Chapter XXI The Leading Propositions of Book VIII
Book VIII treats of the sphere. Just as the circle may be defined either as a plane surface or as the bounding line which is the locus of a point in a plane at a given distance from a fixed point, so a sphere may be defined either as a solid or as the bounding surface which is the locus of a point in space at a given distance from a fixed point. In higher mathematics the circle is defined as the bounding line and the sphere as the bounding surface; that is, each is defined as a locus. This view of the circle as a line is becoming quite general in elementary geometry, it being the desire that students may not have to change definitions in passing from elementary to higher mathematics. The sphere is less frequently looked upon in geometry as a surface, and in popular usage it is always taken as a solid.
Analogous to the postulate that a circle may be described with any given point as a center and any given line as a radius, is the postulate for constructing a sphere with any given center and any given radius. This postulate is not so essential, however, as the one about the circle, because we are not so concerned with constructions here as we are in plane geometry.
A good opportunity is offered for illustrating several of the definitions connected with the study of the sphere, such as great circle, axis, small circle, and pole, by referring to geography. Indeed, the first three propositions usually given in Book VIII have a direct bearing upon the study of the earth.
Theorem. A plane perpendicular to a radius at its extremity is tangent to the sphere.
The student should always have his attention called to the analogue in plane geometry, where there is one. If here we pass a plane through the radius in question, the figure formed on the plane will be that of a line tangent to a circle. If we revolve this about the line of the radius in question, as an axis, the circle will generate the sphere again, and the tangent line will generate the tangent plane.
Theorem. A sphere may be inscribed in any given tetrahedron.
Here again we may form a corresponding proposition of plane geometry by passing a plane through any three points of contact of the sphere and the tetrahedron. We shall then form the figure of a circle inscribed in a triangle. And just as in the case of the triangle we may have escribed circles by producing the sides, so in the case of the tetrahedron we may have escribed spheres by producing the planes indefinitely and proceeding in the same way as for the inscribed sphere. The figure is difficult to draw, but it is not difficult to understand, particularly if we construct the tetrahedron out of pasteboard.
Theorem. A sphere may be circumscribed about any given tetrahedron.
By producing one of the faces indefinitely it will cut the sphere in a circle, and the resulting figure, on the plane, will be that of the analogous proposition of plane geometry, the circle circumscribed about a triangle. It is easily proved from the proposition that the four perpendiculars erected at the centers of the faces of a tetrahedron meet in a point (are concurrent), the analogue of the proposition about the perpendicular bisectors of the sides of a triangle.
Theorem. The intersection of two spherical surfaces is a circle whose plane is perpendicular to the line joining the centers of the surfaces and whose center is in that line.
The figure suggests the case of two circles in plane geometry. In the case of two circles that do not intersect or touch, one not being within the other, there are four common tangents. If the circles touch, two close up into one. If one circle is wholly within the other, this last tangent disappears. The same thing exists in relation to two spheres, and the analogous cases are formed by revolving the circles and tangents about the line through their centers.
In plane geometry it is easily proved that if two circles intersect, the tangents from any point on their common chord produced are equal. For if the common chord is AB and the point P is taken on AB produced, then the square on any tangent from P is equal to PB × PA. The line PBA is sometimes called the radical axis.
Similarly in this proposition concerning spheres, if from any point in the plane of the circle formed by the intersection of the two spherical surfaces lines are drawn tangent to either sphere, these tangents are equal. For it is easily proved that all tangents to the same sphere from an external point are equal, and it can be proved as in plane geometry that two tangents to the two spheres are equal.
Among the interesting analogies between plane and solid geometry is the one relating to the four common tangents to two circles. If the figure be revolved about the line of centers, the circles generate spheres and the tangents generate conical surfaces. To study this case for various sizes and positions of the two spheres is one of the most interesting generalizations of solid geometry.
An application of the proposition is seen in the case of an eclipse, where the sphere O’ represents the moon, O the earth, and S the sun. It is also seen in the case of the full moon, when S is on the other side of the earth. In this case the part MIN is fully illuminated by the moon, but the zone ABNM is only partly illuminated, as the figure shows.92
Theorem. The sum of the sides of a spherical polygon is less than 360°.
