ir MATHEMATICS OF THE MINIMUM 241 



object of thought, by analogy or inference from the 

 combinations of sensible minima in larger composites, 

 the same forms of aggregation being repeated in the 

 higher which occur in the lower forms. 1 



From the consideration of mathematical figures as 

 consisting of minima^ Bruno attempted both to remodel 

 and to simplify the existing mathematical theory, and, 

 unfortunately fell foul of the new analytical mathematics, 

 the theory of rationals and of approximations, which at 

 that time was receiving marked extensions, and which 

 has since justified itself so completely by results. It is 

 true he did not entirely reject it, but he regarded it as 

 merely an artifice for rough practical measurements. 

 The true measure is always the minimum^ inferred by 

 analogy from the combinations of greater parts, which 

 are perceived by sense. Thus the minimal circle, after 

 the atom itself, consists of seven minima, the minimal 

 triangle of three, and the minimal square of four, and 

 as each figure increases not by the addition of one 

 atom merely, but by a number determined by the 

 original number of atoms in the figure, it follows that 

 no one figure is ever equal to another. Thus the 

 second triangle is of six minima, the second square of 

 nine, the second circle of nineteen. The " squaring of 

 the circle " is therefore impossible, 2 although it may 

 be approximately reached through the ultimate coin- 

 cidence of arc and chord, by which the circle becomes 

 equal to a polygon with an infinite number of sides. 3 

 This, however, is only an approximation of sense, 

 which fails to observe the infinitesimal differences that 

 are caused by the existence of a few atoms, more or less, 

 in a figure. They are visible to the eye of reason, 

 which comprehends that no two figures in nature are 



1 De Min. i. ch. 14. p. 184. 23. 2 ii. ch. 8. p. 214. 3 iii. ch. 12. p. 267. 



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