NA TURE 



[May 3, 1906 



tion of angular momentum that the moon must be 

 receding from the earth, and absorbing the spin lost 

 by the earth. This implies that the moon is really 

 moving more slowly. It is impossible to make 

 accurate calculations, for the action of the tides on an 

 earth with oceans and continents of irregular shape 

 cannot be computed, and it is impossible to say how 

 the tidal action varies for different positions of the 

 moon in its elliptic orbit. Hence we cannot say how 

 far the action of the tides is distributed between 

 ciianges in the length of the month and changes in 

 llic eccentricity of the moon's orbit. But it seems a 

 plausible hypothesis that the large eccentricity of the 

 moon's orbit w-as evolved somehow, presumably by 

 tides, and that the eccentricity is therefore increasing, 

 and calculation shows that if the rate of increase 

 assigned to the eccentricity be about one-hundredth of 

 a second of arc a century, the consequent change in 

 the absolute angular velocity of the moon is such as 

 to cancel nine-tenths of the apparent decrease in the 

 length of the month, leaving the remaining one-tenth 

 in agreement with the change inferred from ancient 

 eclipses. This explanation, it should be clearlv under- 

 stood, only shows that certain correlated quantities are 

 of the right order of magnitude : it is unable to prove 

 or disprove an exact numerical relation. 



In the remaining part of this article the basis of 

 the conclusion of the first section is examined. That 

 is the foundation, which must be rendered secure 

 before interest can attach to any superstructure. 



Let us select a definite eclipse, for instance, the 

 eclipse of Thucydidcs in the first year of the Pelopon- 

 nesian War. The record states that stars appeared. 

 It is certain on the other hand that the eclipse, at the 

 mo-,t, could only have been annular. There is there- 

 fore a strong presumption that .Athens was not far 

 from the central line of the eclipse, or in other words, 

 at the time of conjunction in longitude as seen from 

 .-Vthens, the difference of apparent latitudes must have 

 been small. The hypothesis that Athens was the place 

 of observation has been objected to. This however is 

 the natural interpretation of the passage in Thu- 

 cydidcs; let us adopt it for the present and see where 

 it leads. For Athens, therefore, let the difference of 

 apparent latitude for the instant of apparent conjunc- 

 tion in longitude be computed from the present tables. 

 The result is so large as absolutely to negative the 

 possibility that stars could have been seen. Reserving 

 the hypothesis that the record is untrustworthy as a 

 last refuge in case of trouble, let us suppose for the 

 present that the tables require alteration. 



What kind of alteration is permissible? It has been 

 argued in .\s\. Nach..'No. 3682, on physical grounds, 

 that only one unknown quantity may' be inl:roduced. 

 Now against physical reasoning of this kind, strong 

 objections may be urged. It proceeds necessarilv on 

 the assumption that the general nature of the problem 

 of the apparent motions of the sun and moon is fullv 

 understood. It absolutely limits the investigation to 

 the numerical determination of quantities connected 

 with a preconceived theory, and it prevents, at the 

 outset, the attainment of results of a new character. 

 Now as the preconceived theory was entirelv based 

 upon two centuries of observation, there is no im- 

 probability in our knowledge being widened, w-hen the 

 period of observation is largely increased. In the 

 whole of astronomv there is not a single case of a 

 theoretical value of a secular term, that is to sav, a 

 term proportional to the square of the time, being con- 

 firmed by observation. This is because the series of 

 modern observations is not yet long enough. Is it 

 not possible that one or two centuries hence the 

 NO. 1905, VOL. 74] 



observed values of these terms will lay bare a whole 

 series of new phenomena? Physical considerations of 

 the kind alluded to absolutely prevent the achievement 

 of such a result. They may advantageously be re- 

 placed in the following manner by considerations of a 

 purely geometrical character. 



It being, for a time at least, granted that the eclipse 

 of Thucydides suggests that the existing tables, 

 require large modifications, geometrical considerations 

 tell us, that in order to diminish by 200" or there- 

 abouts the difference of latitude at conjunction, we 

 must alter the mean distances of the sun and moon 

 from the node as given by the tables for the 3'ear —430 

 by quantities of the order of 2000". The only geo- 

 metrical alternative is to assume alterations ten times 

 as large in some other quantity such as the position of 

 the perigee, and this alternative may be put aside. 

 Now the mean distances can be expanded in powers 

 of the time, the origin of time being taken near the 

 present day. Then modern observations forbid the 

 correction of the mean motions or of the terms 

 independent of the time. The corrections are there- 

 fore necessarily thrown about the coefficients of the 

 square of the time, that is to say, upon what are 

 called the secular terms, in the mean distances of the 

 sun and moon from the node. Geometrical consider- 

 ations therefore, combined with a becoming modesty 

 as to our powers of applying physical considerations, 

 present us with two unknown quantities for correction, 

 one of which is the quantity admitted in .4^}. Nacli., 

 No. 3682 to be arbitrary, while the other is a new one. 



If the preconceived theory is correct and the records- 

 are trustworrhv the value of the second variable will on 

 solution turn out to be zero or so nearly zero as to- 

 suggest that zero is the true value. If no values 

 satisfv all the equations of condition, then some of the 

 records are untrustworthy or the geometrical consider- 

 ations have been carelessly thought out. If the equa- 

 tions can be satisfied simultaneously, and the value 

 of the second variable is not zero, a very strong case 

 is established against the physical considerations of 

 the preconceived theory. 



If we write down five simultaneous linear equations 

 in two unknown quantities x and y. all satisfied by 

 the same values of the variables, and if we then put 

 y equal to zero, or in other words, rub out the terms 

 in y, we shall of course find the equations in x are 

 inconsistent. If the equations represent historical 

 data, and if, as men of science, we have a proper 

 contempt for literature, we shall no doubt proceed to 

 quarrel with our evidence. This is exactly the way in 

 which astronomers have in the past treated ancient 

 solar eclipses. When, however, equations of condition 

 involving two unknown quantities are formed for all 

 the solar eclipses in which the place of observation 

 appears to have been fairly near the central line, 

 whereas modern tables give residuals of the order of 

 200", that is to say, make the apparent differences of 

 latitude at conjunction in longitude of the order of 

 200", values can be found for the unknown quantities, 

 which will make all the residuals less than 50" ; in 

 other words, whereas the present tables would leave 

 about ten per cent, of the sun's diameter visible, the 

 alterations proposed never leave so much as two per 

 cent, visible. 



Let it be here stated that no solar eclipse is an 

 exception to the above statement. The conclusions 

 rest, not upon the evidence of a majority but upon 

 the unanimous evidence of all eclipses used. A list 

 of these is given in Monthly Notices, Ixv., p. 861, and 

 a reference is given on p. S67 to the eclipse of 

 Agathocles. The eclipse of Thates has not been 



