A LGEBRA depends upon the fame Definitions and Axioms that a confiderable part of Euclid's Elements do, and may therefore very justly be call'd a New Geometry, thô it has no fmall Advantage above that which is commonly called by that Name; fince the Subftance of whole Sciences at once are fometimes comprehended in one fingle Algebraic Theorem; an Inftance of this the Ingenious Mr. Hally gives us in the late Tranfactions of the Royal Society. And in many Geometric Constructions, which otherwife would be defperate, Algebra furnishes us with fuch Laws and Rules, as fail not to effect to what we want. So that no one can be a good Geometrician that is not a good Algebraist;by confequence no tolerable Mathematician without it: And fince Mathematics are now defervedly grown into that Repute, that they begin to be a necessary part of a Gentleman's Education, (as well as a public Encouragement of them, the Intereft of a Kingdom) therefore Algebra, amongst other Parts (which in fome fenfe may be term'd the whole) will certainly come in for a confiderable Share. As to my own Performance herein, I have omitted nothing that I know of which is useful, but have run through a whole System with what brevity and plainnefs I could, advancing nothing which I have not demonftrated where it's necessary, or shew'd the Reason of the Process. I have applied Algebra to Numeral Questions and Geometry, and fhew'd the Method of Geometric Conftructions; and each of these fo far, that afterwards the Learner may proceed himself to what higher Progrefs his own Induftry and Sagacity may prompt him. As for the Bufinefs of Equations, I have made fome little Progrefs, which I have not feen in any other. In Quadratics I find both the Roots at once; and in Cubics I have determin'd all the fixteen Cafes, and the Qualiities of the Roots, as how great (near) if compar'd with each other, how many Pofitive, Negative, or Imaginary, and this by confidering all the poffible Combinations of the Coefficients arifing only from three Genefes, viz. from three pofitive Roots, or two Pofitive and one Negative; or, lastly, from two Negative and one Pofitive. I have alfo given an Account of Mr.Raphfon's Converging Series,with its Demonftration, and an Application of it to feveral Examples, fufficient to flow how all Equations however high or adfected may be easily refolu'd, and have confider'd as previous to this Series the Punctation of Adfected Equations which is very useful in Common Practice; thô now there's no abfolute neceffity of it, for this reaches the Cafe of Anticipations, a thing troublefome enough in any Exegifis Numerofa that has yet been publish'd. I have yet two other Infinite Approximations for all Adfected Equations whatever, built upon very different Principles from that of Mr. Raphfon's, with new Methods of Extracting Roots, and fomething elfe of this Nature, but do not think fit to publish them yet, especially fince I'm inform'd that Mr. Hally has improv'd Lagny's Method for Adfected Equations; which I doubt not but is fomething worthy the Application of fo good a Mathematician as Mr. Hally is, and therefore will be very welcome to the Public when he pleafes to communicate it. As for the Errata's in this Folio, in tranfpofing it to a Quarto, (which I have printed for my own use) I have alter'd most of them; and if there be any other, which in a curfory reading over have escaped me, I will endeavour to fatisfy any Perfon whatever into whofe Hands this fhall fall, if they pleafe to give themselves the trouble of writing to me about it. An . An Explication of the Characters. Signifies The Root of both the Quantities dd+cc. The Universal Root of the Root of bbq added to ddd. The Cubic Root of a+b, if √(4)a+b, it had been the The Cube of the Binomial a+b. " may fignify the Index of any Power whatever of atb. An Unit divided by a+b. An Unit divided 3 times by a+b,or by the Cube of ath. c divided by the Cubic Root of ab. The Biquadratic Root of the Cube of a+b. (1). ALGEBRA is the Science of Quantity in General. More Particularly, 'Tis an Art of Reasoning with unknown Quantities, in order to discover their Ha bitude or Relation to fuch as are known. S.1. LGEBRA is term'd Literal, or Specious Arithmetic, in oppofition to the Numeral, where the Figures firft taken, are loft or swallowed up in others, which by feveral Operations are deriv'd from them; but in Letters the whole Procefs appears at first fight, and gives one general Theorem for all Questions of the fame Nature. §. 2. In any Operation where Letters are alike, they are supposed to be all of the fame Nature, as 4, 4, 34, &c. But different Letters fuppofe Quantities of different Nature, as b, c, d, &c. unless the contrary be expreft, as b & c, may be given for two Lines that are known Sides in a Triangle, but of different length. S. 2. The Roots of Quantities are diftinguishable by Figures prefix'd, va, or √(2)ab, fignifies the Square Root of ab. √(3)ab, expreflès the Cubick Root of ab. V(4)ab the Biquadratic Root, &c. S. 3. Unity is fuppos'd to be prefix'd to every Quantity, as 14, or once 4, is the fame as a it felf. §. 4. If a Quantity have no Sign before it, is fuppos'd to be prefix'd, as + is the fame with 4. §. 