deduced from the Curvature of the Earth. 269 



On the second trial, if his line of vision led beloia its top, then 

 what he had last measured as 3 miles was actually less : this 

 would cause him to lengthen his rod, which for the second trial 

 he had shortened too much, and measure again; and thus 

 would he go on approximating, until at the distance of 2640 

 lengths of his rod his line of vision led exactly to the top of it; 

 when he might rest assured that its length was correcly 1 fathom,' 

 being the unit sought, and the distance 3 miles ; for no other 

 number of lengths will tally ivith the curvature at the same di- 

 stance. To illustrate this doctrine, I shall suppose that at the 

 first trial, his rod, instead of being 72 inches, as it ought, was 

 74 ; 264-0 lengths of it made the distance 5427 yards, at which 

 the straight line of vision would lead to nearly 2 inches above the 

 top of his rod. At the second trial his rod instead of being 72 

 inches was onFy 70 ; 2640 lengths of it made the distance 51 33 

 yards, at which the straight line of vision would lead to more 

 than 2 iiiches belovo the top of his rod. 



This reciprocal self check of the standard can therefore be 

 made at any time with perfect mathematical accuracy'. Its 

 principle depends on the difference of angular convexity between 

 the surface of die water and the straight line of vision, increas- 

 ing, as it does, in a certain geometrical ratio ; while the lineal 

 distance for ascertaining it increases in only an arithmetical 

 ratio. 



I have assumed the above spherical convexity as true : when 

 tried in the way proposed, it may perhaps be found a trifle dif- 

 ferent; but its exact amount will then be accurately ascertained, 

 as may also the true diameter, circumference and shape of the 

 earth. 



To make the cast-iron pipe just 3 miles long, could be effected 

 without a measure by other methods; such as, 1st, with two 

 transit instruments, to note the space that the earth revolved 

 on its axis from west to east, in (say) 5 seconds of time ; 2d, by 

 noting the space that sound travelled, in (say) 14 seconds of time. 

 But r humbly think, that the method I have above suggested, 

 containing as it does within itself a complete check on, and a 

 proof of the result, is the most preferable, simple, and least 

 liable to error. 



The subdivisions of this fathom unit into universal measiues 

 of length, capacity, and weight, may be the subject of a future 

 essay. 



Glasgow, March 1, WZ^. J_ Bq^z. 



LXIl. On 



