As to Hippasus, who was a Pythagorean, they say that he perished in the sea on account of his impiety, inasmuch as he boasted that he first divulged the knowledge of the sphere with the twelve pentagons. Hippasus assumed the glory of the discovery to himself, whereas everything belongs to Him, for thus they designate Pythagoras, and do not call Him by name.
As to Hippasus, who was a Pythagorean, they say that he perished in the sea on account of his impiety, inasmuch as he boasted that he first divulged the knowledge of the sphere with the twelve pentagons. Hippasus assumed the glory of the discovery to himself, whereas everything belongs to Him, for thus they designate Pythagoras, and do not call Him by name.
Iamblichus here refers to the dodecahedron inscribed in the sphere. The Pythagoreans looked upon these five solids as fundamental forms in the structure of the universe. In particular Plato tells us that they asserted that the four elements of the real world were the tetrahedron, octahedron, icosahedron, and cube, and Plutarch ascribes this doctrine to Pythagoras himself. Philolaus, who lived in the fifth centuryB.C., held that the elementary nature of bodies depended on their form. The tetrahedron was assigned to fire, the octahedron to air, the icosahedron to water, and the cube to earth, it beingasserted that the smallest constituent part of each of these substances had the form here assigned to it. Although Eudemus attributes all five to Pythagoras, it is certain that the tetrahedron, cube, and octahedron were known to the Egyptians, since they appear in their architectural decorations. These solids were studied so extensively in the school of Plato that Proclus also speaks of them as the Platonic bodies, saying that Euclid "proposed to himself the construction of the so-called Platonic bodies as the final aim of his arrangement of the 'Elements.'" Aristæus, probably a little older than Euclid, wrote a book upon these solids.
As an interesting amplification of this proposition, the centers of the faces (squares) of a cube may be connected, an inscribed octahedron being thereby formed. Furthermore, if the vertices of the cube areA,B,C,D,A',B',C',D', then by drawingAC,CD',D'A,D'B',B'A, andB'C, a regular tetrahedron will be formed. Since the construction of the cube is a simple matter, this shows how three of the five regular solids may be constructed. The actual construction of the solids is not suited to elementary geometry.[89]
It is not difficult for a class to find the relative areas of the cube and the inscribed tetrahedron and octahedron. Ifsis the side of the cube, these areas are 6s2, (1/2)s2√3, ands2√3; that is, the area of the octahedron is twice that of the tetrahedron inscribed in the cube.
Somewhat related to the preceding paragraph is the fact that the edges of the five regular solids are incommensurable with the radius of the circumscribed sphere. This fact seems to have been known to the Greeks, perhaps
to Theætetus (ca.400B.C.) and Aristæus (ca.300B.C.), both of whom wrote on incommensurables.
Just as we may produce the sides of a regular polygon and form a regular cross polygon or stellar polygon, so we may have stellar polyhedrons. Kepler, the great astronomer, constructed some of these solids in 1619, and Poinsot, a French mathematician, carried the constructions so far in 1801 that several of these stellar polyhedrons are known as Poinsot solids. There is a very extensive literature upon this subject.
The following table may be of some service in assigning problems in mensuration in connection with the regular polyhedrons, although some of the formulas are too difficult for beginners to prove. In the tablee= edge of the polyhedron,r= radius of circumscribed sphere,r'= radius of inscribed sphere,a= total area,v= volume.
Some interest is added to the study of polyhedrons by calling attention to their occurrence in nature, in the form of crystals. The computation of the surfaces and volumes of these forms offers an opportunity for applyingthe rules of mensuration, and the construction of the solids by paper folding or by the cutting of crayon or some other substance often arouses a considerable interest. The following are forms of crystals that are occasionally found:
They show how the cube is modified by having its corners cut off. A cube may be inscribed in an octahedron, its vertices being at the centers of the faces of the octahedron. If we think of the cube as expanding, the faces of the octahedron will cut off the corners of the cube as seen in the first figure, leaving the cube as shown in the second figure. If the corners are cut off still more, we have the third figure.
