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 ^ «.//«*■ '"/ 
 
 MEMOIR 
 
 ON 
 
 A NEW AND CERTAIN METHOD 
 
 OF ASCERTAINING 
 
 THE FIGURE OF THE EARTH 
 
 BY MEANS OF 
 
 OCCULTATIONS OF THE FIXED STAKS. 
 By a. CAGNOLI. 
 
 WITH 
 
 NOTES AND AN APPENDIX 
 By FRANCIS BAILY. 
 
 LONDON 
 1819* 
 
PRINTED BY RICH.IBD AND ARTHUR TAYLOR, 8H01S tANB. 
 
ADVERTISEMENT. 
 
 The following Memoir originally appeared in the 
 Transactions of the Italian Society {Memorie di 
 Matematica e di Fisica della Societa Italiana. 
 Tom. vi. Verona 1792) : but, although it has 
 been so many years before the public, I do not find 
 that the subject of it has been taken up by any 
 persons either here, or on the continent. In fact, 
 I believe that it is not generally known in this 
 country : and it is with a view of drawing the at- 
 tention of astronomers more closely to it, that I 
 now present them with this translation. The ac- 
 knowledged talents and abilities of the Author 
 must at all times ensure it an attentive perusal : 
 but particularly at this period, when a more than 
 ordinary degree of interest is excited towards the 
 subject of the true figure of the Earth, I am 
 aware that there are some apparent difficulties in 
 the method pointed out in the Memoir, yet I am 
 
ADVERTISEMENT. 
 
 not without hope and expectation that they may be 
 eventually overcome : and that the mode herein 
 proposed may at least be brought in aid of the 
 other methods at present adopted for determining 
 that difficult problem. 
 
 To the Memoir itself I have subjoined some Notes 
 connected with the subject ; which, however, may 
 be distinguished from those of the Author by the 
 addition of the letter B. And to the whole I have 
 added an Appendix, wherein I have attempted to il- 
 lustrate the views of the Author ; and ventured to 
 propose such methods as may best tend to carry his 
 object into execution. 
 
 Francis Baily. 
 
 Gray's Inn, 
 May 20, 1819. 
 
 *^* As the present Memoir is not published for sale, I 
 
 shall be happy to furnish such persons, as may send their 
 
 Cards for that purpose, with any number of copies they 
 
 may require. 
 
 F. B. 
 
MEMOIR 
 
 ON 
 
 A new and certain Method of ascertaining 
 the Figure of the Earth. 
 
 By a, CAGNOLI. 
 
 § 1 . The truth of the Newtonian theory of uni- 
 versal gravitation is proved by the wonderful agree- 
 ment in all the celestial phsenomena, which have 
 hitherto been submitted to the tests of calculation 
 and of observation. There probably is not any as- 
 tronomer of the present day who is not convinced 
 that our planet is compressed, or flattened, at the 
 poles, and protuberant at the equator. In fact, if 
 we combine this theory of Newton with the hypo- 
 thesis of the elliptical form of the earth, we shall 
 find that the precession of the equinoxes and the 
 nutation of the earth's axis are sufficiently ac- 
 counted for. Other arguments indeed, in favour 
 of the elliptical form, may be derived from the rota- 
 tion of the earth, which is now generally admitted : 
 
 B 
 
Z CAGNOLIS MEMOIR, 
 
 and others again from the diiferences in the length 
 of the pendukim vibrating seconds in different lati- 
 tudes. Still however we want soijije theory of the 
 internal density of the earth; which, it is feared, 
 will ever remain hid from Iwaman investigation. But 
 if, to the want of this essential knowledge, we add 
 the irregularity of the figure of the earth, as shown 
 by the measurements of the degrees on its surface ; 
 if, to the doubts which some persons may still enter- 
 tain of the truth of the compression of the poles'^, 
 we add the discordant opinions as to the quantity 
 of that compression ; we may surely assert that it 
 will be no small advantage to point out a method 
 (independent of all hypothesis) for determining, 
 with facility and with the greatest precision, the dif- 
 ferences between the terrestrial radii at an indefinite 
 
 number of points on the earth's surface. Such 
 
 indeed is the object of the present memoir. 
 
 ^ 2. It has always been said, when speaking 
 of the compression or flattening of the poles of the 
 earth, that the Parallax of the Moon would afford 
 the best means of ascertaining it ; provided the va- 
 riations, arising therefrom, were of sufficient mag- 
 nitude to be observed with perfect accuracy. But, 
 since, by supposing this compression not to ex- 
 ceed -3-^ of the semidiameter (which is the most 
 received opinion at the present day) there would be 
 
 * See Lorgna " Prmdpj di Geografia ^c." § 31. 
 
CAGNOLI S MEMOIR. 3 
 
 a (Efference of not more than 9'' between the moonV 
 parallax at the equator, and her parallax at a 
 high latitude, for instance 60°; so indeed it is 
 but too true that a difference so small might be 
 easily concealed under the possible errors of obser- 
 vation. It was, on this account, that Manfredi 
 and Maupertuis in vain suggested the determina- 
 tion of the compression by direct observations of 
 the moon's parallax : nor has any advantage been 
 hitherto obtained by this method. 
 
 § 3. But this is not the only instance in which 
 a quantity, extremely minute on one side, has been 
 found to produce effects that are quite discernible 
 on the other. The attention of the observer should 
 be directed to those points where they are most 
 sensible. This has been the object of my research ; 
 and I hope not entirely without success. For, there 
 are circumstances in which scarcely a second in the 
 parallax may cause a difference of 15, 20, or 30 
 seconds of time, or even more, in the duration of 
 the occultation of a star by the moon. But, oc- 
 cultations, more particularly when the immersions 
 take place behind the dark limb of the moon, can 
 easily be observed without committing an error of 
 a single second of time*: so that opportunities^ 
 
 * The duration of an occultation cannot probably be so well ob^ 
 served when the immersions take place behind the dark limb of the 
 moon, as when they take place behind the illuminated side j since 
 the instant of emersion cannot be so well ascertained. B. 
 
 b2 
 
4 CAGNOLIS MEMOIR. 
 
 exist not only of removing all doubt as to the re- 
 ality of the compression of the poles, but also of 
 such frequent occurrence that the gradation of the 
 compression, (or variation of the curvature) at 
 different latitudes, may be accurately distinguished. 
 The importance of the subject induces me to treat 
 it more fully in detail. 
 
 § 4. I shall not take up the time of the reader 
 in demonstrating that no hypothesis is required to 
 deduce the variation of curvature from the observa- 
 tion of the parallax. For, since the variation of 
 curvature is nothing more than the inequality of 
 the terrestrial radii (and that is nothing else than 
 the difference of the parallax), it is evident that 
 finding by observation the variation of parallax at 
 different latitudes is in fact finding the variation of 
 curvature by immediate observation. One involves 
 the other without any intermediate help. The only 
 difficulty consists in freeing from uncertainty those 
 observations which show the unequal parallax : and 
 this is precisely the special character of the parti- 
 cular kind of occultations which I have in view. It 
 is true that astronomers, in calculating occultations, 
 have made the parallax vary in conformity with that 
 quantity of compression which they have adopted 
 as the true theory. Moreover, these variations do 
 not in general produce differences which are dis- 
 cernible in the relative duration of the occultations 
 
CAGNOLl's MEMOIR. J5 
 
 observed in different latitudes. Those differences, 
 which are discernible, arise only in certain circum- 
 stances, to which hitherto no particular attention has 
 been paid. It is on this account that occultations 
 have not, as yet, served at all to determine the 
 amount or quantity of the compression of the polar 
 axis. 
 
 ^ 5. Before I proceed to describe what these fa- 
 vourable circumstances are, it will be proper, in order 
 to appreciate their utility, to define the precise degree 
 of accuracy of which the computation ^nd observation 
 of occultations are capable. With respect to the 
 computation, it is evident that the apparent distance 
 between the star and the moon cannot be accurately 
 determined unless we know exactly not only the 
 apparent place of the star, but also that of the moon 
 and its diameter. These elements are obtained by 
 calculating the phaenomena from observations made 
 in a place (the longitude of which is known) under 
 circumstances where the alteration of the parallax, 
 on account of the variation in the curvature of the 
 earth, does not produce a perceptible difference in 
 the duration of the occultation : conditions not un- 
 common, nor difficult to be obtained. 
 
