Scholium. If a be a finite Number, and an infinite (or in finitely great) Number; then (r being, by a finite Number, y (By 2d Step)ar" must be 1) an infinite Number; and ar" a confequently s= o; ry a z (by the 5th Step) will, in this Cafe, finitely fmall Number, and the only way we have of Writing fuch a Number is by Wherefore s= ry be == ; that is = a finite Number. r - I By this Theorem, Queftions that are ufually propos'd in infinite decreafing Geometrical Proportions are eafily folv'd: As for Inftance, If it were required to find the Sum of this decreasing Geo Again a, y, s being given, in order to find in an infi nite decreafing Series in÷÷ I find, in the 3d. Step, that, in a finite Series in, r = - a S ; Wherefore, in an infinite decreasing one r = Sec. 2. Of Geometrical Propoztion Disjunc. 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 Disjunct or Difcontinued. 4, So,2, 18, 10, 32, or Universally a, ae, d, de are faid to 36, 4, be in Geometrical Proportion Disjunct; for the Ratio which 2 hath to 4 (which is 2) 18 hath to 36; but 4 has not the Part VIIL fame Ratio to 18: And the fame Ratio which a hath to ae (which is e) d hath to de, but ae hath not the fame Ratio 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 a, ae, d, de may represent any four Quantities in :: viz. a .. ae :: d de; it is plain that aex d is a × de = ade. If four Quantities are proportional, they will also be proportional in Alternation, Inverfion, Compofition, Divifion, Converfion and Mixtly. Eucl. 5. Def. 12, 13, 14, 15, 16. that is, If a .. bc.. d be in direct Proportion. Then 24. cb.. d. Alternate. For adbc. bad. c. Inverted. For adbc. 4a+b.. b:: c +d.. d. Compounded. And Alfo 3 a: For 5 da+dbbcbd; that is, ad be, as before. acc::b+d.. d. Alternately Compounded. ad+cdcbcd; that is ad = cb. Or For Again For &a = ab.. b::cd.. d. Divided. adbdbcbd; that is ad cb. Or Ica-ccbd.. d. Alternately Divided. For 11adcdcbcd; that is ad cb. And 12 a b ±a :: c.. dc. Converted. For 13 ad + ac = beac; that is adbc. Laftly 14a+b ..a-b::c+d.. cd. Mixtly. For 15 ac- ad+be bd acad tc bd. that is 16 2be2ad; Confeq. brad; as at firft. Sect. 3. How to turn Equations into Analogies. From the foregoing Section it will be easy to conceive how to turn or diffolve Equations into Anologies or Proportions: For if the Rectangle of the two (or more) Quantities be equal to the Rectangle of two (or more) Quantities; then are those four (or more) Quantities proportional, by 16. 6. Eucl. El. That is, if abde; then, is a .. c::d.. b. or c ..a::b d, &c. Front 105 From whence there arifes this general Rule for turning Equa tions into Analogies, Kule. Divide either Side of the given Equation (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 first was, and let thofe two Parts or Factors be the two Means. For Inftance, fuppofe abad=bd. Then a.. bd.. b+d, or b. a :: b + d.. d, &c. Or, taking ad from both Sides of the Equation, it will be, ab-bd-ad. d::b - a .. b, or b d::b -- A a, &c. Again y and Then a.. Again, fuppofe aa+2ae2by+yy. Here a and a 2e are the two factors of the firft Side in this Equation; for a2ex aaa+2ae. 2 by are the two Factors of the other Side ; Therefore a.. y : 2b + y .. a + 2e, or 26 2e :: a y, &c. y When one side of any Equation cannot be Divided into two Factors as before, and the other Side can be fo Divided; then make ift. Unity and the former Side, or 2dly, the SquareRoot of the faid former Side, either the two Means or the two Extreams. For Inftance, Suppofe be+bd=da+g. Or I b::cd .. dag, &c. Or 2dly. b..: da + g::: √ : da +g: .. c + de da+g, &c. CHA P. III. Of Harmonical Propoztion. 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 thefe following. P Suppofs If there are four Terms in Musical Proportion, the first hath the fame Ratio to the fourth, as the Difference between the first and fecond, hath to the Difference between the third and fourth. That is, Let a, b, c, d be the four Terms, 8. 107 PART IX. Of the Derivation and Compolition of Equations. THE first Sort of Equations which offer themselves to our Confideration are Laterals; fuch as is ab, in which the Quantity a is determin'd to one fingle Value; viz. a in this Equation is equal to b, and to no other Quantity diffe rent from bin value. But in the next Sort of Equations; viz. in Simple Quadraticks, as ab, it may be a = + √ b, or a = vb; for bx-b is +b as well as + bx+ √b. Now if the known Quantities in the precedent Simple Qua dratick, as also in the Values of a therein be tranfpos'd, you'll have abo, a-√bo, and a+b=0; the first of which Equations is Manifeftly the Product of the two last. Hence you have the firft Hint of the Origine of Equations, which Hint, being applyed to all Sorts of Quadratick and Superiour Equations, will be found to fucceed, as will appear fur ther on. Again, If a3k, or a3-ko, we may eafily conceive a to bek, viz. a — 3/ke; But can't, at firft, imagine or conceive that it can have any other Value in that Equation: For fuch other Value can't be Affirmative, neither can it be Negative: Now this Obftacle will be remov'd, and an additional Myf tery reveal'd, by Dividing a3 ko by a 3)/k(=0); for the Quotient thus had, is a+ak + kk = 0, in which Equation the two Values of a will be found, by Part X. to be the two imaginary Quantities, viz. :- + √3: × 2/ R, ÷ v And:xk (for the Cube of either of these Values of a is); confequently a +--:×2/k=0, And a ++ − 3 : × √√ k = 0: And the Product of these two Equations Multiplyed by ako produces a3 — k Ò, as at first. P 2 Fur |