'iSSDemoyistrationqf the greatest Area of IsoperimetricalPolygonS' 



or forest, without having our feelings outraged, or our sym- 

 pathies excited, though we cannot bear either to witness or 

 to contemplate the pains and sorrows of suffering animals but 

 with the tear of sympathy, or the cry of commiseration. 



From the above premises, it follows irresistibly that plants, 

 though exhibiting indubitable indications of such tissual and 

 organic susceptibilities as are proper to their rank in the scale 

 of life, do not after all exhibit any satisfactory indications of 

 the sensible or voluntary susceptibilities of animals, and do 

 not in fact either feel, or will, or desire, or design. The move- 

 ments which they display are singular indeed and surprising ; 

 but they are not such as evince any manifest token whether 

 of pleasure or of pain, or any legitimate evidence whether of 

 sensation or of intellection. These are attributes that reside 

 only in animals, and reach their highest degree only in man. 



Ruckint;e Rectory, Kent, P. Keith. 



June 25, 1831, 



XXXV. Demonstration of the greatest Area of Isoperimetri- 



cal Polygons. By A Correspondent*. 

 'T'HE well-known theorem, that of polygons of the same 

 -*- perimeter, that which has the greatest number of sides 

 has the greatest area, is a proposition of considerable import- 

 ance in reference to several investigations of geometry. But 

 though it has been demonstrated by several methods ; yet all 

 the proofs which have fallen in the way of the writer ot these 

 lines appear to him lengthy and tedious, as well as deficient 

 in directness and simplicity. He is therefore induced to offer 

 the following, which has occurred to him, grounded on a very 

 elementary expression for the area, and depending only on the 

 simplest consideration of ratios, and which he thinks will be 

 found to combine exactness with brevity. 



If in any polygon b = base n side, n = number of sides, 

 and a = area; then by a well-known formula (See Hind's 

 Trigonometry, p. 132.) we have 



nb y "K 



a = — — - b cotang — . 



If in two polygons the perimeter be the same, we have 



nb = n^b^ 



b 71, 



or, -y— = — 



(where n^ > ti and .". 6, < b) 



* Communicated by the Author . 



But 