 § 6. Let us then suppose that the moons dia- 
 meter and the place of the star are well determined 
 (in which elements the least uncertainty exists) ; and 
 
P CAGNOLI S MEMOIR. 
 
 that any small errors, which may occur, be thrown 
 qn the place of the moon. Let us, moreover, en- 
 deavour to discover (by comparing the calculation 
 with the observation) the errors of the tables in the 
 longitude and latitude of the moon, mixed (as we 
 have already stated) with the preceding errors'^. 
 In the present state of astronomy I do not see any 
 great danger to be apprehended from this union of 
 the two errors, notwithstanding the delicacy of the 
 investigation in question. At any rate we may 
 arrive at a sufficient degree of accuracy by verifying 
 the place of the star by a sufficient number of ob- 
 servations, and also the diameter of the moon by 
 means of the telescope which is used for the occuU 
 tations. 
 
 § 7. Having corrected the errors of the lunar 
 tables, we shaH have accurate elements wherewith 
 to represent the state of the heavens : and there can 
 then be no doubt that we may arrive at great ex- 
 actness in our calculation of the duration of an oc- 
 cultation (or of the moment of immersion and 
 emersion) for the place in which we wish to ascer- 
 tain the curvature of the earth : since the geographic 
 cal longitude of the place has been determined by 
 
 ♦ See the Memoir which obtained the prize from the Academy of 
 Copenhagen : " Mithode pour calculer les longitudes geograpk'njiues,'^ 
 
CAGNOLl's MEMOIR. 
 
 means of occultations under circumstances in which 
 (and they are the greater part) the variation in the 
 curvature is not perceptible. 
 
 § 8. With respect to the accuracy of observations 
 of this nature, I have ah'eady said that the immer' 
 sions behind the dark limb are not subject to the 
 least error. Those on the illuminated side may 
 be liable to some slight uncertainty if the star be 
 not of the first or second magnitude, or if the power 
 of the telescope be too small. The emersions frorri 
 the dark limb are in general to be depended upon : 
 whilst those from the illuminated side are the most 
 doubtful of all^'^. But, even supposing that the 
 error, in the last-mentioned case, may amount to 
 8 or even 10 seconds of time, will it be sufficient 
 to conceal the variation of curvature altogether 
 in the circumstances I have contemplated, and 
 where the difference that it will produce may 
 be ten or twenty times greater than the presumed 
 
 * It appears that, in some occultations, the immersions and emer- 
 sions take place wholly on the dark side of the moon. See an ac- 
 count of the occultation of A Virginis observed by Mr. Troughton, 
 on May 22, 1801. Connaissance des Terns, Annee xnl. page 342. 
 Sometimes an occultation may be observed when the moon itself is 
 not visible: as in the occultation of Venus on May 13, 1801, between 
 8 and 9 o'clock in the morning ; the moon being then only a few 
 hours old. Ibid, page 417- A favourable occultation may also occur 
 during a total eclipse of the moon, when the immersions and emer* 
 sions of stars of the 6th or 7th magnitude may be distinctly observed. 
 Ibid. Annie ix. page 335. B. 
 
8 CAGNOLl's MEMOIR. 
 
 error ? Can we, indeed, expect to obtain more cer- 
 tain observations^ ? since a single observation of this 
 kind accurately made in the place, for which the 
 variation of curvature is sought, is sufficient to 
 discover it by means of the comparison of the caU 
 mlated with the observed moment. 
 
 ^9.1 shall now point out what are, in fact, the 
 circumstances in which the differences of the pa- 
 rallax between two latitudes may be shown in an 
 undoubted manner : and, in order virtually to em- 
 brace all the cases, I shall take mean quantities in 
 the elements which enter into this investigation. 
 Let us therefore suppose, first, that the moon's ap- 
 parent semidiameter is 15'. 45", its equatorial pa- 
 rallax 57'. 40", ^nd its horary motion 32'. o^'\h : 
 secondly, that the occultation is observed in N. La- 
 titude 60°, in which parallel there are three cele- 
 brated observatories, viz. Petersburg, Stockholm 
 and Upsal: lastly, that the apparent height of the 
 moon is 10°. If there be no variation in the cur- 
 vature of the earth's surface the parallax of height 
 will be 56'. 47'', 4t: but, if the polar compres- 
 sion amount to -3-^ of the earth's radius, the same 
 
 • I find it difficult here to give a faithful translation of the author's 
 words : the original runs thus, " Senza che ci possiamo attenere alle 
 " osservazioni piu sicure : ed una sola fase, &c. &c." B. 
 
 f Let p == the horizontal parallax of the moon j and h = the 
 height of the moon : then t z=p.cosh = the parallax of height. B. 
 
CAGNOLI S MEMOIR. 9 
 
 parallax will be 56'. 38", 9 ^. The parallax would 
 therefore under these circumstances experience an 
 alteration of 8'',5 ; a quantity certainly too small to 
 be verified with accuracy by means of observations 
 of the height of the moon. But, this slight alte- 
 ration produces, in certain cases, effects that are 
 very visible : a fact which escaped the observation 
 of Maupertuis ; who, it is true, speaks of occulta- 
 tions as a mean of discovering the flattening of the 
 earthf. But, he speaks of them generally; and so 
 slightly as to class them with appulses, as being 
 equally fit to determine it. Now, as appulses are 
 far from being observable with such certainty, in 
 regard to time, as the instantaneous disappearance 
 and reappearance of stars in occultations, it is evi- 
 dent that Maupertuis can never have had in view 
 the particular cases which I am about to point out, 
 and which differ materially from ordinary ones. 
 
 * Let a = the polar compression of the earth, supposed = -j-^^ ; 
 X 1= the latitude of the place ; and the radius of the equator equal 
 to unity : then we shall have the length of any other terrestrial 
 radius = (1— a. «3i"^x) nearly j which, being multiplied by <r, will 
 give sr = * (I— a sm'^>-)= the parallax of height on the supposition 
 that the earth is an oblate spheroid. Whence we also have - = 
 the length of the terestrial radius at any given latitude. And if ^ 
 denote any given increase of the parallax, we shall have - for the 
 corresponding increase in the length of the Earth's radius. B. 
 
 f Preface mi discovrs sur la parallaxe de la lune. 
 
10 
 
 CAGNOLl S MEMOIR, 
 
 ^10. Let US suppose AVB a portion of the cir- 
 cumference of the moon's disc opposite to the ob- 
 
 server ; C the moon's centre ; and that the radius 
 CV, which divides the arc AVB into two equal 
 parts, coincides with the vertical circle of the place 
 of the observer. Let us further suppose that the 
 line MV (or the ver-sine of the arc) is equal to 60'' ; 
 tind that, during the occultation, the chord AB of 
 the moon is that which the star would apparently 
 describe if the earth were spherical; or the chord DE 
 that which it would apparently describe if the axis of 
 the earth were compressed -j-^. Then ML will be 
 equal to 8",5 as before mentioned : and, computing 
 from the m^an rate of the horary motion of the 
 
CAGNOLl's MEMOIR. 1 1 
 
 moon, it will be found that the duration of the oc- 
 cultation behind AB will be 20'. 7'' of time ; and, 
 behind DE, 18'. 41"^. We therefore see how con- 
 siderable is this difference of r.26'^ of time ; and 
 how well such observations are adapted not only to 
 convince those who doubt the reality of the com- 
 pression of the earth's axis, but likewise to show 
 with very great approximation to truth the relative 
 length of the earth's radii in different latitudes. 
 For, in the case just mentioned, the variation of a 
 twenty-thousandth part of that length f will cause 
 a difference of one second of time in the duration 
 of the occultation. 
 
 § 11. Moreover, it is evident that the effect, of 
 which I have been speaking, may be much greater. 
 
 * For, joining C,A, and C,B, we shall have CV=CA=CB:^ 
 15'.45", and CM = (15'.45"-60")=14'.45"; consequentlyAM=MB= 
 1^(AC— CM). (AC+CM)=5'.31",4; which, being multiplied by 
 in order to reduce this distance into time, will give 
 
 32'.S6",5* 
 
 1(K. 3", 6 for the time of the star's passing from A to M : there-r 
 fore the time of the star's passing from A to B will be equal to 
 2x(10'.3",6) =2(y. 7",2. But, if the depression of the earth's 
 axis cause a variation in the apparent height of the moon equ^l 
 to 8", 5, then will CL=14'. 53",5, consequently (as CD==AC) DL=5 
 LE= >v/(AC— CL) . (AC+CL)=5'. 7",7 ; which, being also multi- 
 
 tiplied by iri^» ^^^^^ §^^^ ^'' ^^">^ ^^^' ^^^ ^^^^ °^ *^^ ^^^^-^ 
 passing from D to L : therefore the time of the star s passing from 
 DtoEwillbeequalto2xC9'.2(y',5)==18'.4R B. 
 
 fOr, about 1000 feet. The mean radius of the earth being 
 20;898,240 English feet. B. 
 
