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The Doctrine and Ufe of Infinite Series, one of the greateif Improvements of the prefent Age, we owe to Uic Mercmor of Holfiein, who, however, firtt took it from Dr. Wallis's Arithmetick of Infinites. It takes Place principally in the Quadrature of Curves : Where, as we frequently fall upon Quantities, which cannot be exprefled by any preciie definite Numbers, filch as is the Ratio of the Diameter of a Circle to the Circumference, we are glad to exprefs them by a Series, which, infi- nitely continued, is the Value of the Quantity requir'd : An Idea of the Nature, Ufe, £jfc. of Infinite Series, may be conceived from what follows.
Though Arithmetic furnifh us with very adequate and intelligible Expreffions for all rational Numbers, yet it is very defective as to irrational Ones ; which are infinitely more numerous than the other ; there being, for Inftance, an Infinity of them between t and 2. Were it now required to find a Mean, proportional between 1 and 2, in rational Numbers, which alone are clearly intelligible, (the Root of 2 being cettainly a very obicure Idea) we could frill approach nearer and nearer to the juft Value of the Quantity required, bur without ever arriving at it: Thus, if for the mean Proportional between j and 2, or the Root of 2, we firlt put 1, 'tis evident we have not put enough 5 if we add £ we put too much, for the Square of 1 + 1 is greater than 2. If then we take away £, we fhall find we have taken away too much, and if we return ^, the Whole will be too great : Thus may we proceed, without ever coming at the juft Quan- tity fought. Theft Numbers thus found, and thole found after the fame Manner to Infinity, being difpoftd in their natural Order, make what we call an Infinite Series.
Sometimes the Series do not proceed by alternate Ad- ditions and Subtractions, but by fimple Additions, or an Infinity of Subtractions ; according to the Pofition of the firft Term. In all thefe Infinite Series, 'tis viflble, that as all the Terms are only equal to a finite Magni- tude, they niuft be ftill decreafing ; and 'tis even conve- nient that they be ft, as much as poflible, that one may take only a certain Number of the firft Terms for the Magnitude fought, and neglect all the reft.
But 'tis not irrational Numbers only, that are ex- prefled in rational Ones, by Infinite Series. Rational Numbers themftlves may be exprefled in the lame Man- ner : 1, for Inftance, being equal to the Series ± ^ -^, f>c. but, there is this Difference between them, that where- as irrational Numbers can only be exprefled in rational Numbers, by fuch Series; rational Ones need no fueh Expreflion.
Among Infinite Series, there are fome whofe Terms only make a finite Sum, fuch is the Geometrical Pro- greflion 1 J | £fe. and in general, all Geometrical, decreafing Progreffions : In others, the Terms make an infinite Sum ; fuch is the Harmonica! Progreffion x i 1 i rjfc. 'Tis not that there are more Terms in the Harmonical, than in the Geometrical Progreffion, tho' the latter has no Term which is not in the former, and wants ftveral the former has : Such a Difference would only render the two infinite Sums unequal, and that of the Harmonical Progreffion the greateft ; the Cauft lies deeper. From the received Notion of infinite Divisibility, it follows, that any finite Thing, e. gr. A Foot is a Compound both of Finite and Infinite: Finite, as it is a Foot ; Infinite, as it contains an Infinity of Parts, into which it is divifible. If thefe Infinite Parts be con- ceiv'd, as feparated from one another, they will make an Infinite Series, and yet their Sum only be a Foot 5 only, no Terms are to be here put, but fuch as may, di- ftincT: from each other, be Parts of the fame Finite Whole : Now this is the Cafe in the Geometrical, de- creafing Series \.$\. cifc. for 'tis evident, that if you firft take \ of a Foot, then -J of what remains, or \ of a Foot, then \ of what remains, or •§ of a Foot ; you may proceed to Infinity, ftill taking new decreafing Halves, all diftincr. from each other, and which all toge- ther only make a Foot. In this Example, we not only take no Parts but what were, in the Whole, diftiiitf from each other ; but we take all that were there ; whence it comes to pafs, that their Sum make the preciie Whole again : But, were we to follow the Geometrical Pro- greffion \ J |„ gc. that is, at firft take J- of a Foot, and from what remains ^ of a Foot, and from what ftill remains | 7 of a Foot, £=fc. 'tis true, we fhould take no Parts but what were diftinft from each other in the Foot ; but we ihould not take all the Parts that were there, fince we only take the ftveral Thirds, which are left than the Halves ; Of Conftquence, all thefe de- creafing Thirds, though Infinite in Number, could not make the Whole, and 'tis even demonftrated, that they
