There should be introduced at this time, if it has not already been done, the proposition of the lunes of Hippocrates (ca.470B.C.), who proved a theorem that asserts, in somewhat more general form, that if three semicircles be described on the sides of a right triangle as diameters, as shown, the lunesL+L'are together equivalent to the triangleT.
In the use of the circle in design one of the simplest forms suggested by Book V is the trefoil (three-leaf), as here shown, with the necessary construction lines. This is a very common ornament in architecture, both with rounded ends and with the ends slightly pointed.
The trefoil is closely connected with hexagonal designs, since the regular hexagon is formed from the inscribed equilateral triangle by doubling the number of sides. The following are designs that are easily made:
It is not very profitable, because it is manifestly unreal, to measure the parts of such figures, but it offers plenty of practice in numerical work.
Choir of Lincoln CathedralChoir of Lincoln Cathedral
Porch of Lincoln CathedralPorch of Lincoln Cathedral
In the illustrations of the Gothic windows given inChapter XVonly the square and circle were generally involved. Teachers who feel it necessary or advisable to go outside the regular work of geometry for the purpose of increasing the pupil's interest or of training his hand in the drawing of figures will find plenty of designs given in any pictures of Gothic cathedrals. For example, this picture of the noble window in the choir of Lincoln Cathedral shows the use of the square, hexagon, and pentagon. In the porch of the same cathedral, shown in the next illustration, the architect has made use of the triangle, square, and pentagon in planning his ornamental stonework. It is possible to add to the work in pure geometry some work in the mensuration of the curvilinear figures shown in these designs. This form of mensuration is not of much value, however, since itplaces before the pupil a problem that he sees at once is fictitious, and that has no human interest.
Gothic Designs employing Circles and Bisected AnglesGothic Designs employing Circles and Bisected Angles
Gothic Designs employing Circles and SquaresGothic Designs employing Circles and Squares
Gothic Designs employing Circles and the Equilateral TriangleGothic Designs employing Circles and the Equilateral Triangle
Gothic Designs employing Circles and the Regular HexagonGothic Designs employing Circles and the Regular Hexagon
The designs given onpage 283involve chiefly the square as a basis, but it will be seen from one of thefigures that the equilateral triangle and the hexagon also enter. The possibilities of endless variation of a single design are shown in the illustration onpage 284, the basisin this case being the square. The variations in the use of the triangle and hexagon have been the object of study of many designers of Gothic windows, and someexamples of these forms are shown onpage 285. In more simple form this ringing of the changes on elementary figures is shown onpage 286. Some teachers have used color work with such designs for the purpose of increasing the interest of their pupils, but the danger of thus using the time with no serious end in view will be apparent.
In the matter of the mensuration of the circle the annexed design has some interest. The figure is not uncommon in decoration, and it is interesting to show, as a matter of pure geometry, that the area of the circle is divided into three equal portions by means of the four interior semicircles.
An important application of the formulaa= πr2is seen in the area of the annulus, or ring, the formula beinga= πr2- πr'2= π(r2-r'2) = π(r+r') (r-r'). It is used in finding the area of the cross section of pipes, and this is needed when we wish to compute the volume of the iron used.
Another excellent application is that of finding the area of the surface of a cylinder, there being no reason why such simple cases from solid geometry should not furnish working material for plane geometry, particularly as they have already been met by the pupils in arithmetic.
A little problem that always has some interest for pupils is one that Napoleon is said to have suggested to his staff on his voyage to Egypt: To divide a circle into four equal parts by the use of circles alone.
Here the circlesBare tangent to the circleAat the points of division. Furthermore, considering areas, and takingras the radius ofA, we haveA= πr2, andB= π(r/2)2. HenceB= 1/4A, or the sum of the areas of the four circlesBequals the area ofA. Hence the fourD's must equal the fourC's, andD=C. The rest of the argument is evident. The problem has some interest to pupils aside from the original question suggested by Napoleon.