12 CAGNOLl's MEMOIR. 
 
 It increases greatly as the line MV (or the ver-sine 
 of the arc) becomes smaller: if, for example, it were 
 30", the duration of the occultation under AB would 
 be 14'. 20'' ; and, under DE, 12'. 10", being a differ- 
 ence, in the duration, of 2'. 10" of time *. In such 
 a case, a difference of 500 feet in the length of the 
 eartli's radius might be rendered evident: since 
 such a variation would produce a difference of one 
 second of time in the duration of the occultation t- 
 ^12.1 am well aware that it scarcely ever hap- 
 pens that the apparent path of the star is rectilinear, 
 
 * In this case, CM =: (15'. 45" -30") = 15'. 15"; consequently 
 AM= 3'. 56",2: which being multiplied by -^57-c^E X 2, as in the 
 preceding cases, will give 14'. 20",4 for the time of the star's 
 passing from A to B. But, supposing the depression the same as 
 in the former instance, we should have CL=15'. 23",5 ; consequent- 
 ly DL = LE = 3'. 20",4 : which, multiplied also by 3^^-;^ X 2, 
 will give 12'. 10",2 for the time of the star's passing fromD to E, 
 as mentioned in the text. B. 
 
 f According to my calculation, a difference of only 400 feet in the 
 length of the earth's radius would produce a difference of one second 
 of time in the duration of the occultation. But, if the line MV were 
 only 15", a difference of less than 200 feet would produce a difference 
 of one second of time in the duration. And, agreeably to these prin- 
 ciples, it will be evident that the duration of such occultations will 
 vary according to the elevation of the spot (from which they are ob- 
 served) above the level of the sea; and will sensibly differ when ob- 
 served in valleys, or on the tops of high hills and mountains. A fact 
 which perhaps has not been sufficiently attended to, either in obser- 
 vations of occultations, or in calculations wherein thp parallax of the 
 moon is involved. B. 
 
CAGNOLrs MEMOIR. 13 
 
 and at the same time perpendicular to the vertical 
 circle which passes through it, at the moment of 
 the middle of the occultation, as in the case I have 
 supposed. But such a supposition may be readily 
 conceded, as being a mean between the possible 
 cases. For, if the actual phsenomena do in some 
 eases deviate therefrom, so as to render the effect of 
 the compression less upon the duration of the oc- 
 cultation than when calculated on the above hypo- 
 thesis, it will on the other hand often occur that 
 they are greater^. Hence I have no hesitation in 
 asserting that the means, which I have pointed out, 
 are not dependent on any hypothesis, and are the 
 most certain of any that have yet been suggested 
 for determining the variation in the curvature of the 
 earth's surface. 
 
 § 13. With respect to Xhe^ pendulum, the differ- 
 ences in its length are not only too minute and too 
 difficult to be measured vi^ith precision, but are also 
 too much influenced by the unknown internal den- 
 sity of the earth to be brought at all into comparison 
 with the great differences, above mentioned, in the 
 duration of occultations. 
 ^14. The different measurements of the degrees 
 
 • I confess I do not see the accuracy of this conclusion : for it 
 appears to me that the effect of the compression, on the duration 
 of the occultation, must be greatest when the apparent path of the 
 star is perpendicular to the vertical circle. B. •• 
 
14 
 
 CAGNOLl S memoir; 
 
 of the meridian are likewise so discordant arnongsf 
 themselves, that the respectable author (whom I am 
 about to quote) asserts that the greater the number 
 of degrees that are measured, the more uncertain 
 does the figure of the earth seem to be^. Let us 
 
 * This indeed appears to be the case from some of the late mea- 
 surements. With respect to the meridian of France, M. Delambre 
 states (Tra'ite d'Astronomie, Vol. 3. page 567) that "it was hoped 
 '* that the measurement of that arc (divided nearly into tvvo equal 
 ** parts by the parallel of Evaux) would have given the quantity of 
 ** the compression of the earth's axis. The calculation, however, 
 " makes it = T3-g ; a quantity much too great. The whole arc, corn- 
 spared with that of Bouguer at Peru, makes it = ji^ ; which \s 
 *' too small." And he goes on to observe that, from a revision of the 
 calculations of Lacondamine, and taking a mean between him and 
 Bouguer, he has deduced the quantity -g-^ ; which he considers as 
 the most probable one, atid which he proposes to be adopted in all" 
 future calculations. Nevertheless he afterwards states (^Ibid. page 
 572) that a ** compression of xVo ^'^P^'^^^"^'' very well the arcs be- 
 *' tween Greenwich and Paris, between Dunkirk and Paris, between 
 *' Greenwich and Evaux, between Greenwich and Barcelona, and' 
 " between Carcassonne and Mount-Jouy. It appears then (he adds) 
 ** that, without much variation, the arc of the meridian from Green- 
 *' wich to Bai'celona, indicates (in the whole and in its several parts) 
 ** a compression which differs very little from -j-|f5-" And the conclu- 
 sion, which he draws from the whole, is that "the arc of France 
 *' indicates a compression more considerable than that of the globe 
 ** in general." It is worthy of observation that Maupertuis, who was 
 one of the astronomers that measured the arc of the meridian in 
 Sweden in 1736, makes the compression = p-f^, by comparing it 
 with the arc then measured in France. But, M. Svanberg who, 
 inl805, remeasured the same arc, deduces the compression =-3^.4 
 when compared with the arc recently measured in France ; and 
 ~"Sins-7 when compared with that of Bouguer at Peru. Tliis is 
 
CAGNOUS MEMOIR. 15 
 
 moreover bear in mind the enormous expence and 
 labour required in measuring a degree of the earth's* 
 surface. Let us recollect the great uncertainty that 
 may attend such a measurement from the deviation 
 of the plumb-line made use of in the instruments^ 
 employed for the astronomical observations, occa- 
 sioned by mountains and valleys, subterraneous 
 cavities and other irregularities in the upper strata 
 
 however on the presumption that the standard measure, used in the 
 survey, was equal to the double metre at zero of the thermometer : 
 but, if it was so only at the temperature of + lG'^f, the corre- 
 sponding compressions would be -j^-s- 9» «^iid -g-jr t* Other results 
 are obtained according to the different tables of refraction made use 
 of in the calculations; so that, after all, considerable doubts exist as to 
 the exact quantity of the compression, as deduced from that survey. 
 M. Delambre, from the measurements, given by Major Lambton, of 
 an arc in the East Indies, stated the compVession = ■^-^t--j ' but, 
 from the subsequent operations of the same gentleman, he afterwards 
 deduces it = -^1^. Major Lambton himself, however, from a more 
 recent measurement has stated the compression to be = 5-^, 
 but, that on comparing the whole of his arc with the whole of the 
 French arc, tbe compression would be = -j^-^. A remarkable discor- 
 dance in the results. The measurement of Lacaille, at the Cape of 
 Good Hope, compared with that at the equator, gives -^(j. Lastly, 
 I shall add that the recent measurement of the ait;, in England, 
 seems to indicate a prolongation, instead of a compression of the polai* 
 axis. These discordant conclusions leave a very unsatisfactory im- 
 pression on the mind, notwithstanding the acknowledged talents of 
 the persons employed in those surveys 3 and prevent us from placing 
 too great reliance on a method which produces results, differing so 
 widely from each other. The learned Boscovich might well exclaim 
 that " the greater the number of degi'ees that are measured, the 
 ** more uncertain does the figure of the earth seem to be." B. 
 
16 CAGNOLl's MEMOIR. 
 
 of the earth : and, of what great importance such 
 anomahes may he, we may learn in the celebrated 
 \V'ork of Boscovich *. 
 
 § 15. It may perhaps be objected that the two 
 conditions necessary for my purpose may not fre- 
 quently occur together ; namely, that the altitude of 
 the moonf, and the chord of the lunar disc which 
 passes over the star, must be small. This is pro- 
 bably the reason why this method of ascertaining 
 the compression of the earth's axis has not hitherto 
 been practised. It is true that the greater the alti- 
 tude of the moon above the horizon, and the nearer 
 to its centre the apparent path of the star, so much 
 the smallerwillbe the difference of parallax caused by 
 a sphere and an oblate spheroid J: and consequently 
 the more easily might the errors of the elements 
 
 • De Expeditione Litteraria, Lib. v. § 230, et seq. 
 
 t It does not appear that the altitude of the moon should be limited 
 to the quantity mentioned by the author in the text, since the varia- 
 tions, alluded to, will be perceptible at the height of 20, and even 
 30 degrees. Thus, in the case mentioned in § 9, it appears that 
 the variation in the parallax of the moon at the height of 10'' is 8",5 : 
 but, if the moon were at the height of 20^ above the horizon, the 
 variation would be 8",1 j and at 30" it would be 7",4. B. 
 
