will be equal to the Rectangle of the Means.

As in thefe, a ae.ace. aeee; here a x a

e e eee

; here alfo a x

Faexall.

Confequently, If there are never fo many Terms in the Series of, the Rectangle of the Extreams will be equal to the Rectangle of any two Means that are equally diftant from thofe Extreams.

· ae. aee

aeee.aet.ae

As in these, a viz. a e3 xa=a etxa e. Or a e13 × a = q e e e × α è e = a a è 1

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

a:ae::a+ae+ace + ac3 +ae+: ac+ace+ae + ac2 + aç3

xa+ae+ace+ac3 +act.

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

Note, The Ratio of any Series in

viding any of the Confequents by it's Antecedent.

Thus, a) ae (e

increafing, is found by di

Orae) a ee (e, &c.

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

CONSECTARY.

Thefe Things being premifed, fuch Equations may be deduced from them, as will folve all fuch Questions as are ufually propofed about Quantities in Geometrical Proportion. In order to that,

let

a= the firft Term.
e the common Ratio.

y = the laft Term.

S= the Sum of all the Terms.

Then Sythe Sum of all the Antecedents.
And Sa the Sum of all the Confequents.

Analogy. Ija: ae::S-y: Sa per III. of this Sea. 1'.' 2 Sa➡aa=aes—aey

2 a 3 S

a=es-ey

3+ey 4 Steyaes

4-S
·S 5ey

61

ye

5

=S, the Sum of all the Series.

of

4+ a 10 S+ey = es+a

10-S11S+ey—es=a, the firft Term,

Nate, The fet in the Margin at the fecond Step, is instead ergo; and imports that the Rectangle of the two Extreams in the firft Step, is equal to the Rectangle of the Means. And fo for any other Proportion.

Sect. 3. Of harmonical Proportion.

HARMONICAL or Mufical Proportion is, when of three Quantities (or rather Numbers) the firft hath the fame Ratio to third, as the Difference between the first and fecond, hath to the Difference between the fecond and third. As in thefe folJowing.

Suppofe a, b, c, in Mufical Proportion.

Then Ia:c::b-a:c-b

If there are four Terms in Mufical Proportion, the first hath the fame Ratio to the fourth, as the Difference between the firft and fecond hath to the Difference between the third and fourth.

That is, let a, b, c, d, be the four Terms, &c.

Of Proportion Disjunct, and how to turn Equations inte Analogies, &c.

PROPORTION Disjunct, or the Rule of Three in Numbers, is already explained in Chap. 7. Part 1. And what hath been there faid, is applicable to all Homogeneous Quantities, viz. of Lines to Lines, &c.

Sect.

I

SECT. I.

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 Inftance, Suppofe, a. b. c. d. to reprefent the four Homogeneal Quantities in Proportion, vix a:b::c: d; then will ad bc. For suppose b = 2 a, then will d= 2c, and it will be a : 2 a::c: 2 c. Here the Ratio is 2. But a × 2 c 2 ax c. viz. 2 ca=2 ac. Or fuppofe 34, then will d≈ 3c, and it will be a 3a::c: 3c. Here the Ratio is 3. But ax 3c =3axc. viz. 3ca=3ac. Or univerfally putting e for the Ratio of the Proportion, viz. making bae, then will dce, and it will be a: ae::c: ce. But ax cea ex c, viz. ace =aec. Confequently, a dbe which was to be proved.

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

If four Quantities are Proportionals they will also be Proportionals in Alternation, Inverfion, Compofition, Divifion, Converfion, and Mixtly, Euclid 5. Def. 12, 13, 14, 15, 16.

That

That is, if

Then

2

And

3

Alfo

4

Or

IO ...

And

12.

a:b::c:d be in direct Proportion, as before.
a:c:: bd, alternate. For a dbc.
b: a::d: c, inverted. For a dbc.

a+b:b::c+d:d; compounded.

5 da+bd=bc+bd, that is, ad bc, as before. 6a+c:c::b+d: d; alternatively compounded. ad+cd=bd+ed, that is, ad=bc. 8abb::cd: d, divided.

7

ΙΟ

9 ad bd=bcbd, that is, a dbc.
a-c:c::b-d: d, alternately divided.
II a dcd=bc-cd, that is, a d=bc.
12 a:bad+c, converted.

13

Laftly 14

ad±ac=beac, that is, a d=be. a+ba-b::c+d: cd, mixtly. 15 acad+bc-bd=ac+ a d = bcbd. 15+ 16 26c 2 a d, that is, a d=bc; as at firft.

14:

Note; What has been here done about whole Quantities in Simple Proportion, may be eafily performed in Fractional Quan tities, and Surds, &c.

d+ and if it be required to

Means; which being divided by the firft Extream

с

the fourth Term.

Or if b: bdbc:: √bd+bc: to a fourth Term. Then is, ✔bd+bc × √b d+bcbd+be the Rectangle of the Means; and b) bd+bc (d+c the fourth Term. That is, bbd + bo :: √ b d + b c d + c, &c.

Sect. 2. Of Duplicate and Triplicate Propoztion.

THE

HE Proportions treated of in the laft Section, are to be understood 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. '

But