236 W. Ilarkness — Magnitude of the Solar System. 



already given a summary sketch of the trigonometrical meth- 

 ods, as applied by the ancient astronomers to the dichotomy 

 and shadow cone of the moon, and by the moderns, to Venus, 

 Mars and the asteroids, and we shall next glance briefly at the 

 gravitational and photo-tachymetrical methods. 



The gravitational results which enter directly or indirectly 

 into the solar parallax are six in number, to wit : first, the rela- 

 tion of the moon's mass to the tides ; second, the relation of 

 the moon's mass and parallax to the force of gravity at the 

 earth's surface ; third, the relation of the solar parallax to the 

 masses of the earth and moon ; fourth, the relation of the solar 

 and lunar parallaxes to the moon's mass and parallactic inequal- 

 ity ; fifth, the relation of the solar and lunar parallaxes to the 

 moon's mass and the earth's lunar inequality ; sixth, the rela- 

 tion of the constants of nutation and precession to the moon's 

 parallax. 



Respecting the first of these relations it is to be remarked 

 that the tide-producing forces are the attraction of the sun 

 and moon upon the waters of the ocean, and from the ratio of 

 these attractions the moon's mass can readily be determined. 

 But unfortunately the ratio of the solar tides to the lunar tides is 

 affected both by the depth of the sea and by the character of the 

 channels through which the water flows, and for that reason the 

 observed ratio of these tides requires multiplication by a cor- 

 recting factor in order to convert it into the ratio of the forces. 

 The matter is further complicated by this correcting factor 

 varying from port to port, and in order to get satisfactory 

 results long series of observations are necessary. The labor 

 of deriving the moon's mass in this way was formerly so great 

 that for more than half a century La Place's determination 

 from the tides at Brest remained unique, but the recent appli- 

 cation of harmonic analysis to the data supplied by self-regis- 

 tering tide gauges is likely to yield abundant results in the 

 near future. 



Our second gravitational relation, viz : that connecting the 

 moon's mass and parallax with the force of gravity at the 

 earth's surface, affords an indirect method of determining the 

 moon's parallax with very great accuracy if the computation is 

 carefully made, and with a fair approximation to the truth 

 even when the data are exceedingly crude. To illustrate this, 

 let us see what could be done with a railroad transit such as is 

 commonly used by surveyors, a steel tape, and a fairly good 

 watch. Neglecting small corrections due to the flattening of 

 the earth, the centrifugal force at its surface, the eccentricity 

 of its orbit, and the mass of the moon ; the law of gravitation 

 shows that if we multiply together the length of the seconds 

 pendulum, the square of the radius of the earth, and the square 



