PROGRESS OF GEOMETRICAL ASTRONOMY. 213 



combined use of equal magnitudes and equal relations, by 

 which all exact science, all quantitative prevision, is reached, 

 For the relation observed was between the length of the 

 sun s shadow and his position in the heavens ; and the in 

 ference drawn was that when, next year, the extremity of 

 his shadow came to the same point, he occupied the same 

 place. That is, the ideas involved were, the equality of the 

 shadows, and the equality of the relations between shadow 

 and sun in successive years. As in the case of the scales, 

 the equality of relations here recognized is of the simplest 

 order. It is not as those habitually dealt with in the higher 

 kinds of scientific reasoning, which answer to the general 

 type the relation between two and three equals the rela 

 tion between six and nine ; but it follows the type the re 

 lation between two and three, equals the relation between 

 two and three ; it is a case of not simply equal relations, 

 but coinciding relations. And here, indeed, we may see 

 beautifully illustrated how the idea of equal relations takes 

 its rise after the same manner that that of equal magnitude 

 does. As already shown, the idea of equal magnitudes 

 arose from the observed coincidence of two lengths placed 

 together ; and in this case we have not only two coincident 

 le ngths of shadows, but two coincident relations between 

 sun and shadows. 



From the Tise of the gnomon there naturally grew up 

 the conception of angular measurements ; and with the 

 advance of geometrical conceptions there came the hemi 

 sphere of Berosus, the equinoctial armil, the solstitial armil, 

 and the quadrant of Ptolemy all of them employing shad 

 ows as indices of the sun s position, but in combination 

 with angular divisions. It is obviously out of the question 

 for us here to trace these details of progress. It must suf 

 fice to remark that in all of them we may see that notion 

 of equality of relations of a more complex kind, which ia 

 best illustrated in the astrolabe, an instrument which con- 



