VOL. LXI.] PHILOSOPHICAL TRANSACTIONS. 2l'l 



of the parallel lines; in which case, the figure, in which they are included, must 

 be a parallelogram. The number of circles, ranged along the parallel right lines 

 AG, HP, being equal, the number of interstices in each of the rows, a, b, c, d, 

 &c. a, (3, y, i, &c. is less by unity than the number of circles on either line, ag, 

 or HP, be that number what it will. Thus the 2 circles a, b, on the line ag, 

 with the 2 circles hk, on the line hp, have the single interstice a, in the row 

 a, b, c, d, &c. and the single interstice «, in the row a, (3, y, S, &c. Again, the 

 3 circles a, b, c, on the line ag, with the 3, h, k, l, on the line hp, have the 

 2 interstices a, b, in the row a, b, c, d, &c. and the 2 «, (3, in the row a, (3, y, S, 

 &c. And universally, if the number of circles in each row be m, the number 

 of interstices, in each of the 2 rows of interstices, will be m — 1. Consequently, 

 the whole number of interstices formed by these 2 rows of circles is 1m — 2. 

 In like manner, the 2 rows of circles hp, qx, form 2 more rows of interstices. 

 And the number of circles on each line, hp, qx, being m, the number of inter- 

 stices in each row is m — 1, and the whole number in both rows 2m — 2. 

 Therefore, the whole number of interstices formed by the 3 rows of circles, ag, 

 HP, QX, is 2m — 2 twice taken, or {2m — 2) X 2. By the same reasoning if a 

 4th row of VI circles, r£ be added, the number of interstices formed by the 4 

 rows is {2m — 2) X 3. And universally, if there be n rows of equal circles, 

 and m circles in each row, the number of interstices formed by all the rows is 

 (m — 2) X (n — 1). Now, when the circles are infinitely small, their diameters 

 are infinitely small. Therefore, the space which they cover being of finite mag- 

 nitude, it is necessary, that both the number of circles in each row, and the 

 number of rows, that is, that each of the numbers, m and n, should be infinitely 

 great. But when m and n are each infinitely great, (2m — 2) X (w — l), that 

 is, the number of interstices, becomes ultimately 2»m; and the interstices being 

 all equal one to another, if the area of one be called p, the sum of their areas 

 will be 2mn X p. But the number of circles in n rows, each row consisting of 

 m circles, is mn ; and the circles being equal, if the area of one be called a, the 

 sum of their areas will be mn X a. Hence the space covered by all the circles, 

 is to the space covered by all their interstices, when the magnitude of each circle 

 is infinitely diminished, and the number of them so infinitely augmented, as 

 that they shall cover a space of finite magnitude, ultimately, as mw X a to 2mn 

 X P, that is, as a to 2p, or as J-a to p, that is, as half the area of one circle to 

 the whole area of one interstice. 



Case 2. Now, suppose that unequal numbers of circles are ranged along the 

 several lines ag, hp, qx, &c. which must always be the case, if the figure of the 

 space, in which they are contained, be any other than a parallelogram ; and let 

 the number on ag be the greatest of all, and call that number, as before, m. 

 If from the row hp, the extreme circle p be taken away, all the rest being left, 



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