2d Rank, and the remaining two of the 2d Rank common to the 3d Rank, and so on, then the other Couplets in each Rank will be :: 7.

DEMONSTR. All this is the fimple and immediate Application of Axiom 3. and Theorem III. and needs no more Explication but a few Examples, where you may fee the different Forms in which these things may appear.

IF there are two Ranks of 4 Numbers, which have two comparative Terms common to both, the Sums or Differences of the Antecedents and Confequents of the two different Couplets are in the fame Ratio with the Antecedent and Confequent of the common Couplet. Thus:

:

If A B C : D, and E B:: F : D; then A+E: B::C+F:D; alfo A-E: B:: C-F: D.

viz.

DEMON. By the preceding A:C::E: F; whence A+E, or A-E:C+F, or C-F:: B; D; or alternately, as in the Margin.

Again; Because A+B:C+D::B; D, and E+B: F+D::B: D; therefore thefe Proportions are also true,

A+E:C+F:: A+B:C+D::E+B:F+B, and A-E: C-F:: A-B:C-D:: E-B: F-B.

Alfo any Couplet of the firft Rank is ::7 with any of the fecond; being all as B: D.

Again; It is alfo A+E:B+B::C+F:D+D. Since 2 B:2D:: B:D.

THEOREM IX.

If there are two Ranks of four Numbers :: 1, whereof the Extremes or Mears of the one are the fame as the Extremes or Means of the other; or if they are reverfly the Means or Extremes of the other; then the remaining four Terms are reciprocally :: 1. i.e. make the remaining two of the one Rank the Extremes, and those of the other the Means, and these four are ::. Thus:

If A:B::C: D, and E:B::C: F; then A: E:: F: D.

If A: B::C: D, and A:E:: F: D; then B: E:: F: C.

If A: B:: C: D, and BEF: C; then A: E:: F: D.

DEMON. From the equal Products of Extremes and Means, it

in which all these Conclufions are comprehended.

If A:B::C:D, and F: A:: D:F; then B: E::F:C. is A D=BC=EF;

THEO

THEOREM X.

IF four Numbers are :: I, and if any two of the comparative Terms are equally multiplied or divided; or if the one Extreme or Mean is multiplied, and the other equally divided: or again, if the one Extreme is multiplied, and the other equally divided; and at thefame time the one Mean multiplied by any other, or the fame Number, and the other equally divided; the Proportionality ftill remains, tho' in the fecond Cafe the Ratio is changed; and will be alfo in the third Cafe, when two different Multipliers are employed. Thus: If A: B:: C: D, then these Proportions follow, viz.

Again; Inftead of dividing them, we may apply the given Numbers as Divifors; which will make the following Proportions:

C

DEMON. In all these Conclufions, and many more that may be contrived of this Nature, the Truth of the Proportion is evident from the equal Product of the Extremes and Means; founded all upon this, that A D=BC, nr=nr, You'll find alfo other complex ways of arguing with proportional Numbers in the next Chapter.

B D'

S. 2. Of Geometrical Progressions.

Obferve, By the Distance of one Term of a Series from another, is meant the Number of Terms from the one exclufive to the other inclufive; or including both, it is the Number of Terms lefs 1. So if the Number of Terms is n, the Distance of the Extremes is ¤-1.

PROBLEM III.

Having the firft Term and Ratio to raise a Geometrical Series.

RULE. IF the given Ratio is a whole Number or mixt, then for an increafing Series multiply, and for a decreasing divide the first Term by the Ratio, the Product or Quote is the fecond Term; which multiplied or divided by the Ratio, gives the third Term, and fo on. But if the Ratio is a proper Fraction, this of itself determines that the Series ought to increase, and we muft multiply by the Reciprocal of the Ratio.

Example 1. First Term 2. Ratio 3, the increafing Series is 2:6:18:54, &c. and the decreafing Series 2:

2 2 2

3

9:27.06.

Book IV. Example 2. First Term 2, Ratio 4, the Series is 2: 8 : 16 : 32, &c. Or thus, 2: I I I &c.

28 32

DEMON. The Reafon of this Rule is manifeftly contained in the Definition of Geometrical Relation; fee the Coroll. after it: for when the Ratio is a whole or mixt Number, which foever of the two Views there explained it is taken in, the Rule is good; because for an increasing Series, a whole or mixt Number is in the first View the Ratio, and in the fecond it is the reciprocal Ratio; and by the Coroll. referred to, the first Term or Antecedent ought to be multiplied by this; and for a decreafing Series, a whole or mixt Number is the Ratio in both Views, and Divifion is the Rule. Laftly, If the given Ratio is a proper Fraction, it is taken in the second View, and neceffarily infers an increafing Series; and therefore the Antecedent multiplied by the reciprocal Ratio produces the Confequent.

SCHOLIUM S.

