[ 55 + ] - 
fore, let the figure of the finite fpace. Including the 
circles thus clofely arranged, with their interftices, 
be what it will, the proportion of the fpace covered 
by all the circles, to the fpace taken up in interftice, 
is ultimately that of | the area of one circle to 
.the whole area of one interftice. 
Now, that this is the proportion of 39 to 4, very 
nearly, will appear by computing one ot the interfti- 
tial areas. 
The method of computing the interftitial area is 
obvious. Let A, B, H be the centers of the three 
circles, which clofe the interftice TOT. Joit^ AB, 
A H, B H. The right lines A B, AH, B H, pals 
through the points of contad: T, O, T, refpedively. 
and each of thefe numbers tOibe infinitely increafed. Then, k 
beino- the number of rows, the whole number of circles will be 
o . A 
nm — a y. n — i b X n — 2 — c X. n — d X n 4, &c. 
Number the interftices formed by every two contiguous rows,, 
and add them all together, and the whole number of interftices s 
will be found to be ' , 
2711 X I— 2^ X «— I -p 2‘bxn- 5 — 2cX n~^ + zd xn— 5, See. . 
Now, by comparing thefe two expreflions, it appears, that 
both confift of the fame number of terms : That the fame 
numerical terms in order from the firft, have the fame figns 
That the firft term of the latter (zw X « — i) is ultimately 
the double of the firft term, of the former, when w is infinitely 
increafed : That of the terms following the firft, the negative 
terms of the latter are each double the correfponding negative 
terms of the former : and each pofitive term of the latter differs 
from the double of>hc correfponding pofitive term of the former., 
"by a number wUi^vaniflies with refpedl to either of thole cor- 
refponding terms, when it becomes infinite. Therefore, when 
M becomes infinite, the whole of the latter expreflion becomes . 
'the double of the whole of the former. Hence the conclufioii, 
is a4 before. 
The 