In all such cases the relation to the polyhedral angle should be made clear. This is done in the proofs usually given in the textbooks. It is easily seen that this is true only with the limitation set forth in most textbooks, that the spherical polygons considered are convex. Thus we might have a spherical triangle that is concave, with its base 359°, and its other two sides each 90°, the sum of the sides being 539°.
Theorem. The sum of the angles of a spherical triangle is greater than 180° and less than 540°.
It is for the purpose of proving this important fact that polar triangles are introduced. This proposition shows the relation of the spherical to the plane triangle. If our planes were in reality slightly curved, being small portions of enormous spherical surfaces, then the sum of the angles of a triangle would not be exactly 180°, but would exceed 180° by some amount depending on the curvature of the surface. Just as a being may be imagined as having only two dimensions, and living always on a plane surface (in a space of two dimensions), and having no conception of a space of three dimensions, so we may think of ourselves as living in a space of three dimensions but surrounded by a space of four dimensions. The flat being could not point to a third dimension because he could not get out of his plane, and we cannot point to the fourth dimension because we cannot get out of our space. Now what the flat being thinks is his plane may be the surface of an enormous sphere in our three dimensions; in other words, the space he lives in may curve through some higher space without his being conscious of it. So our space may also curve through some higher space without our being conscious of it. If our planes have really some curvature, then the sum of the angles of our triangles has a slight excess over 180°. All this is mere speculation, but it may interest some student to know that the idea of fourth and higher dimensions enters largely into mathematical investigation to-day.
Theorem. Two symmetric spherical triangles are equivalent.
While it is not a subject that has any place in a school, save perhaps for incidental conversation with some group of enthusiastic students, it may interest the teacher to consider this proposition in connection with the fourth dimension just mentioned. Consider these triangles, where ∠A = ∠A’, AB = A’B’, AC = A’C’. We prove them congruent by superposition, turning one over and placing it upon the other. But suppose we were beings in Flatland, beings with only two dimensions and without the power to point in any direction except in the plane we lived in. We should then be unable to turn ⧍A’B’C’ over so that it could coincide with ⧍ABC, and we should have to prove these triangles equivalent in some other way, probably by dividing them into isosceles triangles that could be superposed.
Now it is the same thing with symmetric spherical triangles; we cannot superpose them. But might it not be possible to do so if we could turn them through the fourth dimension exactly as we turn the Flatlander’s triangle through our third dimension? It is interesting to think about this possibility even though we carry it no further, and in these side lights on mathematics lies much of the fascination of the subject.
Theorem. The shortest line that can be drawn on the surface of a sphere between two points is the minor arc of a great circle joining the two points.
It is always interesting to a class to apply this practically. By taking a terrestrial globe and drawing a great circle between the southern point of Ireland and New York City, we represent the shortest route for ships crossing to England. Now if we notice where this great-circle arc cuts the various meridians and mark this on an ordinary Mercator’s projection map, such as is found in any schoolroom, we shall find that the path of the ship does not make a straight line. Passengers at sea often do not understand why the ship’s course on the map is not a straight line; but the chief reason is that the ship is taking a great-circle arc, and this is not, in general, a straight line on a Mercator projection. The small circles of latitude are straight lines, and so are the meridians and the equator, but other great circles are represented by curved lines.
Theorem. The area of the surface of a sphere is equal to the product of its diameter by the circumference of a great circle.
This leads to the remarkable formula, a = 4πr2. That the area of the sphere, a curved surface, should exactly equal the sum of the areas of four great circles, plane surfaces, is the remarkable feature. This was one of the greatest discoveries of Archimedes (ca. 287-212 B.C.), who gives it as the thirty-fifth proposition of his treatise on the “Sphere and the Cylinder,” and who mentions it specially in a letter to his friend Dositheus, a mathematician of some prominence. Archimedes also states that the surface of a sphere is two thirds that of the circumscribed cylinder, or the same as the curved surface of this cylinder. This is evident, since the cylindric surface of the cylinder is 2πr × 2r, or 4πr2, and the two bases have an area πr2 + πr2, making the total area 6πr2.
