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simple construction of a regular Pentagon in a Circle ; the princi- 

 ples, however, from which this construction is derived, are less evi- 

 dent, involving some peculiar properties of the chords subtending 

 1, 3, 5, &c. of the equal arches arising from tiie division of the 

 semicircumference into seventeen equal parts, or which is the same, 

 of the Supplemental Chords of the progressive arches arising from 

 the division of the whole circumference into 17 equal parts. These 

 properties form the five Theorems in the Analysis, and which com- 

 prehend all the necessary data for the solution of the problem. The 

 1st is an obvious deduction from the 2d Lemma, and the 5th, the 

 immediate application of the 4th : — the 2d, 3d and 4th properties are 

 derived by a more extended application of the 3d and 4th Lemmas, 

 as shewn in the notes ; of these, the 2d property is very remark- 

 able, and of great importance in enabling us at the commence- 

 ment of the Analysis, combined with Theorem I. to effect the de- 

 composition of the chords as therein specified ; and without which 

 we might perhaps in vain attempt the solution : — the 3d and 4th 

 properties are equally valuable in effecting a further decomposition 

 of these chords ; leading finally to the entire separation of the 

 supplemental chord of twice the arch, and chord of half the arch 

 subtended by the side of the Polygon, and thence, by an obvious 

 construction, to the side of the Polygon : and it might perhaps be 

 worth while to enquire how far the principles employed in this 

 Analysis might be attended with advantage in their application to 

 the solution of similar problems. 



The five Theorems in the Analysis may be conveniently exhibited 

 under the following abridged form : 



