F R A
the number r ; but, according to Mr. Stirling 3 , the rule glv en by the doctor fails fometimes. [ J Lineae tert. ordin. New toman, p. 28.]
Mr. Stirling found a correction of Dr. Taylor's rule, but fays he cannot affirm it to bt; univerfal, having only found it by chance. 'S Gravefandc obferves, that though he thinks Mr, Stirling's rule never leads into error, yet that it is not perfect. See '5 Gravefande^ Ue determin. form, feriei infinit. printed at the end of his Mathefeos univerfalis elementa, Lugd. Bat. ,172.7. This learned profeflbr has endeavoured to rectify the rule. But Mr. Cramer has fhewn it to be frill defective ia feveral refpc£ts ; and be himfelf, to avoid the inconveniences to which the methods of thofe who wrote before him arc fubject, has afcended to the nrft principles of the method of infinite feries, and has entered into a more exact and inftruiftive detail of the whole method, than is to be met with elfewhere ; for which, and for many other reafons, his treatife deferves to be particularly recommended to beginners. But it mult be obferved, that in determining the value of a quantity by a converging feries, it is not always necellary to have recourfe to an indeterminate feries : for it is fometimes more expeditious to find it by common divifion, or by ex- traction of roots. See Newton, Meth. of Fluxions and inf. Series, above cited. Thus, If it were required to find the arc of a circle from its given tangent, that is, to find the value x
of v in the fluxional equation, ■£' = — ; , by an infinite fe-
n 1 -J-**' *
ties: divide x by i-j-jcv, the quotient will be the feries
- — x z x-\-.v + x — * ;6 *-4", c?V. And taking the fluents of
each term, we fhall have v~x — ^3-j-^ A *5-j-.: A -7j_ j & Ct which is the feries often ufed for the quadrature of the circle. If x .1, that is, if x be the the tangent of 45 , then will
- v=j — ^-|-4-_^_f_ 3 tjfe. = the length of an arc of 45 °,
■that is, l of the circumference of a circle, whofe radius = 1 ; or J of the circumference, if the diameter = 1. Confcquenly, if 1 be the fquare of the diameter, 1 — 4 + T — --f-,c?Y. =: the area of the circle : becaufe £ of the circumference multi- plied by the diameter, gives the area of the circle. And this is Leibnitz's feries, as alfo James Gregory's.
FORMEDySWj, in natural hiflory. See the article Formed Stones, Suppi.
FORMICA, among fportfmen, the name of a difeafe incident -to fpaniels. See the article Spaniel, Suppi.
FORMICATION (Dpi)— This term is alfo ufed amon- builders, for an arching or vaulting.
FORMS, among fportfmen, is faid of a hare when (he fquats in any place. Diet. Ruft. in voe.
FOUCADE. See the article Fougade, Cycl.
FOULDAGE, the fame with foldage. See the article Fold- age, Append.
FOUNDATION (Cycl.) — Architefls ought to ufc theutmoft diligence in regard to foundations, fince of all errors which may happen in building, an error in this point is moll perni- cious.
The ground fit for building upon is of various kinds ; fome- times it is fo hard as fcarce to be cut with iron ; in other places it is ftift", blackifh, or whitifh. This laft is reckoned the weakeft ; and, in general, that is the beft which requires mofl labour in cutting or digging.
When the ground is very bad, you muft get large oaken piles of fuch a length as may reach the found ground, and whofe diameter muft be about one twelfth part of their length : thefe muft be driven down with a machine, as clofe to one another as poffible, and that under the middle walls as well as the outer ones ; and upon their tops large planks are to be pinneddown. But if the ground be only faulty in fome places, arches may be turned over them, by which means no part of the weight of the building will reft upon them. As to the rules neceiTary to be obferved in conftruffinc the ground-work, they are thefe: i. That .the bottom of the trench be made exaflly level. 2. That the Weft ledge or row be all of ftone, laid clofe together. 3. That the breadth of the ground- work be at lead double that of the wall that is to he raifed on it. However, art ought always to give way to difcretion, for the breadth may be regulated according to the goodnels of the ground, and the weight of the intended edifice. 4. That the foundation be made to diminifh as it rifes, only care muft be taken that" it do fo equally on both fides. 5. That you ought never to build upon the ruins of an old foundation, unlels well affined of its depth and raod- nefs. Build. Dift. in voc. e
Foundation of bridges. Seethe article Bridge, Suppi.
FOX, wipes, in zoology, an animal of the dog-kind. See the articles Canis and Vulpes, Suppi.
PfiVf' 7 ^"^' '" bota "J'- Sce the article Gkass, Append.
b OY LING of land, among farmers, is the fallowing it in the fummer or autumn. Diit. Ruft. in voc.
Fovling, among fportiinen, a term ufed for the footfteps of
™ , Sfe,^ L thc S ralii or Jeaves - Dia. Ruft. in voc.