 \ The smaller the chord of the lunar disc (or the smaller the ver- 
 sine of the arc) the more perceptible will be the difference in the 
 duration of the occultation in such cases j as will appear from the 
 following Table ; which shows the duration of an occultation of the 
 star on the supposition that the place of the star, when it passed the 
 vertical circle, would be 60", 30", 15", 10" and 8",5 wdthin the moon's 
 
CAGNOLIS MEMOIR. 
 
 17 
 
 made use of in the calculation be confounded with 
 that difference. But these are precisely the occasions 
 most favourable for the observations with which 
 these particular occultations may be compared ; and 
 by which (as I have said) the errors of the tables are 
 corrected, and the geographical longitude deter- 
 mined. 
 
 ^ 16. It is true that the conditions here required, 
 for ascertaining the flattening of the poles, cannot 
 be very frequently obtained : but, if we look out for 
 them with the diligence that the importance of the 
 
 disc, provided the earth were a perfect sphere : together with the 
 corresponding durations provided the earth were an oblate spheroid^ 
 haying the axis compressed = -j-^-o". Latitude of the place 60" : 
 height of the moon 10". 
 
 Earth = Sphere. 
 
 Earth = ^^ 
 
 Difference 
 
 of 
 Duration. 
 
 ^ within 
 D's disc. 
 
 Duration. 
 
 iS- within 
 })'s disc. 
 
 Duration. 
 
 // 
 60 
 30 
 15 
 10 
 8,5 
 
 20:7 
 
 14.20 
 
 10.11 
 
 8.19 
 
 7.41 
 
 // 
 
 51,5 
 
 21,5 
 
 6,5 
 
 1,5 
 
 
 
 i^.4r 
 12.10 
 6.43 
 
 3.14 
 Appulse 
 
 1 II 
 1.26 
 2.10 
 3.28 
 5. 5 
 7.41 
 
 In the Appendix I have given other tables, showing the differ- 
 eitces that would arise from varying the latitude of the place, the 
 height of the moon, and the quantity of the earth's compression : 
 whereby the reader may be better able to judge of the maximum of 
 difference which would arise under the most favourable circum- 
 stanges. B. 
 
 C 
 
18 CAGNOLl's MEMOIR. 
 
 object demands, we shall perhaps meet with them 
 more frequently than we imagine. For, I may 
 venture to assert that there does not pass a month 
 without the occurrence of an occultation of some 
 star whose position is well known : and there is no 
 occultation that will not afford, to some part or 
 other of the earth's surface, the conditions required * . 
 If only once out of twenty times they should occur, 
 in a place where there is an astronomical observa- 
 tory, the question as to the compression, and 
 also as to its quantity, would be very soon deter- 
 mined. 
 
 § 17. For the solution of such questions, for 
 which immense sacrifices have hitherto been made 
 
 * Since this was written, the positions of most of the zodiacal stars 
 have been determined with a degnie of accuracy suflScient for the 
 purposes detailed in the memoir. The labours of Cagnoli himself* 
 of Piazziy Harding, Zach, and Bessell have contributed much to this 
 end : so that we may now safely assert that scarcely a night passes 
 " without the occun*ence of an occultation of some star whose posi- 
 ** tion is well known." Nearly forty years ago Messier made the fol- 
 lowing remark, at the close of a numerous list of observed occulta- 
 tions : ** We see by this collection of occultations how many new ones 
 '* I have observed, in the first quarter of the moon, on the dark 
 " limb J which are distinguished with the greatest precision. They 
 " are frequent, and much preferable to the observations of Jupiter's 
 " satellites, or lunar eclipses, for determining the longitude. It were 
 *' much to be wished that the conductors of our Ephemerides should 
 " announce, for the first part of each lunation, the immersions of 
 *' stars of even the 7th, 8th and 9th magnitude, which are as readily 
 ** observed as those of the 1st, 2nd and 3rd magnitude." Connais^ 
 mnce des Terns, Annie viii. page 319. B. 
 
CAGNOU'S MEMOIR. 19 
 
 (for instance, in the measurement of the degrees of 
 the meridian) it surely is not requiring too much 
 that the trifling expense should be incurred of ena- 
 bling astronomers to travel to places more favoura- 
 bly situated for making observations of such occul- 
 tations. This would be an undertaking worthy of 
 a sovereign who wishes to distinguish himself as a 
 patron of science. We should, by such means, 
 gradually arrive at a knowledge of the relative length 
 of the terrestrial radii, in a great number of places : 
 and, it is most probable that we might thereby be 
 enabled finally to deduce the true and exact figure 
 of the earth. 
 
 § 18. Although it appears^ from the different 
 measurements of the degrees of the meridian, that 
 the figure of the earth is not regular, still it is pos- 
 sible that the irregularities do not belong so much 
 to the figure, or the radii, as to the nature of the 
 upper strata, the different density of which may 
 occasion the concealed error in the perpendicularity 
 of the instruments : an error which (as I have 
 elsewhere shown) may be quite sufficient to recon- 
 cile all the disagreements between the measure- 
 ments hitherto taken =*. Consequently the varia- 
 
 • Independent of the deviations arising from the causes here al- 
 luded to, the plumb-line has been sometimes known to be attracted 
 towards the sides of the glass vessel, containing the weight, as 
 powerfully as gold-leaf towards an electrical tube. The remark ap- 
 
 c 2 
 
20 CAGNOLl's MEMOIR. 
 
 tions in the parallax in different latitudes might 
 very well proceed with as much regularity as appears 
 to exist in the variation of gravity, and in the 
 length of the pendulum. 
 
 § 19. It is in the powerof every principal Academy 
 materially to assist in such a discovery, by two 
 methods. First, in regard to times past^ to collect 
 together, from all quarters, the observations of oc- 
 cultations stated to have been made in a given in- 
 terval ; for instance, in the last ten years : and to 
 employ some calculator to select and compute all 
 those which are proper for showing the variation of 
 curvature at different places. Secondly, with re- 
 spect to Xhe future, to insert in the Ephemerides 
 
 pears to have been made by M. Flaugergues. '* Astronomers ought 
 *' to avoid using a glass vessel for the water in which the weight of 
 " the plumb-line is suspended ; for, I have observed twice, in one 
 ** year, a singular deviation in the plumb-line occasioned by the at- 
 " traction of the ball of the plumb-line towards the side of the vessel 
 *' in which it was suspended. This ball was drawn towards the side 
 " with as much rapidity as gold-leaf is attracted by an electrified tube : 
 '^ and I was obliged (in order to destroy the effect of this spontane- 
 *' ous electricity, so as to enable me to take equal altitudes) to put 
 *' a coating of sealing-wax upon the ball. But, since I have substi- 
 " tuted a metal vessel, this singular phsenomenon has not again oc- 
 *' curred." Connaissance des Terns, Annee xiii. pageAl^. Although 
 such a powerful impulse as this may not often have occurred, yet it 
 is possible that slight deviations from the perpendicular may fre- 
 quently have arisen from the cause here alluded to ; and which may 
 account for some anomalies which have been remarked in the obser- 
 vations made in the course of the surveys. B. 
 
CAGNOLl's MEMOIR. 21 
 
 accurate notices of those places or districts where 
 it would be most important that any occultation 
 should be observed (particularly of the principal 
 stars), in order that it might serve to apprize, and 
 excite the attention of such astronomers as might 
 be favourably situated themselves, or contiguous to 
 more advantageous situations. It appears to me 
 that such notices would be more important and use- 
 ful than those of the phases of solar eclipses, about 
 which the calculators of Ephemerides are in the 
 habit of taking so much trouble. 
 
 END OF THE MEMOIR. 
 
APPENDIX 
 
 By F. BAILY. 
 
