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rupted Geometrical Proportion 5 as are 1 : 4. : 3 : 6 5 where s is to 4 as 3 to 6" 5 but not fo as 4. to 3.
A Series or Progreffion of more than four Geometrical proportionals, is call'd a Geometrical Progreffion. See Progression.
^Properties of Geometrical Proportion.
i°. If three Quantities be in continual Geometrical Pro- portion, the Product of the two Extremes is equal to the Square of the middle Term. Thus, in 6 : 12 : ; 12 : 24, the Product of 6 and 24 is equal to the Square of 12, viz. 144. Hence we have a Rule,
i°. To find a mean Geometrical Proportional between two Numbers, e.gr. 8 and 72.
Multiply one of the Numbers by the other, and from the product 575, extract the Square Root 24. This will be the Mean required.
3 . To find a fourth Proportional to three given Numbers, e-gr. 3, is, 5 5 or a third Proportional to two given Num- bers.
Multiply the fecond 12 into the third 5, in the firft Cafe ; and in the latter, multiply the fecond into itfelf. Divide the Product by the firft 3, the Quotient 20 is the fourth Proportional fought in the one, or the third in the other.
The Solution of this Problem is what we popularly call the Rule of (Proportion, or the Golden Rule, or Rule of Three. See Rule.
4°. If four Numbers be in Geometrical (Proportion, the Product of the Extremes is equal to the Product of the two middle Terms. Thus in 2 : 5 : : 4 : 10, the Product of 10 and r 2 is equal to that of 5 and 4, viz. 20. Hence,
5 . If four Numbers reprefented a : hi : c s 4 be either in Arithmetical, or Geometrical (Proportion ; they will alfo be in the fame, if taken inverfely, viz. d: c ; 1 h : a $ or alternately, as a : c : : h : 4 j or alternately and inverfely, as d : C : : b : a.
6 Q . If the two Terms of a Geometrical Ratio be added to, orfubftracted from, other two in the fame Ratio, the lefs to or from the lefs, i£c. the Sums, or Differences, are in the fame Ratio. Thus, in 6 : 3 : 10 : 5, where the com- mon Ratio is 2 -j 6 added to 10, makes 16", as 3 to 5 makes 8 j and 16 : 8 are in the fame Ratio as 6 : 3, or 10 : 5. Again, 16 being to 8 as 6 to 3, their Differences 10 and 5 are in the fame Ratio.
The Reverfe of which Propofition is likewife true ; viz. if to or from any two Numbers be added or fubftradted other two, if their Sums, or the Differences, be in the fame Geometrical Ratio as the firft two, the Numbers added or fubftracted are in the fame Ratio. Hence,
7 Q . If the Antecedents, or the Confequents of two equal Geometrical Ratios 3 : 6 and 12: 24 be divided by the fame 3 j in the former Cafe, the Quotients 1 and 4 will have the fame Ratios to the Confequents 5 viz. 1 : 6 : : 4 : 24 5 and in the latter, the Antecedents the fame Ratio to the Quotients, viz. 3 : 1 : : 12 : 4.
S Q . If the Antecedents, or Confequents of fimilar Ratios a : 6 and 3 : 9 be multiply'd by the fame Quantity 6 5 in the former Cafe the Facta 12 andiS have the fame Ratio to the Confequents, viz, 12:6:: 18:9; and in the latter, the Antecedents have the fameRatio to the Products, viz. 2 : 6 :: 3 -.9-
9°. If in a Geometrical Proportion 3 : 6 : : 12 : 24, the Antecedents be multiply'd or divided by the fame Number 2 ; or divided by the fame Number 3 5 in the former Cafe, the Facta j in the latter, the Quotients will be in the fame Proportion, viz. 6 : 18 : : 24 : 72, and 1 : 3 : : 4 : 12.
io g . If, in a Proportion 4 : 2 : : jo : 5, the Antecedent of the firft Ratio be to its Confequent as the Antecedent of the fecond to its Confequent 5 then, by Composition, as the Sum of the Antecedent and Confequent of the firft Ratio, is to the Antecedent or Confequent of the firft ; fo is the Sum of the Antecedent and Confequent of the fecond, to the Antecedent or Confequent of the fecond 5 viz. 6:2:: 15 : 5, ord : 4: ; 15 : 10.
ii q . If, in a Proportion 5:4:: 15 : 10, as the Antece- dent of the firft Ratio is to its Confequent ; fo is the Ante- cedent of the other to its Confequent \ then, by (Divifon, as the difference of the Terms of the firft Ratio is to its Antecedent, or Confequent, fo is the difference of the Terms of the fecond Ratio to its Antecedent or Confequent 3 viz. z: 4 : : 5 :io 5 or, 2:5:: 5:15.
1 2 . If, in a Proportion 4 : 2 : : 6 : 3, as the Antecedent of the firft Ratio is to its Confequent, fo is the Antecedent of the fecond to its Confequent: And as the Confequent °f the firft is to another Number S, fo is the Confequent of the fecond to another Number 12 5 viz. 2:81:3:125 then will the Antecedent of the firft be to 8, as the Ante- cedent of the fecond to 12 5 viz. 4: 8 : : 6": 12.
