The Sphere

Excerpts

• The Sphere

  The spherical degree differs fundamentally from the units of measure hitherto used. This unit is a certain fraction of the surface of the sphere and hence its actual size depends upon the size of the sphere.

• The Sphere

  A plane tangent to a sphere is perpendicular to the radius from the point of tangency; and conversely, a plane perpendicular to a radius at its extremity is tangent to the sphere.

• The Sphere

  The shortest distance on a sphere between two of its points is measured along the minor arc of a great circle passing through these points.

• The Sphere

  It follows from the preceding theorem and corollaries that, if a *spherical* blackboard is at hand, circles may be constructed on it by means of crayon and string the same as on a *plane* blackboard. Likewise, curve-legged compasses may be used.

• The Sphere

  $PB$ and $BD$ being known, we may now *compute* $PD$ from the right triangle $PBD$, and then compute $PP'$ from the similar triangles $PBD$ and $PP'B$, for the latter using the relation $PD : PB = PB : PP'$ or $PD × PP' = \overline{PB}^2$.

• The Sphere

  The number of degrees by which the sum of the angles of a spherical triangle exceeds $180$° is called the *spherical excess* of the triangle.

• The Sphere

  A sphere has a definite area which is less than the surface of any circumscribed figure and greater than the surface of any inscribed convex figure

• The Sphere

  The student should note that while the statement just preceding is obviously true, it is not capable of proof by pure deduction.

• The Sphere

  The triangle $A'B'C'$ as thus described is *the polar triangle* of $ABC$.

• The Sphere

  This theorem was discovered by Cavalieri.

• The Sphere

  The area of a sphere whose radius is $r$ is $4\pi r^2$.

• The Sphere

  The volume of a sphere whose radius is $r$ is $\tfrac{4}{3} \pi r^3$.

• The Sphere

  The sphere is covered with a network of spherical quadrilaterals. If these are taken small enough, they may be regarded as approximately *plane surfaces*.

• The Sphere

  Hence, their combined volume is $\frac{1}{3} r × (\text{area of sphere})$ or $\tfrac{1}{3} r \cdot 4\pi r^2$. That is, the volume is $\tfrac{4}{3} \pi r^3$.

• The Sphere

  This is of course obvious at a glance, though a formal deductive proof is very difficult.

• The Sphere

  The total area of the sphere is $4\pi × 6^2 = 452.3904$ sq. in., and one spherical degree is $\tfrac{1}{720}$ of $452.3904 = .62832$ sq. in.

• The Sphere

  He was one of three commissioners who introduced the metric system in France, having also been a member of the commission for determining the length of the meter.

• The Sphere

  The portion of a sphere included between two parallel planes cutting it is called a *spherical segment*, and the two circular sections made by the parallel planes are its *bases*.

• The Sphere

  The perpendicular distance between the planes is the *altitude* of the zone and of the corresponding segment.

Equations

• The Sphere

  PD : PB = PB : PP'

  On the sphere, the ratio of PD to PB equals the ratio of PB to PP', a relation the book uses to find the diameter from a point P on a circle and its pole-to-point distance.

• The Sphere

  PD × PP' = \overline{PB}^2

  The product of PD and the sphere's diameter PP' equals the square of the segment PB, which lets the diameter be computed from measured segments.

• The Sphere

  \wideparen{AD} + \wideparen{DC} + \wideparen{CB} > \wideparen{AB}

  A path made of great-circle arcs from A through D and C to B on a sphere is longer than the minor great-circle arc AB, so the shortest distance between two points is the arc of a great circle.

• The Sphere

  A + a' &= 180\text{°}

  An angle of a spherical triangle and the corresponding side of its polar triangle together make 180 degrees.

• The Sphere

  \angle A + \angle B + \angle C + a' + b' + c' = 6 \text{ rt.\ } \Angles

  The angles of a spherical triangle plus the sides of its polar triangle together total six right angles.

• The Sphere

  \angle A + \angle B + \angle C < 6 \text{ rt.\ } \Angles

  The sum of the angles of a spherical triangle is less than six right angles.

• The Sphere

  \angle A + \angle B + \angle C > 2 \text{ rt.\ } \Angles

  The sum of the angles of a spherical triangle is greater than two right angles.

