F R A
odd numbers 3, 5, 7, 9, BV. is to the product of the fame fquares feverally diminifhed by unity, as the fquare of the dia- meter is to the area of the circle b . — [ "Arithmet. infinit. prop. exci. Oper. vol. i. p. 469. r Id. Oper. vol. ii. p. 819.]
Thefe produfis of fractions, and the like quantities arifing from the continued multiplication of certain factors, have been particularly confidered by Mr. Euler, in his Analyfis infinit. vol.1, chap. XV. p. 221, feq. Continued Fraction is ufed for a fraction the denominator of which is a whole number with a fraction, the denomina- tor of which is again a whole number and a fraction, and fo on, whether this affection be continued ad infinitum, or whe- ther the feries breaks off after a finite number of terms. Thus,
2-1 , 1
3 + T+-
continued fraclions.
If we make ufe of letters inftead of numbers, we fhall have
general expreflions of thefe fraclions, thus,
3 1+—
f+4
+TJ.2
d+-
and
■ + 7 + ,
'+7+2
'+/+:
&v.
fcfv.
The reduction of thefe continued fractions to thofe of a com- mon form, is not difficult by the ufual rules of arithmetic and algebra. Thus, to give an example, which may at the fame time mew the ufe of thefe continued fractions, fuppofe the con- tinued fraction 1
3 +7+±j
+ i+, if*
which expreffes the circumference of a circle, when the dia- meter is one ; if we flop at -, we fhall have 7 4- - z= 2 i_Xi 7 7 7
22 1
55 £•• If we flop at — , we fhall have
J 7 + - JT io6-3 + I0 6 To6~-Io6
'5 But if we flop at 4-, which is convenient on account of the fmall fraliim T ^, added to the laft denominator 1, we mall then find, 3+ j ,_£. = 3 f*. The firft of the
"113-
reduflions gives the proportion of Archimedes, and the laft that of Adrian Metius.
But as beginning at the laft denominator of the continued frac- tion makes the computation fomewhat tedious, fhorter methods have been contrived for the rcduflion of thefe fractions, and for a continued approximation to their true value. And Mr Cotes's method for the reduffion of ratios to fmaller terms' may be here applied. (See the article Ratio, Suppl ) For it is to be obferved, that when the numerators of the conti- nued ■ fraCl,on are each units, the denominators will be the quotients arifing from the continued divifions in Mr Cotes's method j or in the common, for reducing /W?»w to a lower denomination, which is in effect the fame as Euclid's method for finding the greateft common meafure of two magnitudes lib. x. prop. 3.
But a detail of thefe things would lead us too far, we there- fore refer the reader for a farther account of thefe continued fractions to Dr. Walhs's Anthmet. infinit. prop. 19,. oper. vol. 1. p. 469, feq. Huygens, Defcript. automat, planetar. in oper pofthum. p.l 7 3»A- edit. Amftelod. 1728. and parti- cularly to Mr. Ws Analyf. infinit. vol. I. cap. xviii. p. 295, feq. who has fhewn the ufe and application of this do- ctrine in many mitances.
Lord Brouncker feems to have been the firft who confidered contmued fractions, or at leaft who firft applied them to the quadrature of curves The hint feems ufeful, but has been pretty much neglected excepting in approximations to frac- tions or ratios exprefied ,n great numbers. See the article Ra- tio, Suppl. His feries for the quadrature of the circle, is
1
□ = 1— 9
2 — 25 2 — 49 2 -Rr 2 V, Vc. 2
in his and Dr. Wallis's notation, which amounts to the fame thing as
FRO
in the notation of Huygens and Euler.
The fymbol D denotes, as before, the ratio of the fquare
of the diameter to the area of the circle.
Repetend of a decimal Fraction, See the article Repetend append. f
FRAMBOISE, a name ufed by fome for the rubus, or black- berry-bum. See the article Rubus, Suppl.
FRANK (Cycl.) — Frank, among farmers, denotes a place to feed a boar in. Diet. Ruft. in voc.
FRAY, among fportfmen. A deer is faid to fray her head when fhe rubs in againft a tree to renew it, or caufe the pills of her new horns to come off.
