Arcs drawn to fixed points on the Surface of a Sphere

Excerpts

• Arcs drawn to fixed points on the Surface of a Sphere

  Thus, whatever may be the position of $T$, the sum of the cosines of the arcs which join $T$ to the fixed points varies as the cosine of the single arc which joins $T$ to a certain fixed point $U$.

• Arcs drawn to fixed points on the Surface of a Sphere

  A sphere is described about a regular polyhedron; from any point on the surface of the sphere arcs are drawn to the solid angles of the polyhedron: to shew that the sum of the cosines of these arcs is zero.

• Arcs drawn to fixed points on the Surface of a Sphere

  *Thus the sum of the squares of the cosines of the arcs which join any point on the surface of the sphere to the solid angles of the regular polyhedron is one third of the number of the solid angles.*

• Arcs drawn to fixed points on the Surface of a Sphere

  We leave to the student the exercise of shewing that the formul of the two preceding Articles are perfectly general for all positions of $T$ and $U$, outside or inside the triangle $ABC$: the demonstrations will remain essentially the same for all modifications of the diagrams.

Equations

• Arcs drawn to fixed points on the Surface of a Sphere

  \cos^2{TA}+\cos^2{TB}+\cos^2{TC}=1

  For any point T on the sphere, the squares of the cosines of the arcs from T to the three vertices of the quadrantal triangle ABC sum to 1.

• Arcs drawn to fixed points on the Surface of a Sphere

  \cos TU = \cos TA \cos UA + \cos TB \cos UB + \cos TC \cos UC

  The cosine of the arc TU equals the sum of the products of the cosines of the arcs from T and from U to each vertex of the quadrantal triangle ABC.

• Arcs drawn to fixed points on the Surface of a Sphere

  \Sigma = \cos TH_1 + \cos TH_2 + \cos TH_3 + \ldots

  Sigma is defined as the sum of the cosines of the arcs joining T to the fixed points H1, H2, H3, and so on.

• Arcs drawn to fixed points on the Surface of a Sphere

  G=\surd{(P^2+Q^2+R^2)}

  G is defined as the square root of the sum of the squares of P, Q and R.

• Arcs drawn to fixed points on the Surface of a Sphere

  \cos \alpha = \frac{P}{G}

  The arc alpha is defined by the cosine equal to P divided by G (the source line, as transcribed, omits the equals sign before Q/G and R/G in the same display).

• Arcs drawn to fixed points on the Surface of a Sphere

  \cos^2\alpha +\cos^2\beta+\cos^2\gamma = 1

  The squares of the cosines of the three arcs alpha, beta, gamma sum to 1.

• Arcs drawn to fixed points on the Surface of a Sphere

  \Sigma=G \cos TU

  Sigma, the sum of the cosines of arcs from T to the fixed points, varies as the cosine of the arc from T to a fixed point U, with factor G.

• Arcs drawn to fixed points on the Surface of a Sphere

  \cos TU= \lambda \cos \alpha + \mu \cos \beta + \nu \cos \gamma

  The cosine of the arc TU equals the sum of the products of the cosines of T's arcs to A, B, C with the cosines of alpha, beta, gamma.

• Arcs drawn to fixed points on the Surface of a Sphere

  G = 0

  For a regular polyhedron inscribed in a sphere, or a rectangular parallelepiped inscribed in a sphere, G must be zero, since symmetry gives more than one position of T with the greatest value of Sigma.

• Arcs drawn to fixed points on the Surface of a Sphere

  \Sigma = \cos^2 TH_1 + \cos^2 TH_2 + \cos^2 TH_3 + \dots

  Sigma is defined as the sum of the squares of the cosines of the arcs joining T with the fixed points.

• Arcs drawn to fixed points on the Surface of a Sphere

  \Sigma = P\lambda^2+Q\mu^2+R\nu^2+2p\mu\nu+2q\nu\lambda+2r\lambda \mu

  The sum of the squares of the cosines to the fixed points is a quadratic form in lambda, mu, nu, with cross-coefficients p, q, r.

• Arcs drawn to fixed points on the Surface of a Sphere

  \Sigma=P\lambda^2+Q\mu^2+R\nu^2

  With the triangle ABC placed so that p, q, r vanish, the sum of the squares of the cosines equals P lambda^2 + Q mu^2 + R nu^2.

• Arcs drawn to fixed points on the Surface of a Sphere

  0 = 2p\mu\nu + 2q\nu\lambda + 2r\lambda\mu

  When P=Q=R, the cross-term relation forces the cross-coefficients p, q, r each to vanish for every position of T.

• Arcs drawn to fixed points on the Surface of a Sphere

  \lambda\lambda'P + \mu\mu'Q + \nu\nu'R

  The sum of the products of the corresponding cosines to two points T and U equals lambda lambda' P + mu mu' Q + nu nu' R when p, q, r vanish.

• Arcs drawn to fixed points on the Surface of a Sphere

  P = Q = R = \dfrac{S}{3}

  For a regular polyhedron inscribed in a sphere, P, Q and R are each one third of S, the number of solid angles of the polyhedron.

• Arcs drawn to fixed points on the Surface of a Sphere

  \dfrac{S}{3} (\lambda\lambda' + \mu\mu' + \nu\nu') = \dfrac{S}{3} \cos TU

  For a regular polyhedron inscribed in a sphere, the sum of products of corresponding cosines equals one third of the number of solid angles times cos TU.

Problems

No exercises in this chapter.