78 PHILOSOPHICAL TRANSACTIONS. [aNNO 1791. 



For the sake of workmen and other persons not conversant in astronomy, Mr. 

 C. describes an easy and accurate method of ascertaining the vakie of the divi- 

 sions of the micrometer ; viz. mark on a wall, or other place, the length of 6 

 inches, which may be done by making 2 dots or lines 6 inches asunder, or by 

 fixing a 6 inch ruler on a stand ; then place the telescope before it so that the 

 ruler or fj-inch length may be at right angles with the direction of the telescope, 

 and just 57 feet 3-f inches distant from the object-glass of the telescope: this 

 done, look through the telescope at the ruler or other extension of 6 inches, 

 and observe how many divisions of the micrometer are equal to it ; then that 

 same number of divisions is equal to half a degree, or 30'; and this is all that 

 needs be done for the required determination ; the reason of which is, because 

 an extension of 6 inches subtends an angle of 30' at the distance of 57 feet Sc- 

 inches, as may be easily calculated by the rules of plane trigonometry. In one 

 of Dollond's 14-inch pocket telescopes, if the divisions of the micrometer be 

 the lOOths of an inch, 1 1-J of those divisions will be found equal to 30', or 23 

 to a degree. When this value has been once ascertained, any other angle mea- 

 sured by any other number of divisions is determined by the rule-of-three. Thus, 

 suppose that the diameter of the sun, seen through the same telescope, is found 

 equal to 12 divisions, say as 11-|- divisions are to 30', so are 12 divisions to 3l'.3, 

 which is the required diameter of the sun. 



JlX. a New Method of Investigatmg the Sums of Infinite Series. By the Rev. 

 S. Fince, A. M., F. R. S. p. 295. 

 The summation of infinite series is a subject, not only of curious speculation, 

 but also of the greatest importance in the various branches of mathematics and 

 philosophy; in consequence of which it has always claimed a very considerable 

 share of attention from the most celebrated mathematicians. Mr. V. therefore 

 makes no apology for offering to the public the following new and very expedi- 

 tious method, by which we may obtain the sums of a great variety of series, most 

 of which have never before been treated of. As the summation depends on the 

 sums of the reciprocals of the powers of the natural numbers, tables of such 

 sums are given as far as the 40th power to 1 2 places of decimals, by which the 

 sums of the series will be found true to 10 or 11 places; and if greater accuracy 

 were required, which is a case that can very rarely happen, it might easily be ob- 

 tained by continuing the tables. The 1st and 2d columns show the sums, and 

 the 2d and 4th the powers corresponding. 



