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 s.p.a.ce of two dimensions), and having no conception of a s.p.a.ce of three dimensions, so we may think of ourselves as living in a s.p.a.ce of three dimensions but surrounded by a s.p.a.ce 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 s.p.a.ce. 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 s.p.a.ce he lives in may curve through some higher s.p.a.ce without his being conscious of it.
So our s.p.a.ce may also curve through some higher s.p.a.ce 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 [L]_A_ = [L]_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 [triangle]_A"B"C"_ over so that it could coincide with [triangle]_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.
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[Ill.u.s.tration]
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 cla.s.s 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. Pa.s.sengers 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 lat.i.tude 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 circ.u.mference of a great circle._
This leads to the remarkable formula, _a_ = 4[pi]_r_^2. 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 circ.u.mscribed cylinder, or the same as the curved surface of this cylinder. This is evident, since the cylindric surface of the cylinder is 2[pi]_r_ 2_r_, or 4[pi]_r_^2, and the two bases have an area [pi]_r_^2 + [pi]_r_^2, making the total area 6[pi]_r_^2.
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)[pi]_r_^3. 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 circ.u.mscribed cylinder. This is easily seen, since the volume of the cylinder is [pi]_r_^2 2_r_, or 2[pi]_r_^3, and (4/3)[pi]_r_^3 is 2/3 of 2[pi]_r_^3. 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 a.s.serts 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 alt.i.tude, _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 alt.i.tude, 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.
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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 alt.i.tudes are equal.
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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)[pi]_r_^3, and _OPRQ_ will generate a cylinder of volume [pi]_r_^3. Hence the figure generated by _ORQ_ will have a volume [pi]_r_^3 - (1/3)[pi]_r_^3, or (2/3)[pi]_r_^3, 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 [pi](_AD_)^2 - [pi](_AB_)^2 = [pi](_AC_)^2.
But [pi](_AD_)^2 - [pi](_AB_)^2 is the area of the ring generated by _BD_, a section of _x_, and [pi](_AC_)^2 is the corresponding section of the hemisphere. Hence, by Cavalieri"s Theorem,
(2/3)[pi]_r_^3 = the volume of the hemisphere.
[therefore] (4/3)[pi]_r_^3 = the volume of the sphere.
In connection with the sphere some easy work in quadratics may be introduced even if the cla.s.s 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_[sqrt]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_[sqrt]2)/2.
Hence, by the Pythagorean Theorem,
_r_^2 = _x_^2 + ((_x_[sqrt]2)/2)^2 = (3/2)_x_^2
[therefore] _x_ = _r_[sqrt](2/3),
and the total surface is 6_x_^2 = 4_r_^2,
and the volume is _x_^3 = (2/3)_r_^3[sqrt](2/3).
FOOTNOTES:
[92] The ill.u.s.tration is from Dupin, loc. cit.
L"ENVOI
In the Valley of Youth, through which all wayfarers must pa.s.s on their journey from the Land of Mystery to the Land of the Infinite, there is a village where the pilgrim rests and indulges in various excursions for which the valley is celebrated. There also gather many guides in this spot, some of whom show the stranger all the various points of common interest, and others of whom take visitors to special points from which the views are of peculiar significance. As time has gone on new paths have opened, and new resting places have been made from which these views are best obtained. Some of the mountain peaks have been neglected in the past, but of late they too have been scaled, and paths have been hewn out that approach the summits, and many pilgrims ascend them and find that the result is abundantly worth the effort and the time.
The effect of these several improvements has been a natural and usually friendly rivalry in the body of guides that show the way. The mountains have not changed, and the views are what they have always been. But there are not wanting those who say, "My mountain may not be as lofty as yours, but it is easier to ascend"; or "There are quarries on my peak, and points of view from which a building may be seen in process of erection, or a mill in operation, or a ca.n.a.l, while your mountain shows only a stretch of hills and valleys, and thus you will see that mine is the more profitable to visit." Then there are guides who are themselves often weak of limb, and who are attached to numerous sand dunes, and these say to the weaker pilgrims, "Why tire yourselves climbing a rocky mountain when here are peaks whose summits you can reach with ease and from which the view is just as good as that from the most famous precipice?" The result is not wholly disadvantageous, for many who pa.s.s through the valley are able to approach the summits of the sand dunes only, and would make progress with greatest difficulty should they attempt to scale a real mountain, although even for them it would be better to climb a little way where it is really worth the effort instead of spending all their efforts on the dunes.
Then, too, there have of late come guides who have shown much ingenuity by digging tunnels into some of the greatest mountains. These they have paved with smooth concrete, and have arranged for rubber-tired cars that run without jar to the heart of some mountain. Arrived there the pilgrim has a glance, as the car swiftly turns in a blaze of electric light, at a roughly painted panorama of the view from the summit, and he is a.s.sured by the guide that he has accomplished all that he would have done, had he laboriously climbed the peak itself.
In the midst of all the advocacy of sand-dune climbing, and of rubber-tired cars to see a painted view, the great body of guides still climb their mountains with their little groups of followers, and the vigor of the ascent and the magnificence of the view still attract all who are strong and earnest, during their sojourn in the Valley of Youth.
Among the mountains that have for ages attracted the pilgrims is Mons Latinus, usually called in the valley by the more pleasing name Latina.
Mathematica, and Rhetorica, and Grammatica are also among the best known. A group known as Montes Naturales comprises Physica, Biologica, and Chemica, and one great peak with minor peaks about it is called by the people Philosophia. There are those who claim that these great ma.s.ses of rock are too old to be climbed, as if that affected the view; while others claim that the ascent is too difficult and that all who do not favor the sand dunes are reactionary. But this affects only a few who belong to the real mountains, and the others labor diligently to improve the paths and to lessen unnecessary toil, but they seek not to tear off the summits nor do they attend to the amusing attempts of those who sit by the hillocks and throw pebbles at the rocky sides of the mountains upon which they work.
Geometry is a mountain. Vigor is needed for its ascent. The views all along the paths are magnificent. The effort of climbing is stimulating.
A guide who points out the beauties, the grandeur, and the special places of interest commands the admiration of his group of pilgrims.
One who fails to do this, who does not know the paths, who puts unnecessary burdens upon the pilgrim, or who blindfolds him in his progress, is unworthy of his position. The pretended guide who says that the painted panorama, seen from the rubber-tired car, is as good as the view from the summit is simply a fakir and is generally recognized as such. The mountain will stand; it will not be used as a mere commercial quarry for building stone; it will not be affected by pellets thrown from the little hillocks about; but its paths will be freed from unnecessary flints, they will be straightened where this can advantageously be done, and new paths on entirely novel plans will be made as time goes on, but these paths will be hewed out of rock, not made out of the dreams of a day. Every worthy guide will a.s.sist in all these efforts at betterment, and will urge the pilgrim at least to ascend a little way because of the fact that the same view cannot be obtained from other peaks; but he will not take seriously the efforts of the fakir, nor will he listen with more than pa.s.sing interest to him who proclaims the sand heap to be a Matterhorn.