1. If the Ratio is expreffed by two Numbers ordered in a certain Comparison, the one as Antecedent, and the other as Confequent, then, that determines whether the Series decreases or increases; and accordingly, if we make a Fraction of the Antecedent fet over the Confequent, (which is the Ratio in the fecond View) and by it divide, we fhall raise an increafing or decreafing Series, acccording as the Ratio is a proper or improper Fraction. For Exam. If the Ratio is thus expreffed, viz. the Ratio of 2 to 3, or 2:3, then dividing by makes an increasing Series. But if it's the Ratio of 3:2, then dividing by 23 makes a decreasing Series.

2. Again: If the two firft Terms of a Series are given, (which do contain the Ratio) then the Series is continued by the common Rules of finding a third or fourth ÷l, viz. finding a third to the two given ones, and then a fourth to the fame two, and the Term last found. And fo a Series from a:b, whether increasing or decreafing, will be thus re62 63 14 &c. But when a Number diftinct from the two first Terms a a2

presented, a: b: -:

is given for the Ratio, then

3. Any Geometrical Progreffion, whether increasing or decreafing from a given Numbera, may be clearly reprefented thus, a: ar2: ar3, &c. adding still an Unit at every Step to the Index of r: for according as r is fuppofed greater or leffer than 1, fo is the Series increafing or decreafing. If the Series increases, then is the Ratio in one of the two Senfes explained, or the Reciprocal of it in the other. For it is the Quote of ar, the greater Term divided by a the leffer; but the reciprocal Quote of a divided by ar, which is the Ratio in another View. Again; If the Series decreases, 7 is a proper Fraction, and is the Reciprocal of the Ratio taken in either View, which are in this Cafe the fame, viz. the Quote of a divided by ar, which is, whofe Reciprocal is r.

ar

Wherefore, if we take the Ratio always under the Notion of the Quote of the Antecedent divided by the Confequent; then is to be understood as the Reciprocal of the Ratio; which makes the Series either increafing or decreafing, according as is greater or leffer than I And therefore tho' we call the Ratio, yet if we understand it as the reciprocal Quote of the Antecedent divided by the Confequent, there will be no Ambiguity or Hazard of Error; and hereby we fhall fave the Trouble of making different Reprelentations for nc cafing and decreafing Series. Yet in fome Cafes it will be convenient to reprefent them differently; and then taking the Ratio always as the Quote of the greater

Term

Term divided by the leffer, which must be a whole or mixt Number; and calling that Ratior, an increafing Series will be thus represented, a; ar: ar2, &c. and a decreafing

a

thus, a:-:

a

,&c. Wherefore if we reprefent a Series in the first manner, it's supposed to be taken indifferently for increafing or decreafing, unless it's exprefsly faid to be increafing; but the other manner does always exprefs a decreating Series.

From these Rules and Expreffions of Geometrical Series we have the following

COROLLARIES.

1. If the leffer Extreme of a Series is the Ratio, then the other Terms are the feveral Powers of the Ratio, thus: r:r2: r3; r4, &c. or A: A2: A3 : A4, &c. For here r or A is the Ratio, and rxr=r2; r2 × r=r3, and so on: Whence it's plain, that the Series of the Powers of any Number is a continued Geometrical Progreffion.

2. If I is the first Term of a Series, the fecond Term is the Ratio; and all the fucceeding Terms are the feveral Powers of the Ratio. Thus, 1:r:r2, &c. for the Ratio of I tor being r, the second Term must be r2, the third r3, and so on.

3. Betwixt 1, and any Power of a Number as ", there fall as many Geometrical Means as the Index of that Power lefs 1, viz. n— 1, in the Ratio of 1 to the Koot; for every Term after I is fuch a Power of the fecond, whole Index is equal to the Number of Terms after 1 to that Power; the thing is evident in the universal Series, Ir: r2: r3:14,

&c.

4 Every Geometrical Series, whose first Term is not 1, is equal to fuch a Series multiplied by the given firft Term; the Ratio of that Series being the fame with that of the given Series. Thus: A: Ar: Ár2: Ar3: Art, &c. is no other than the Products of this Series, 23:4, &c. multiplied by A.

5. As any Progreffion may be thus reprefented, A: Ar: Ar2: Ar3:, &c. it's manifeft that the Index of the Ratio in every Term expreffes the Distance of that Term from the first A: And hence it follows immediately, that

6. Every Term is equal to the Product of the first, by that Power of the Ratio whose Index is the Distance of that Term from the firft; and reverfly, the firft Term is equal to the Quote of any greater Term divided by that Power of the Ratio, whofe Index is the Distance of that Term from the firft. So, for Example, the first Term being A, the Ratior, and the Distance of any Term from A being d, that Term is Ard; and if you call that Term L, then is L-Ard; and reverfly, A=L÷÷rd.

