208 0;/ the Regular inscribed Polygon of Twenty Sides. 



greater than twenty-five diameters, has not perhaps hitherto 

 appeared in any scientific* work or periodical; and if a page of 

 the Philosophical Magazine can be spared for its insertion, some 

 readers may find it interesting from its extreme simplicity. In 

 fact, for completely understanding it, no preparation is required 

 beyond the four first Books of Euclid, and the few first Rules 

 of Arithmetic, together with some rudimentary knowledge of 

 the connexion between arithmetic and geometry. 



1. It follows from the Fourth Book of Euclid's < Elements/ 

 that the rectangle under the side of the regular decagon in- 

 scribed in a circle, and the same side increased by the radius, 

 is equal to the square of the radius. But the product of the 

 two numbers, 791 and 2071, whereof the latter is equal to 

 the former increased by 1280, is less than the square of 1280 

 (because 1638161 is less than 1638400). If then the radius be 

 divided into 1280 equal parts, the side of the inscribed decagon 

 must be greater than a line which consists of 791 such parts ; 

 or briefly, if the radius be equal to 1280, the side of the decagon 

 exceeds 791. 



2. When a diameter of a circle bisects a chord, the square of 

 the chord is equal, by the Third Book, to the rectangle under 

 the doubled segments of that diameter. But the product of the 

 two numbers, 125 and 4995, which together make up 5120, or 

 the double of the double of 1280, is less than the square of 791 

 (because 624375 is less than 625681). If then the radius be 

 still represented by 1280, and therefore the doubled diameter 

 by 5120, and if the bisected chord be a side of the regular deca- 

 gon, and therefore greater (by what has just been proved) than 

 791, the lesser segment of the diameter is greater than the line 

 represented by 125. 



3. The rectangle under this doubled segment and the radius, 

 is equal to the square of the side of the regular inscribed poly- 

 gon of twenty sides. But the product of 125 and 1280 is equal 

 to the square of 400 ; and if the radius be still 1280, it has been 

 proved that the doubled segment exceeds 125 ; with this repre- 

 sentation of the radius, the side of the inscribed polygon of twenty 

 sides exceeds therefore the line represented by 400; and the 

 perimeter of that polygon is consequently greater than 8000. 



4. Dividing then the numbers 1280 and 8000 by their 

 greatest common measure 320, we find that if the radius be 

 now represented by the number 4, or the diameter by 8, the 

 perimeter of the polygon will be greater than the line repre- 



* A sketch of the proof was published, at the request of a friend, in an 

 eminent literary journal last summer, but in a connexion not likely to 

 attract the attention of mathematical readers in general. At all events, it 

 pretends to no merit but that of brevity, and the simplicity of the principles 

 on which it rests. 



