Then will NL = 2 S. Confequently, 1⁄2 NL=S. viz. one Half of fo many times the greatest Term as there are Number of Terms in the Series. == Thus 0+1+2+3+4 To the Sum of the Series INL And this will always be fo, how many Terms foever there are, by Confect. 1, Page 185. LEMMA III. If a Series of Squares whofe Sides or Roots are, in Arithmetick Progreffion, beginning with a Cypher, &c. (as in the last Lemma) be infinitely continued; the laft Term being multiply'd into the Number of Terms will be Triple to the Sum of all the Series, viz. NL L= 3S, or NLL=S. That is, the Sum of fuch a Series will be one Third of the laft or greatest Term, fo many times repeated as is the Number of Terms in the Series. 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 I decrease, the said Excess always being which, if we fup 6N-6; pofe the Series to be infinitely continued, will then become infinitely small, viz. in Effect nothing at all. Confequently, NLL may be taken for the true or perfect Sum of fuch an infinite Series of Squares. LEMMA IV. If a Series of Cubes whofe Roots are in Arithmetick Progreffion, beginning with a Cypher, &c. (as above) be infinitely continu'd, the Sum of all the Series will be NLLLS. That is, one Fourth of the laft or greatest Term fo many times repeated as is the Number of Terms, Infiances Then I. 2. 3. Inftances in Cube Numbers. If o.1.2.3. &c. be the Roots of the Cubes. 0+ 1+ 8+27 36 4 I (27+27+27+27 108 12 ΙΟ 4 S 0+ 1+ 8+27+64_100 5 { { I 十一 12 I I 64+64+64+64+64 320 32 16 4 16 ot + 8+ 27+ 64+125 225 45 3 6 I + 125+125+125+125+125+125 750 150 10 20 4 20 From thefe Examples it plainly appears, that, as the Number of Terms in the Series encreases, the Fraction or Excess above de44 creases, the Excess being always 4; which, if we suppose the Series to be infinitely continued, will become infinitely small or rather nothing; as in the laft Lemma. Confequently, NL LL may be taken for the true and perfect Sum of all the Terms in fuch an infinite Series of Cubes. 4N 44 If a Series of Biquadrates, whofe Roots are in Arithmetick Progreffion, beginning with a Cypher, &c. (as before) be infinitely continued, the Sum of all the Terms in fuch a Series will be NL4. The Truth of this may be manifefted by the like Process as in the foregoing Lemma's, and fo on for higher Powers. But if any one defires a farther Demonftration of thefe Series, he may (I prefume) meet with ample Satisfaction in Dr. Wallis's Hiftory of Alge bra, Chap. 78 and 79, within the Doctor concludes with these Words: "Thus having fhew'd, that in a Progreffion of Laterals (or "Arithmetical Proportions) beginning at o. the Sum of 2. 3. 4, 5.6 Terms, is always equal to half of fo many times the great"eft; and there being no Pretence of Reason why we should "then doubt it in a Progreffion of 7. 8. 9. 10. &c. we conclude it fo to be, tho' fuch Number of Terms be fuppos'd infinite. "Again; in a Progreffion of their Squares having fhew'd, that "in 2. 3. 4. 5. 6 Terms the Aggregate is always more than one "Third of fo many times the greateft, and the Excefs always fuch " aliquot 66 "aliquot Part of the greateft, as is denominated by fix times the “Number of Terms wanting 1. (As, if the Terms be 2, "it is+; if three it is+; if 4, it is +; if 5, "it is of fo many times the greatest Term, and so on"ward) we may well conclude (there being no Pretence of "Reafon why to doubt it in the reft) that it will be fo, how ma"ny foever be fuch Number of Terms. And because fuch Excefs, "as the Number of Terms do increafe will become infinitely "fmall (or lefs than any affignable) we conclude (from the Me"thod of Exhauftions) that, if the Number of Terms be suppos'd « infinite, fuch Excefs must be suppos'd to vanifh, and the Ag"gregate of fuch infinite Progreffion fuppos'd equal to of fo "many times the greatest. "In like manner having prov'd that fuch Progreffion of Cubes "doth (as the Number of Terms encreases) approach infinitely near "to of fo many times the greatest, and of Biquadrates to, and "fo of Surfolids to of fo many times the greateft, and so on"wards as we please to try; and there being no Pretence of Rea"fon why to doubt it as to the reft, we may take it as a fufficient "Discovery, that (univerfally) the Aggregate of fuch infinite "Progreffion is equal (or doth approach infinitely near) to fuch a "Part of fo many times the greateft, as is denominated by the Exponent (or Number of Dimenfions) of fuch Power (as is "that according to which the Progreffion is made) encreas'd by 1. namely, of Laterals; of Squares; of Cubes; of Biquadrates; (of fo many times the greateft) and fo onwards 66 infinitely. 