Chase.] 414 [Oct. 6, 



c. Tlic coeilicient of tj, in the above approximation, is tlie pliyllotactic 

 product, 1 X 2 X B. It is also very nearly equivalent to the square 

 root of the quotient of Laplace's solar limit by Sun's semi-diameter, -svliich 

 would give, ^3= 80.619 a. 



d. The mass of Mars is very nearly a mean proportional between the 

 masses of Earth and Moon ; (3,093,500 -- 331,776)2 _ 86.938. 



e. An approximation similar to b is given by the proportion : 



month : day : : m^ : 3 /j. : : 81.965 : 3. 

 /. Moon's locus of subsidence, or aphelion (s), and the mass of Venus 

 (Wa), furnish the following approximation : 



m.^ :fi::s:ro,: : 63.593 : 1. 

 Substituting the observed basis of the second approximation to m., in Note 

 250, this gives, w, = 82.119 /j.. 



Many other approximations might doubtless be found which would 

 represent obvious harmonic tendencies within the belt of greatest conden- 

 sation. 



257. Simplicity and Conciseness of Harmonic Calculus. 



The range of estimates in the foregoing note is about 8| per cent., and 

 the mean of all the estimates is about 3 per cent, greater than the most 

 recent astronomical estimates. These deviations are four times as great as 

 in my extreme estimates of solar distance, and twelve times as great as 

 in the estimates which have been based upon the latest determinations of 

 the harmonic elements. If these approximations are compared with those 

 which had been made T)y astronomers, a hundred 3'ears after Xewton had 

 published the laws of gravitation, the indications of superiority in the har- 

 monic methods become verj' striking. The difficult}' of finding the har- 

 monic influences which are most important, is incomparably less than that 

 of determining the corresponding gravitating influences, and the saving of 

 labor is obvious to every one who has ever solved astronomical problems 

 by the ordinary processes of mathematical analysis. Doubts as to the 

 degree of certainty which attaches to purely harmonic results will natur- 

 ally arise, in the minds of those who have never carefully inquired 

 into the necessity of clastic rhythm, but I believe that such doubts will 

 gradually yield to the fast accumulating evidences of its universal sway. 

 Astronomical, chemical and mechanical science may all be challenged to 

 produce a series of connected fundamental determinations that are com- 

 parable, in precision and in intrinsic mathematical probability, with those 

 which are embodied in Note 168 and in the three solutions of Note 254. 



258. Needless Obscurity. 



In Sir John Leslie's Dissertation on the Progress of Mathematical and 

 Physical Science {Encyc. Brit., 8th. Ed., i, 732), after referring to the 

 "maze of intricate and abstruse formuW in which Laplace had involved 

 the phenomena of capillary attraction, the following reflections of Dr. 