Similarly, an octahedron may be inscribed in a cube, and by letting it expand a little, the faces of the cube will cut off the corners of the octahedron. This is seen in the following figures:
This is a form that is found in crystals, and the computation of the surface and volume is an interestingexercise. The quartz crystal, an hexagonal pyramid on an hexagonal prism, is found in many parts of the country, or is to be seen in the school museum, and this also forms an interesting object of study in this connection.
The properties of the cylinder are next studied. The word is from the Greekkylindros, fromkyliein(to roll). In ancient mathematics circular cylinders were the only ones studied, but since some of the properties are as easily proved for the case of a noncircular directrix, it is not now customary to limit them in this way. It is convenient to begin by a study of the cylindric surface, and a piece of paper may be curved or rolled up to illustrate this concept. If the paper is brought around so that the edges meet, whatever curve may form a cross section the surface is said to inclose acylindric space. This concept is sometimes convenient, but it need be introduced only as necessity for using it arises. The other definitions concerning the cylinder are so simple as to require no comment.
The mensuration of the volume of a cylinder depends upon the assumption that the cylinder is the limit of a certain inscribed or circumscribed prism as the number of sides of the base is indefinitely increased. It is possible to give a fairly satisfactory and simple proof of this fact, but for pupils of the age of beginners in geometry in America it is better to make the assumption outright. This is one of several cases in geometry where a proof is less convincing than the assumed statement.
Theorem.The lateral area of a circular cylinder is equal to the product of the perimeter of a right section of the cylinder by an element.
For practical purposes the cylinder of revolution (right circular cylinder) is the one most frequently used, and the important formula is thereforel= 2πrhwherel= the lateral area,r= the radius, andh= the altitude. Applications of this formula are easily found.
Theorem.The volume of a circular cylinder is equal to the product of its base by its altitude.
Here again the important case is that of the cylinder of revolution, wherev= πr2h.
The number of applications of this proposition is, of course, very great. In architecture and in mechanics the cylinder is constantly seen, and the mensuration of the surface and the volume is important. A single illustration of this type of problem will suffice.
A machinist is making a crank pin (a kind of bolt) for an engine, according to this drawing. He considers it as weighing the same as three steel cylinders having the diameters and lengths in inches as here shown, where 7-3/4" means 7-3/4 inches. He has this formula for the weight (w) of a steel cylinder wheredis the diameter andlis the length:w= 0.07πd2l. Taking π = 3-1/7, find the weight of the pin.
A machinist is making a crank pin (a kind of bolt) for an engine, according to this drawing. He considers it as weighing the same as three steel cylinders having the diameters and lengths in inches as here shown, where 7-3/4" means 7-3/4 inches. He has this formula for the weight (w) of a steel cylinder wheredis the diameter andlis the length:w= 0.07πd2l. Taking π = 3-1/7, find the weight of the pin.
The most elaborate study of the cylinder, cone, and sphere (the "three round bodies") in the Greek literature is that of Archimedes of Syracuse (on the island of Sicily), who lived in the third centuryB.C.Archimedes tells us, however, that Eudoxus (bornca.407B.C.) discovered that any cone is one third of a cylinder of the same base and the same altitude. Tradition says that Archimedes requested that a sphere and a cylinder be carved upon his tomb, and that this was done. Cicero relates that he discovered the tomb by means of these symbols. The tomb now shown to visitors in ancient Syracuse asthat of Archimedes cannot be his, for it bears no such figures, and is not "outside the gate of Agrigentum," as Cicero describes.
The cone is now introduced. A conic surface is easily illustrated to a class by taking a piece of paper and rolling it up into a cornucopia, the space inclosed being aconic space, a term that is sometimes convenient. The generation of a conic surface may be shown by taking a blackboard pointer and swinging it around by its tip so that the other end moves in a curve. If we consider a straight line as the limit of a curve, then the pointer may swing in a plane, and so a plane is the limit of a conic surface. If we swing the pointer about a point in the middle, we shall generate the two nappes of the cone, the conic space now being double.
