30. Metrical Theory.—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 OO′ by lines perpendicular to OO′, then the length of the projection P′Q′ is equal to the length of PQ into 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. The principle of § 16, that the algebraical sum of the projections of the sides of any closed polygon on any line is zero, or that the two sets of sides of the polygon which connect a vertex A and a vertex B have the same sum of projections on the line, in sign and magnitude, as we pass from A to B, is applicable when the sides do not all lie in one plane.
31. Consider the skew quadrilateral QMNP, the sides QM, MN, NP being respectively parallel to the three rectangular axes Ox, Oy, Oz; let the lengths of these sides be ξ, η, ζ, and that of the side QP be = ρ; and let the cosines of the inclinations (or say the cosine-inclinations) of ρ to the three axes be α, β, γ; then projecting successively on the three sides and on QP we have
ξ, η, ζ = ρα, ρβ, ργ,
and
ρ = αξ + βη + γζ,
whence ρ² = ξ² + η² + ζ², which is the relation between a distance ρ and its projections ξ, η, ζ upon three rectangular axes. And from the same equations we obtain α² + β² + γ² = 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 α′, β′, γ′, and δ be its cosine-inclination to QP; also let ρ be the length of the projection of QP upon QT; then projecting on QT we have
ρ = α′ξ + β′η + γ′ζ = ρδ.
And in the last equation substituting for ξ, η, ζ their values ρα, ρβ, ργ we find
δ = αα′ + ββ′ + γγ′,