176 



Arithmetics, in which he has shown that all such regular polygons 

 as have the number of their sides expressed by 2" + 1, and being at 

 the same time a prime number, admit of being inscribed geometri- 

 cally in a circle: and hence since 5, 17, 257, 65537, are prime 

 numbers of this form, consequently regular polygons of these sides, 

 admit of geometrical construction. 



As the construction, however, of none of these polygons, except 

 of the first, or regular pentagon, has yet been publicly given, I beg 

 leave to submit that for the second, or a regular polygon of 17 sides, 

 together with the analysis. The former will be found exceedingly 

 simple, but as the latter depends on the application of some general 

 properties of the circle, I shall previously demonstrate these in the 

 four following Lemmas. 



SAMUEL JAMES. 

 • Dublin, \7th October, 1819. 



LEMMA L 



If on each side of a given point in the circumference of a circle, 

 equal arches be taken, parallel right lines drawn from their extre- 

 mities, to terminate in a diameter passing through the given point, 

 are equal to each other. See Plate, Fig. 1 . 



Let AB and x\C be equal arches taken at each side of the 

 given point A in the circumference BAG ; and BD and CE two 

 right lines drav^n from their extremities B and C parallel to each 

 other, terminating in the diameter FA passing through A : then 

 will BD and CE be equal to each other. 



For if BC be drawn cutting the diameter FA in I, it is well known, 

 because the diameter FA bisects the arch BC, in A, that it also 

 bisects its chord BC, in I ; and because BC meets the parallels BD, 