In practice the right circular cone, or cone of revolution, is the important type, and special attention should be given to this form.
Theorem.Every section of a cone made by a plane passing through its vertex is a triangle.
At this time, or in speaking of the preliminary definitions, reference should be made to the conic sections. Of these there are three great types: (1) the ellipse, where the cutting plane intersects all the elements on one side of the vertex; a circle is a special form of the ellipse; (2) the parabola, where the plane is parallel to an element; (3) the hyperbola, where the plane cuts some of the elements on one side of the vertex, and the rest on the other side; that is, where it cuts both nappes. It is to be observed that the ellipse may vary greatly in shape, from a circle to a very long ellipse, as the cutting plane changes from being perpendicular to the axis to being nearly parallel to an element. The instant itbecomes parallel to an element the ellipse changes suddenly to a parabola. If the plane tips the slightest amount more, the section becomes an hyperbola.
While these conic sections are not studied in elementary geometry, the terms should be known for general information, particularly the ellipse and parabola. The study of the conic sections forms a large part of the work of analytic geometry, a subject in which the figures resemble the graphic work in algebra, this having been taken from "analytics," as the higher subject is commonly called. The planets move about the sun in elliptic orbits, and Halley's comet that returned to view in 1909-1910 has for its path an enormous ellipse. Most comets seem to move in parabolas, and a body thrown into the air would take a parabolic path if it were not for the resistance of the atmosphere. Two of the sides of the triangle in this proposition constitute a special form of the hyperbola.
The study of conic sections was brought to a high state by the Greeks. They were not known to the Pythagoreans, but were discovered by Menæchmus in the fourth centuryB.C.This discovery is mentioned by Proclus, who says, "Further, as to these sections, the conics were conceived by Menæchmus."
Since if the cutting plane is perpendicular to the axis the section is a circle, and if oblique it is an ellipse, a parabola, or an hyperbola, it follows that if light proceeds from a point, the shadow of a circle is a circle, an ellipse, a parabola, or an hyperbola, depending on the position of the plane on which the shadow falls. It is interesting and instructive to a class to see these shadows, but of course not much time can be allowed for such work. At this point the chief thing is to havethe names "ellipse" and "parabola," so often met in reading, understood.
It is also of interest to pupils to see at this time the method of drawing an ellipse by means of a pencil stretching a string band that moves about two pins fastened in the paper. This is a practical method, and is familiar to all teachers who have studied analytic geometry. In designing elliptic arches, however, three circular arcs are often joined, as here shown, the result being approximately an elliptic arc.
HereOis the center of arcBC,Mof arcAB, andNof arcCD. SinceXYis perpendicular toBMandBO, it is tangent to arcsABandBC, so there is no abrupt turning atB, and similarly forC.[90]
HereOis the center of arcBC,Mof arcAB, andNof arcCD. SinceXYis perpendicular toBMandBO, it is tangent to arcsABandBC, so there is no abrupt turning atB, and similarly forC.[90]
Theorem.The volume of a circular cone is equal to one third the product of its base by its altitude.
It is easy to prove this for noncircular cones as well, but since they are not met commonly in practice, they may be omitted in elementary geometry. The important formula at this time isv= 1/3πr2h. As already stated, this proposition was discovered by Eudoxus of Cnidus (bornca.407B.C., diedca.354B.C.), a man who, as already stated, was born poor, but who became one of the most illustrious and most highly esteemed of all the Greeks of his time.
Theorem.The lateral area of a frustum of a cone of revolution is equal to half the sum of the circumferences of its bases multiplied by the slant height.
An interesting case for a class to notice is that in which the upper base becomes zero and the frustum becomes a cone, the proposition being still true. If the upper base is equal to the lower base, the frustum becomes a cylinder, and still the proposition remains true. The proposition thus offers an excellent illustration of the elementary Principle of Continuity.
Then follows, in most textbooks, a theorem relating to the volume of a frustum.
