WILSON AND LEWIS. — RELATIVITY 



397 



Figure 4. 



VII. The diagonal of a parallelogram divides it into two equal 

 areas. 



For if the sides of the parallel- 

 ogram be divided by repeated bi- 

 section into 2 " parts, there will 

 be an equal number of equal 

 parallelograms on each side of 

 the diagonal (Figure 4), and in 

 the limit the total area of these 

 parallelograms approaches the 

 area of the triangles. 



VIII. If from any point in 



the diagonal of a parallelogram lines be drawn parallel to the sides, 



the two parallelograms formed on either side of 

 the diagonal are equal in area (Figure 5). 



IX. Two parallelograms between the same 

 parallel lines and with congruent bases are equal 

 in area. 



Cor. Two triangles having congruent bases on 

 one line and vertices on a parallel line have equal areas. 



Cor. The diagonals di\ide a parallelogram into four equal triangu- 

 lar areas. 



Proofs may be given by obvious and familiar methods. 

 X. Of all parallelograms having two sides coinmon to two sides of 

 a given triangle and a vertex on the third side of the triangle, that one 

 has the greatest area whose vertex bisects that third side. 



For in the figure (Figure 6), where ABC is the triangle and E is the 

 middle point of the third side, the difference of the two parallelograms 

 is 



IIBFE — IBGD = MGFE — IHMD = KM EL — I HMD 



= KMEL — KDXL = DMEX. 

 Propositions IV and VIII are used in the proof. 



Figure 5. 



dxdij 



dx'dij.' 



the value of the area, in terms of the urea measured w ith icfcrenee to tlie new 

 axes, is 



a h 



a b 



Hence if the measure of area is to be the same, that is, if the unit parallelogram 

 on the new axes is to have a unit area referred to the old a.xes, the determinant 

 of the transformation must be unity. This implies a relation between the 

 choice of unit intervals on the new axes. Indeed when the unit interval on 

 one of the new axes has been arbitrarily chosen, the unit interval on the other 

 is determined. In other words the unit intervals on the new axes must each 

 varv inversclv as the other. 



