PEN
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PEN
wherein the feveral Ofcillations areperformM; and therefore reciprocaJIy as the Squares ot the Numbers of Ofcillations preform 'd in the fame Time.
On this fame Principle, M. Monton, Canon of LyonS t has a pretty Treatife, de menfura fofteris tranfmittenda.
Mr. Huygens lays down the Length of a 'pendulum that fhallfwing Seconds, to be % Feet, 3 Inches, and 2 Tenths or an Inch 5 according to Sky, Moor's Reduction ; which agrees perfectly with M. Mouton's 'Pendulum 8 Inches 1 Tenth long, to Vibrate 132 times inaMinute: fo that this may be relied on asafure Meafure.
Note, the Lengths of pendulums are ufually meafur'd from the Centre of Motion.
The Firft who obferv'd this noble Property, the Ifochronifm of 'Pendulums, and made Ufe thereof in mcafuring Time, Stur- mius tells us, was Ricciolus ; after him Tycho, Langrenus, Wendeline, Merjenne, Kircher and others hit on the fame Thing ; tho' without any Intimation of what Ricciolus had done.
Huygens firft applied the 'Pendulum to Clocks. See c Pen- duluta Clock.
<pendtilnms are either Simple or Co?npotffid.
Simple Pendulum, isthat confuting of alingle Weight, as A, confider'd as a Point ; and an inflexible right Line, as AC, confider'd as void of Gravity, fufpended on a Centre C, and voluble about it. fTab. Mechanicks Fig. $6)
Corn-found Pendulum, is that which confifts of feveral Weights, fo fix'd as to retain the fame Diftance both from one another, and from the Centre about which they vibrate.
The Dotlrine and Laws of Pendulums.
A 'Pendulum rais'd to B, thro' the Arch of the Circle S A ; will fall, and again rife, thro 1 an equal Arch, to a 'Point equally high, 2) ; and thence fall to A, and again rife to B 5 and thus continue rifwg and falling reciprocally, for ever.
For fuppofe H 1 a horizontal Line, and B 2) parallel theretos it the Bail A, which we here confider as a Point . be rais'd to B ; the Line of Direction B H, being Perpendicu- lar from the Centre of Gravity B to the Horizontal Line H J, falls without the Bafe, which is in the Point C.
The Ball therefore cannot reft, but muft defcend. See Descent.
But beingretain'd by the Thread B C, from falling perpen- dicularly thro' B H; it will fall thro' the Arch B A. Con- fequently, when the Centre of Gravity arrives at the Bottom ; A has the lame force, it wou'd have acquired in falling from K' and will therefore be able to rife equally high as if it had, i.e. in defcending thro' the firft half of its Vibration, it acquires a Velocity by the continual Acceleration of its Fall 5 and as this Velocity is always proportionable to the Height whence it falls, as being in fome meafure the Effect thereof ; it is {till able to make it: remount to the fame Height, fup- pofing according to the Syltem of Galileo, that the Veloci- ties arc always the fquare Roots of the Heights. See Acce- leration.
Since then the Thread prevents the 'Pendulum going off in the Tangent A I, it muft afcend thro' the Arch A D, equal to that AS.
All the Force therefore which it had acquired by falling, being exhaufted; it will return by the force of Gravity thro' the fame Arch A C D, and again rife from ^to S 5 and thus for ever. ^ E. T). See Gravity.
Experience confirms this Theorm, in any finite Number of Ofcillations; but if they be fuppofed infinitely continued, there will arife a Difference. For the Refiftence of the Air, and the Friction about the Centre C, will take off part of the Force acquired in failing ; whence it will not rife precifely to the fame Point whence it fell.
Thus the Afcent continually diminiflring; the Ofcillation will be at laft ftopp'd, and the 'Pendulum hang at reft. See Resistance and Friction.
II. If a fimple 'Pendulum befufpended between two Semi-cy- chidsCBand CD.fTab Mechanicks Y\g.^ )Whcfe generating Circles have their diameters C F equal to half the length oj the thread C A ; fo as the 'Thread in Of Mating be wound about 'em ; all the Ofcillations, however unequal in Sface, will be jfochrcnal, or perform' d in equal Times 5 even in a rejipng Medium.
For fince the Thread ot the, 'Pendulum CE, is wound about the Semi-cycloid B C; the Centre of Gravity of the Ball E, which is here confider'd as a Point, by its Evolution, will defcribe a Cycloid B EA ffi; as is fhewn from the Doc- trine of Infinites; but all Afcents and Defcents in a Cycloid are Ifochronal, or equal in Time : Therefore the Ofcillations of the 'Pendulum are alfo equal in Time £KE.Z). See Cy-
Hence,if with the length of the 'pendulum CA, a Circle be defcribed from the Centre C ; fince a Portion of the Cycloid near the Vertex A, is almoft defcribed by the fame Motion; a fmall Arch of the Circle will almoft coincide with the Cycloid.
In little Arches of a Circle, therefore, the Ofcillations of 'Pendulums will be Ifochronal astoSenfe; however, unequal in 'emfelvcs ; and their Ratio to the Time of perpendicular Defcent thro* half the length of the 'Pend.umm, is the fame with that of the Circumference of a Circle to its Diameter.
