210 PHILOSOPHICAL TRANSACTIONS. [aNNO 1771. 



observations which were taken before the middle of the transit, or such as could 

 not, through some impediment, be observed before, may be proper to be ob- 

 served .after the middle of the transit. 7^h. It will be adviseable to practise ob- 

 servations similar to those here recommended, previous to the transit of Venus, 

 by means of spots in the sun. •. 



L. A Supplement to a former Paper, concerning Difficulties in the Newtonian 

 Theory of Light. By the Rev. S. Horsley, LL.B., F.R.S. p. 547. 



Prob. I. — A parcel of equal circles being disposed on a plane surface, of any 

 figure whatever, in the closest arrangement possible; to determine the ultimate 

 proportion of the space covered by all the circles, to the space occupied by all 

 their interstices, when each circle is infinitely small, and the space, over which 

 they are disposed, is of a finite magnitude. 



The closest manner in which a parcel of equal circles can be disposed on a 

 plane, is when the centres of every 3 contiguous circles are situated at the angles 

 of an equilateral triangle, which has each of its sides equal to a diameter of any 

 one of the circles. A number of circles, thus disposed, may be divided, as fig. 

 13, pi. 5, shows, into several rows of circles, having their centres ranged on 

 parallel right lines, AG, hp, ax, rs, &c. Every circle, which is not in an outer- 

 most row, or at the extremity of any other row, touches 6 others, namely, 2 in 

 its own row, and 2 in the row on either side of its own : and each adjacent pair 

 of these 6 do also touch each other. In the outer rows, every circle, which is 

 not at one extremity of its row, touches 4 others, 2 in its own row, and 2 in the 

 row next beside it: which last 2 do likewise touch each other. A circle at either 

 extremity of an outer row, touches only a single circle in its own row, but either 

 1 or 2 in the row next beside it. The bare inspection of the figure will make 

 these assertions manifest. 



Now, imagine the equal circles, exhibited in the figure, to be each infinitely 

 : small, the number of them being infinitely great, and the whole space over which 

 they are disposed being of a finite magnitude. The ultimate proportion of the 

 space covered by all the circles, to the space occupied by all their interstices, is 

 that of half the area of one of the circles to the whole of one interstitial area, 

 i. e. the proportion of 39 to 4 very nearly. 



Demonst. The circles ranged along the parallel right lines ab, hp, form 2 

 rows of interstices ; the row marked a, b, c, d, &c. and the row marked a, |3, y, S, 

 &c. and, in like manner, 2 rows of interstices are formed by every 2 contiguous 

 rows of circles. Now, the numbers of the circles ranged along the several parallel 

 right lines, ag, hp, qx, &c. are either equal or unequal, according to the figure 

 of the space over which they are disposed. 



Case 1. First suppose, that an equal number of circles is ranged along each 