SCHOLIUM. Because the greater Extreme of a Series is, by what's now fhewn, equal to Ad, or Ar-, (A being the leffer Extreme, r the common Ratio, and d the Distance of the Extremes equal to -1, the Number of Terms lefs 1) and every Term below having the Ratio involved in it to one degree lefs than the preceding greater Term; therefore a decreasing Series, which, when the greater Extreme is called, is represented thus, 1: &c. may also be reprefented thus: Ad: A,d-: A1d-2:, &c. Or thus, A: Ap12: Ar-:, &c. (because d=n-1) going on fo till the Index of r be equal to 1, and then we have Ar the Term next to Á.

72 73

7 But again more univerfally: From the fame Expreffion of a Series, it's manifeft that the Difference of the Indexes of the Ratio in any two Terms expreffes the Distance of thefe two Terms; for in every Step afcending, the Ratio is involved once more than in the preceding; and therefore from any Term to any other, it's as much oftner involved in the greater than in the leffer, as their Distance expreffes; that is, the Difference of their Indexes is their Distance; and hence it follows immediately, that

8. Any Term of a Geometrical Series is equal to the Product or Quote of any other leffer or greater Term multiplied or divided by fuch a Power of the Ratio, whofe Index is the Distance of these Terms; and any Term divided by any other leffer Quotes fuch a Power of the common Ratio, whofe Index is the Distance of these Terms. For any Term being expreffed Ard, (d being the Distance of this Term from A) any greater Term must have a Power of r, whofe Index exceeds d by the Distance of thefe Terms (by the laft) so, that Distance being m, the greater Term is Ad+m; but rx pm; (Tb.6. Ch.1.B.3.) confequendly, Ard+mArxrm; and reverfly, Ard+mm Ard, and Ad+Apdrm. Or this Truth may be deduced from Coroll. 6. Thus: Any Part of a Series, i. e. from any Term to another is ftill a Geometrical Series, whereof these two Terms are the Extremes; which being called A, L, and their Diftance d, it's fhewn that A=Lr, and L-Ar1; and laftly, LA.

Exam. In this Series, 2:6:18:54:162:486. If we compare 6 and 162, whose Diftance is 3; then is 6 162 Cube of 3, or 27; and reverfly, 1626x27; and lastly 162 627.

SCHOL. The immediate Ufe and Application of this last Truth we have in the Solution of these Problems.

(1.) Having any one Term of a Series and the Ratio, to find a Term at any Distance from the given one, without finding all the intermediate ones: The Solution of which is plainly contained in this Coroll. Thus: Take that Power of the Ratio, whofe Index is the given Distance; and by it multiply or divide the given Term, and you have the Term fought above or below the given one. Thus: Any Term of a Series being multiply'd or divided by the fourth Power of the Ratio, gives a Term, which is the fourth above or be low after the given one.

(2.) Having any two Terms of a Series and their Distance, to find any other Term at any Distance from either of the given ones; which is folved thus: Divide the greater by the leffer of the given Terms, the Quote is a Power of the Ratio, whofe Index is the Diftance of the given Terms. Suppofe this Distance to be d, and extract the d Root of that Quote, (by the Methods explained, or referred to in Book III) it is the Ratio fought: (all which is immediately contained in this Coroll.) Then having the Ratio, by it and any given Term we can find any other Term at any given Distance from the given Term, as in the preceding Problem.

Obferve, For fome Cafes of this Problem there is a more eafy Solution, which you'll find in Theo. XX. fee the 3d Article of the Schol. after the 4th Coroll. as particularly, fuppofe the Term fought is leffer or greater then either of the Terms given, and its Distance from the leffer if it's greater than either of them, or from the greater if it's leffer, is a Multiple of the Distance of the given Numbers.

(3.) Of these three things, viz. the common Ratio of a Series, the Distance of any two Terms, and the Ratio of thefe two Terms; having any two we can find the third: for any Term being called A, r the common Ratio, and d the Distance of any greater Term from A, that Term is Ard, and the Ratio of A to Ard is rd; which if we call R, then (1.) if d, r, are given, R (rd) is also known. (2.) If R and d are given, r is allo known, viz. by extracting the d Root of Rd. (3. If r and R are given, d is alfo known, viz, by raising r to a Power which is equal to R (rd) and the Index of it is d.

9. The Sum of the Extremes of a Geometrical Series is equal to the Product of the leffer Extreme multiplied into the Sum of 1, and fuch a Power of the Ratio whofe Index is the Distance of the Extremes. Thus, A being the leffer, L the greater Extreme, and d the Distance of the Extremes; A+LA+ Ard, for L-Ard; and confequently A+LA +Ard=A+rd+1. Again; Becaufe any two Terms of a Series may be confidered as Extremes of a leffer Series, the fame Rule will be good for expreffing their Sum.

Thus,