772 PROCKKDINfJH OF TIIK AMKKK'AN A(MI)KMY. 



aroa of any cloHod ]t()\y^oi\ &h the Hum of the areas of the triaiig](w into 

 which it can h(5 diviihid tlio area of any [joly^'on will he its i)erinieter. 



32. Similarly w(! (hilino th(! atif/lc air/i of a trian^de a,s a .scalar (|iian- 

 tity (leturminod hy throe lines taken in a dclinite onUir and Huch that 

 triangle.H having,' the Hame vertex and hases on tlio Hame line haveareaH 

 proportional to the angles at the vertex. Hy a pnjjier ehoiee of units 

 the angle area takes the lurm 



Area ahc = A -\- B -\- C. (19) 



Defining the angle area in general aw liaving tlio sum pro])erty we see 

 that any cloHed jxjlygon has an angle area equal to the Hum of the angles 

 betw(!en c(jnHecutive sifleH. 



33. 'I'he formulae for Ihc two areaH may he written in other forms that 

 hIiovv more chvuly the analogy with the ordinary trigonometric formulae 

 for area. 'J'h uh I'rom the equation 



. __ a + h -\-c 



^- Fc 



wo have 



Area ABC = a -\- + c = bcA 



analogouH to the formula 



2 Area ABC = ho sin A 



in trigonometry. Our foruiula thuH corroHponds to tho formula for 

 twice the ordinary area.^ 

 Again from the equation 



_ A -^ n + o 



"*" JJC 

 wo have 



Area abc = A + /i -^ (.' = aliO 



which is to be comi)ared with tho ordinary formula 



2 Area abc = « sin B sin C 



In both of thoHO i)airH of formulae for area angle replaces tho sine of an 

 angle of trig(jnometry. 



• Ah (he prcMcnt, Hcheme of HiHianrc and anKle in entirely diHtJncI from tho 

 ordinary diHliiiKu; and un)^l<', llif one Ixinj^ lincur, I he oilier (luiulriil ie, il. (Idch 

 not Hoem udviHuble to complicate! our formuhm hy the introduotJon of tho 

 multi|)lier \ neocHHary to make t}ie analogy eoinplctc 'I'he only c-aHe eoinrnon 

 to I he. two HyHtcniH in that in wliicli Mm' (^rcular j)oiuLH uL infinity coincide, 

 which corrcHpond.s t,o our Hchemu wh(!n 1<' iH <jn f. 



