100 COSMIC PHILOSOPHY. [ft. i\ 



drawn to tlie moon must make at the two ends of his long 

 base-line from London to Cape Town, he is at once enabled, 

 like the land-surveyor, to calculate by trigonometry the 

 lengths of these ideal lines, and thus to ascertain the moon's 

 distance. "What, now, is the essential characteristic of the 

 process which the astronomer goes through ? Or, in other 

 words, what is the fundamental psychical process by the mani- 

 fold compounding of which is built up this highly-complex 

 series of inferences ? 



From beginning to end, the fundamental process is the 

 cognition of the equality of sundry relations. The thought 

 which underlies and determines the whole calculation is the 

 cognition that the relations between the sides and angles of a 

 great triangle, having for its apex the moon, and for its base 

 the chord of the arc of the meridian of London measured to 

 a point in the southern hemisphere upon the same parallel 

 with Cape Town, are equal to the relations between the sides 

 and angles of a similar small triangle, having an inaccessible 

 tower for its apex and a measured line of five or six rods for 

 its base ; and that these relations, in turn, are equal to the 

 relations between the sides and angles of a still smaller and 

 similar triangle which may be drawn on a sheet of paper, and 

 of which the sides and angles may, if necessary, be directly 

 measured. Now, this cognition implies the previous establish- 

 ment, in the calculator's mind, of sundry cognitions of the 

 equalities and inequalities of certain relations between the 

 sides and angles of triangles. To show briefly how such 

 cognitions have been established, let ns cite the simplest case 

 — that in which the two angles at the base of an isosceles 

 triangle are recognized as equal to each other. Euclid es- 

 tablishes this point by supposing two similar and equal 

 isosceles triangles, of which the one is turned over and placed 

 upon the other, so that the apex and one side of the one will 

 coincide with the apex and opposite side of the other. 

 Then the other sides and the ba^es must respectively coincide 