• The Sphere

  \text{area } \triangle ABC = \text{area } \triangle A_1B_1C_1

  Two symmetrical spherical triangles have equal areas.

• The Sphere

  \triangle ABC = \angle A + \angle B + \angle C - 180

  The area of a spherical triangle, in spherical degrees, equals its spherical excess: the sum of its angles minus 180 degrees.

• The Sphere

  2\pi r × AB = 2\pi r × 2r = 4\pi r^2

  The area of a sphere of radius r is 4 pi r squared, found as the limit of circumscribed surfaces with AB = 2r.

• The Sphere

  \tfrac{4}{3} \pi r^3

  The volume of a sphere of radius r is four-thirds pi times r cubed.

• The Sphere

  4\pi r^2

  The surface of a sphere of radius r has area four pi r squared.

• The Sphere

  s = 2\pi rh

  The area of a zone is two pi times the radius of the sphere times the altitude of the zone.

• The Sphere

  v = \dfrac{r}{3} \cdot s

  The volume of a spherical cone is one third the radius times the area of the zone cut out of the sphere by the cone.

• The Sphere

  v = \dfrac{r}{3} \cdot 2\pi rh = \dfrac{2\pi}{3} r^2h

  Substituting the zone area into the spherical cone formula gives volume two pi r squared h over three.

• The Sphere

  v = \dfrac{2\pi}{3} r^2h

  The volume of a spherical sector is two pi over three times the square of the radius times the altitude of its zone.

• The Sphere

  v = \dfrac{\pi h}{2} (r_1^2 + r_2^2) + \dfrac{\pi}{6} h^3

  The volume of a spherical segment is pi h over two times the sum of the squares of the base radii, plus pi h cubed over six.

• The Sphere

  v = \pi h^2 \left( r - \dfrac{h}{3} \right)

  The volume of a spherical segment of one base is pi h squared times the quantity r minus h over three.

• The Sphere

  d &= \dfrac{r_2^2 - r_1^2 - h^2}{2h}

  The distance d from the center of the sphere to the plane of the smaller base is expressed in terms of the base radii and the altitude h.

• The Sphere

  r^2 = \dfrac{r_2^4 + r_1^4 + h^4 - 2r_1^2r_2^2 + 2h^2r_2^2 + 2h^2r_1^2}{4h^2}

  The squared sphere radius r is written in terms of the base radii r_1, r_2 and the altitude h of the segment.

• The Sphere

  r^2 = r_2^2 + d^2

  By the Pythagorean theorem, the squared sphere radius equals the squared base radius r_2 plus the squared distance d.

• The Sphere

  r^2 = r_1^2 + (h+d)^2

  By the Pythagorean theorem, the squared sphere radius equals the squared base radius r_1 plus the squared distance h plus d.

• The Sphere

  V = abc

  The volume of a rectangular parallelepiped with dimensions a, b, c is the product abc.

• The Sphere

  V = hb

  The volume of a prism or cylinder is its base area times its altitude.

• The Sphere

  S = pe

  The lateral surface of a prism or cylinder is the perimeter of a right section times the lateral edge or element.

• This equation is in Pyramids and Cones (Pyramids and Cones)

• The Sphere

  S = \tfrac{1}{2} pl

  The lateral area of a regular pyramid or cone is one half the base perimeter times the slant height.

• The Sphere

  V = \tfrac{1}{3} h (b + b' + \sqrt{bb'})

  The volume of a frustum of a pyramid or cone is one third its altitude times the sum of the two base areas plus the square root of their product.

• The Sphere

  S = \dfrac{a}{720} \cdot 4 \pi r^2

  The area of a spherical polygon is its spherical excess in degrees divided by 720 times the total area of the sphere.

• This equation is in Prisms and Cylinders (Prisms and Cylinders)

• The Sphere

  V = \dfrac{2\pi}{3}r^2h

  The volume of a spherical cone is two pi over three times the square of the sphere's radius times the altitude of the cone's zone.

• The Sphere

  S = 4 \pi r^2

  The surface of a sphere of radius r is four pi r squared.

Problems

No exercises in this chapter.