1 a : b :: c : d be in direct Proportion, as before. 2 a: c::b: d, alternate. For ad=bc. That is, if And Alfo 4 4 Or 6 Again, 8 Or 10. 3b:a::d:c, inverted. For ad=bc. a+b:6::c+d:d; compounded. 5 da+bd=bc+bd, that is, ad=br, as before. 6a+c:c::b+d: d; alternatively compounded. 7 ad+cd=bd+cd, that is, ad=bc. 8a-b:b:: cd: d, divided. 9 ad-bd-bc-bd, that is, ad=bc. 10 a-c:c::b-d: d, alternately divided. II ad-cd=bc-cd, that is, ad=bc. And 12 a:b+a:: c:d+c, converted. 12. 13 ad+ac=bc + ac, that is, ad=bc. Lastly 14 a + b : a-b:c+d:c-d, mixtly. 14 15 ac-ad+bc-bd=ac+ad-bc-bd. 15+1612bc=2ad, that is, ad=bi; as at first. Note; What has been here done about whole Quantities in Simple Proportion, may be easily performed in Fractional Quan tities, and Surds, &c. For Instance, If find the fourth Term, it will be +, and if C it be the Rectangle of the a b required to ddcc fc Means; which being divided by the first Extream come ab = abf will be the fourth Term. dd-ccf ddc-ccc_dd-c fc abfc Or if b:bd+bc :: √bd+bc: to a fourth Term. Then is, √bd+bcx b d + b c = b d + be the Rectangle of the Means; and b) bd+bc (d+c the fourth Term. That is, b: √bd+bc ::√bd+bc:d+c, &c. Sect. 2. Of Duplicate and Triplicate Proportion. THE Proportions treated of in the laft Section, are to be understood 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, such Comparisons are diftinguished from the former, by the Names of Duplicate, and Triplicate, (&c.) Proportions; so that Simple, Duplicate, and Triplicate, &c. Proportions are to be understood in a different Sense from Simple, Double, Treble, &c. Proportions, which are only as 1, 2, 3, &c. to I; but those of Simple, Duplicate, Triplicate, &c. Proportions are those of a. aa.aaa., &c. to 1. Or if the Simple Proportions be that of a to b, whose Ratio or Exponent is a or b a Triplicate Proportions, &. 16 And if there are three, four, or more Quantities in, as I.a.aa.aaa.a.a', &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.at. a') is Duplicate, Triplicate, Quadruplicate, &c. of the first to the second (viz. of I to a ;) and by Inversion, that of the third, fourth, fifth, is Duplicate, Triplicate, &c. of that of the second to the first (a to 1) per Def. 10. Eucl. 5. But the Name of these Proportions will appear more evident, and be easier understood when they are applied to Practice, and illustrated by Geometrical Figures, further on. Sect. 3. How to turn Equations into Analogies. FROM the first Section of this Chapter, it will be easy to conceive how to turn or dissolve 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 those four (or more) Quantities Proportional. By the 16 Eucl. 6. That is, if abcd, 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 Instance, Suppose a b + ad=bd. Then a : b :: d:b+d, orb:a::b+d: d, &c. Or taking a d from both Sides of the Equation, and it will be ab=bd-ad; then a:d::b-a: b, or, b:d::b-a:a, &c. Again, suppose aa+2ae=2by+yy. Here a and a + 2 e are the two Factors of the first Side in this Equation; for a + 2e xa=aa+2ae. Again, y and 26 + y are the two Factors of the other Side; therefore, a: y:: 2b+y: a + 2 e, or 2b+y: a+2e::a:y, &c, When one Side of any Equation can be divided into two Factors, as before; and the other Side cannot be so divided, then make the Square Root of that Side either the two Extreams or the two Means. For Instance, Suppose bc + b d = da+g, then b: √da+g:: √da+g:c+d, or √ da +8 : b :: c + d : √da+g,&c. CHAP. VIII. Of Substitution, and the Solution of Quadratick Equations. Sect. 1. Of Substitution. WHEN new Quantities not concerned in the first Stating of any Question, are put instead of some that are engaged in it, that is called Substitution. For Instance, If instead of ✓ bc-dc you put z, or any other Letter; that is, make z = √bc-dc. Or suppose aa+ba-ca+da=dc, instead of b-c + d puts, or any other Letter not engaged with the Question, viz.s=b-c+d, then aa+sa=dc. That is, if c be greater than bd, it is aa-sa=dc; but if b + d be greater than c, then it is a a +sa=dc. dc+ch And this way of substituting or putting of new Quantities instead of others, may be found very useful upon several Occasions; viz. in Order to make some following Operations in the Question more easy, and perhaps much shorter than they would be without it, as you may observe in some Questions hereafter pro-. posed in this Tract. And when those Operations, in which the substituted Quantities were affifting or useful, are performed according as the Nature of the Question required, you may then (if there be Occafion) bring the original or first Quantities into the Equation, in the Place (or Places) of those substituted Quantities, which is called Restitution, as you may fee further on. Sect. 2. The Solution of Duadratick 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 whose 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 some known Co-efficients; you must reduce them by Division; as in Sect. 4. of Chap. 5. and for the Fractional Quantities that may arife by those Divisions, substitute another Quantity doubled. For Instance, let baa+caa-ca-da=dc+cb, then aa ca-da__dc+cb for b+c put z. Then will aa-2xa=z 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 simple Powers by casting off the fecond or lowest Term of the unknown Quantity; which is done by Substitution; thus, always take half the known Coëfficient, and add it to (Cafe 1.) or fubtract it from (Case 2.) it's fellow Factor; and for their Sum, or Difference, Substitute another Letter; as in these. Let 1 aa+2ba=dc Cafe r. 223aa+2ba+bb=ec 3-14bbee-dc 4+dc5ee=bb+dc 5 m2 6=√bb+dc. 2 and 67 a+b=√bb + dc, per Axiom 5. 7-b8a=√ bb + dc:-b : In Cafe 3. From Half the known Co-efficient substract it's fellow Factor. |