FOR
FOR
horfe's force, is by making him draw up out of a well, over a fingle pully or roller ; and in fuch a cafe one horfe with an- other will draw 200lb. as already obferved. Five men are found to be equal in ftrength to one horfe, and can, with as much eafe, pufll round the horizontal beam of a mill, in a walk 40 feet wide ; whereas three men will do it in a walk only 19 feet wide.
The worft way of applying the force of a horfe, is to make him carry or draw up hill ; for, if the hill be fteep, three men will do more than a horfe, each man climbing up fafter with a burden of 100 lb. weight, than a horfe that is loaded with 300 lb. a difference which is owing to the pofition of the parts of the human body being better adapted to climb than thofe of a horfe.
On the other hand, the beff. way of applying the force cf a horfe, is in an horizontal direction, wherein a man can ex- ert leftft force ; thus a man drawing a boat along, by means of a rope coming over his fhoulders, cannot exert above one fe- venth part of the force of a horfe employed to the fame purpofe. The very belt and moft effectual pofture in a man, is that of rowing ; wherein he not only acts with more mufcles at once for overcoming the refinance, than iri any other pofition ; but as he pulls backwards, the weight of his body affifts bv way of leaver. Defiguliers, Exp. Phil. vol. i. p. 241, feq. where we have feveral other obfervations relative to force acquired by certain pofitions of the body, from which that author accounts for moft feats of ftrength and activity. Ibid. p. 254. feq.
FORCER, in mechanics, is properly a pifton, without a valve. See the article Piston, Cyd,
There are feveral ways of nmk'm^ forcers : the moft common of all confifts of a hrafs cylinder, a very little lefs in diameter, at its top and bottom, than the bore of the barrel of the pump, and turned Mill lefs at the middle, in order to let in a leathern ring or collar {made cf a thick leather put round the brafs cylinder) which makes it juft equal to the bore of the barrel, lb as to fit it quite when it is put into it. The fecond fort of forcers confifts of three brafs cylinders which can be fcrewed together. The middle one ought to be almoft equal in diameter to the bore of the pipe, fo as to Hide in it without any friction. The upper cylinder and the lower muft be a little lefs, and equal to one another. There are two leathers which muft be put between them when they are unferewed : then it is evident, that if the cylinders be fcrewed
" together, and the leathers, which ought to be a little bigger thaa the brafs cylinders, apply themfelves folding upwards round the upper cylinder, and downwards round the lower, they will become juft equal to the bore of the barrel ; and coafequently they will hinder any air from getting throuo-h the fides of the forcer, when it moves up and down in the . barrel. ,The ufe of the'middle brafs cylinder is to hinder the leathers from turning themfelves back by the motion. This kind of forcer has, above the other, the advantage of having a great deal lefs friction ; and bcfides, as the leathers, which are applied to it, may be thin ones, they are much fmoother than thick ones, which are ufed in the other. But the beft way of making farcers, in, to have a plunder, or folid brafs cylinder, equal in length to the barrel of the pump, and a little lefs in the diameter than the bore ; (b that it may move freely in it without any friction. There muft be two hollow, Alert, brafs cylinders, or rather rin°"s, at the top of the barrel, which can be fcrewed together ; the upper one muft be equal in bore to the barrel, and the lower a little ■ lefs : there are two leathers, both having in the middle a lefs hole than the bore of the pipe ; the one muft be applied be- tween the barrel and the lower ring, and the other between the fame ring and the upper one ; and the whole muft be fcrewed together. Then if the folid cylinder or forcer, be put into it, and moved up and down, it is evident that the two before-mentioned leathers, which are applied the one to the barrel and the other to the infide of the hollow cylinder or ring, will hinder any air from getting between them and the folid cylinder.
The advantage of this kind of forcer, is, that it has no other friction but at the top of the barrel, and that the infide of the barrel need not be fmooth, as in other kinds of pumps ; but only the outfide of the force r muft be turned true and polifhed, which can be done with much more cafe. See DefaguUers Courfe of Experim. Philof. p. 161, 162.
FORCING pump. See the article Pump, Append.
Forcing pipe. See the article Pipe, Append.
FORFICULA, the car-wig, in zoology, the name of a genus
' of infers, the tail of which forms a kind of forceps, capable of pinching; the exterior wings are very fhort, but dimidiat- ed ; and the antennas are fctaceous. See the article Ear- ivig, Append.