In the case of a cone of revolutionv= (1/3)πh(r2+r'2+rr'). Here ifr'= 0, we havev= (1/3)πr2h, the volume of a cone. Ifr'=r, we havev= (1/3)πh(r2+r2+r2) = πhr2, the volume of a cylinder.
In the case of a cone of revolutionv= (1/3)πh(r2+r'2+rr'). Here ifr'= 0, we havev= (1/3)πr2h, the volume of a cone. Ifr'=r, we havev= (1/3)πh(r2+r2+r2) = πhr2, the volume of a cylinder.
If one needs examples in mensuration beyond those given in a first-class textbook, they are easily found. The monument to Sir Christopher Wren, the professor of geometry in Cambridge University, who became the great architect of St. Paul's Cathedral in London, has a Latin inscription which means, "Reader, if you would see his monument, look about you." So it is with practical examples in Book VII.
Appended to this Book, or more often to the course in solid geometry, is frequently found a proposition known as Euler's Theorem. This is often considered too difficult for the average pupil and is therefore omitted. On account of its importance, however, in the theory of polyhedrons, some reference to it at this time may be helpful to the teacher. The theorem asserts that in any convex polyhedron the number of edges increased by two is equal to the number of vertices increased by the number of faces. In other words, thate+ 2 =v+f. On account of its importance a proof will be given that differs from the one ordinarily found in textbooks.
Lets1,s2, ···,snbe the number of sides of the various faces, andfthe number of faces. Now since the sum of the angles of a polygon ofssides is (s- 2)180°, therefore the sum of the angles of all the faces is (s1+s2+s3+ ··· +sn- 2f)180°.Buts1+s2+s3+ ··· +snis twice the number of edges, because each edge belongs to two faces.∴ the sum of the angles of all the faces is(2e- 2f)180°, or (e-f)360°.Since the polyhedron is convex, it is possible to find some outside point of view,P, from which some face, asABCDE, covers up the whole figure, as in this illustration. If we think of all the vertices projected onABCDE, by lines throughP, the sum of the angles of all the faces will be the same as the sum of the angles of all their projections onABCDE. CallingABCDEs1, and thinking of the projections as traced by dotted lines on the opposite side ofs1, this sum is evidently equal to(1) the sum of the angles ins1, or (s1- 2) 180°, plus(2) the sum of the angles on the other side ofs1, or (s1- 2)180°, plus(3) the sum of the angles about the various points shown as inside ofs1, of which there arev-s1points, about each of which the sum of the angles is 360°, making (v-s1)360° in all.
Lets1,s2, ···,snbe the number of sides of the various faces, andfthe number of faces. Now since the sum of the angles of a polygon ofssides is (s- 2)180°, therefore the sum of the angles of all the faces is (s1+s2+s3+ ··· +sn- 2f)180°.
Buts1+s2+s3+ ··· +snis twice the number of edges, because each edge belongs to two faces.
∴ the sum of the angles of all the faces is
(2e- 2f)180°, or (e-f)360°.
Since the polyhedron is convex, it is possible to find some outside point of view,P, from which some face, asABCDE, covers up the whole figure, as in this illustration. If we think of all the vertices projected onABCDE, by lines throughP, the sum of the angles of all the faces will be the same as the sum of the angles of all their projections onABCDE. CallingABCDEs1, and thinking of the projections as traced by dotted lines on the opposite side ofs1, this sum is evidently equal to
(1) the sum of the angles ins1, or (s1- 2) 180°, plus
(2) the sum of the angles on the other side ofs1, or (s1- 2)180°, plus
(3) the sum of the angles about the various points shown as inside ofs1, of which there arev-s1points, about each of which the sum of the angles is 360°, making (v-s1)360° in all.
Adding, we have(s1- 2)180° + (s1- 2)180° + (v-s1)360° = ((s1- 2) + (v-s1))360°= (v- 2)360°.Equating the two sums already found, we have(e-f)360° = (v- 2)360°,ore-f=v- 2,ore+ 2 =v+f.
