1843.] Treatment of Geometry as a branch of Analysis. 123 



If ac = a'c' and B = B' ; A = A' (VI. 15). If again equal trian- 

 gles have each a pair of sides reciprocally proportional, or A = A' 

 and ac = a' c' ; then sin B = sin B', or the angles contained by those 

 sides are equal or supplementary. Also if B only = B' ; A : A = 

 ac : a'c'. This extended to parallelograms is (VI. 23), as (III. 15) 

 may be extended into (VI. 14). 



Again, since two rectangles are as their products ab : a' b', the 

 truth of (VI. 16 and 17) is evident. 



19. Considering the area P of a polygon in the light of a function 

 of sides and angles, we have 



p_ a -! #, by c A, B, C. ...... V or in a numerical form 



— = 6 J ~, ~, .... A, B, C I , a being taken as linear and a 2 



a 2 r [a a ) 



as superficial unit. Hence in all similar polygons P : P' = a 2 : a' 2 - 



(VI. 19, 20.) If further P" : P'" = a " 2 : a"' 2 and it be given a : a' = 



a" : a'" then ex cequali 



P : F = P" : P'" (VI. 22.) 



Likewise if P : P' : P" = a 2 : a' 2 : a" 2 and a 2 = a' 2 + a" 2 ; then 



P = P'+ F'...(VI. 31.) 



20. As we have treated areas, we might treat volumes. The right 

 solid being of three dimensions V = <j> (a, b t c). Increasing a jo-fold, b 

 9-fold and c r-fold V is increased pqr-fo\& and <j> («, b, c) is shewn 

 to be kabc. The solid unit then assumed is the cube on the linear 

 unit, and V = abc = altitude X base. Hence the right prism is also 

 altitude X triangular base. The oblique parallelopiped is also altitude 

 X base. By these principles we see at once the truth of (XI. 25 f 28, 

 29, 30, 31, 32, 33, D, 34, 40.) 



21. The examination of the circle is divided into the consideration 

 of angles, of chords, secants, and tangents (which have one general 

 analytical character,) and of areas as connected with the circle. 



Laying down the angle at the centre double of that at the circum- 

 ference on the same arc, as in Euclid, it will apply even if the former 

 be 7r or a reverse angle, (III. 20). One consequence is — all angles at 

 the circumference on the same arc are equal, (III. 21). Another, that 

 they will be J (tt — x), ^ tt> or i (tr -f x) as the angle at the centre 

 is less, =, or greater than 7r; (III. 31). Lastly, if an angle at centre 



