178 PROCEEDINGS OF THE [Sept, 1887. 



Note on the usual methods of demonstrating the arithmetical rule 

 for finding The Area of a Triangle when the three sides are given. 



BY PROF. LEWIS K. GIBBE9. 



The well known arithmetical rule for the solution of this problem is as fol- 

 lows : Find the sum of the'three given sides, and take its half ; from this half 

 sum take each of the mdes successively, obtaining three remainders ; take these 

 three remainders and the half sum as four factors, and find their continued 

 product ; take the square root of this product, and the result will be the area 

 of the given triangle, in square units of that kind used in expressing the 

 lengths of the sides. 



One hardly seems warranted in bringing forward for discussion a subject so 

 trite as this, presented in every text book on mensuration and surveying, and 

 even in many on arithmetic; but in none of the text books on geometry known 

 to us, at present in use, can we find a geometrical demonstration of the theorem 

 on which this rule is based. It is usually demonstrated mialytically, either in 

 the body of the work or in an appendix, by the application of algebra to some 

 of the geometrical theorems in the text, or else by the use of some trigonomet- 

 rical theorems, given later in the course of instruction. It would seem but 

 proper that a geometrical theorem should have a geometrical demonstration, 

 and at least two such were to be found in elementary treatises pubhshed 70 

 years ago ; one of these was given in the appendix to the American edition of 

 Lsgendre's Oeomeiry, in use 50 years since, but has been omitted in the la- 

 ter editions. Perhaps the complex form in which they originally appeared 

 seemed too difficult for use in an elementary work on geometry, like that of 

 Legendre or the many works based upon it, and the appearance, in the last 

 mentioned one, of the ungeometrical phrase, " the square of the area, " may 

 have deterred editors from continuing the use of that demonstration. We 

 shall endeavour to obviate these difficulties, by using the other, and breaking it 

 up into several propositions, the demonstration of each being simple enough 

 to form an easy step toward the required final one. We can lay no claim to 

 novelty, nor even to originality, in the method employed in the following de- 

 monstrations, but only desire to facilitate the apprehension of the closing prop- 

 osition, and so promote the introduction into geometrical treatises, of this geo- 

 metrical demonstration of an important geometrical theorem. 



1. Let ABC be the given triangle ; draw AK, bisecting the angle A, and BD 

 bisecting the angle B ; from D, the intersection of these two bisectrices, draw 

 the three perpendiculars, DE, DH, DF, one to each side thus dividing each side 

 into two segments ; lastly draw CD. 



2. Next, prolong the sides AB, AC, the first forming an external angle at B, 

 the second another external angle at C ; draw the line BK, bisecting the exter- 

 nal angle at B, and from K, the point where this bisectrix intersects the bisect- 

 rix AK, draw the three perpendiculars KI, KL, KM, one to each side ; thus will 



