i :? 



NAVIGATION-NAUTICAL ASTRONOMY. [rorr.u-i.T's 



jataral tendency of the pendulum tuflieiently overrule* 

 th*o hindrance* to render tlie deviation % of its ] .nh 



r.U tli.- wo*t always palpable, and always accumula- 

 tive.* The rotation of the earth from we*t to oiut U the 

 only way of accounting for this apparent deviation f the 

 path of the bob ; what we really toe is the mot inn of the 

 lino PP', receding from tin- track of the bob, with an 



.!.ir movement about tin- centre P, towards tlie east. 

 If the pendulum can be kept - .ilicieiitly long, it 



>ui tlmt, an thu angular deviation U always accumu- 

 lative. :i inn.- will come wlii-n it (hall have Described a 

 . or 3tH) ; or, to speak more accu- 

 nt.-ly, when the radius 1' 1' of tlie table shall have re- 

 TO!TM completely round. Let us seek to determine this 

 time. 



TIME IN WHICH THR MERIDIAN LINE COMPLETES \ HK- 

 As the absolute direction of the oscillations 

 remains invariable, it follows that, when the cone of 

 latitude has tunied through any angle, the original hori- 

 zontal lino must have turned through an equal angl< 

 that, when tin- cone has made a complete revolution, the 

 deviation of tile path of the bob, from the horizontal line, 

 amounts to an angle equal to the plane angle given by 

 'ping tho conical angle. t Thin, it is obvious, can 



r amount to so much as 360, or a complete revolu- 

 tion of the table, except when the bob is suspended over 

 one of the ]< iles of the earth, the c-inic surface then be- 

 coming a jiliine, tangential to the sphere of the polo. 

 Proceeding from this extreme limit towards the equator, 

 them -becomes loss and less, till, on ivach- 



iii4 the e.|ii:iti.r, it vanishes altogether,' as before re- 

 nicked ; the cone then become* a cylinder, and no 

 deviation can take place. 



It thus appears that thu angle of deviation in any time, 

 at any place, is a [.lane angle exactly equal to the deve- 

 lopment of the convspon ling conical angle, turned 

 through in that time by the rotation of tin- cone <if lati- 

 tude It remains to be ascertained what part this angle 

 of drvi.nion is of a complete rotation, or 300. In other 

 c.r<ls, we have to ascertain what part of 300 the 

 developed angle at the apex of the cone of latitude 

 amounts to. 



..vivo an upright cone, unconnected with the globe, 

 mid from its apex let any two straight lines be drawn 

 along the slant side, intercepting an arc of the base ; this 

 arc will measure a certain angle at the centre of the base. 

 Measure from the apex along either of the two lines 

 drawn from it, a length equal to the radius of the base, 

 and describe with this length, as radius, a circular arc, 

 limited by the two straight lines on the cone : this arc 

 will evidently measure the conic angle ; it will therefore 

 be to the corresponding arc of the base, as the angle at 

 the a p. t to the corrcs] Minding angle at thu centre of the 

 base ; but the developed arcs are also to one another as 

 their distance* from the apex : for these distances are 

 tln-ir r-idii. Hence, the conic angle is to the correspond- 

 ing plane angle or angle at the base on the horizontal 



as the rndius of the base of the cone to the 



b of the slant side ; that is, as the sine of half the 

 plane angle of the cone to unity. Consequently, the 

 whole conic angle is to 360 a* the sine of half tin; plain! 

 angle of tin' one that i, as the nine of the latitude is 

 to unity. Hence, from what is shown above, the angle 



viatioii, or of the path of the bob, in one eutn 

 tation of the earth, U to 300 as the sine of the latitude 

 U to unity. The time of tills rotation is 24 hours ; cou- 



In tb experiment u performed at the Royal limitation of Great Bri- 

 Uln, the impulaa wae not titrn by the hind ; the bob wo* drawn nut of 

 the rertical br a lkrn thrrad, Ibc mil of which was fattened to a peg in 

 Hi floor : a flame wa> then applied to the thread, and the ball ael free. 

