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II. If four Quantities are in the Rectangle of the Extreams will be equal to the Rectangle of the Means.

As in these, a .ae, qel. 2.60e; here a xa es Flex24 . Ora. . ; here also a x

&c. éc Consequently, If there are never so many Terms in the Series of the Rectangle of the Extreams will be equal to the Rectangle of any two Means that are equally distant from those Extreams. As in these, a

aepe. a et.ae viz. a esxara etxae. Or al xaraeeexaer=aac

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III. If never so many Quantities are in : it will be, as any one of the Antecedents is to it's Consequents; fo is the Sum of all the Antecedents, to the Sum of all the Consequenis.

a.eeb. aeece:a6', &c. increasing. As in these.

&c. decreasing.

eee q:ae::a+aetare tail tact:retacetad tad taps

::attato not ++viz. a = a + a + a + a + a ) xataetaee tae taet.

That is, the Rectangle of the Extreams is equal to the Rect. angle of the Means; per Second of this Sect.

Note, The Ratio of any Series in increasing, is found by dividing any of the Consequents by it's Antecedent.

Thus, a) a e le Or a e) acę le, &c.

But if the Series be decreafing, then the Ratio is found by dividing any of the Antecedents by it's. Consequent.

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CONSECTARY. These Things being premised, fuch Equations may be deduced from them, as will solve all such Questions, as are usually proposed, about Quantities in Geometrical Proportion ; In order to that, the first Term.

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as before. let

= the common Ratio.
y = the last Term.
Sthe Sum of all the Terms.
Then Sy=the Sum of all the Antecedents.
And S-= the Sum of all the Consequents,
Analogy. Tla:ae::S-y:Sa per III. of this Se&.

2 S@aa=aes-aey
2° a 3.S-a=0.8-ey
3+ 48+

Steyrares
4 S sley-a=es-S

ye 51 Il 6

=S, the Sum of all the Series.

S
3:

=e, the common Ratio,
5 +al 8ey=estas
estas

=y, the last Term
4+ a 10 Stey=esta
10.Sulstey-S=a, the firft Term,

I.

او- S :3

Sy

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Nate, The :;: set in the Margin at the second Step, is inftead of ergo; and imports that the Rectangle of the two Extreams in the firft Step, is equal to the Rectangle of the Means. And so for any other Proportion.

Sect. 3. Of Harmonical Proportion.
HA ARMONICAL or Musical Proportion is, when of three

Quantities (or rather Numbers) the first hath the same Ratio
to third, as the Difference between the first and second, hath
to the Difference between the second and third, As in these fol-
lowing.
Suppose a, b, c, in Musical Proportion.
Then I

a:6::ba:1-6 1 :: 21cb-44 346-b4

2+cal 31cb=29c-ba

cb 3-25-64

21-7 = a, the first Term.
3+bal 520c=cbt.ba
51.ta 6 =b, the second Term.

ota
5-c 6 712 4 6=6

ba
7+ 24-48 =1, the third Term.

2 ac

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ca

If there are four Terms in Musical Proportion, the first hath the same 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 four Terms, &c.

Then Ija:d::ba:d

I :: 2 db da=da
2-da

3
db 2 da 6a

db 3: 2004

2 d.

2 daca 35d5b=

d
3tca 6db +ca=2 da
6-db7ca=2da-dh

?
2 da-db
75 a 8 =

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Of Proportion Disjund, and how to turn Equations into

Analogies, &c. PROPORTION. Disjunet, or the Rule of Three in

Numbers, is already explained in Chap. 7. Part 1. And what hath been there said, is applicable to all Homogeneous Quantities, viz. of Lines to Lines, 896

Sect.

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IF

F four Quantities, (viz. either Lines, Superficies, or Solids) be

Proportional: the Rectangle comprehended under the Extreams, is equal to the Rectangle comprehended under the two Means. (16 Euclid 6.)

For Instance, Suppose, a.b.6.d. to represent the four Homogeneal Quantities in Proportion, viz a:b::c:d; then will ad=bc. For Suppose b = 2 a, then will d=2c, and it will be a : 22::C:26. Here the Ratio is 2. But a x20=2@xc. viz. 2 ca=2ac. Or suppose b=34, then will d=36, and it will be a : 3a::6:36. Here the Ratio in 3. But ax 3c

= 3axc. viz. 3ca=3 ac. Or universally putting for the Ratio of the Proportion, viz. making b= a e, then will d=ie, and it will be a: a e:::ce. But axcesa exc, viz. are =q2c. Confequently, a d=bc which was to be proved.

Whence it follows, that if any three of the four Proportional Quantities be given, the fourth may be easily found; thus,

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If four Quantities are proportionals they will also be Proportionals in Alternacion, Inversion, Composition, Division, Conversion, and Mixtly, Euclid 5. Def. 12, 13, 14, 15, 16.

That

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That is, if a:b::c:d be in direct Proportion, as before

Then 2 a:0:: 6:2, alternate. For adbc.
And 36:a::d:c, inverted. For a d=bc,
Also 41 atb:b:::+d:d; compounded.
4 5 da tbd=bctbd, that is, adabc, as before.

Or 6 atc::: b+d:d; alternatively compounded.
6.

7 ad tod=bd tod, that is a d=bc. Again, 8 at bib::0*d:d, divided.

91 ad-bd=bbd, that is; ad=bs.
Or Q1::: b-did, alternately divided.
JO •.

ad-cd=b-cd, that is, a d=bc.
And 12 a:bta:0:d+c, converted.
12 ,

13 adtac=bct ac, that is, ad=bl.
Laftly 14 a +6:26::6+:-d, mixtly.
14 15ac-adtbc-bd=actad-bcbd.

16 | 2bc82ad, that is, ad=bc; as at first,

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Note; What has been here dore about whole Quantities in Simple Proportion, may be easily performed in Fractional Qualne tities, and Surds, &c.

d For Instance, If

and if it be required to

dd CC find the fourth Term, it will be the Rectangle of the

fo

I b Means ; which being divided by the first Extream will be dd

ddccc dd-CC come

the fourth Term. fo

abfc abf Or if b:vbd+bc:: Vod+bc: to a fourth Term. Then is, vbd+bcxvba tbc=bdibc the Rectangle of the Means ; and b) bd+bc(d+c the fourth Term. That is, b: wod+bo :: V.bdt6c:dt 6, &c.

at

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Sect. 2. Of Duplicate and Tripticâte Propoztion. THE Proportions treated of in the laft Section, are to be un

derstood 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. ?

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