Theorem. The area of a spherical triangle is equal to the area of a lune whose angle is half the triangle’s spherical excess.
This theorem, so important in finding areas on the earth’s surface, should be followed by a considerable amount of computation of triangular areas, else it will be rather meaningless. Students tend to memorize a proof of this character, and in order to have the proposition mean what it should to them, they should at once apply it. The same is true of the following proposition on the area of a spherical polygon. It is probable that neither of these propositions is very old; at any rate, they do not seem to have been known to the writers on elementary mathematics among the Greeks.
Theorem. The volume of a sphere is equal to the product of the area of its surface by one third of its radius.
This gives the formula v = (4/3)πr3. This is one of the greatest discoveries of Archimedes. He also found as a result that the volume of a sphere is two thirds the volume of the circumscribed cylinder. This is easily seen, since the volume of the cylinder is πr2 × 2r, or 2πr3, and (4/3)πr3 is 2/3 of 2πr3. It was because of these discoveries on the sphere and cylinder that Archimedes wished these figures engraved upon his tomb, as has already been stated. The Roman general Marcellus conquered Syracuse in 212 B.C., and at the sack of the city Archimedes was killed by an ignorant soldier. Marcellus carried out the wishes of Archimedes with respect to the figures on his tomb.
The volume of a sphere can also be very elegantly found by means of a proposition known as Cavalieri’s Theorem. This asserts that if two solids lie between parallel planes, and are such that the two sections made by any plane parallel to the given planes are equal in area, the solids are themselves equal in volume. Thus, if these solids have the same altitude, a, and if S and S’ are equal sections made by a plane parallel to MN, then the solids have the same volume. The proof is simple, since prisms of the same altitude, say a/n, and on the bases S and S’ are equivalent, and the sums of n such prisms are the given solids; and as n increases, the sums of the prisms approach the solids as their limits; hence the volumes are equal.
This proposition, which will now be applied to finding the volume of the sphere, was discovered by Bonaventura Cavalieri (1591 or 1598-1647). He was a Jesuit professor in the University of Bologna, and his best known work is his “Geometria Indivisilibus,” which he wrote in 1626, at least in part, and published in 1635 (second edition, 1647). By means of the proposition it is also possible to prove several other theorems, as that the volumes of triangular pyramids of equivalent bases and equal altitudes are equal.
To find the volume of a sphere, take the quadrant OPQ, in the square OPRQ. Then if this figure is revolved about OP, OPQ will generate a hemisphere, OPR will generate a cone of volume (1/3)πr3, and OPRQ will generate a cylinder of volume πr3. Hence the figure generated by ORQ will have a volume πr3 - (1/3)πr3, or (2/3)πr3, which we will call x.
Now OA = AB, and OC = AD; also (OC)2 - (OA)2 = (AC)2, so that
(AD)2 - (AB)2 = (AC)2,
and π(AD)2 - π(AB)2 = π(AC)2.
But π(AD)2 - π(AB)2 is the area of the ring generated by BD, a section of x, and π(AC)2 is the corresponding section of the hemisphere. Hence, by Cavalieri’s Theorem,
(2/3)πr3 = the volume of the hemisphere.
∴ (4/3)πr3 = the volume of the sphere.
In connection with the sphere some easy work in quadratics may be introduced even if the class has had only a year in algebra.
For example, suppose a cube is inscribed in a hemisphere of radius r and we wish to find its edge, and thereby its surface and its volume.
If x = the edge of the cube, the diagonal of the base must be x√2, and the projection of r (drawn from the center of the base to one of the vertices) on the base is half of this diagonal, or (x√2)/2.
Hence, by the Pythagorean Theorem,
r2 = x2 + ((x√2)/2)2 = (3/2)x2
∴ x = r√(2/3),
and the total surface is 6x2 = 4r2,
and the volume is x3 = (2/3)r3√(2/3).