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Then will NL= 2 S. Consequently, įNL=S. viz, one Half of so many times the greatest Term as there are Number of Terms in the Series.

oti+2+3+4 10 = the Sum of the Series = NL Thus

4+4+4+4+4 20 = NL. And this will always be so, how many Terms foever there are, by Confeet. 1, Page 185.

LEMMA III. If a Series of Squares whose Sides or Roots are, in Arithmetick

Progression, beginning with a Cypher, &c. (as in the last Lemma) be infinitely continued; the last Term being multiply'd into the Number of Terms will be Triple to the Sum of all the Series, viz. NLL= 3S, or ļNLL=S.

That is, the Sum of such a Series will be one Third of the la or greatest Term, so many times repeated as is the Number of Terms in the Series.

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Infiances in Square Numbers, otit4 5

II

=ct
4+4+4 12 3
otit4+ 9 14 7

II 2.

---9+9+9+9 30 18 3

18 otit4+ 9+16 30 3 9 I I 3.

to & C.
16+16+16+16 80 8 24 3 24
From these Instances 'tis evident, that, as the Number of Terms
in the Series does encreafe, the Fraction or Excess above } does
decrease, the faid Excess always being oN_6; which, if we sup-
pose the Series to be infinitely continued, will then become infa-
nitely small, viz. in Effect nothing at all. Consequently, NLL
may be taken for the true or perfect Sum of such an infinite Sc.
sies of Squares.

LEMMA IV.
If a Series of Cubes whose Roots are in Arithmetick Progreffion,

beginning with a Cypher, &c. (as above) be infinitely continu’d,
the Sum of all the Series will be NLLL=S.
That is, one Fourth of the last or greatest Term fo many

times repeated as is the Number of Terms,

Infiances

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{

12

12

IO

{

Instances in Cube Numbers,
If 0.1.2.3.& c. be the Roots of the Cubes.

of it 8+27 36 4 Then i.

ニー十一 27+27+27+27 108

4
ot it 8+27+64 100 5 I I
2.

-=-+-
64764764764+64 320 32 16 · 4 16
S ot it 8+ 27+ 64+125 225 45 3 6 !
3

+ 2125+125+125+125+125+125 750 150 10 4 From these Examples it plainly appears, that, as the Number of Terms in the Series encreafes, the Fraction or Excess above decreases, the Excess being always which, if we suppose the Series to be infinitely continued, will become infinitely small or rather nothing; as in the last Lemma. Consequently, NL LL may be taken for the true and perfect Sum of all the Terms in such an infinite Series of Cubes.

=-=

20

20

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LEMMA V.

If a Series of Biquadrates, whose Roots are in Arithmetick Pro

gression, beginning with a Cypher, &c. (as before) be infinitely continued, the Sum of all the Terms in fuch a Series will be ŽNL 4.

The Truth of this may be manifested by the like Process as in the foregoing Lemma's, and so on for higher Powers. But if any one desires a farther Demonstration of these Series, he may (I presume) meet with ample Satisfaction in Dr. Wallis's History of Algebra, Chap. 78 and 79, within the Doctor concludes with these Words :

“ Thus having shew'd, that in a Progression of Laterals (or « Arithmetical Proportions) beginning at o. the Sum of 2. 3. 4, “ 5.6 Terms, is always equal to half of so many times the great« eft ; and there being no Pretence of Reason why we should “ then doubt it in a Progression of 7. 8. 9. 10. &c. we conclude it " fo to be, tho' such Number of Terms be suppos'd infinite.

“ Again ; in a Progression of their Squares having shew'd, that “ in 2. 3. 4. 5. 6 Terms the Aggregate is always more than one “ Third of so many times the greatest, and the Excess always such

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“ aliquot Part of the greatest, as is denominated by fix times the “ Number of Terms wanting 1. (As, if the Terms be 2, “ it is sti; if three it is to; if 4, it is ; +++; if 5, “ it is of so many times the greateft Term, and so on“ ward) we may well conclude (there being no Pretence of “ Reason why to doubt it in the rest) that it will be fo, how ma“ ny foever be such Number of Terms. And because fuch Excels, " as the Number of Terms do increase will become infinitely “ small (or less than any affignable) we conclude (from the Me " thod of Exhaustions) that, if the Number of Terms be fuppos'd « infinite, such Excels must be suppos’d to vanifh, and the Ag

gregate of such infinite Progreffun fuppos'd equal to į of lo 5 many times the greatest.