Flenceaifo, the longer the 'Pendulums are, that ofcillate in Arches of a Circle ; the more Ofcillations arc Ifochronal ; which agrees with Experiment 5 for in two 'Pendulums of equal lengths, but ofciilating in unequal Arches, provided neither Arch be very great, you'll fcarce difcern any inequa- lity in an hundred Ofcillations.
Hence alfo, we have a Method of determining the Space which a heavy Body, falling perpendicularly, pafies over in a given Time. For ihe Ratio which the Time of one Ofcilla- tion has to the Time of the Fall thro' half the Length of the 'pendulum, being thus had ; and the Time wherein the feveral Vibrations of any given c P?-iidulimi being found 5 we have the Time ot the Fall thro' half the length of the 'Pendulum, And hence may collect the Space it will pafs over in any other Time.
The whole Doctrine of 'Pewkthtms ofciilating between two Semi-cycloids, both Theory and Practice, we owe to the great Huygens ■, who firft publifh'd the fame in his HorcL Ofcillatcr. five demon-fir ationei de Mow 'Pendulomm , &c.
Ill The Atlicn of Gravity > is lefs in thofe 'Parts of the Earth where the Ofcillations of the Jame 'Pendulum are fiower, and greater where they arefwifter.
For the Time of Ofcillation in a Cycloid, is to the Time of Perpendicular Defcent thro' the Diameter of the generating Circle, as the Periphery of the Circle to the Diameter.
If then, the Ofcillation of the fame 'Pendulum be flower 5 the Perpendicular Defcent of heavy Bodies islikewife flower 5 i. e. the Motion is lefs accelerated, or the Force of Gravity is lefs: and converfely. See Gravity.
Hence, as 'tis found by Experiment, that the Ofcillations ot the fame 'Pendutum are flower near the Equator, than in Places lefs remote from the Pole ; the force of Gravity is lefs towards the Equator than towards the Poles. And con- fequcntly the Figure of the Earth is not a juft Sphere, but a Spheroid. See Earth and Spheroid.
This M. Richie? found by an Experiment made in the In- land Cayenna, about four Degrees from the Equator ; where a 'Pendulum 3 Foot, S Lines § long, which at 'Paris Vibrates feconds, was to be fliorten'd a Line and a Quarter to reduce its Vibrations to Seconds.
M. des Hayes, in a Voyage to America, confirms the Ob- fervationof Richier ; but acids, that the Diminution eftab- lifh'd by that Author, appears too little.
M.Couflet the younger, upon his return from a Voyage to Brafil and 'Portugal, falls in with M. des Hayes, as to the Ne- ceffity of fhortening the 'Pendulum towards the Equator more than Richier has done. He obferv'd, that even at Lisbon the 'pendulum which beats Seconds, muft be two Lines-|. fhorter than ihat of 'Paris$ which is fhorter than that of Ca- yenna, as fix'd by Richier 5 tho' Cayenna be in 24 Degrees lefs Latitude than Lisbon.
The Truth is, this Diminution does not proceed regularly : Mefs. tpicart and de la Hire, found the Length of the 'Pendu- lum which beats Seconds exactly the fame at Bayonne, at 'Pa- ris, and at Uranisbcurg in tCenmark ; tho' the firft be in 43^ ■| of Latitude, and the laft in the Latitude 55 3'.
Hence M de la ILre takes cccafion to fufpect that the Di- minution is only Apparent; and that M.gr. the Iron Yard, whercwirh M. Rtchier meafur'd his 'pendrluw, .might be lengthen'd by the great Heats of the Ifle of Cayenna ; not the 'Pendulum fhortcn'd by the approach towards the Line.
To confirm this, he tells us he found, by very careful Expe- riments, that an Iron Bar, which expofed to the Froft was 6 Foot long; was lengthen'd \ of a Line by the Summer's Sun. See Dilatation, Heat, Thermometer.
IV. If two 'Pendulums vibrate in fimilar Arches, the Times of the Ofcillations are in the Subduflicate Ratio of their Lengths.
Hence the Lengths of Pendulums vibrating in fimilar Arch- es, are in a Duplicate Ratio of the Times wherein the Of- cillations are perform 'd.
V. The Numbers rf Ifochronal Ofcillations ferfcrm'd in the fame Time by two 'Pendulums, are reciprocally as the Times wherein the feveral Ofcillations are perform' d.
Hence, the Lengths of 'Pendulums vibrating in fimilar and, fmall Arches,are in the Duplicate Ratio of the Numbers of Of- cillations perform 'd in the fame Time, but reciprocally taken.
VI. The Lengths of 'Pendulums fuff ended between Cycloids, are in a duplicate Ratio oftheTimes wherein the feveral Ofcil- lations are ferform'd.
And hence they are in a Duplicate Ratio of the Numbers of Ofcillations perform'd in the fame Time, but re- ciprocally taken : and the Times of Ofcillations indifferent Cycloids are in a Subduplicate Ratio of the Lengths of the <Pendulmns. VII. To
