Mr. J. Snart on the Qiiadratuie of the Circle. 341 



by the test of numbers, not only generates an area, or super- 

 ficies ; but, to produce a plenary effect, as indicated by mul- 

 tiplication, must be rectangular as well as rectilineal, or else 

 the superficies so generated would be defective in the propor- 

 tion of such aberration from the rectangle; and instead of 

 producing 81 tetragons (fig. 2.), by squaring the nine, we 

 should obtain but 81 lozenges or rhombi (fig. 3,), each of 

 which would be minus in proportion to the deflection from 

 the right angle; and which, if the declension were carried to 

 60 degrees from the perpendicular, would sacrifice half the 

 surface, as in fig. 4", producing only 40*5, although the peri- 

 meter of the rhombus (fig. 3), whose two segments generated 

 it, was equal to the tetragon (fig. 2). For when that rhombus 

 is bisected in the perpendicular a a, and united by the diago- 

 nal bb, it forms the diminished parallelogram (fig. 4.) Q.E.D. 

 Fis 2. 



Fis4- 



The third, or cubic power of figures, is a compound of the 

 first and second modes, and therefore so similar (indeed Men- 

 ticaV) in its rectangidar operation, that it is almost supei-fluous 

 to say any thing about it ; as those who ax-e convinced of the 

 sufficiency of the former arguments will scarcely withhold 

 their assent to the latter. For although 729 cubes arise from 

 the primitive root (as 9x9x9 = 729) when the operations 

 have had their plenary or numerical effect, yet if the third 

 power of the same root in figures could be alike deflected 

 from the pei-pendicular, and applied to the oblique angled 

 parallelogram or rhombus above spoken of, the solid pro- 

 duct would be only 729 parallelopipeda, oblique in all their 

 angles, and whose whole cubic content, therefore, would be 

 but 182-25 solid feet ( = 729-f-4). In which erroneous opera- 

 tion the loss would be no less than 3-4ths of the whole solid 

 mass. 



But the integrity of figures cannot be so grossly vitiated, 

 nor is it presumed that their misapplication, in the mensura- 

 tion of the circle, has been so glaring as the possibilities sup- 

 posed in this extreme case: but that their indispensable pro- 

 perties have been invaded and violated, and incompatibilities 

 expected from them, by all those who sought either to mcn- 

 surate or quadrate the circle to perfection by either of their 

 two first powers, is quite plain. — And that they did expect 



to 