FORM(CycL) — Form of a fries, in algebra, is ufedfor that affection of an undeterminate ferics, fuch as, A^H-B* n + r - r -C*" + 2r + D* n + 3 % &. which arifes from the different values of the indices of x. Thus if « = 1, 3ndr=i, the feries will affume the farm* Ax+Bx* + Cxi+Dx* + y C5V. If »— 1, and r=2, the form will be s Ax + Bxi + Cxi+Dxi-t-, &c. If « — f, and r==I 3 the form is, Append.
nil be
Again, if n = o, and x=z— 1, the form o£ the feries v A + B.y-' + C*~ 2 + D*-3-|-, ^ f _
When the value of a quantity cannot be found exactly, it is of ufe in algebra, as well as in common arithmetic, to feek an approximated value of that quantity fufficient for practice. Thus in arithmetic, as the true value of the fquare root of 2 cannot be affigned, a decimal fraction is found to a fufficient degree of exaclnefs in any particular cafe. And this decimal fraction is in reality no more than an infinite feries of fractions converging or approximating to the true value of the root required. For the exprcflion ^2 = 1.41421356, &c, is equi- valenttothis */2=i-f- SV+T& + wfc + iQmnr+-noBW:b &c. Or fuppofing
■ 4 , J 4- 2 1
V ' X'XX ' X* ' X*~ X* ~ 9
&c
= 1+4* I -f-,-- I -f-4*-3 + 2*-4 + *-5_|_ >
EsV. which laft feries is a particular cafe of the general inde- terminate feries Aa-* -}-B.v k + ' -f-Cv"' 1 " 2 -^ , fefV, when
= 4, C :
» = o, r= — 1, and the coefficient Ar 1, D = 4, E = 2, &c> But the application of the notion of approximations in num- bers, to Ipecies, or to algebra, is not fo obvious. Sir Mac Newton, with his ufual fagacity,- took the hint, profecuted it, .and thereby difcovered general methods in the doctrines of infinite feries, which before him had only been treated in a particular manner^ though with great acutenefs, by Dr. Wal- Iis and a few others. See Newton's Meth. of Fluxions and infinite Series, with Mr. Colfon's Comment; as alfo the A- nalyfis per :equationes numero terminorum infinitas, publifhed by Mr. Jones, in 171 1, and fince tranflated and explained by Mr. Stewart, together with Sir Ifaac's Quadrature of Curves, Lond. 1745, quarto. To thefe may be added Mr. Mac Laurin's Algebra, part II. chap. x. p. 244. and Cramer's Analyfe des Lignes Courbes Algebriques, ch. vii. p. 148. Among the various methods for determining the value of a quantity by a converging feries, that fecms, on many ocea- fions, preferable to the reft, which confifts in afluming an in- determinate feries equal to the quantity, the value of which is fought, and afterwards determining the values of the terms of this affumed feries.
For inftancc, fuppofe a logarithm being given, it were re- quired to find its number.
Let the number be = i-f-A-, and the given logarithm s*z. Then, by the nature of logarithms and fluxions (fee the ar-
ticle Fluxion and Logarithm, Suppl.) 5*:=
— i — *
I+A 1
and
z-\-xz;=x. Suppofe,
- = Az- r -Bz*-|-Cz'- r -Dz + - r -, &c. confequenfly,
= i + Aai-f-Bz 3 i + Cz'i-f- Dz*~ + , fcfr.
Now if the correfponding terms of thefe two equal feries 's (each being zzzx) be compared, we fhall have A=i, B=|, C = £, D = -j' T , &c. which values beinfr fiib- ftituted in the feries Az -f-Bz^Cz < + , fcf>. give,
^Z+I^ + ^Z' + ^Z^t^Z'+^^-L, C5V. II X I
T j.a ^1.2.3 ^1.2.3.4 r r. 2. 3.4.5* -f-, l3c. And consequently i-f-.r, the number fought, will
be
- «-»+*«*.
But the indeterminate feries Az -f- Bzz-f-Cz 1 -}- , &c. was here arbitrarily aflumed, and will not fucceed in all cafes. For inftancc, if from an arc given it were required to find the tangent. Let * = tangent, <z;=arc, the radius = 1 . Then,
from the nature of the circle, we fhall have
1 -f-A-A-
x — v-\-xx<i: Now if to: find the value of x we fuppofe A-^Aiz-f-B^-f-C?; 3 -}-, isc, and operate as before, wc fliall find all the coefficients B, D, F, of the even powers of V, each =0. Therefore the feries affumed is not of a proper form. But fuppofing a- = Ai> -j-B-y 3 -f C-y 5 -f-Dy 7 -f, ts\. we fliall find A = i, B=4, C=V T > Dss-^-H &c. and confequendy x = v±^vi -L._i_ w s_L..JLZ_.y - .4.- & Cm Now to find a proper indeterminate feries inall cafes, tentative- ly, would often be very laborious, and often an impracticable work. Mathematicians have therefore endeavoured to find out a general rule for this purpofe. But, till lately, the me- thod has been but imperfectly underftoou and delivered. Moft authors, indeed, have explained the manner of finding the coefficients A, B, C, D, &c* of the indeterminate feries Aa-x+B*" r _j_CA-' : ' i ' 2 + , &c. which 'is eafy ; but the values of r. and r, wherein the main difficulty lies, have been affigned by many, as if they were felf-evident, of at leaft dis- coverable by an eafy trial or two, as in the laft example. Sirlfaac Newton himfclf has (hewn the method of determin- ing the number n, by his rule for finding the fifft term of a converging feries, by the application of bis parallelogram and ruler. For the particulars of this method, fee the authois above cited. See alfo the article Parallelogram, in- fra.
Dr. Taylor, in bis Methodus incrementorum, invcftlgatcs K the
