And by Subftituting o for a, in each of the foregoing X Propofitions, you'll have Canons for any Propofition in infinite Decreafing Geometrical Proportion continued. As for Inftance; a, y, s, being given in order to find r in an infinite Decreafing Series in ގ 1 find in Propofition IV. that in a finite Series in ÷, r a ; wherefore in an infinite Decreasing oner = fic de Ceteris. Sect 2. Of Proportions Disjuna. Et When the first Term has the fame Ratio to the fecond, that the third hath to the fourth Term; but not the fame Ratio which the fecond hath to the third Term: that Proportion is faid to be Dilcontinued or Disjunct. Are faid to be in Geometrical Proportion Disjunct; for the Ratio which 2 hath to 4, (which is 2); 18 hath to 36; but 4 hath not to 18: and the fame Ratio which a hath to ae (which is e) d hath to de, but ae hath not to d. Theorem. In any Geometrical Proportion Disjunct, the Product of the Means is equal to the Product of the Extreams; that is to fay fince ae::d I fay a x de Note that Euclid in his 5th Book expreffes the Ratio of Proportionals in this manner, viz. the Ratio of a to b, thus . a If 4 Quantities are Proportional, they will also be Proportional in Alternation, Inverfion, Compofition, Divifion, Conver fion and Mixtly. Eucl. 5. Def. 12, 13, 14, 15, 16. And 3 316 For 5 da + db be+ bd; that is, ad be, as before. 6a+c.. cbd.. d. Alternately Compounded. For ad cd cbcd; that is, ad cb. Or And 12 a. b + a::c, d+c Converted. For 13 ad ac = bc tac that is, ad — bc. Laftly 14 ab..a-bic + d..ed Mixtly. For 15 ac ad + bc= bd: - ac + ad That is 162bczad Confequently be bc-bd. ad. As at first. Sect. 3. How to turn Equations into Analogies. From the next foregoing Section, it will be eafy to conceive how to turn or Diffolve Equations into Analogies or Proportion. For if the Rectangle of two (or more) Quantities, be equal to the Rectangle of two (or more) Quantities; then are thote 4 (or more) Quantities Proportional. By the 16. 6. Euclid. That is, if I abdc. Then 2 4..c: d. .b. Orl 31c a:: b..d. &c. From whence there arifes this general Rule for turning Æquations into Analogies. Kule. Divide either Side of the given Æquation (if it can be done) into two fuch Parts or Factors, as being Multiplyed together, will produce that Side again, and make thofe two Parts the two Extreams, then Divide the other Side of the Equation (if it can be done) in the fame manner as the firft was, and let those two Parts or Factors be the two Means. For Inftance, fuppose ab -†ad — bd. Then a..bi: d.. btd, orb.. a:: bd.. d, c. Or, taking ad from both Sides of the Equation, it will be, abbdad. Then Then ad::bai.. b, or b.. d::ba.. a. &c. Again fuppofe aa+2ae2by+y. Here a and a + ze are the two Factors of the firft Side in this Equation; for a + ze × a = aa +2ae. Againy and 2b y, are the two Factors of the other Side. Therefore a..y :: 26+y. : 26 ty..a + 2e. Or 26 +9..+ze:: a.. y. &c. When one side of any Æquation 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 instance, Suppose be + bd=da +g. CHA P. III. Of Harmonical Propoztion. Definition. MUfical, or Harmonical Proportion, is when of three Quantities (or rather Numbers) the firft hath the fame Ratio to the third, as the difference between the first and fecond hath to the Difference between the fecond and third. As in these following. Suppofe a, b, c, in Musical Proportion. B C A Lino AD is said to bodi= Adod harmonically in A, B, C, D; when the whole LD has the same protion to either extreme part "D, that the other extreme AB has to the middle BC; 'sas AD to CD・・ AD-BD 1 BD-CD D If If there are 4 Terms in Mufical Proportion, the First hath the fame Ratio to the Fourth, as the Difference between the First and Second hath to the Difference between the Third and Fourth. That is, Let a, b, c, d, be the 4 Terms, &c. hop B x x I have seen ax.ax. ax‚ax, to ie. I have seen a xaxx yet none sox as w Too PART arlä PART IX. The Derivation and Composition of Quadratick, Cubick, &c. Æqua tions. CHAP. I. The Origination of Quadzatick Equations, is thus Deriv'd by Mr. Harriot. = Cafe 1. If{4}then by Transpofition {4} and by Multiplying one by the other, you'll have C + b Sa Cafe 3. If then by Tranfpofition {46=0} And by Multiplying them by one another, the Product is and by Multiplying them by one another, you'll have aa + c + b xa + be=0. Hence |