Sg2 PHILOSOPHICAL TRANSACTIONS. [aNNO 1722. 



rational and part irrational. He has also added general notes on the chief 

 difficulties in the book, with a method of composing synthetical demonstra- 

 tions of logometrical and trigonometrical constructions, illustrated by various 

 examples. 



The first treatise in the miscellaneous works, is concerning the estimation of 

 errors in mixt mathematics. It consists of 28 theorems, to determine the pro- 

 portions among the least contemporary variations of the sides and angles of 

 plane and spherical triangles, while any two of them remain invariable. An 

 example will show their great use in astronomy. The time of the day or night 

 is frequently to be determined by the altitude of some star. Let it then be 

 proposed to find the error that may arise in the time, from any given error in 

 taking the altitude. By applying the 22d theorem to the triangle formed by the 

 complements of the star's altitude and declination, and by the complement of 

 the pole's elevation, the author shows, that the variation of the angle at the 

 pole, and consequently the error in time, will be as the error in the altitude 

 directly, as the sine complement of the pole's elevation inversely, and as the 

 sine of the star's azimuth from the meridian inversely. Consequently, if the 

 error in the altitude be given, under a given elevation of the pole, the error in 

 time will be reciprocally as the sine of the azimutli contained by the meridian 

 and the vertical which the star is in. This error therefore will be the same, 

 whatever be the altitude of the star in the same vertical ; and will be least when 

 the vertical is at rigVit angles to the meridian. But will be absolutely the least 

 in the same circumstance, if the observer be under the equator. In which case, 

 if the error in the altitude be one minute, the error in the time will be 4 seconds. 

 If the observer recedes from the equator, towards either pole, the error will be 

 increased in the proportion of the radius to the sine complement of the latitude: 

 so that in the latitude of 45*^, it will be 5-|- seconds ; and in the latitudes of 50 

 and 55, it will be 6^ and 6-§-J seconds respectively. If the star be in any other 

 vertical, oblique to the meridian, the error will still be increased in the propor- 

 tion of the radius to the sine of that oblique angle. Lastly, if the error in the 

 altitude be either more or less than one minute, the error in time will be more 

 or less in the same proportion. Much after the same manner may the limits of 

 errors be computed in other cases, which arise from the inaccuracy of observa- 

 tions, and from hence the most convenient opportunities for observing are also 

 determined. 



The second treatise, is concerning the differential method. The author, 

 having written it before he had seen Sir Isaac Newton's treatise on that subject, 

 has handled it after a manner somewhat different. 



The title of the 3d treatise, is Canonotechnia, or concerning the construction 