5. The Signx has reference to the whole Quantity that follows or precedes it, if a Line be drawn over every Member thereof, as a xb+c+d, or b+c+dxa, where a is fuppofed to be multiplied into each fimple Quantity b, c, d. 5.6. A Quantity drawn or multiply'd into it felf,is a Squareas axaaa; if multiply'd into its felf three times, 'tis a Cube or third Power, as axaxaaaa; if four times, a Biquadrate, or fourth Power,as a×a×a×aaaaa, &c. And fo in Figures, 9 is the Square of 3, 64 the Cube of 4, &c. Thefe literal Powers, when they rife high, are more commodiously exprefs'd by the Index of the Power it felf fet over them inftead of aad, we write a', and a instead of aaaa; fuppofe the fame in the 5th, 6th, 7th, and higher Powers. §. 7. Quantities are Simple or Compound; Simple, when there's but one Member, as b, or bcd, or ddgg, &c. Compounds, when connected by the Signs-or-, a+b, or ad-ee+ 99, &c. ADDITION of Algebraic Integers. DDITION finds the Sum of two or more) given Quantities, I. Where Quantities have the fame Sign prefix'd, and are of the fame Nature. The Reason of 2 It's felf-evident from the ordinary way of Notation, that 2 and the Procefs.3 make 5, whatever the things be that are added, provided they be of like Nature; as two Men, two Lines, two like Pofitions, (or like B like Quantities with the Sign + prefix'd) two like Negations, (or like Quantities with the Sign-prefix'd ). II. Where Quantities are of the fame Value and Nuture, and have different Signs. The Reason of the Procefs. If has 5000l. and owes yaod it's evident that the Sam (or his whole Estate) is o. So allo in Quantities, if the Line AB be equal to the Line BC, and the laft be fubducted from the firft, (or the first leffened by the last) it's manifeft that the Whole is taken away, or is equal to o. III. Where Quantities have not the fame Value, yet are of like Nature, and have different Signs. Which is very little different from the former Cafe. The Reason of 2 If A has 5000, and owes soo the Whole of his Eftate is only the Process. lellen'd by fo much as the Debt, that is 5000-500, or 4500; but if the Debt had been greater, his Eftate had been-4500, or 4500 worse than nothing. Suppose the fame in Quantities. The Reason of the Procefs. 1. To 366 + 190 — ad ef To 296b —¿3 + ď3—16q” + 17 IV. Where Quantities are of a differem Nature. 13. 13 94s que 4 Where Quantities are of different Species, (whether the Signs be like or unlike) they are incapable of further Connection, and muft therefore be fet down with their own Signs prefix'd; two Men and three Horfes, make not five Men or five Horfes. To bb-cc Add q+d To d3 + qq + dd-b Add rr+ss To 13+d Add 3gg-bb ・Sam bb cc + q td Sum dad -qq+dd-b+rrts Sum 13+d+-388-bb S SUBTRACTION in Algebraic Integers. UBTRACTION finds the Difference (whether Defect or Excess) between any two propos'd Quantities. The three Cafes that occur in this Rule, are, Subducting Quantities, whofe prefix'd Signs are, (1.) +and+. (2.) and (3.)-and The The Reafon of the } I. To take a Pofitive Quantity out of a Pofitive One, is the fame thing as to fubjoin the Defect of the Quantity taken; thus 3 taken out of 5, leaves 5-3, or 2. II. To fubtract a Negative Quantity out of a Pofitive One, is the fame thing as to fubtract the Subtraction, or take away the Defect of the Quantity taken; and to take away the Defect of a Quantity, is to put it in a positive State: Thus 3 taken out of 5, gives 53, or 8. III. To take a Negative Quantity out of a Negative, is (as before) to take away the Negation of fuch Quantity. Thus-3 taken out of-5, gives-5+3, or-2. Hence arifes this general Rule for Subtraction, both in whole Numbers and FraЄtions. Rule. Change the Sign or Signs of the Quantity fubtracting. From 54 From 3bb Take-2bb From-3cd From-dd--ee-gg Rem.-dddd-tee-qq-88 Note. That when it's doubtful whether Quantity is greater, the Difference of them is usually expreffed by this Character ab, and ab-dodd-e. MULTIPLICATION in Algebraic Integers. M ULTIPLICATION finds the Product or Rectangle of any two propos'd Since Multiplication is nothing else but a taking the Multiplicand, fo often as there are Unites in the Multiplier. Therefore, As to the Multiplier::, fo the Multiplicand to the Product. Here also occur three Cafes. (1.) When multiplies +. (2.) When + multiplies, or the contrary. (3.) When multiplies The Reafon of the 1. In multiplying 4 into-5, I put or repeat the Pofition Procefs in each. S of 5, fo often as there are Unites in the Multiplier; therefore the Total will be positive, viz. 20. 71 2. In multiplying + 4 into -5, I put the Negation of 5 four times; therefore 4x-5=-20. And in the Converfe, in multiplying 4 into +5, I deny the Pofition of 5 four times; therefore -4x5-20. 3. In multiplying 4 into-5, I deny the Negation of four times; therefore it's made Pofitive four times, viz. +20. 4. Multiplication is either of Simple Quantities into Simple Quantities, or Sim ple Quantities into Compound ones; or, laftly, Compound Quantities into Gempound. 1. Simple Quantities are immediately connected. If one Number multiplies another, the Product fall be equal to two Products made of the Multiplication of the first Number, into two parts of the fecond divided at pleasure: Let 4 multiply 7 for a Product, divide 7 into any two parts, +2, and multiply each part by 4 for two more Products; Then, Bź |