C S49 3 
Cafe I. Flrfl fuppofe, that an equal number of 
circles is ranged along each of the parallel lines ; in 
which cafe, the figure, in which they are included^ 
mufi: be a parallelogram. The number of circles, rang- 
ed along the parallel right lines A G, H P, being equal, 
the number of interftices in each of the rows, h, 
Cy dy &c. ay I3y y, Sy 6cc. IS Icfs by unity than the 
nurpber of circles upon either line, AG, or HP, be 
that number what it will. Thus the two circles A, B, 
upon the line AG, with the two circles H, K, upon, 
the line HP, have the fingle interfiice in the row 
a, by Cy dy 6cc. and the fingle interftice a, in the ^ 
row a, /Q, yy $y &c. Again, the three circles 
A, B, C, upon the line AG, with the three, H, K, L, 
upon the line H P, have the two interftices Cy by in 
the row by c, dy &c. and the two a, (3, in the^ 
row ccy (3, yy J', &c. And univerfally, if the num- 
ber of circles in each row be w, the number of in- 
terftices, in each of the two rows of interftices, will, 
be w — I. Confequently, the whole number of 
interftices formed by thefe two rows of circles is. 
' 2 w — 2 . In like manner, the two rows of circles 
HP, QX, form two more rov/s of interftices. Andt 
the number of circles upon each line, H P, QX,, 
being w, the number of interftices in each row is , 
m— I, and the whole number in both rows 2w— 2,. 
Therefore, the whole, number of interftices formed by 
the three rows of circles, AG, HP, QX, is zm — 2 
twice taken, or 2 m — 2 x By the fame reafoning, 
if a fourth row of m circles, rX be added, the num- 
ber of interftices formed by the four rows is^, 
— 2 X 3^ And univerfally, if there be 7r.rows - 
off 
