or broken Quantities. Like Quantities are thole which are exprefs'd by the fame Letters under the fame Power; as b and b, a a and a a, cd b and 6cdb, &c. Unlike Quantities are fuch as are exprefs'd by different Letters, or by the fame Letters under different Powers, as a and b, cdf and cd, b3 and bs &c. Sect. 2. Of Tracing the Steps used in bringing Quantities to an Equation. The Method of tracing the Steps ufed in bringing the Quantities concern'd in any Queftion to an Equation, is beft perform'd by Regiftring the feveral Operations with Figures and Signs pla ced in the Margent of the Work, according as the feveral Operations require; being very useful in long and tedious Operations. For Inftance, if it be required to fet down and Register the Sum of the two Quantities a and b, the Work will ftand Thus, 14 2 b First fet down the propos'd Quantities a and b, over against the Figures 1 and 2, in the fmall Column (which are call'd Steps) and againft 3, (the third Step) fet down the Sum, viz. ab; then against the third Step, fet down +2, in the Margin; which denotes that the Quantities against the first and fecond Steps are added together, and that thole in the third Step are their Sum. To illuftrate this in Numbers, fuppofe a=9, and b=6, then it will be 1+23a+6=9+6=15 Again, if i were required to fet down the Difference of the fame two Quantities, Then it will be Thus, Or if it were required to fet down their Product, then it will be Note, Letters Set, or join'd immediately together, (like a Word) fignifie the Rectangle, or Product of thofe Quantities they Reprefent; as in the laft Example, wherein a 6 = 54, is the Product of a = 9, and b = 6. Axioms. 1. If equal Quantities be added to equal Quantities, the Sums of thofe Quantities will be equal, As if a beb, and ed, then a fe will be=b+d, or a+d=b+c, by the firft Axiom. 2. If equal Quantities be taken from equal Quantities, the Quantities remaining will be equal; fo if a beb, and cd, then ac, will be bd, or adbc, by the fecond Axiom. 3. If equal Quantities be Multiplied by equal Quantities, the Products will be equal. So if a beb, and c➡d, then a c will be bd, or a dbc, by the third Axiom. 4. If equal Quantities be Divided by equal Quantities, their Quotients will be equal. Example, If a beb, and cd, then b d b (or ac) will be (or b÷d ;) or 1: by the fourth Axiom, C and 5. Those Quantities that are equal to one and the fame thing, are equal to one another. As for Inftance, if a be= q, then a will be b, by the fifth Axiom. PART PART I Of whole Duantities. CHA P. II. Addition of whole Quantities. ADdition in Algebra may be eafily 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 Coefficients, 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. The Reafon of thefe Additions is evident from the Work of Common Arithmetick; for fuppofe b to represent 1 Crown, to which if I add 1 Crown, the Sum will be 2 Crowns, or & b, as in Ex. 1. Or if we fuppofea to reprefent the Want, or Debt of one Crown, to which if another Want, or Debt ef 1 Crown be added, the Sum must needs be the Want or Debt of z Crowns, as in Ex. 2. And fo for all the reft. 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 Negative ones into another, (by the firft Rule) then prefix the Difference of the |