“ In like manner having prov'd that such Progression of Cubes “ doth (as the Number of Terms encreases) approach infinitely near “ to of so many times the greatest, and of Biquadrates to , and « so of Sursolids to ¢ of so many times the greatest, and so on“ wards as we please to try; and there being no Pretence of Rea. “ Ton why to doubt it as to the rest, we may take it as a sufficient " Discovery, that (universally) the Aggregate of such infinite “ Progression is equal (or doth approach infinitely near) to such a “ Part of so many times the greatest, as is denominated by the “ Exponent (or Number of Dimenfions) of fuch Power (as is “ that according to which the Progression is made) encreas'd by

1. namely, of Laterals ; ; of Squares ; of Cubes ; of Biquadrates š š (of so many times the greatest) and fo onwards infinitely.

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This Discourse of the Doctor's I thought convenient to insert, fuppofing it may give fome Satisfaction to the Learner, to hear la Great a Man as Dr. Wallis's Argument about the Truth of these Series, which I have briefly deliver'd in the 'foregoing Lemna's.

LEMMA VI. If any two Series or Ranks of Proportionals bave the fame Num. ber of Ternis (whether Finite or Infinite) it will always

As the firft Term of one Serics : is to the first Term of the be other Series :: fu is the Sum of all the Terms in the ene See ries : to the Sum of all the Terms in the other Series,

(12, e. 5.)

As

2

412

As in these Numbers, Il 3 Or these Numbers 4

5 12

15 31 9

36

45 108

135 515

324 | 405 6118

972 |1215 That is, 1 : 3 :: 21:63 And 4:5:: 1456 : 1820&c.

The Application of these Lemma's to Geometrical Quantities, viz. to Lines, Superficies, and Solids, wholly depends upon granting the following Hypotheses.

The Hypotheses.

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i. That every Line is suppos'd to confift (or be compos'd) of an infinite Series of equidistant Points.

2. A Surface (viz. the Area of any Figure) to confift of an infi. nite Series of Lines, either streight or crooked, according as the Figure requires.

3. A Solid to consist of an infinite Series of Plains, or Superficies, according as its Figure requires.

Not that we suppofe Lines, which have really no Breadth, can fill a Space or Superficies ; or that Plains, which have not any Thickness, can conftitute a Solid : But by what we here call Lines are to be understood small Parallelograms (or other Superficies) infinitely narrow, yet so as that their Breadths, being all taken and put together, must be equal to the Figure they are suppos’d to fill up. And those Plains or Superficies, which are here said to con fitate a Solid, are to be understood in Sinitely thin; yet so as that their Depths or Thicknesles (which are hereafter also called Lines) being all taken together, must be equal to the Height of the propos'd Solid. Now, in order to render chis Hypothesis as eafy for a Learner to underftand as I can, I fall here propose a very plain and familiar Example; Viz. Let us suppose any Book to be compos'd (or made up) of 100, 200, 300, (more or less) Leaves of fine Paper ; such a Book, being close put together, will have Length, Breadth, and Depth or Thickness, and therefore may (not improperly) be called a Solid; and each of its Edges (being evenly cut) will be a Superficies compos'd of a Series of small Parallelograms, every one of their Breadths being only the Edge of a single Leaf of Paper; and if we conceive the Thickness of every one of those Leaves to be divided into 10, or 100, or ioco, &c. they will

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then become such a Series of infinitely small Lines as are (by the Hypothelis) said to compose or fill up a Superficies. And all the Súperficies of those infinitely thin or divided Leaves of Paper will be come such a Series of Plains, or Superficies, as are said to constitute a Solid, viz. fuch a Solid as the Bigness and Figure of that Book.

Now according to this Idea of Lines, Superficies, and Solids, oge may, without the leaft Prejudice to any Demonstration, admis of the following Definitions and Theorems.

Definitions.

1

I. The Area's of Squares, and all other Parallelograms, are compos'd or filld up with an infinite Series of equal Right Lines.

II. The Area of every plain Triangle is compos'd of an infinite Series of Right Lines parallel to its Base, and equally decreasing until they terminate in a Point at the vertical Angle.

III. The Area of a Circle may be compos'd either of an infinite Series of concentrick or parallel Circles, or of an infinite Series of Chord Lines parallel to its Diameter, or of an innumerable Multitude of Sectors.

IV. The Area of an Ellipfis may be compos'd either of an iofi. nite Series of Ordinates rightly apply'd, or of an infinite Series of Right Lines parallel to its Transverse Diameter.

V. The Area's of the Parabola and Hyperbola are compas'd of an infinite Series of Ordinates; or may also be compos'd of Right Lines parallel to its Axis, &c.

VI. A Prism is a solid Body' contain’d or included within feveral equal Parallelograms, having its Bases or Ends equal and alike; and it is generally nam'd according to the Figure of its Base: That is,

VII. A Cube (or Solid like a Dye) is a Prism bounded or included with fix equal square Plains.

VIII. A Parallelopipedon is a Prism that hath its Sides bounded or included within four equal Parallelograms and two square Bases or Ends.

IX. A Cylinder (or Solid, like a Rolling-stone in a Garden) is only a round Prism, having its Bases or Ends a perfect Circle.

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