J 843.] Treatment of Geometry as a branch of Analysis. 121 



15. With regard to the linear properties of parallels. If a straight 

 line cut the sides of a triangle or these produced, parallel to the base, 

 a triangle is formed of the same species, and hence the sides are divid- 

 ed proportionally. The converse is similarly true (VI. 2). The base 

 of the new triangle will also bear the same proportion to that of the 

 primitive. 



If now the base angles of the primitive triangle increase, so that the 

 sides approach parallelism, the sides of the two triangles increase with- 

 out limit, approaching equality as they do so, without limit. Hence 

 when the sides do become parallel, the ratio is one of equality, and the 

 frustrum of the triangle having become a parallelogram, it follows that 

 the opposite sides of a parallelogram are equal (I. 34). If the paral- 

 lelogram be rectangular, each pair of sides will be the distances between 

 the other pair, hence parallels are equidistant. 



The two very elegant propositions (VI. 3, A,) are fragments of an 

 entire series relating to the segments of sides by lines drawn from the 

 opposite angles. It is not the intention of this paper to touch on 

 supplemental trains of inquiry, but only to sketch those on which the 

 rest may be scaffolded with ease. The propositions in question may, 

 however, be simply proved thus : If a line be drawn from A to a and 

 making with c an angle called 0, the segment on a between this line 



and B is m e ™° . & that between the line & C is 6 ^ ( f R H * >- 

 sin (B + 0) sln ( B + 0) 



Their ratio is consequently always c sin 0: b sin (A r* 0) 9 and 

 will be reduced to that of c : b, when sin = sin (A H 0). If the 

 cutting line fall within the triangle, this gives = A — or^ = !A; 

 (VI. 3). If without, = it — (0 — A) or 9 =L (tt + A); 

 (VI. A). 



16. The area of any plane figure is a function of its sides and angles. 

 But the sides can be projected on two rectangular axes by help of 

 what precedes, hence the area is also determinable by means of these 

 projections and the angles. The simplest area to consider is that of 

 the rectangle, because if the origin be at one of its angles and the includ- 

 ing sides be the axes, they are also the projections of the others. The 

 angles are besides equal, and natural constants. Let the ratio of the 

 sides to the linear unit be a and b, and that of the area to the super- 

 ficial unit be A, then A = <p (a, b). Inspection and our previous 



