THE LAW OF EQUILIBRIUM 151 



of the diameter of the earth's orbit, the principle of fall- 

 ing bodies, the velocity of the sun as deduced from the 

 direct study of the proper and the radial motions of the 

 stars, and, finally, the parallactic motion of the Vertex. 



Given the length of the arc of a circle in degrees and 

 also in linear measure, it is only a matter of high-school 

 mathematics to find the total length of the circumference 

 and, from that, all the remaining functions of the circle. 

 Again, if we know the length of an arc and the amount it 

 deviates from its tangent (technically called the curva- 

 ture), we can, by a simple formula, ascertain the radius 

 in the first instance, and so on to the rest. In the case 

 under consideration we virtually know these two things : 

 the arc 's length, and its curvature in miles. The latter is 

 quickly ascertained in this way : 



If you have been following this somewhat intricate 

 explanation closely, you should be able to see with little 

 effort, that, inasmuch as the sun has at the instant of the 

 second vernal equinox failed to get back fully into line 

 with the same star it was in line with at the preceding 

 equinox, the earth must continue on in her orbit beyond 

 her equinoctial position until the old star, sun, and earth 

 are all brought again into alignment. Astronomers tell 

 us that it takes 20ms. 23 sees, for the earth to move far 

 enough in her orbital journey to correct this discrepancy 

 this curvature of the arc. Now, the earth's average 

 velocity being known (18.5 miles per second), to reduce 

 this curvature to terms of miles we need only multiply 

 this quantity by the number of seconds of time to obtain 

 the answer, 22,625 miles. 



As said before, given the length of arc in seconds and 

 the curvature in miles, it is easily possible to ascertain 

 the radius of the circle. The rule, stated in the form of 

 an equation with the initial letters of the things referred 

 to (see Young's Gen'l Astr., Art. 420) is: 



c : a : : a : 2 r 



Now the length of the radius in terms of degrees of 

 circumference is called a radian, and is known to be 