I 3 9 * If, in a. Proportion^ i 4:: 12 : 6, as the Antecedent of the firft Ratio is to its Confequent ; fo is the Antecedent of the fecond to its Confequent 5 and as the Confequent of
the firft is to another Number 16 5 fo is another Number 3s to the Antecedent of the fecond, viz. 4: rtfcstia. Then, will the Antecedent of the firft be to id, as 3 to the Confe- quent of the fecond, viz. S : 16 : -. 3 ; s.
14?. Suppofe any four proportional Quantities, viz. 3 : 6: : 12 : 24, and any ether four proportional Quanti- ties 1 : 3 :_:9 =27 ; if you multiply the feverat Terms of the latter into thofe of the former, the Products will like- wile be (Proportional, viz. 3 : 18 : : 108 : 548. < 15 . If there be feveral Quantities" continually (Propor- tional A, B, C, D, & c . the firft, A, is to the third, C in a dup icate Ratio;'; to the fourth, D, in a triplicate Ratio, We. of the firft A to the fecond B.
t69 -J? there be thr-ce Numbers in continual Proportion the djfterence of the firft and fecond will be a mean Proportional between the difference of the firft and fecond lerm, and the difference of the fecond and third, and the firft Term.
Harmonical, or ATuftcal? ro? oktiox, is a third kind of Proportion, form'dout of the other two; thus: Of three Numbers, if the firft be to the third, as the difference of the firft and lecond to the difference of the fecond and third; the three Numbers are in Harmonical 'Proportion.
Thus, 2 : 3 : 6 are Harmonical, becaufe 2 : 6 : : 1 : 3. And four Numbers are Harmo?ikal, when the firfl is to the " fourth as the difference of the firft and fecond to the diffe- rence of the third and fourth.
Thus 24 : 16 : : 12 : 9 are Harmonical, becaufe 24:9:: 8: 3.
By continuing the (Proportional Terms in the firft Cj.Cq, there arifes an Harmonical Progreffion, or Series. See Se- ries.
Properties of Harmonical or Mufical Proportion.
i°. If three or four Numbers in Harmonical (Propor- tion be multiply'd or divided by the fame Number; the Produfts, or Quotients, will alfo be in Harmonical (Propor- tion. Thus, if 5, 8, 12, which are Harmonical, be divided by 2 ; the Quotients 3, 4, 6, are alfo Harmonical 5 and re- ciprocally their Products by 2, viz. 6, 8, iz.
^ 2 Q . To find an harmonical Mean between two Numbers given.
Divide double the Product of the two Numbers by their Sum, the Quotient is the Mean required. Thus, fuppofe 3 and 6 the Extremes ; the Produa of thefe is 18, which doubled gives 36" ; this divided by 9 (the Sum of 3 and 6) gives the Quotient 4. Whence, 3 : 4 : 4" are Harmonical.
3«. To find a third Harmonical Proportional to two Numbers given.
Call one of them the firft Term, and the other the fe- cond ; multiply 'em together, and divide the Product by the Number remaining after the fecond is fubftracte.1, from double the firft ; the Quotient is a third Harmonical 'Pro- portional. Thus fuppofe the given Terms 3 : 4, their Pro- duct 12 divided by 2 (the Remainder after 4 is taken from o~, the double of the firft) the Quotient is 6 j the Harmo- cal third fought.
4°. To find a fourth Harmonical Proportional to three Terms given.
Multiply the firft into the third, and divide the Product by the Number remaining after the middle or fecond is fubftrafted from double the firft ; the Quotient is a fourth Term in Harmonical Proportion. Thus, fuppofing the Numbers given 9 : 12: 16 3 a fourth will be found by the Rule to be 24.
5°. If there be four Numbers difpofed in Order ; whereof one Extreme and the two middle Terms are in Arithmeti- cal Proportion ; and the fame middle Terms with the other Extreme, are in Harmonical Proportion j the four are in Geometrical Proportion : As here, 2 : 3 : : 4: 6, which are Geometrical j whereof 2 : 3 : 4 are Arithmetical s and 3:4:6".
6°. If betwixt any two Numbers you put an Arithmeti- cal Mean, and alfo a Harmonical one 5 the four will be in Geometrical Proportion. Thus, betwixt 2 and 6, an Arith- metical Mean is 4, and an Harmonical one 3 ; and the four 2 : 3 : : 4 : 6 are Geometrical.
We have this notable difference between the three kinds of 'Proportion ; That from any given Number wecan raife a continued Arithmetical Series increafing in infinitum, but not decreasing ; the Harmonical is decreafable in infinitum* but not increafable ; the Geometrical is both.
Contra-harimnical Proportion, is thatRelation of three Terms, wherein the Difference of the firft and fecond, is to the Difference of the fecond and third, as the third to the firft.
Thus, 3,5,6", are Numbers xncontra-ftarmonical Propor- tion, becaufe 2 : 1 : : 6 : 3.
To find a Mean in Contra-harmonical Proportion between two Numbers : Divide the Sum of two Squares by the Sum of the Roots, the Quotient is the Mean requir'd- Thus
the