 Since the original publication of the preceding 
 Memoir, I am not aware that any attempts have 
 been made to derive any practical results from the 
 theory of the learned author ; notwithstanding the 
 simplicity of the method which he has proposed. 
 Perhaps no practicable mode, which has been hi- 
 therto suggested for determining the true figure of 
 the earth, or the precise quantity of the compres- 
 sion of the polar axis, is entirely free from errors : 
 but, as those errors probably arise from different 
 sources, according to the methods adopted, it is de- 
 sirable that the modes of investigating it should be 
 varied as much as possible, in order that the exist- 
 ing discordances may be ultimately reconciled or 
 removed. 
 
 M. Lalande has stated {Bibliographie Astrono- 
 mique, page 613) that in the Ephemeris of Vienna 
 for 1791 there is "a dissertation on the figure 
 " of the earth by M. Triesnecker ; who has de-^ 
 
24 APPENDIX. 
 
 " duced, from sixteen occultations, the compress 
 " sion = -34-5-." Whether the occultations, which 
 were used by M. Triesnecker in his calculations, 
 were the peculiar sort of occultations alluded to 
 by M. Cagnoli, I am unable to ascertain ; as I have 
 not been able to procure a sight of M. Triesnecker's 
 labours on this subject : but, as that work was pub- 
 lished two years before this Memoir, I much doubt 
 whether his method is precisely the same as that 
 laid down by M. Cagnoli. However this may be, 
 the design of the present translation is not at all 
 affected thereby ; as my object is to recall the at- 
 tention of the public to the subject, in order that 
 the benefit and advantage of the method may not 
 be wholly lost ; and that such scientific persons, as 
 may be induced to co-operate therein, may look out 
 for those peculiar occultations which are described 
 in the preceding pages, and note down their obserr 
 vations thereon accordingly. 
 
 For this purpose, nothing more is requisite than 
 a telescope sufficiently powerful to see the star di- 
 stinctly, when close to the illuminated side of the 
 moon*s disc; together with a good clock, or watch, 
 beating seconds. And although it would be de- 
 sirable, in all observations of this kind, to have the 
 exact mean time of the immersion and emersion 
 of the star, yet, as it is the duration only from 
 which the consequences are to be deduced, it will 
 
APPENDIX. 25 
 
 be sufficient (the latitude and longitude of the 
 place being well ascertained) if the clock or watch 
 can be depended upon for the few minutes that the 
 star is hid behind the moon : since the duration 
 will be the same whether the clock is set to mean 
 time, or not*. 
 
 In observing occultations, similar to those which 
 have been the subject of the preceding Memoir, 
 the observer should be careful not to withdraw his 
 eye from the telescope before he is fully satisfied 
 that the star is completely hid by the body of the 
 moon. For the star passes behind so small a seg- 
 ment of the moon's disc, that it may undergo a 
 partial occultation by the projecting mountains of 
 the moon, before it is wholly/ hid by that body ; 
 as appears from the following singular phsenome- 
 non, noticed by M. Koch at Dantzic, when ob- 
 serving the occultation of Aldebaran on March 7, 
 1794. "He was looking out for the immersion 
 " of the star, near the upper crescent of the moon. 
 " It disappeared at first : but 10'' afterwards it re- 
 '^ appeared suddenly in all its brilliancy. It was soon 
 " afterwards hid a second time. It re-appeared how- 
 
 * In calculating the exact time of the immersion or emersion of a 
 star, in order to deduce any practical and veiy accurate results there- 
 from, I do not find that any allowance is ever directed to be made for 
 the time that light takes to travel from the moon to the earth 5 which 
 is about one second and a quarter. 
 
26 APPENDIX. 
 
 '* ever again : but, presently after, its immersion 
 " took place for the third time at^ 8^. 28'. 21" ap- 
 " parent time. The observer then counted 30" ; 
 *^ and, finding that the star did not make its ap- 
 " pearance again, quitted the telescope in order to 
 ** write down the observation. He came back 
 " immediately ; but the star had already emerged 
 " from behind the moon. It is to be regretted 
 '' that he was not present at the instant of emer- 
 " sion, in order to complete an observation which 
 " had never been made before. However, it was suf- 
 *' ficientto show that the star did not pass a single 
 ** second of a degree within the moon's disc. For, 
 " the semidiameter of the moon being 15'. 52'', 
 " and its apparent motion 29" in a minute, it is 
 '^ found that, if the duration of the occultation 
 " had been one minute , the line joining the centre 
 " of the moon and the middle of the chord (tra- 
 '^ versed by the star) ought to be 15'. 51"." Con- 
 naissance des Tems, Anme vi. page 253. The 
 writer of that article justly adds that " this is one 
 ^' of the very interesting but very rare cases that 
 " M. Cagnoli has proposed to select for the pur- 
 ** pose of determining the true figure of the earth." 
 It is not necessary to inform the practical astro- 
 nomer that in many occultations the star, imme- 
 diately before its immersion, and immediately 
 
APPENDIX. 27 
 
 after its emersion, is observed to change its co- 
 lour*, and for many seconds to appear not only 
 adhering to the circumference, but oftentimes pro- 
 jected 07^ the disc, of the moon. Whether this opti- 
 cal deception is caused by an atmosphere surround- 
 ing the moon, or by the instrument used for the 
 observation t, or by what other means, I shall not 
 stop to discuss : but, as it is probable that its ef- 
 fects may be more discernible in those peculiar oc- 
 cultations which have been the subject of the pre- 
 ceding Memoir, it is desirable that the particular 
 circumstances of the case in each occultation should 
 be noted down as they occur ; together with a de- 
 
 * Messier, who observed the occultation of Aldeharan on Sept. -25, 
 1765, states that just before the immersion, on the illuminated side 
 of the moon's disc, |he saw the star sensibly diminish in light, 
 change its colour and become white : and that for 10" before its ac- 
 tual immersion it appeared adhering to the border of the moon. Con. 
 naissance des Terns, Annie 1810, page 336. In a subsequent occulta- 
 tion of the same star, on July 11, 1757> he states that, just before 
 its immersion, also on the illuminated side of the moon's disc, it ap- 
 peared adhering to the border of the moon for 7", and afterwards re- 
 mained 2" on the moon s disc, previous to its total disappearance: 
 and that the star was easily discernible on the face of the moon by a 
 reddish colour which it had acquired on approaching thereto. Ibid. 
 page 339. 
 
 t In the occultation of Aldeharan on July 5, 1 755, Messier, with 
 a 4| feet Newtonian reflector magnifying 140 times, saw the star for 
 near 2" on the moon's disc : Le Gentil, with a refracting telescope 
 of 6 or 7 feet, saw it adhering to the border of the moon, for 4" 
 or b" : whilst Cassini, with his 18 feet telescope, saw it detached 
 from the border, to -the very moment of immersion. Ibid, page 335 
 
28 APPENDIX. 
 
 scription of the telescope used for the observation. 
 Astronomers indeed now adopt the method of noting 
 down the moment of apparent contact^ as well as 
 the moment of the disappearance of the star : and 
 when it emerges again from behind the moon, they 
 likewise note down not only the moment of its re- 
 appearance, but also the moment of its separation 
 from the moon's disc. Indeed every remarkable 
 phaenomenon which may occur during any obser- 
 vation ought to be carefully registered : as it is 
 only by a comparison of such cases that we can 
 ultimately expect to discover the causes which in- 
 fluence or produce them. 
 
 M. Cagnoli, in his Memoir, seems to have ad- 
 dressed his remarks chiefly to the principal observa- 
 tories of Europe. But, however desirable it maybe to 
 have the co-operation of those scientific bodies, it is 
 evident that many private individuals may do much, 
 towards accomplishing the object and design of the 
 author. In many cases, they may perhaps be able 
 to do more than can be effected in any fixed obserr 
 vatory : since they may be more favourably situ- 
 ated, on the earth's surface, for the observation, or 
 may enjoy the advantage of a more clear and fa- 
 vourable atmosphere. Indeed, the greater the num- 
 ber of observers attached to this duty, the greater 
 will be the chance of an accurate result : for, inde- 
 pendent of taking a mean of the observations made 
 
APPENDIX. 29 
 
 near the same spot, we might be enabled to detect 
 the errors of the lunar tables by means of simulta- 
 neous observations made in a favourable situation 
 in a distant part of the world ; agreeably to what 
 the author has suggested in ^ IS of his Memoir. 
 In fact^ by increasing the number of observers to 
 an indefinite extent, over the whole surface of the 
 globe, almost every occulta tion that occurs might 
 be brought in aid of the method proposed by M. 
 CagnoH. 
 
