VOL. LVII.] PHILOSOPHICAL TRANSACTIONS. 43^ 



»•^If, besides these examples that are obvious to the naked eye, we extend the 

 same argument to the smaller stars, as well those that are collected together in 

 clusters, such for example, as the Praesepe Cancri, the nebula in the hilt of 

 Perseus' sword, &c. as to those stars which appear double, treble, &c. when 

 seen through telescopes, we shall find it still infinitely more conclusive, both in 

 the particular instances, and in the general analogy, arising from the frequency 

 of them. ■ --"^ 



We may hence therefore, with the highest probability conclude, (the odds 



. )'•',■ 

 whole heavens, would be within the distance of 1 1 minutes from the star Maia. From the log. of 

 35068 1 therefore subtract the log. of 3.'i06'S2, as in the former example, and the remainder will be 



— .00000! 11, the proportional part for a unit in the nvimber 3.')068.', which multiplied by 1500 

 will give us — .( :Ol66'50 for the log. of the probability sought. In like manner from 7.6746086 

 subtract 2 580069'^, twice the log. of I9J (the number of minutes between Electra and Maia) 

 and the remainder will be S-O^iSS^i, the proportional part for a unit corresponding to the natural 

 number of which will be — .0000>');:>49 ; and 1500 times this quantity, or — .O05'-'3.')0, will be the 

 log. of the quantity, representing th« probability, that no star out of 1500 scattered by chance would 

 be within the distance of IpJ minutes from Maia. If we follow the same rule for the three remain- 

 ing stars Merope, Alcyone, and Atlas, we shall find the similar logs, corresponding to these to be 



— .0076650, —.0100395, and — .03-">030O respectively. The natural numbers corresponding to 

 these five logs, taken in the same order, are .99''173, 988018, .982506, .977148, and .926766, 

 which severally express the respective probabilities, that no stars out of 1500 scattered by chance, 

 would be within the same distances, at which the five stars above mentioned are found to be, from 

 Maia. The complements of these quantities to unity 003827, .011,982, . 017^.94, .0.2852, and 

 .073234, which severally express the respective probabilities, on the contrary, that such stars would 

 be found within the distances above specified from the star Maia, must all be multiplied together, to 

 determine the probability, that these events should all take place at the same time. The sum of the 

 logs, of these numbers is — 9 + •'279139or — 8. 8720b6l, which is therefore the log. of all these 

 numbers multiplied together; and subtracting this number from 0, or, what amounts to the same 

 thing, changing the sign, we shall have the log. of the number of times that this quantity is contained 

 in unity, that is, about 7*4880000 times; a fraction therefore, whose denominator is this number, 

 and its numerator unity, will represent the probability in favour of all these events taking place to- 

 gether; and a fraction, whose denominator is the same number, and its numerator the same number 

 less by a unit, will represent the probability of the contrary. But as this event might as well have 

 happened to any other star as to Maia, we must multiply this last fraction into itself 1500 times, ac- 

 cording to the supposed nnmber of stars, to find the probability that it should not have happened any 

 where in the whole heavens. Now the proportional part for a unit on this number is — 

 .0000000005S25, which multiplied by 1500 gives us —.0(:0!)00.s7377, the proportional part for ^ 

 unit in somewhat more than 49'iOOO. 



But it must be observed, that this number is smaller than it ought to be on two accounts ; for, in 

 the first place, this method of compulation gives only the probability, that no 5 stars would be within 

 the distance above specified from a Clh, if they occupied the largest space they possibly could do, 

 under that limitation; and secondly, we have made no allowance on account of the different magni- 

 tudes, which, if it had been attended to, wouW have given a somewhat greater result. These con- 

 siderations however would have made the reasoning a good deal more intricate ; and we have no need 

 to descend to minutiae, a difference in the proportion of 10 to 1 not at all affecting the general con- 

 clusion. — Orig. 



