Chapter 20

Given the hypothesis of the acute angle, both Saccheri and Lambert showed that the sum of the angles of any triangle is less than two right angles. Let the difference between \(180^{\circ}\) and the angle sum of a triangle be the defect of the triangle. Suppose triangle \(A B C\) is split into two triangles by line \(B D\) (Fig. 20.16). Show that the defect of triangle \(A B C\) is equal to the sum of the defects of triangles \(A B D\) and \(B D C\).

Given that the angle sum of a triangle made of great circle arcs on a sphere (a spherical triangle) is greater than two right angles, define the excess of a triangle as the difference between its angle sum and \(180^{\circ} .\) Show that if a spherical triangle \(A B C\) is split into two triangles by an arc \(B D\) from vertex \(B\) to the opposite side, then the excess of triangle \(A B C\) is equal to the sum of the excesses of triangles \(A B D\) and \(B D C .\)

Given the relationships \(\cosh i x=\cos x\) and \(\sinh i x=\) \(i \sin x\), determine \(\cosh x\) and \(\sinh x\) in terms of the cosine and sine functions and show that \(\cosh ^{2} x-\sinh ^{2} x=1\).

Clairaut developed the method of finding the length of a space curve by the use of the integral calculus, namely, by integrating \(d s=\sqrt{d x^{2}+d y^{2}+d z^{2}} .\) Use this result to find the length of the curve given by the intersection of the cylinders \(a x=y^{2}\) and \((9 / 16) a z^{2}=y^{2}\), between the origin and the point \(\left(x_{0}, y_{0}\right)\).

Calculate the length of the perpendicular from a point \(P\) on the curve defined by \(a x=y^{2}, b y=z^{2}\) to the \(x z\) plane, where the perpendicular is also perpendicular to the plane defined by the tangent and subtangent to that curve.

Prove that the angle \(\theta\) between the plane \(\alpha x+\beta y+\gamma z=a\) and the \(x y\) plane is given by \(\cos \theta=\gamma / \sqrt{\alpha^{2}+\beta^{2}+\gamma^{2}}\) Determine the cosine of the angles this plane makes with the other two coordinate planes.

Show that Euler's result relating the curvature of any section of a surface made by a plane at angle \(\phi\) to the principal plane is equivalent to the modern formulation \(\kappa_{\phi}=\kappa_{1} \sin ^{2} \phi+\) \(\kappa_{2} \cos ^{2} \phi\)

Show that the plane \(z=\alpha y-\beta x+\gamma\) is perpendicular to the surface \(z=f(x, y)\) if \(\beta \frac{\partial z}{\partial x}-\alpha \frac{\partial z}{\partial y}=1\). (Show that the plane contains a normal line to the surface.)

Find the normal line to the plane \(A x+B y+C z+D=0\) that passes through the point \(\left(x_{0}, y_{0}, z_{0}\right)\).

Convert Monge's form of the equations of the normal line to the surface \(z=f(x, y)\) into the modern vector equation of the line.

Show that an "Euler path" over a series of bridges connecting certain regions (a path that crosses each bridge exactly once) is always possible if there are either two or no regions that are approached by an odd number of bridges.

Find the numbers of vertices, edges, and faces for each of the five regular polyhedra and confirm that Euler's formula holds in these five cases.