 In the present contracted state^ however, of the 
 science of Astronomy, we must confine our views 
 to a more limited scale of assistance. In this 
 country, indeed, there are many persons scattered 
 throughout the kingdom who have the means and 
 opportunity of making observations of this kind : 
 and, amongst the numerous occultations which 
 they observe, if they should find only one that be- 
 longs to that peculiar class of occultations, which is 
 alluded to in the preceding pages, it would be a va- 
 luable discovery; and ought to be noted down ac- 
 cordingly, with all the circumstances attending it, 
 in order that it may be compared with others of a 
 similar kind. 
 
 It is much to be lamented, however, that in this 
 country there is no association of scientific persons 
 formed for the encouragement and improvement of 
 Astronomy. In almost all the other branches of 
 
30 APPENDIX. 
 
 the Arts and Sciences, institutions have been formed 
 for the purpose of promoting and diffusing a ge- 
 neral knowledge of those particular subjects ; such 
 as Botany, Agriculture, Chemistry, Anatomy, Geo- 
 logy, &c. &c. : the beneficial effects of which are 
 too evident to be insisted upon in this place. But, 
 Astronomy, the most interesting and sublime of all 
 the sciences (and, to our country, certainly the most 
 useful) cannot claim the fostering aidof any society. 
 It is however well known that many individuals, 
 in various parts of the kingdom, have of late years 
 erected and furnished private observatories at a very 
 considerable expense. Nevertheless the utility of 
 those establishments must be greatly circumscribed 
 through the want of some mode of general commu- 
 nication amongst observers, by means of which 
 their labours might be collected and registered ; and 
 thus rendered permanently useful. The formation 
 of an ASTRONOMICAL SOCIETY would not 
 only afford this advantage, but would in other re- 
 spects be attended with the most beneficial conse- 
 quences. It would induce many intelligent persons 
 (who are at present restrained by want of due encou- 
 ragement) to come forward as the patrons and fol- 
 lowers of the science ; and excite the more experi- 
 enced to further exertions and new discoveries*. 
 
 * The name of the Royal Society will naturally occur to the 
 reader on this occasion : but that society wm formed for the promo- 
 
APPENDIX. 31 
 
 Although much has been ah-eady done towards per- 
 fecting our present system of astronomy, much still 
 remains to be effected, both in theory and practice. 
 Not only would the interchange of sentiments and re- 
 marks on the various celestial phaenomena, which 
 such a Society would afford, lead to the advancement 
 of the science in general ; but the comparison and 
 discussion of the merits of the several instruments 
 used for astronomical purposes would also tend 
 either to their improvement or to the discovery of 
 new ones. By means also of a society of this kind 
 an active communication might be kept up with the 
 principal astronomers in different parts of the world: 
 and, thus, continual subjects of research or obser- 
 vation be submitted to public attention. The funds 
 of such a society might likewise be usefully employ- 
 ed in the formation of an Astronomical Library^ 
 consisting of the most rare and valuable books 
 on the science ; and in collecting and transcribing 
 the observations of astronomers in various parts 
 of the world*. But, even without any views or pre- 
 
 tion and encouragement of science in general; and the subject of 
 Astronomy appears to form but a small portion of its labours. Never- 
 theless the Astronomical Society would, in common with the Linncean, 
 Geological and other scientific societies, hope for the co-operation of 
 its learned members. 
 
 * The observations and remarks of many eminent astronomers, 
 both ancient and modern, still exist only in manuscript, in the hands 
 of private individuals ; and ought, if possible, to be rescued from their 
 present precarious situation, and deposited in a place of safety and 
 
32 APPENDIX. 
 
 tensions to new discoveries, or to the improvement 
 of optical instruments and time-keepers, or to the 
 formation of such a library, an Astronomical Society 
 might render very important benefits to Geography 
 and Navigation by simultaneous observations of 
 lunar distances, eclipses, occultations, and various 
 other phsenomena : which are at present in a great 
 measure neglected because the public attention is 
 not specially directed thereto. And the more nume- 
 rous and widely diffused the members of such a so- 
 ciety might be, the greater the probability of obvi- 
 ating the baneful effects of our cloudy atmosphere ; 
 an occurrence which often renders the efforts of the 
 finest observatory totally unavailing. 
 
 convenient reference. BuUiald made an immense collection of ob- 
 servations of this kind, many of which came into the possession of 
 Lcmonnier ; who put them into the hands of Pingre, for the purpose 
 of publication. P'mgri had himself collected also an astonishing num- 
 ber of rare and valuable manuscripts from all parts of Europe, and 
 had not only calculated and reduced their contents ; but had likewise 
 discussed and compared the observations, so as to render it a work 
 of the highest utility to astronomers. It was the labour of thirty 
 years ! and the National Assembly in 1791 issued a decree for print- 
 ing it, under the title of Annales Celestes du dia^-septume siecle. In 
 the year 1/94, about 360 pages of this work were printed ; being one 
 third only of the intended publication. The learned author died in 
 1796 at the advanced age of 85 : since which period I do not find 
 that the progress of the work has been at all advanced. Many va- 
 luable observations of the more recent astronomers, such as Halley, 
 JDe L'isle, Lemonnier, Godin and others, it is well known, exist also 
 only in manuscript j and are now mouldering in the dust, a disgrace 
 to the nineteenth century. 
 
APPENDIX. 33 
 
 There IS yet another source of assistance which 
 we may hope for in this country. It is well known 
 that, on the continent, great advantage is rendered 
 to astronomy by means of the Ephemerides, which 
 are pubhshed under the authority of Government: 
 and those annual publications contain a vast fund 
 of valuable information on the subject of astro- 
 nomy, which otherwise might be for ever lost to 
 the world. The Board of Longitude in this country 
 have now the power and the means * of affording 
 
 * Extract from the recent Act, 58 Geo. III. cap. 20, §6. "And: 
 *' whereas it is expedient that the said Commissioners should be en- 
 *■* abled to expend certain suras towards making experiments of in- 
 " struments, modes or proposals, and for making and publishing 
 " observations, calculations, and tables for ascertaining the longitude, 
 " or towards improving or correcting such as may have been already 
 *' made, or for other purposes useful to navigation ; be it enacted that 
 " they may pay or expend any sura or sums of money, not exceed- 
 *' ing a^lOOO in any one year, towards the making, correcting or 
 " publishing any such experiments, modes, observations, calcula- 
 *' tions, or tables. 
 
 § 7. " And whereas it is expedient that the said Commissioners 
 '• should be enabled to cause to be ascertained, as accurately as 
 *• may be, the latitude and longitude of places, whereof the exact 
 *' situation hath not been already sufficiently asceitained j be it en- 
 '* acted that they may expend or cause to be expended any sum, not 
 " exceeding, in the whole ^1000, in any one year, for that purpose. 
 
 § 8. "And whereas it may happen that proposals, inventions and 
 *' tables, or corrections and amendments of former proposals, inven- 
 " tions or tables, ingenious in themselves and useful to science, and 
 '* which may deserve encoitrragement, (though they do not come within 
 ** the limits and conditions specified for the above-mentioned rewards) 
 ** may be made to the said Commissioners j and it is expedient that 
 JO 
 
34 APPENDIX. 
 
 similar assistance by enlarging the original plan 
 and design of the Nautical Almanac, and by as- 
 similating it to those which are published at Paris, 
 Berlin, Vienna, and other places: a measure which 
 would tend to retrieve the character of the work, 
 and redound to the honour of the country ='^. 
 
 With the very ample funds (<3£4000 per annum) 
 which Government has placed at the disposal of 
 
 " they should be enabled to bestow such moderate rewards upon the 
 ** person or persons who may have made such proposal, invention, 
 ** or correction j be it therefore enacted that the said Commissioners 
 *' iriay pay or cause to be paid such sum, not exceeding ^500 to 
 '* any one person for any one proposal or invention, or^2000 in one 
 ** year, as they may consider the said proposals, inventions, tables or 
 ** corrections to deserve." 
 
 * Amongst the several amendments which might be made to that 
 useful work, I would suggest the propriety and advantage of having 
 the Right Ascension and Declination of the moon inserted to seconds 
 of a degree ; similar to the method now adopted in the Connaissance 
 des Terns. This would facilitate the finding of the apparent place of 
 the moony for the purpose of comparing it with the place of any 
 given star, the position of which is always given in Right Ascension 
 and Declination : and would moreover obviate the necessity of finding 
 the position of the Nonagesimal, — an unnecessary and troublesome 
 process. As the original calculations of the computer must extend 
 to seconds, there is no good reason for omitting them in the publica- 
 tion. I would likewise (as connected with the subject of this Me- 
 moir) take the liberty of suggesting the propriety of publishing an- 
 nually in that work, an accurate list of all the Zodiacal stars (including 
 even those of the 9th magnitude) not only in Right Ascension and 
 Declination, but also in Longitude and Latitude, with their annual 
 and secular variations, corrected from the latest observation?. The 
 expense would be trifling, compared with the advantage to be de- 
 rived from the result. 
 
