VOL. XLVril.] PHILOSOPHICAL TRANSACTIONS. 405 



centres of the segments, from the focus of the eye-glass, where the edges are 

 seen in contact, is equal to the angle subtended by the diameter of the object 

 from that same point. 



Demons. Let the line ab, fig. 7> ph 8, represent the diameter of the object to 

 be measured ; and the points c, d the centres of the two glass segments : also g 

 the focus where the images of the extremities of the object coincide. It is evi- 

 dent, from Obs. 3, that ag and bg are straight lines, that pass through the 

 centres of the segments, and connect the extreme points of the object with their 

 corresponding points in the images; and therefore, as the diameter of the object 

 and the distance between the centres of the segments, are both inscribed between 

 these two lines, they must needs subtend the same angle from the point where 

 those lines meet ; which is at g. 



The focal distance cg, or dg, is variable, according to the distance of the 

 object from the glass : so that it decreases as the distance of the object from the 

 glass increases ; and when the object is so far off, that the focal length of the 

 glass bears no proportion to its distance, then will it be least of all ; as cf or df; 

 and the point f is called the focus of parallel rays. Any other focus, as g, being 

 the focus of a near object, is called a respective focus ; as it respects a particular 

 distance : but the focus of parallel rays respects all objects that are at a very great 

 distance; such as is that of all the heavenly bodies. 



Prop. 2. — ^The distance he of the object from the glass, is to ef, the focal 

 distance of parallel rays, as the distance hg of the object from its image, is to 

 EG, the distance of the image from the glass : that is, he : ff :: hg : eg. 



The demonstration of this proposition may be gathered from any ireatise of 

 dioptrics ; it being a general rule for finding the respective focus to any given 

 distance, when the focus of parallel rays is known. 



Prop. 3. — ^The angle subtended by the diameter of the object, from the glass, 

 is equal to that subtended by the opening of the centres of the segments, from 

 the focus of parallel rays. That is, the angle aeb equal to the angle cfd. 



Demons. — It appears, by inspection of the figure, that ab : cd :: hg : eg. 

 And by the last prop, he : ef :: hg : eg. Then, as the two last terms of tliese 

 two analogies are alike ; the two first terms of one will be in the same proportion 

 as the two first terms of the other ; which gives the following proportion : ab : 

 CD :: HE : ef. Whence the truth of the proposition is evident. 



From this proposition it appears, that the angle subtended by the diameter of 

 the object from the glass, is found without any regard to the distance of the 

 object, or to the distance of the respective focus, where the image is seen ; as 

 the measure depends entirely on the focus of parallel rays, and the opening of 

 the segments. We may hence also derive a rule for the quantity of the angle, 

 without considering the length of the glass. Let an object, whose diameter is 



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