717 



PRECESSION AND NUTATION. 



PREDICABLES. 



718 



*he end of 1747. In 1749 appeared the ' Recherches sur la Precession,' 

 &c. of D'Alembert, in which the phenomenon was shown -to be the 

 necessary consequence of the moon's attraction upon the earth. Newton 

 had already, in the ' Principia,' given the general explanation of the 

 subject, and had even foretold, without assigning magnitudes, the 

 existence of those terms of nutation which depend upon twice the 

 true longitudes of the sun and moon ; but the most important terms, 

 those depending on the moon's node, appear to have been altogether 

 unsuspected by him. 



We now come to such a physical explanation of the cause of pre- 

 cession and nutation as can be given without mathematical analysis. 

 On looking at the motion of the equator arising from precession and 

 nutation, we see that it precisely resembles in character some of 

 the alterations which take place in a planet's orbit, the precession 

 answering to the regression of the nodes, the equation of the equinoxes 

 to the variation of that regression, and the remaining part of the 

 nutation, or the variation of the obliquity, to the variation of the 

 inclination to the ecliptic. It was soon seen by Newton, that on the 

 supposition of the mutual attraction of all the particles of matter, the 

 action of the heavenly bodies on the protuberant parts of the earth 

 must produce exactly that sort of effect on the motion of the equator 

 which the disturbing force of the sun, for instance, produces on the 

 moon. He thus explains, firstly, the precession ; secondly, that part 

 of the nutation of the inclination which depends upon twice the 

 longitude of the disturbing body. This explanation (prop. 66, corol- 

 laries 18-22) is substantially as follows : 



If a sphere in rotation be attracted by another body, the axis of 

 rotation must remain unaltered : for since a plane drawn through any 

 attracting point and the centre of the attracted sphere cuts the sphere 

 into two perfectly similar halves, there is no effect upon the rotation 

 (or tendency to an effect) arising from the attraction upon one half of 

 the sphere which is not destroyed by the tendency to the exactly 

 opposite effect arising from the attraction upon the other half. If 

 then the earth were a perfect sphere, whatever motion of translation 

 the whole sphere might receive, the axis would always remain parallel 

 to its first position, and there would be neither precession nor nutation. 

 Again, let the earth be a solid of revolution, protuberant for example 

 at the equator, as is the case, and let an attracting point be situated in 

 the plane of the equator ; the symmetry just alluded to still exists, 

 and the result is the same. But if an attracting point be not situated 

 in the plane of the equator, the plane passing through the attracting 

 point aud the centre divides the spheroid into parts which, though 

 equal, are no longer similarly situated with respect to the attracting 

 point. The alteration of the axis which would take place if one half 

 only were attracted, is no longer counterbalanced by the attraction on 

 the other half : the direction of the axis is therefore continually 

 changed. 



Fig. 1. 



To get a specific idea of the nature of the change, first suppose the 

 spherical part of the earth only to exist, the protuberance being 

 removed; and the solidity of the sphere still remaining, let all its 

 matter be supposed to be removed to the centre at c (Fig. 2). The 

 diagram shows the spheroid of the earth, distinguishing the inscribed 

 sphere from the protuberant part; the solidity of both parts Jg 

 supposed to remain, but the matter of the internal sphere is removed 

 to c, that of the protuberant part is not yet introduced : M is the 

 attracting body, and the plane of its orbit is given, while the directions 

 of the earth's rotation and of M'S orbital motion are denoted by arrows. 

 At present x produces no effect on the rotation ; now let a small mass 

 of matter be affixed to the equator at z, which will therefore move 

 round the mum c in and with the equator. The consequence will be 

 [GBAVITATION], that the node of this orbit (the equinox A) will regress, 

 or move in the direction opposite to that of the arrows, while the 

 inclination of the orl.it will alternately increase and diminish, being 

 greatest when the line c M passes through one of the equinoxes. If 

 we put such satellite* all round the equator, the effect will not be 

 altered in quality, but increased in magnitude ; and if we fill up all 

 the protuberant part of the spheroid, the effect will still lie of the same 

 sort, though further increased in magnitude. The effect of the parts 

 of the protuberance nearer to the pole is, for a given mass, less than 

 that of the parts near the equator. Finally, if we rertore the mass of 

 the internal sphere to its proper place, the effect will be less than 

 before ; for since no motion of the protuberant part can take place 

 without one of the whole sphere, and since rotation is more difficult 

 to produce, the greater the distance of the masses moved from the 

 axia, the distribution of the mass at p over all parts of the sphere will 

 render M less efficient in the alteration of the direction of the a 



Thus it appears that the phenomena of precession and nutation may 

 arise from the consideration of the protuberant part of the spheroid as 

 a fixed satellite to the internal part ; but the proof that the precession 

 and nutation do so arise consists in taking a strict mathematical 

 process, investigating the precession and nutation in quantity as well 

 as quality, and showing that the results agree with those of obser- 

 vation. 