APPENDIX. 35 
 
 this new BDard, for scientific purposes, united 
 to the known abilities and zeal of its several 
 members, we may confidently hope that a new im- 
 pulse will soon be given to the progress of astro- 
 nomy in this country: and that, by emulating our 
 scientific neighbours on the continent, we may 
 avoid the lamentable necessity of resorting to them 
 for almost all our means of deducing the situa- 
 tion of the heavenly bodies, and of investigating 
 the laws which govern their motions'^. Probably 
 the object of the legislature would be more effec- 
 tually answered if the new^ Board of Longitude 
 would undertake to lead and direct the attention 
 .of the scientific world to such particular objects as 
 they might think most worthy of encouragement ; 
 rather than to wait for proposals on every sub- 
 ject (useful or visionary) that may be laid before 
 them. The publication of specific rewards — for 
 the formation of new Tables of the Sun, Moon, or 
 •any of the Planets f; — for the best Essays on any 
 particular subject of Practical or Theoretical astro - 
 •nomy; — for any valuable improvement in Time- 
 pieces, Telescopes, Micrometers, or other astro- 
 nomical instruments ; — for the best Engravings of 
 
 * It is too well known that all our astronomical Tables, all our 
 Catalogues of stars, and most of .our astronomical Formulae, in pre- 
 sent use, are the production of the continent. 
 
 1 1 believe there are no tables whatever of two of the nevyly dis* 
 covered planets. 
 
 D 2 
 
36 APPENDIX. 
 
 any portion of the celestial sphere, and particularly 
 of the zodiacal stars ^ ; — for reducing the Observa- 
 tions of any celebrated astronomer ; — for the Dis- 
 covery of any New planet, comet, fixed star, or 
 other remarkable celestial phsenomenon ; — for the 
 Translation of any valuable astronomical treatise 
 into the English language ; — and, in general, for 
 any other object which may be " useful to science 
 and which may deserve encouragement;'' — ^would 
 necessarily excite and fix the public attention to 
 those subjects, and perhaps more effectually pro- 
 mote the views of a liberal and enlightened Go- 
 vernment. 
 
 But to return to the subject of the present Me- 
 moir. — It has been stated by M. Cagnoli, ^ 4, 
 that the difference of parallax arises from the in- 
 equality of the terrestrial radii ; and that the one 
 is the measure or consequence of the other. 
 Therefore, in order to determine the parallax at 
 any given point of the earth's surface, it is neces- 
 sary to know the distance of that point from the 
 centre of the earth. But, that the reader may see 
 the anomalies which arise on this subject, (from 
 the actual measurement of the degrees of the me- 
 ridian in different countries, and from different the- 
 
 • Accurate engravings of the Zodiacal stars, upon a large scale, 
 would enable astronomers with greater certsdnty to look out for oc- 
 cultations of fixed stars by the moon j the observations of which ar« 
 undoubtedly the best method of determining the longitude of place*. 
 
APPENDIX, 
 
 37 
 
 ©ries which have been assumed) I have calculated 
 the following table, which shows the radius of the 
 earth in the latitude of Greenwich (51°. 28'. 40"), 
 according to the several hypotheses of the com- 
 pression of the earth's axis therein stated. 
 
 No. 
 
 Hypothesis. 
 
 Radius in feet. 
 
 Difference. 
 
 1 
 
 TW 
 
 20 827 239 
 
 
 
 
 
 6609 
 
 2 
 
 I b 
 
 20 833 848 
 
 14376 
 
 3 
 
 1 
 
 20 848 224 
 
 
 
 
 
 7115 
 
 4 
 
 '2 
 
 20 855 339 
 
 8350 
 
 5 
 
 «-3 
 
 20 863 689 
 
 4455 
 
 6 
 
 •5 50- 
 
 20 868 144 
 
 8675 
 
 7 
 
 I 
 
 3 
 
 20 876 819 
 
 1101 
 
 8 
 
 3 p-9- 
 
 20 877 920 
 
 2640 
 
 9 
 
 '3 3 6- 
 
 20 880 560 
 
 
 The first value here given arises from the recent 
 measurement of the arc in France, by taking the 
 middle arc at Evaux, which makes the compression 
 of the earth's axis = -^^ ; and the length of the 
 earth's radius in the latitude of Greenwich equal to 
 20 827 239 feet. By comparing this value with 
 the l^st in the table, which is one of the hypotheses 
 
38 APPENDIX, 
 
 assumed by the Bureau des Longitudes in the 
 recent tables of the Moon by M. Burg, it will be 
 found that there is a difference of no less than 
 53 321 feet, or more than 10 miles ; a quantity 
 sufficiently great to be detected by the observation 
 of the moon's parallax, and which would produce a 
 very sensible effect on the duration of such occulta- 
 tions as have been the subject of the preceding Me- 
 moir : since we have seen that, in some cases, a 
 difference of less than 200 feet will produce a dif- 
 ference of one second in the duration. 
 
 But, although such occultations may clearly show 
 that the earth is compressed at the poles, and 
 although they may be effected by quantities which 
 differ so widely from each other as in the case just 
 mentioned; yet it maybe doubted whether they are 
 capable of determining the precise quantity of that 
 compression ; and therefore whether they are more 
 eligible for that purpose than the methods at pre-» 
 sent in use. In reply to which, it maybe repeated 
 that this plan is at present proposed only in aid of 
 the modes hitherto adopted for determining the 
 true figure of the earth ; and by no means as sup- 
 plying the place of other methods for deciding that 
 difficult problem. 
 
 In order, however, that the reader may have a 
 more comprehensive view of this subject, and be 
 enabled to see the differences which would arise in 
 
APPENDIX. 8ft 
 
 various latitudes and under various circumstances, 
 according to two nearly similar hypotheses of the 
 earth's compression compared with the hypothesis 
 of its being a perfect sphere, I have inserted the 
 three following tables : which show the alteration in 
 the position of the star within the moon's disc, and 
 its consequent effect on the duration of the occulta- 
 tion, according to the three several hypotheses of 
 the earth being considered as a perfect sphere, and 
 of its being compressed at the poles 3-^, and -^^ ; 
 at the several heights of lO"*, 20" and 30" of the moon 
 above the horizon ; and at the several latitudes of 
 50", 60°, and 70°. The chord of the star's ap- 
 parent path behind the moon being, in all these 
 cases, assumed to be perpendicular to the vertical 
 circle of the place. 
 
 Thus, by an inspection of Table II, it will be 
 seen that in latitude 60"^, when the moon is 10° 
 high, and the star 60" within the moon's disc, the 
 duration of the occultation would be 20'. 7", if the 
 earth were a perfect sphere. But, if the polar axis 
 were compressed -j~-^, the star would then be only 
 5V\5 within the moon's disc, and the duration of 
 the occultation only 18', 41" : which is the case 
 alluded to by M. Cagnoli § 10. Or, if the polar 
 axis were compressed only ^-5^, the star would 
 then be 5 2 '',3 within the moon's disc, and the 
 duration of the occultation would be 18'. 50''; 
 
40 APPENDIX. 
 
 being a difference of 9'^ between the results of the 
 two hypotheses. This difference would be the 
 same, if the moon were 20*^ high ; as may readily 
 be seen by an inspection of the same table : it 
 would also be the same under corresponding cir- 
 cumstances, if the observation were made in lati- 
 tude 70° : as may be seen by an inspection of 
 Table III. 
 
 This difference however may probably be con- 
 sidered as too small to enable us to derive any sa- 
 tisfactory result as to the precise quantity of the 
 compression of the earth's axis. But, if we take 
 the position of the star within the moon's disc 
 equal to 1 0'^ only, and the moon at the height of 
 10^ as before, we shall have the duration of the 
 occultation 8'. 19'' on the supposition that the 
 earth is a perfect sphere. Then, supposing the 
 axis to be compressed -^-^y the star would be only 
 2", 3 within the moon's disc, and the duration of 
 the occultation only 4'. 0" : or, supposing the axis 
 to be compressed ^-fg-, the star would be only r',5 
 within the moon's disc, and the duration only 
 3', 14'' ; being a difference of 46" in the result of 
 the two hypotheses. If the moon were 20^ high, 
 the difference would be only 42". If however the 
 observation were made in latitude 70°, and the 
 moon at 10° high, the difference would be much 
 greater : since, on the supposition of the conipres" 
 
appendix; " 41 
 
 sion = .j4-5-, the duration would be 2\ 30'' ; whereas, 
 on the supposition of the compression = -^-^ the 
 star would not undergo any occultation at all, but 
 would merely touch the moon's disc. Nearly simi- 
 lar results might be obtained, under particular cir- 
 cumstances, in lower latitudes ; as maybe seen from 
 an inspection of the Tables. And it will thus be evi- 
 dent that a veri/ small difference in the ellipticity of 
 the earth, may, under some very favourable circum- 
 stances, be rendered sensible to observation. 
 