FREAM, among farmers, arable or ploughed land worn out of heart, and laid fallow till it recover.
Fream, among fportfmen, a term ufed for the noife of a boar in rutting time. Diet Ruft. in voc.
FRET, in mufic, is ufed for the flops ufed on fome inftru- ments, elpecially bafs viols and lutes : it confifts of firings tied round the neck of the inftrument at fuch diftanceF, with- in which fuch and fuch notes are to be found. Thefe firing! or frets are fometimes, yet feldom, put on the bafs violin for learners, and taken off again when they can find the notes without them. On lutes and viols they always remain. Graf- fin. Muf. Diet, in voc.
FRESCO (Cycl.) — The brufhes and pencils for this work ought to be long and foft, otheiwife they will rake and raife the painting. The colours fhould be full, and flowing from the brum ; and the delign perfect : for, in this work, you cannot alter or add upon any colour.
FRIAR's cowl, in botany, a name by which fome call the ari~ farum of botanical authors. See the article Arisarum, Suppl.
FRICTION (Cycl.) — It Ls hardly poffible to give general and exact rules concerning/W^w, fmce it depends upon the ftruc-
■ ture of bodies, the form of their prominent parts and cavities, and upon their rigidity, their clafticity, their coherence, and other circumftances. Some authors have made friclion upon a ho- rizontal plane equal to one third of the weight ; but others have found that it was only one fourth of it, and fometimes only I or y of it. Of late, authors have told us, that friclion depends not on the furface of the body, but its weight only ; but neither is this found to be accurately true. In lefter velocities, the fritlion is nearly in the fame ratio as the velocities ; but in greater velocities the friStton increafes in higher proportion, whether the bodies are dry or oiled. It being of great importance to diminifh this friction, feveral contrivances have been invented for that purpofe. In wheel- carriages, the friclion is transferred from the circumference of the wheel (where it would act if the wheel did not turn round) to the circumference of the axis ; and, confequently, is dimi- nished in the proportion of the radius of the axis to the radius of the wheel. In thefe, therefore, the friclion is always df- minifhed by diminifhing the diameter of the axis, or by in- creafing the diameter of the wheel. The friclion is likewife diminifhed by making the axis of an engine to reft upon the circumferences of wheels that turn round with it, inftead of reftlng in fixed grooves that rub upon it ; for, by this con- trivance, the friclion is transferred from the circumference of thofe wheels to their pivots ; and the friclion may be ftill diminifhed farther, by making the axles of thofe wheels reft upon other/r/'^?/;»-wheels that turn round with them. Dr. Defaguliers has treated fully of the friclion of engines, carriages, &c. See his Courfe of Experim. Philof. vol. i. p. '33 to 138. and p. 182 to 254. fee alfo p. 458 to 460.
FRIGERATORY, among builders, denotes a place defigned to keep things cool in fummer. Build. Diet, in voc.
FRILL, in falconry. When a hawk trembles or fhivers, they fay fhe frills. Di&. Ruft. in voc.
FRINGE-tore, the Englifh name of a genus of plants called by botanical authors chionanthus. See the article Chionanthus, Append.
FRINGILLA, in zoology, the name t of a bird, known in Englifh by the name of the chaffinch, and called fpina by the antient naturalifts.
The fame word is alfo fometimes ufed as the name of the brambling, a bird more frequently known by the name of the montifringilla, and called orofp'ma by the old authors. See the article Montifringilla, Suppl.
FRITTILLARIA craffa, in botany, a name fometimes given to the afclepias, or fwallow-wort. See the article Asclepias, Suppl.
FROG, among farriers, the fame with frufh. See the article Frush, Append.
FRONTALIS verus mitfculus, in anatomy, a name given by Douglas, in his Myography, to one of the mufcles of the face, called alfo by himfelf, and, fince his time, by Albinus, corrugator fupcrcilii. See the article Corrugator, Suppl.
FROUNCE, a difeafe in horfes, when fmall warts or pimples arife in the midft of the palate, which are very foft and fore, and fometimes breed in the lips and tongue.
This
