That is, if And Alfo Or 6... Again, 8 abcd be in direct Proportion, as before. 2 a:c::b: d alternate. For a d=bc. 3b:a::d: c, inverted. For a d=bc. 4a+bb::c+d:d; compounded. 5 da+bd=bcbd, that is, ad=bc, as before. 6a+c:c::b+d: d; alternatively compounded. 7 ad+cd=bd+cd, that is, a dbc. abb::cd: d, divided. 8 9 Or 10 II And 12 12. 13 ad-bd-be-bd, that is, a dbc. ad±ac = bc±ac, that is, a d=bc. Laftly 14 a+b: a b c +d: cd, mixtly. 15 acad+bc-bd=ac+ad-bc-bd. 15162bc2ad, that is, a dbe; as at first. 14. Note; What has been here done about whole Quantities in Simple Proportion, may be eafily performed in Fractional Quan tities, and Surds, &c. Means; which being divided by the firft Extream will be Or if bbdbc::✔bd+bc: to a fourth Term. Then is, ✔bd+bcx√bd+bcbd+be the Rectangle of the Means; and b) bd+bc (d+c the fourth Term. That is, bbd + b c :: ✔b d + b c d + c, &c. Sect. 2. Of Duplicate and Triplicate Propoztion. THE Proportions treated of in the laft Section, are to be un derftood when Lines are compared to Lines, and Superficies to Superficies; or Solids to Solids, viz. when each is compared to that of it's like Kind, which is only called Simple Proportion. But But when Lines are compared to Superficies, or Lines are compared to Solids, fuch Comparisons are diftinguished from the former, by the Names of Duplicate, and Triplicate, (&c.) Proportions; fo that Simple, Duplicate, and Triplicate, &c. Proportions are to be understood in a different Senfe from Simple, Double, Treble, &c. Proportions, which are only as 1, 2, 3, &c. to I; but thofe of Simple, Duplicate, Triplicate, &c. Proportions are those of a.aa.aaa., &c. to I. Or if the Simple Proportions be that of a to b, whofe Ratio or Exponent is Triplicate Proportions, &c. And if there are three, four, or more Quantities in, as I.a.aa.aaa. a. a3, &c. (as in the first Series, Sect. 2. of the last Chapter.) Then, that of the first to the third, fourth, and fifth, &c. (viz. I to aa aaa. a. a) is Duplicate, Triplicate, Quadruplicate, &c. of the firft to the fecond (viz. of I to a ;) and by Inverfion, that of the third, fourth, fifth, is Duplicate, Triplicate, &c. of that of the fecond to the firft (a to 1) per Def. 10. Eucl. 5. But the Name of these Proportions will appear more evident, and be eafier understood when they are applied to Practice, and illuftrated by Geometrical Figures, further on. Sect. 3. How to turn Equations into Analogies. FROM the firft Section of this Chapter, it will be easy to con ceive how to turn or diffolve Equations into Analogies or Proportions. For if the Rectangle of two (or more) Quantities, be equal to the Rectangle of two (or more) Quantities; then are thofe four (or more) Quantities Proportional. By the 16 Eucl. 6. That is, if a b c d, then is a :c::d: b, or c: a :: b: d, &c. From whence there arifes this general Rule for turning Equations into Analogies. RULE. Divide either Side of the given Equation (if it can be done) into two fuch Parts, or Factors, as being multiplied together will produce that Side again; and make those two Parts the two Extreams. Then divide the other Side of the Equation (if it can be done) in the fame Manner as the first was, and let those two Parts or Factors be the two Means. For Inftance, Suppofe ab+ad=bd. Then a : b: :d : b + d, orba: b+d: d, &c. Or taking a d from both Sides of the Equation, and it will be a bbdad; then a: d::b-a: b, or, b: dba: a, &c. Again, suppose a a + 2 a e = 2 by +yy. Here a and a +2e are the two Factors of the firft Side in this Equation; for a + 2e xa=aa+2 a e. Again, y and 2 b+y are the two Factors of the other Side; therefore, ay::2b+ya+2e, or 26+ya+2e::a:y, &c. When one side of any Equation can be divided into two Factors, as before; and the other Side cannot be fo divided, then make the Square Root of that Side either the two Extreams or the two Means. For Inftance, Suppofe b c + b d = da+g, then b: √da+g::√da+g: c+d, or √da +g : b :: c+d: √da+g,&c. CHA P. VIII. of Subftitution, and the Solution of Quadzatick Equations. Sect. 1. Of Subffitution. WHEN new Quantities not concerned in the firft Stating of any Question, are put inftead of fome that are engaged in it, that is called Subftitution. For Inftance, If instead of Nbc-de you put z, or any other Letter; that is, make z= Nbc-dc. Or fuppofe aa+ba-ca+dade, inftead of b-c +d put s, or any other Letter not engaged with the Queftion, viz.s=b—i+d, then a asa=de. That is, if c be greater than bd, it is a a-sa-de; but if b+d be greater than c, then it is a a+sa=dc. And this way of fubftituting or putting of new Quantities inftead of others, may be found very ufeful upon feveral Occafions; viz. in Order to make fome following Operations in the Queftion more easy, and perhaps much shorter than they would be without it, as you may obferve in fome Questions hereafter pro-. pofed in this Tract. And when thofe Operations, in which the fubftituted Quantities were affifting or ufeful, are performed according as the Nature of the Queftion required, you may then (if there be Occafion) bring the original or firft Quantities into the Equation, in the Place (or Places) of thofe fubftituted Quantities, which is called Reftitution, as you may fee further on. Sect. 2. The Solution of Duadatick Equations. WHEN the Quantity fought is brought to an Equality with those that are known, and is on one Side of the Equation, in no more than two different Powers whofe Indices are double one to another, those Equations are called Quadratick Equations Adfected; and do fall under the Confideration of three Forms or Cafes. When there happens to be more Terms in one of these Kind of Equations than two, and the highest Power of the unknown Quantity is multiplied into fome known Co-efficients; you must reduce them by Divifion; as in Sect. 4. of Chap. 5. and for the Fractional Quantities that may arifé by thofe Divifions, substitute another Quantity doubled. For Inftance, let baa+caa-ca-da=dc+cb, then aa— ca_da_dc + c b 2x, and if you please, 6+6 = d Make put z. Then will a a- 2x az be the new Equa tion, equal to the other, being now fitted for a Solution. Now any of these three Forms of Equations being thus prepared for a Solution, may be reduced to fimple Powers by cafting off the fecond or loweft Term of the unknown Quantity; which is done by Substitution; thus, always take half the known Coefficient, and add it to (Cafe 1.) or fubtract it from (Cafe 2.) it's fellow Factor; and for their Sum, or Difference, Substitute another Letter; as in these. Let 1aa2ba de Cafe 1. 223 aa+2 ba+bb=ee 2 and 67 a+b=vbb+dc, per Axiom 5. 7b8a=√bb + dc: — b 3 Again. -b Let Ia a 2 bad c Cafe 2. 223aa2 ba+b bee - 4dc5ee dc + b b 5 w2 2 and 6 6e = v dc + b b 1 7 ab= v dc + b b 7+b8a=b+ √ dc + bb v In Cafe 3. From Half the known Co-efficient fubftract it's fellow Factor. |