 It is unnecessary however to dwell farther upon 
 this subject at present : a fair experiment does not 
 appear to have ever yet been made of the ingenious 
 mode proposed by the illustrious author of the 
 Memoir, for determining the true figure of the 
 earth. It seems to me to be highly deserving of 
 encouragement : and my object will be fully an- 
 swered if any practical good should result from 
 the public attention which I may have excited in 
 its favour. 
 
 F. B. 
 
42 
 
 APPENDIX. 
 
 Table I. 
 
 Latitude 50\ 
 
 Height 
 of the 
 Moon. 
 
 Earth = Sphere 
 
 Earth = ^^ 
 
 Ea^-t^=3iir 
 
 * within 
 
 3) 's disc. 
 
 dura- 
 tion. 
 
 ^within dura- 
 
 i)'sdisc. tion. 
 
 1 
 
 ^within 
 i) 's disc. 
 
 dura- 
 tion. 
 
 10^. 
 
 60 
 
 30 
 15 
 10 
 6,6 
 
 / II 
 20. 7 
 
 14.20 
 
 10.11 
 
 8.19 
 
 6.46 
 
 II 
 54,0 
 
 24,0 
 
 9>o 
 
 4,0 
 0,6 
 
 19. 7 
 
 12.51 
 
 7.54 
 
 5.16 
 
 2.2 
 
 53,4 
 
 23,4 
 
 8,4 
 
 3,4 
 
 
 
 19. 1 
 
 12.41 
 
 7.38 
 
 4.52 
 
 Appulse 
 
 20°. 
 
 6o' 
 30 
 1.5 
 10 
 6,3 
 
 1 II 
 20. 7 
 
 14.20 
 
 10.11 
 
 8.19 
 
 6.37 
 
 54,3 
 24,3 
 
 9.3 
 
 4,3 
 
 0,6 
 
 / // 
 19.10 
 
 12.56 
 
 8. 2 
 
 5.28 
 
 2.2 
 
 53,7 
 
 23,7 
 
 8,7 
 
 3,7 
 
 
 
 19. 4 
 
 12.46 
 
 7.46 
 
 5. 4 
 
 Appulse 
 
 30«. 
 
 6o" 
 30 
 15 
 10 
 5,8 
 
 / 1/ 
 20. 7 
 
 14.20 
 
 10.11 
 
 8.19 
 
 6.21 
 
 II 
 54,7 
 
 24,7 
 
 9>7 
 
 4,7 
 
 0,5 
 
 / // 
 19-14 
 
 13. 2 
 
 8.12 
 5.43 
 1.52 
 
 II 
 
 54,2 
 
 24,2 
 
 9>2 
 
 4,2 
 
 
 
 / // 
 19. 9 
 
 12.54 
 
 7.59 
 5.24 
 
 Appulse 
 
APPENDIX. 
 
 43 
 
 Table IL 
 
 Latitude 60". 
 
 Height 
 of the 
 Moon. 
 
 Earth = 
 
 sphere. 
 
 Earth = ^^ 
 
 Earth = ^ 
 
 ^within 
 ]) 'sdisc. 
 
 dura- 
 tion. 
 
 ^within 
 D 'sdisc. 
 
 dura- 
 tion. 
 
 ^within 
 3) 'sdisc. 
 
 dura- 
 tion. 
 
 
 60 
 
 / // 
 20. 7 
 
 n 
 
 32,3 
 
 18.50 
 
 II 
 51,5 
 
 * 1' 
 18.41 
 
 
 30 
 
 14.20 
 
 22,3 
 
 12.23 
 
 21,5 
 
 12.10 
 
 10°. 
 
 15 
 
 10.11 
 
 7,3 
 
 7. 7 
 
 6,5 
 
 6.43 
 
 
 10 
 
 8.19 
 
 2,3 
 
 4. 
 
 1,5 
 
 3.14 
 
 
 8,5 
 
 7.41 
 
 0,8 
 
 2.22 
 
 
 
 Appulse 
 
 1 
 
 // 
 60 
 
 20. 7 
 
 // 
 52,7 
 
 / // 
 18.54 
 
 11 
 
 51,9 
 
 1 u 
 
 18.45 
 
 
 30 
 
 14.20 
 
 22,7 
 
 12.30 
 
 21,9 
 
 12.17 
 
 2o^ 
 
 15 
 
 10.11 
 
 7,7 
 
 7.19 
 
 6,9 
 
 Q.55 
 
 
 10 
 
 8.19 
 
 2,7 
 
 4.20 
 
 1.9 
 
 3.38 
 
 
 8,1 
 
 730 
 
 0,8 
 
 2.22 
 
 
 
 Appulse 
 
 
 60 
 
 1 II 
 20. 7 
 
 53",2 
 
 18.59 
 
 52^6 
 
 18.52 
 
 
 30 
 
 14.20 
 
 23,2 
 
 12.38 
 
 23,6 
 
 12.29 
 
 30". 
 
 15 
 
 10.11 
 
 8,2 
 
 7.33 
 
 7>Q 
 
 M6 
 
 
 10 
 
 8.19 
 
 3,2 
 
 4.43 
 
 2,6 
 
 4.15 
 
 
 1A 
 
 7.10 
 
 0,6 
 
 2. 2 
 
 
 
 Appulse 
 
44 
 
 APPENDIX. 
 
 Table III. 
 
 Latitude 70*'. 
 
 Height 
 of the 
 
 Eaith — 
 
 Sphere. 
 
 Earth =^^^ 
 
 Earth: 
 
 -7^Tr 
 
 
 
 1 
 
 
 
 Moon. 
 
 * within 
 3) 's disc. 
 
 dura- 
 tion. 
 
 ^within' dura- 
 }) 'sdisc.l tion. 
 
 ^within 
 }) 'sdisc. 
 
 dura- 
 tion. 
 
 
 60 
 
 20. 7 
 
 II 
 50,9 
 
 / // 
 18.34 
 
 II 
 50 
 
 / II 
 18. 2 
 
 10°. 
 
 30 
 
 14.20 
 
 20,9 
 
 12. 
 
 20 
 
 11.44 
 
 
 15 
 
 10.11 
 
 5,9 
 
 6 24 
 
 5 
 
 5.54 
 
 
 10 
 
 8 1.9 
 
 0,9 
 
 2.30 
 
 
 
 Appulse 
 
 
 60 
 
 / It 
 20. 7 1 
 
 II 
 51,3 
 
 1 n 
 I8.39 
 
 11 
 50,5 
 
 18.30 
 
 
 30 
 
 14.20 
 
 21,3 
 
 12. 7 
 
 20,5 
 
 11.53 
 
 20°. 
 
 15 
 
 10.11 
 
 6,3 
 
 6.37 
 
 5,5 
 
 6.11 
 
 
 10 
 
 8.19 
 
 1,3 
 
 3. 
 
 0,5 
 
 1.52 
 
 
 9,5 
 
 8. 7 
 
 0,8 
 
 2.22 
 
 
 
 Appulse 
 
 
 60 
 
 / // 
 20. 7 
 
 52 
 
 i ii 
 18.46 
 
 II 
 51,2 
 
 18.38' 
 
 
 30 
 
 14.20 
 
 22 
 
 12 18 
 
 21,2 
 
 12. 5 
 
 30°. 
 
 15 
 
 10.11 
 
 1 
 
 6.58 
 
 6,2 
 
 6.34 
 
 - 
 
 10 
 
 8.19 
 
 2 
 
 3.44 
 
 1,2 
 
 2.54 
 
 
 8,8 
 
 1A^ 
 
 0,8 
 
 2.22 
 
 
 
 Appulse 
 
 THE END. 
 
 tiinUa ly ti. and A, Tayorj Hhoeluiiey Lonatjn. 
 
Ce3M 
 
 \ 
 
mi