. , C 548 ] 
circle, which is not in an outermoft row, or at the 
extremity of any other row, touches fix others, 
namely two in its own row, and two in the row on 
either fide of its own ; and each adjacent pair of 
thefe fix do alfo touch each other. In the outer 
rows, every circle, which is not at one extremity of 
its row, touches four others, two in its own ; row, 
and two in the row next befide it ; which laft two 
do Hkewife touch each other. A: circle at either 
extremity of an outer row, touches only a fingle 
circle in its own row, but ^either one or two in the 
row next befide it. The. bare infpeftion of the figure 
(Tab, XVII.) will make thefe aflertions manifefi-. 
Now, imagine the equal circles, exhibited in the 
figure, to be each i infinitely fmall, the number of 
them being infinitely great, and the whole fpace 
over which .they are difpofed being of a finite magni- 
tude. . The ultimate proportion of the fpace covered 
by all the Circles, to the fpace occupied by all their 
Tnterfiices, is that of | the area of one of the circles 
to the^ whole of one interftitial area, /. e. the pro- 
portion. of 39 to 4 very nearly. 
Demonstration. 
The circles ranged along the parallel right lines 
A B, H P, form two rows of interfaces ; the row 
marked a, c, d, 6 cc. and the row marked a, / 3 , y, S, 
&c. and, in like manner, two rows of interftices are 
formed by every two contiguous rows of circles. 
Now, the numbers of the circles ranged along the 
feveral parallel right lines, AG, HP, Q^X, &c. are 
either equal or unequal, according to the figure of 
the fpace over which they are difpofed. 
I 
Cafe I. . 
