647 
of Edinburgh, Session 1879 - 80 . 
Also Euclid I. 21. 
Euclid I. 24 and 25. 
Euclid I. 26 (the second part). 
Also the usual propositions concerning the perpendicular and the 
obliques drawn from a given point to a given straight line. 
The amount by which the sum of the three angles of a triangle 
falls short of 2R is called the defect of the triangle. This is the same 
as the excess of the sum of its exterior angles over 4R. If we take 
the latter statement of the definition, we may talk of the defect of any 
plane rectilineal figure. In forming the external angles of figures 
generally, we must go round, producing all the sides in the direction 
of our progress, assigning the positive or negative sign according as 
the angle is not or is re-entrant. 
Thus in figure 2 the defect is 
a + /3 — y + 8 + e — 4R . 
Defining defect in this way, it is easy to prove that 
The defect of any rectilineal figure is equal to the sum of the defects 
of any rectilineal figures of which it may he supposed to he composed. 
Cor. Hence if one rectilineal figure lie wholly within another the 
defect of the former is not greater than that of the latter. 
Hence follows at once the following important proposition :■ — 
If the defect of any triangle whose sides are finite he zero, then the 
defect of every finite triangle must he zero. 
For if ABC (fig. 3) he a triangle whose defect is zero, then, 
by applying to its sides three triangles, each congruent with itself, 
as shown in the figure, we evidently construct a triangle A'B'C', 
having the same angles as ABC, and hence zero defect, each of 
whose sides is double a corresponding side in ABC. We may repeat 
this process with A'B'C', and so on. Hence we may construct a 
triangle, having zero defect, large enough to contain within it any 
finite triangle whatever. But the defect of any triangle cannot be 
greater than that of a triangle within which it is contained, and the 
defect cannot he less than zero; hence the defect of every finite 
triangle must he zero, if the defect of any one finite triangle he zero. 
Thus in hyperbolic space , as defined above, we are shut up to one 
or other of two alternatives. Either the defect of a triangle is 
always positive or it is always zero. 
4 H 
VOL. X. 