would only make Half: In like Manner, all the Fourths decreafing to Infinity, would only make one Third and all the Hundredths, only the Ninety-ninth Part'- ) that the Sum of the Terms of an Infinite Series' fa creafing geometrically, is not only always finite, but may even be lefs than any finite Quantity thar can be a% n ed If an Infinite decreafing Series exprefs Parts which cannot fubfift in the Whole, diltinflly from each other; but f uc h as to take their Values, we mull fuppofe the j ani j Quantity taken ftveral Times, in the lame Whole; then will the Sum of theft Parts make more than a Whole nay, infinitely more ; that is, the Series will be Infinite' if the fame Quantity be taken, an Infinity of Times! Thus, in the Harmonical Progreffion t t 4i &&■ if we take ■f- of a Foot, or 6 Inches, then -J 4 Inches, 'tis evident we cannot take further ^ of a Foot, or 3 Inches, with. out taking i Inch more than was left in the Foot. Sj nC e then the Whole is already exhaufted by the Three firft Terms, we can take no more of the following Terms without taking fomething already taken : And, fince thofe Terms are infinite in Number, 'tis very pofTible that the fame finite Quantity may be repeated an ing, nite Number of Times, which will make the Sum of the Series infinite. We fay foffibk ; for though of two M. nite Series, the one may make a finite Sum, and the other an Infinite; 'tis true, that there may be fuch a Series, where the Finites, having exhaufted the Whole, the following ones, though infinite in Number, fhall only make a finite Sum: And, in Effect, fince 'tis de- monftrated, by Geometrical Progreffions, that there are Series, whole Sums are lefs than the Whole, nay, infi. nitely lefs ; it follows, that there niuft likewife be Series which make infinitely more.
There are two further Remarks neccflary to be made on Series in general. 1°. That there are lbme, wherein, aftef a certain Number of Terms, all the other Terms, though infinite in Number, become each a Cypher. Now, 'tis evident, that the Sum of thefe Series is finite, and e'aftly found; they having only an Appearance of Infinity. 2°. That the fame Magnitude may be exprefled by different Series 1 and may be exprefled both by a Series, whole Sum may be found, and by another., whofe Sum cannot be found.
Geometry does not labour under the fame Difficulty as Arithmetic. It expreffes irrational Numbers exactly in Lines, and needs not have Recourfe to Infinite Series: Thus the Diagonal of a Square, whofe Side is , is known to be the Root of 2 . But, in fome other Cafes Geometry it felf is under the like Embarrafs ; there be- ing fome Right Lines, which cannot be exprefled other- wife than by an Infinite Series of fmaller Lines, whofe Sum cannot be found : Of which Kind are the Right Lines equal to Curves ; fo that in fteking, for Inftance, a Right Line equal to the Circumference of a Circle, we find, that the Diameter being 1, the Line fought will be f Minus, I Plus, < Minus, * Plus, *, £r?c.
As to the finding of Infinite Series, to exprefs Quantifies fought ; Mercator, the firft Inventor of the Method, did it by Divifion : But Sir I. Newtm and M. Leibnitz, have improved the Doctrine very confiderably ; The Firft find- ing his Series by the Extraflion of Roots and the Second by another Series prefuppofed.
To find a Series expejjing a Quantity fought, fy iDwiJion.
Suppofc a Series required to exprefs the Quotient of i divided by a -\- c. Divide the Dividend by the Divifor, as in common Arithmetic ; continuing the Divifion till the Quotient fliew the Order of the Progreffion, or the Law, according to which the Terms proceed to Infinity : Still observing the Rules of Subtraaion, Multiplication, and Divifion about the changing of the Signs. The
Procefs carried on, the Quotient will be found A~* +
j, }[1 1 1'
—. l$c. in Infinitum. Thefe Four or Five Terms
a* a*
thus found, both the Quotient and Manner of the Divifion fhew, That the Quotient confifts of an Infinite Series of Terms, whofe Numerators are the Powers of c, whole Exponents differ from the Number of the Order by U n |" ty, whofe Denominators, are the Powers of a, and their Denominators equal to the Number of the Order of the Terms, li.gr. In the Third Term, the Power of c is the Second in the Numerator ; and the Power of a the Third in the Denominator.
Hence, 1°. If b=i and a=i fubftituting this Value for that, we have, in that Quotient 1 — c-^-c 1 — ci -> &' to Infinity. Wherefore -J— =i— c-\-c'— £ J > & e - '"
infinite —
