570 
GEOMETRY. 
[790 
Metrical Theory. 
30. The foundation in solid geometry of the metrical theory is, in fact, the before- 
mentioned theorem that, if a finite right line PQ be projected upon any other line 
00' by lines perpendicular to 00', then the length of the projection P'Q' is equal to the 
length of PQ multiplied by the cosine of its inclination to P'Q'—or (in the form in 
which it is now convenient to state the theorem) the perpendicular distance P'Q' of two 
parallel planes is equal to the inclined distance PQ into the cosine of the inclination. 
Hence also the algebraical sum of the projections of the sides of a closed polygon 
upon any line is = 0; or, reversing the signs of certain sides and considering the 
polygon as made up of two broken lines each extending from the same initial to the 
same terminal point, the sum of the projections of the one set of lines upon any 
line is equal to the sum of the projections of the other set of lines upon the same 
line. When any of the lines are at right angles to the given line (or, what is the 
same thing, in a plane at right angles to the given line), the projections of these 
lines severally vanish. 
31. Consider the skew quadrilateral QMNP, the sides QM, MN, NP being 
respectively parallel to the three rectangular axes Ox, Oy, 0z\ let the lengths of these 
sides be £, p, £, and that of the side QP be = p; and let the cosines of the inclin 
ations (or say the cosine-inclinations) of p to the three axes be a, /3, 7; then pro 
jecting successively on the three sides and on QP, we have 
V, P/5, py> 
and 
P = «£ + &V + y£ 
whence p 2 = | 2 + rf 4- £ 2 , which is the relation between a distance p and its projections 
£, p, £ upon three rectangular axes. And from the same equations we obtain a 2 + /3 2 + 7 2 = 1, 
which is a relation connecting the cosine-inclinations of a line to three rectangular axes. 
Suppose we have through Q any other line QT, and let the cosine-inclinations of 
this to the axes be ct, /3', 7', and 8 be its cosine-inclination to QP; also let p be the 
length of the projection of QP upon QT; then projecting on QT, we have 
p — a'% + /3'p + y'%, = p8. 
And in the last equation substituting for £, p, £ their values pa, p/3, py, we find 
8 = act' + /3/3' + yy', 
which is an expression for the mutual cosine-inclination of two lines, the cosine- 
inclinations of which to the axes are a, /3, 7 and a', /3', y respectively. We have of 
course a 2 + /3 2 + 7 2 = 1, and a' 2 + /3' 2 + 7 /2 = 1, and hence also 
1 - S 2 = (a 2 + /3 2 + 7 2 ) (a' 2 + /3' 2 + 7 ' 2 ) - (aa' + /3/3' + yyj 
= - /3'y) 2 + (y^ - Y a ) 2 + («£' - a'/3) 2 ; 
so that the sine of the inclination can only be expressed as a square root. These 
formulae are the foundation of spherical trigonometry.