r 553 ] 
n being the number of rows, the number of the In- 
terflices will be 
zm — iaX7i — I — zbxn — 2 — icxn — 3, &c. That 
is, zmx n — i — 2 a x n — i — zbx n — 2 — 2 c x n — 3 , 
&c. By comparing this exprelTion with the former 
expreffion of the number of the circles, it will ap- 
pear, that when riy the number of the rows of circles, 
is infinitely augmented, the number of interflices is 
to th« number of circles, ultimately, as 2 to i. For 
the two expreffions always confifi; of an equal num- 
bers of terms. The fame numerical terms in both 
are affe< 5 ted with the fame figns. The firfl term of 
the latter {zmxn — 1) is ultimately double the firfl: 
term of the former {mn)y when n is infinitely in- ' 
creafed, and each fucceeding term of the latter is 
double the correfponding term of the former. There- 
fore, the whole of the latter expreffion is ultimately 
to the whole of the former, as 2 to i. That is, the 
number of interfhees is ultimately double the num- 
ber of circles : whence it follows, as in the former 
cafe, that the whole fpace covered by the circles is 
to the whole fpace occupied by the interfiices, as 
I the area of one circle to the whole area of one in- 
terfhee. 
In this Demonflration I have fuppofed the num- 
ber of circles upon the feveral lines AG, HP, QX, 
&c. to decreafe continually. Had I fuppofed them to 
decreafe by fits, and in any manner imaginable, fliill 
the conclufion would have been the fame {a), There- 
(a) Suppofe the number of circles upon the ifl: row to be m, 
upon the 2d, m — <7, upon the 3d, m — upon the 4th, 
m — — r, upon the 5th, m — a + b-c-\-dy and fo on, and 
VoL. LXI. 4 B fore. 