FRACTIONCC),-/.;- /»>fe/„,W ./Fractions. The fums, or rather the limit of the fums of infinite feries's of fraBimt, has been one of the principal objefls of the modern method- of computation ; and thefe fums may oftenlbe found. Thm the fum of i + i+|-t- T ' T + , tsV. ad infinitum, or rather, I 7
F R A
the limit to which this fum may approach nearer than by a given difference is 1. So likewife the limit of the fum 4- + -0- + -17-+TT + , &f- is i. And thefe andthelike fums of geo- metric progreflions may be readily found, by applying the com- mon rule for determining the fum of geometric progreflionsfrom the firft and laft terms and the ratio given. For in thefe con- verging infinite geometric progreflions, the laft term muft be conlidered as ; fo that the fums of the antecedents of thefe progreflions are confidered as the fum itfelf, becaufe they dif- fer from the fum of the progrcflion by lefs than any affi>ned quantity; but in every geometric progreflion the fum of the antecedents is to the fum of the confequents as one antece- dent to one confequent. Hence calling the fum S, the firft term a, and the ratio r, we mall have S : S — a::a:ar::i:r.
Therefore Sr=zS— a, orS=-^— . Thus if a — \, and
r=i, S will =i divided by I, or 1. And if a = {-, and r — ,, S will = .; divided by §, or § ; and fo of the reft. But the feries's of fractions that occur in the folution of pro- blems are rarely reducible to geometric progreflions; nor can any general rule, in cafes fo infinitely various, be given. The art here, as in moft other cafes, is only to be Acquired by examples, and by a careful obfervation of the arts ufed by great authors in the inveftigation of the feries's of fraclions they have confidered. And the general methods of infinite feries which have been carried (o far by Mr. de Moivre ', Mr. Stirling', and Mr. Euler', are often found necenary to determine the fum of a very fimple feries of fractions — [ ■ Mifcel. Analyt. paffim. ' Method. Differentialis. <Ana- lyf. infinitorum, & Aa. Pctropol. paflim.] The fum of a feries offracjions decreafing continually, is not always finite, but fometimes infinite, that is, no limit can be afligned but what may be exceeded by the fum of a certain number of the terms of the feries. This is the cafe of the feries 1+ i + \ +I+-J-+4, &c. called the harmonic feries, the fum oi which (as has been faid under the article Pro- gression) exceeds any given number, and the analogy of this progreflion with the (pace comprehended between the Apollonian hyperbola and its afymptote, fhews this. But the fame may be fhewn independently of the hyperbola from the nature of progreflions. See Jac. Bernoulli, De Seriebus infinit.
The foundation of Mr. Bernoulli's demonftration is, that a number of terms, beginning from any part of the feries, may be found, the fum of which fhall always exceed unity, ana confequently the number of terms of the feries being fuppofed infinite, as many partial fums as we pleafe, each exceeding unity, may thus be taken out of die feries, which therefore may be continued till it exceed any given number. But if the denominators of this harmonic feries, A, ~, ' ' CiV. befquared, that is, if we form the feries;, i,'-,"tjc the common numerator of which is I, and the denominato-s" of which are the fquares of the natural numbers I, 2, 3 4 feV. the fum of this feries of fractions will not only be limit- ed, as was faid under the head Pkocression, Suppi but this fum will be precifely equal to the fixth part of the num- ber, which ejrprefles die ratio of the fquare of the circum- ference of a circle to the fquare of its diameter. That is if the circumference be 3.14159, jSfc. and th e diameter 1, then will
±+i+i+-^+^+, lSc.=U ^f^- \ Thispro-
pofition was firft difcovered by Mr. Euler, and his inveftiga- tion may be feen in the A3a Petropolit. vol. vii. Mr. Mac Laurin " has fince obferved, that this may eafily be deduced from his Fluxions, art. 822. — [ aPhilof. Tran'f. N°. 469.] It would require a treatife to enumerate the various kinds of feries's offracjions which may be fummed. Sometimes the fum or limit of the infinite feries cannot be afligned, either becaufe it is infinite, as in the harmonic feries A+i-l-.'.-i-i SV. or although this fum be finite, as in the 'feries ~+t+t+tV+j 6fc yet its fum cannot be afligned in fi- nite terms, or by the quadrature of the circle, or hyperbola, which was the cafe of this feries before Mr. Euler's difcovery ; but yet the fum of any given number of the terms of the fe- ries may he expeditioufly found, and the whole fum may be afligned by approximation, independently of the circle. See Stirling, Method. Different, and De Moivre, Mifcel. Anal. Befides the feries's ol fractions, the fums of which converge to a certain quantity, there fometimes occur feries's of frac- tions, which converge by a continual multiplication. Of this kind is the feries found by Dr. Wallis ', for the quadrature of the circle, which he expreffes thus, D _ 3*5x5X5x7X 7 X 9 x 9 x Isle.
~2X 4 X4X6X6x SXSxitJxW- Where ° gn ^ eS the ratio of the fquare of the diameter to the area of the circle. Hence the denominator of this fraction, continued ad infini- tum, is to its numerator as the circle is to the fquare of its diameter. It. may be obferved, that this feries is equivalent 3X3 5*5 7X7 3' <>
t0 ; -r ■* -jt* ir*' ■*'■ or t0 y—i x 5^5 *
T^ — X , bV. that is, the product of the fquares of ajl .the
odd