Adding, we have
(s1- 2)180° + (s1- 2)180° + (v-s1)360° = ((s1- 2) + (v-s1))360°
= (v- 2)360°.
Equating the two sums already found, we have
(e-f)360° = (v- 2)360°,
ore-f=v- 2,
ore+ 2 =v+f.
This proof is too abstract for most pupils in the high school, but it is more scientific than those found in any of the elementary textbooks, and teachers will find it of service in relieving their own minds of any question as to the legitimacy of the theorem.
Although this proposition is generally attributed to Euler, and was, indeed, rediscovered by him and published in 1752, it was known to the great French geometer Descartes, a fact that Leibnitz mentions.[91]
This theorem has a very practical application in the study of crystals, since it offers a convenient check on the count of faces, edges, and vertices. Some use of crystals, or even of polyhedrons cut from a piece of crayon, is desirable when studying Euler's proposition. The following illustrations of common forms of crystals may be used in this connection:
The first represents two truncated pyramids placed base to base. Heree= 20,f= 10,v= 12, so thate+ 2 =f+v. The second represents a crystal formed by replacing each edge of a cube by a plane, with the result thate= 40,f= 18, andv= 24. The third represents a crystal formed by replacing each edge of an octahedron by a plane, it being easy to see that Euler's law still holds true.
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. Itis 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 isABand the pointPis taken onABproduced, then the square on any tangent fromPis equal toPB×PA. The linePBAis sometimes called theradical 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 commontangents 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 sphereO'represents the moon,Othe earth, andSthe sun. It is also seen in the case of the full moon, whenSis on the other side of the earth. In this case the partMINis fully illuminated by the moon, but the zoneABNMis only partly illuminated, as the figure shows.[92]
An application of the proposition is seen in the case of an eclipse, where the sphereO'represents the moon,Othe earth, andSthe sun. It is also seen in the case of the full moon, whenSis on the other side of the earth. In this case the partMINis fully illuminated by the moon, but the zoneABNMis 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 fourthdimension 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 shipscrossing 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.287-212B.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 formulav= (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)πr3is 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 212B.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 ifSandS'are equal sections made by a plane parallel toMN, then the solids have the same volume. The proof is simple, since prisms of the same altitude, saya/n, and on the basesSandS'are equivalent, and the sums ofnsuch prisms are the given solids; and asnincreases, 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 quadrantOPQ, in the squareOPRQ. Then if this figure is revolved aboutOP,OPQwill generate a hemisphere,OPRwill generate a cone of volume (1/3)πr3, andOPRQwill generate a cylinder of volume πr3. Hence the figure generated byORQwill have a volume πr3- (1/3)πr3, or (2/3)πr3, which we will callx.NowOA=AB, andOC=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)2is the area of the ring generated byBD, a section ofx, and π(AC)2is 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.
To find the volume of a sphere, take the quadrantOPQ, in the squareOPRQ. Then if this figure is revolved aboutOP,OPQwill generate a hemisphere,OPRwill generate a cone of volume (1/3)πr3, andOPRQwill generate a cylinder of volume πr3. Hence the figure generated byORQwill have a volume πr3- (1/3)πr3, or (2/3)πr3, which we will callx.
NowOA=AB, andOC=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)2is the area of the ring generated byBD, a section ofx, and π(AC)2is 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 radiusrand we wish to find its edge, and thereby its surface and its volume.Ifx= the edge of the cube, the diagonal of the base must bex√2, and the projection ofr(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 isx3= (2/3)r3√(2/3).
For example, suppose a cube is inscribed in a hemisphere of radiusrand we wish to find its edge, and thereby its surface and its volume.
Ifx= the edge of the cube, the diagonal of the base must bex√2, and the projection ofr(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 isx3= (2/3)r3√(2/3).
In the Valley of Youth, through which all wayfarers must pass 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 canal, 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 pass 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 assured 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 masses 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 bestraightened 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 assist 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 passing interest to him who proclaims the sand heap to be a Matterhorn.