6 



EXACT ASTRONOMY. 1 7 



Completing the square and extracting root : 

 Cu'rv. 2 + 2 curv. + i = i. 000001538892788352. 

 Curv. + 1 = 1.0000007694462. 



Curvature = 0.0000007694462 is identical with the fall 

 of a weight (all resistance being eliminated) in one second 

 at sea level on the equator ; because its deflection in equal 

 times from the tangent to its path is the same, whether 

 dropped from rest or projected laterally with any velocity. 

 The acceleration of gravity at London has been determined 

 with the required accuracy : 32.1928306 feet per second. 



Gravity is as the square of the number of vibrations 

 by the same pendulum in equal times. 



A pendulum that beats seconds at London loses 136 

 beats in 24 hours at sea level on the equator. On the 

 equator gravity is diminished ?£* part of itself by the 

 momentum of the earth's rotation. All resistance elimi- 

 nated, the fall of a weight in 1 second : 32 - I9 ^ 3 ° 6 x (g^) 2 x 

 2g = 16. 101 4958005563 feet. 



Length of the equatorial radius : 16. 10 14958005563 -7- 

 0.0000007694462 = 20926084 feet. 



The Figure of the Earth Dynamically Determined. 



Pendulum observations have demonstrated the regular 

 increase of gravity with increase of latitude ; equal to rii 

 part of equatorial gravity at the pole. 



The distance a weight would there fall in 1 second : 

 32,1938306 x Qg^ t x p _ I 6.i 2 849i4950i25 feet. 



The earth's oblateness, concluded from the geodetical' ; 

 measurement of meridional arcs, is about *i« part of the 

 equatorial radius. I will now demonstrate that the terres- 

 trial radii are in the inverse duplicate ratio of gravity at 