 The author of thU treatUc flr-t witneMcd the experiment at tho houee of 

 ttlrMinc friend. W. II f lliirhif.it.- in th.- different re- 



ettttoai of It. the rii>p"n.linir wire, nerly twenty feet in length, wa> of 

 ditrnm metal, and the hand wan employed to let the bob go; the de- 

 Utio wae quite palpable in eight mlnuu->. 



t The proper diMtnetion mint, of enurae, be obarrred hr the reailcr 

 Wtwee the anile of the eooe awl the conical angle. The angle at tin- 

 MM le the pUne angle of the laoaeelea triancle, which the *"i nf tin- 

 one thnmirh Ita uU pre*enU; the ennical nIe U the anile at the 

 Tenet, formed hr the .urface of the cone : If tbia nirface be cut along the 

 etratchl IUe which mj be eoncetr4 to generate It, and then the turface 

 ilerifaaeil or sn/old.-rt Into a plane, the oonlcal angle will become an 



,tly, 'Jl hours divided by tho sine of the latitude, 

 is the time in which tho path of the pendulum make* a 



iet.- revolution in that latitude. Thus, assuming 

 the deviation to be 3(in , we have tho angle of rotation 

 of the earth in any time being x 



ion 

 angle of rotation 



".ri i 



3GO . x sin. lat., .'.x . , in degrett. 



sin. Ut 



24 hours . 

 ' 



And all tho more carefully conducted experiment* jus- 

 tify this result, within those limits of dill'erenee that they 

 K-.-LMinably be attributed to thu disturbing causes 

 adverted to aliVO. 



\\ e have further obtained, from a few dimple 

 considerations, thu following ' proposition, 



n imely : 



The length of the arc of the rim of the table, subtend- 

 ing the angle of deviation at its centre, which a pendulum 

 oscillating over it makes during one rotation of the earth, 

 is exactly equal to the difference between the parallel of 

 latitude described by that centre, and the parallel de 

 scribed by the extremity 1" of the meridional diameter of 

 tho table. 



Draw Pr, (Fig. 22), PV perpendiculars upon the 

 of the earth, and P I) parallel to the axis N S. It has 

 been proved above that the angle of deviation in one 

 revolution of the earth is 



360 sin. N- 30 



360 



P'D 



This angle, multiplied by the radius P P* of the table, 



and by II 1411'., and the product divided by 1X0', is the 

 arc of tho rim of the table subtending it ; that is, tho 

 measure of this arc is 2 1" D X 3-1410. But twice 1' D 

 is the difference between the diameters of the two paral- 

 '. <scribed by P and P'. Hence the arc that measures 

 the an^lu of deviation in one rotation of the earth, is 

 e.|u.il to the diiFurence between the two circumfei. 

 described by P and P'. And the arc of deviation, duo 

 to any portion of a complete rotation of the earth, is 

 inal to the difference between the two portions of 

 parallels described by P and F. 



The same conclusion may be obtained by aid of con- 

 siderations still more simple. It is plain that the dif- 

 ference between the circumferences of any two equidis- 

 tant circles on the surface of a cone is always the same ; 

 hence, if a circle bo described about the apex, with a 

 radius equal to P P*, the circumference of it will be equal 

 to the cliii'eiviice between the two circumferences de- 

 scribed by 1'and I'. But this same circumference, when 

 the cone is developed, is the arc of deviation, on the 

 table, due to a complete rotation of the earth: }>< 



ire must IKS equal to the difference between the two 

 .lula described by P and P' in a complete rota 

 lion. 



It thus appears, not only that the pendulum experi- 

 ment aM'ords ocular demonstration of the rotation of the 

 earth, but that it moreover exhibits to us the actual 

 velocity, ill linear measure, with which the point P' pro- 

 ceeds in advance of P. It is the velocity with which the 

 arc of deviation increase*. 



If the length of this arc, described in any interval of 

 time, be measured, we may readily deduce tlie arc that 

 would be deserilied in a complete rotation of the earth. 

 If the length of this arc be taken for the circumference 

 of an entire circle, the diameter of that circle may be 

 inferred ; this diameter, applied as a chord to the circle 

 of deviation, will subtend an arc of it, tho degrees and 

 minutes of which will be double the latitude of the place. 

 And thus we may conceive it possible that a person, 



conveyed to a dungeon in s e unknown part of the 



world, with a piece of string ami a weight at hand, might 

 forman estimate of the latitude of nis position. The 

 following diagram shows tho method of trying the 