But, as before noticed, the largest part of the nutation] depends, not 

 on the place of the moon in its orbit, but on the position of the orbit, 

 that is, on the node of the orbit. Supposing the moon's orbit circular, 

 imagine the mass of the moon to be distributed in a ring all round its 

 orbit. If this ring were simply to revolve in its own plane, the pre- 

 cession and nutation produced by it in the earth, though materially 

 altered in quantity, would be of the same sort as before, and in both 

 cases very small. But suppose the ring to shift its position, as does 

 the moon's orbit, its nodes slowly regressing at the rate of a revolution 

 in eighteen years. This shifting of the position of the ring will of 

 course produce an alteration in the phenomena, and the substitution 

 of the moon revolving in a shifting orbit in place of this ring must 

 now be made. That the effect of the change of the orbit should be 

 greater than that of the planet itself in a fixed orbit ought not to be 

 surprising, since there is no d priori reason why it should be either 

 greater or less. 



Throughout the solar system there is no action of one planet upon 

 a second, without a corresponding action of the second upon the first. 

 The protuberance of the earth, by which the planets produce preces- 

 sion and nutation, attracts those planets, and slightly varies their 

 motion. In the case of the moon, sensible irregularities, both in 

 longitude and latitude, amounting at the maximum to about 7" in 

 each, were found by Mayer, before Laplace showed them to be the 

 consequences of the earth's protuberance. These inequalities may be 

 made the means of calculating the amount of that protuberance, or, 

 as it is technically called, the ellipticity of the earth : and it is a fact 

 not a little remarkable, that the amount of this ellipticity, as calculated 

 from its effect upon the moon's longitude, agrees with the same, as 

 calculated from its latitude, better than actual measurements of the 

 earth generally agree with one another, while both agree very nearly 

 with the best of the latter. This sort of result had been anticipated 

 as to quality by Newton, who showed that the motion of the equi- 

 noxes, being retrograde, proves the earth to be protuberant at the 

 equator, and that if it had been protuberant at the poles (as many then 

 thought was the case), the precession would have been in the contrary 

 direction. 



PRECIPITATE. A name given in chemistry to any comparatively 

 insoluble solid or liquid matter separated from a liquid by chemical 

 action. It is ako, though more rarely, applied to solid or liquid matter 

 deposited from a gas ; thus snow and rain may be termed precipitates 

 from the atmosphere. 



PREDESTINATION. [ELECTION; FREE- WILL; and CALVIN, in 

 Bioo. Div.] 



1'REDICABLES. The term predicable (aarriyopiKif, prcedicabile) is 

 applied in logic to general names, considered as capable of being the 

 predicates of propositions. (On Predication, see OROANON.] The 

 classes of predicables usually recognised by logicians are five, namely, 

 1 , Genus ; 2, Species ; 3, Differentia ; 4, Proprium ; 5, Accidens, which 

 Latin names are translated from the Greek, 1, yitos ; 2, el5oj ; 3, Sicupopd ; 

 4, ISiov ; 5, (ru/tij3^7)K(!i. 



The five-fold classification of the predicables does not occur in 

 Aristotle's ' Organon,' nor in any other of his extant writings ; and it 

 probably did not occur in any of his lost writings. The word yevos is 

 often used by Aristotle to signify a class : and the word tlSos in the 

 sense of a logical species. The word in/w8cj87)/c4s (or xari <ru/i/8f/37)(tis) is 

 often used by Aristotle to signify that which is contingent or acci- 

 dental, in opposition to that which is necessary (amyKalov). 



The earliest work in which the received classification of the predi- 

 cables occurs, is an_ Introduction; to Aristotle's Categories, written by 

 Porphyry of Tyre, the well-known heathen philosopher of the 3rd 

 century (bora 283 A.D.) and the author of other extant works. (Printed 

 in Bekker's Aristotle, vol. iv., p. 1-6). Porphyry states, at the outset of 

 this treatise, that a knowledge of the five predicables is necessary for 

 the proper explanation of Aristotle's work on the categories ; and he 

 therefore addresses to a certain Chrysaorius a popular account of them, 

 derived from the ancient philosophers, especially the Peripatetics. 

 The five predicables (at -nivrt <puval, as they were originally styled) are 

 not however mentioned in Aristotle's work on the categories, as is 

 incorrectly stated in Hermeias, ib., p. 10, b. 14 ; and it is probable that 

 the " ancient philosophers " alluded to by Porphyry were of con- 

 siderably later date than Aristotle. An abridgment of Porphyry's 

 treatise on the predicables, by Michael Psellus, of Constantinople, who 

 lived in the llth century, has also been preserved ; and it is reprinted 

 in the beginning of the small Oxford edition of 'Excerpts from 

 Aristotle's Orgauon ' (Clarendon press, 1802). 



From this treatise of Porphyry the classification and explanation of 

 the predicables have passed into the various treatises of the Aristotelian 

 logic, and have been repeated in them with some variations and de- 

 velopments, but with little substantial change up to the present day 

 (Whately's ' Logic,' part i.) 

 The explanation of the predicables which is usually given in treatises 