66 This Difcourfe of the Doctor's I thought convenient to infert, fuppofing it may give fome Satisfaction to the Learner, to hear fa Great a Man as Dr. Wallis's Argument about the Truth of these Series, which I have briefly deliver'd in the 'foregoing Lemma's. LEMMA VI. If any two Series or Ranks of Proportionals have the fame Num ber of Ternis (whether Finite or Infinite) it will always SAs the firft Term of one Series: is to the firft Term of the be other Series: fo is the Sum of all the Terms in the one Se ries to the Sum of all the Terms in the other Series. (12. e. 5.) As As in these Numbers, I 36 Or thefe Numbers 4 That is, 13: 2 39 4 12 515 6/18 5 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. 1. That every Line is fuppos'd to confift (or be compos'd) of an infinite Series of equidiftant Points. 2. A Surface (viz. the Area of any Figure) to confift of an infinite Series of Lines, either ftreight or crooked, according as the Figure requires. 3. A Solid to confift of an infinite Series of Plains, or Superficies, according as its Figure requires. Not that we fuppofe 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 fmall Parallelograms (or other Superficies) infinitely narrow, yet fo as that their Breadths, being all taken and put together, must be equal to the Figure they are fuppos'd to fill up. And thofe Plains or Superficies, which are here faid to con Atitute a Solid, are to be understood infinitely thin; yet so as that their Depths or Thickneffes (which are hereafter alfo called Lines) being all taken together, must be equal to the Height of the propos'd Solid. Now, in order to render this Hypothefis as eafy for a Learner to underftand as I can, I fhall here propofe a very plain and familiar Example; Viz. Let us fuppofe any Book to be compos'd (or made up) of 100, 200, 300, (more or lefs) Leaves of fine Paper; fuch a Book, being clofe 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 fmall 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 to, or 100, or 10co, c. they will then become fuch a Series of infinitely fmall Lines as are (by the Hypothefis) faid to compofe or fill up a Superficies. And all the Superficies of thofe infinitely thin or divided Leaves of Paper will become fuch a Series of Plains, or Superficies, as are faid to conftitute a Solid, viz. fuch a Solid as the Bignefs and Figure of that Book. Now according to this Idea of Lines, Superficies, and Solids, one may, without the leaft Prejudice to any Demonftration, admit of the following Definitions and Theorems. Definitions. I. The Area's of Squares, and all other Parallelograms, are compos'd or fill'd 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 Bafe, and equally decreafing 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 infinite Series of Ordinates rightly apply'd, or of an infinite Series of Right Lines parallel to its Tranfverfe Diameter... V. The Area's of the Parabola and Hyperbola are compos'd of an infinite Series of Ordinates; or may also be compos'd of Right Lines parallel to its Axis, &c. VI. A Prifm is a folid Body contain❜d or included within feveral equal Parallelograms, having its Bafes or Ends equal and alike; and it is generally nam'd according to the Figure of its Bafe: That is, VII. A Cube (or Solid like a Dye) is a Prism bounded or included with fix equal fquare Plains. VIII. A Parallelopipedon is a Prifm 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-ftone in a Garden) is only a round Prifm, having its Bafes or Ends a perfect Circle. |