OF ARTS AND SCIENCES: MAY 10, 1864. 289 



If we knew the tensoi's, but not the signs, of the errors (ci, . . €^) of 

 (xi, .. X/.), the consideration that all the 2* sign-combinations are 

 equally i)robable, would give for that average or probable increase in 

 the squared error of x due to the inaccuracy of (mU : •• m\), the 

 value 



/m\ _ m^Y ^ 2 I (f^k _ ^Y . 2 , 



e 



k J 



but since of (e{^ , . . e/) we know only that their probable values are 

 inversely as (toj : . . m^.), on the smallness of 



J_ _ J_ huj _ m^Y I J_ /m^ _ mA^ 



W nil V^" ^V nik \S' *SV 



must depend the required criterion. Since — is here proportioned to 



the square of the probable effect of the inaccuracy of (m\ : . . tn\) 

 upon X, IT may be called the weight of the approximation (?«\ : . . m\) 

 to {m^ : . . mf.) . 



It remains to examine the properties and results of the criterion 



F ( — , nil, .. m\) = min., where F depends merely on the gen- 

 W 



eral magnitudes of — , nii, &c., . . ; and to compare them with those of 



^^ I— , nil, . . ni\ j = min., &c. 



It may be said that Nature, while obeying with absolute precision 

 the resultant of her efficient laws, such as Inertia, Attraction, &c., falls 

 into forms which commonly both utility and taste, often independently 

 of each other, would prescribe ; that some of her iBsthetic laws can be 

 learned from the simple integers (or other simple members of a series 

 whose form is prescribed a priori), whose ratios nearly represent those 

 of certain natural constants ; and that the probability that such sesthetifc 

 laws are intentionally observed may be roughly computed from the 

 constants. Hence the need of some criterion to guide beforehand our 

 selection of approximate ratios, unbiassed by the interests of the special 

 assthetic law we are testing. 



The phyllotaxis, and still more the planetotaxis noticed by Peirce, 

 illustrate one such aesthetic law. The mutual configurations of two 

 planets, and even their separate motions, whether we regard the dis- 

 turbed orbits as variable ellipses or as higher curves, are patterns 

 containing the element time ; and their character must depend mainly 