Here the circlesBare tangent to the circleAat the points of division. Furthermore, considering areas, and takingras the radius ofA, we haveA= πr2, andB= π(r/2)2. HenceB= 1/4A, or the sum of the areas of the four circlesBequals the area ofA. Hence the fourD's must equal the fourC's, andD=C. The rest of the argument is evident. The problem has some interest to pupils aside from the original question suggested by Napoleon.
At the close of plane geometry teachers may find it helpful to have the class make a list of the propositions that are actually used in proving other propositions, and to have it appear what ones are proved by them. This forms a kind of genealogical tree that serves to fix the parent propositions in mind. Such a work may also be carried on at the close of each book, if desired. It should be understood, however, that certain propositions are used in the exercises, even though they are not referred to in subsequent propositions, so that their omission must not be construed to mean that they are not important.
An exercise of distinctly less value is the classification of the definitions. For example, the classification of polygons or of quadrilaterals, once so popular in textbook making, has generally been abandoned as tending to create or perpetuate unnecessary terms. Such work is therefore not recommended.
There have been numerous suggestions with respect to solid geometry, to the effect that it should be more closely connected with plane geometry. The attempt has been made, notably by Méray in France and de Paolis in Italy, to treat the corresponding propositions of plane and solid geometry together; as, for example, those relating to parallelograms and parallelepipeds, and those relating to plane and spherical triangles. Whatever the merits of this plan, it is not feasible in America at present, partly because of the nature of the college-entrance requirements. While it is true that to a boy or girl a solid is more concrete than a plane, it is not true that a geometric solid is more concrete than a geometric plane. Just as the world developed its solid geometry, as a science, long after it had developed its plane geometry, so the human mind grasps the ideas of plane figures earlier than those of the geometric solid.
There is, however, every reason for referring to the corresponding proposition of plane geometry when any given proposition of solid geometry is under consideration, and frequent references of this kind will be made in speaking of the propositions in this and the two succeeding chapters. Such reference has value in the apperception of the various laws of solid geometry, and it also adds an interest to the subject and creates someapproach to power in the discovery of new facts in relation to figures of three dimensions.
The introduction to solid geometry should be made slowly. The pupil has been accustomed to seeing only plane figures, and therefore the drawing of a solid figure in the flat is confusing. The best way for the teacher to anticipate this difficulty is to have a few pieces of cardboard, a few knitting needles filed to sharp points, a pine board about a foot square, and some small corks. With the cardboard he can illustrate planes, whether alone, intersecting obliquely or at right angles, or parallel, and he can easily illustrate the figures given in the textbook in use. There are models of this kind for sale, but the simple ones made in a few seconds by the teacher or the pupil have much more meaning. The knitting needles may be stuck in the board to illustrate perpendicular or oblique lines, and if two or more are to meet in a point, they may be held together by sticking them in one of the small corks. Such homely apparatus, costing almost nothing, to be put together in class, seems much more real and is much more satisfactory than the German models.[87]
An extensive use of models is, however, unwise. The pupil must learn very early how to visualize a solid from the flat outline picture, just as a builder or a mechanic learns to read his working drawings. To have a model for each proposition, or even to have a photograph or a stereoscopic picture, is a very poor educational policy. A textbook may properly illustrate a few propositions by photographic aids, but after that the pupil should use
the kind of figures that he must meet in his mathematical work. A child should not be kept in a perambulator all his life,—he must learn to walk if he is to be strong and grow to maturity; and it is so with a pupil in the use of models in solid geometry.[88]
The case is somewhat similar with respect to colored crayons. They have their value and their proper place, but they also have their strict limitations. It is difficult to keep their use within bounds; pupils come to use them to make pleasing pictures, and teachers unconsciously fall into the same habit. The value of colored crayons is two-fold: (1) they sometimes make two planes stand out more clearly, or they serve to differentiate some line that is under consideration from others that are not; (2) they enable a class to follow a demonstration more easily by hearing of "the red plane perpendicular to the blue one," instead of "the planeMNperpendicular to the planePQ." But it should always be borne in mind that in practical work we do not have colored ink or colored pencils commonly at hand, nor do we generally have colored crayons. Pupils should therefore become accustomed to the pencil and the white crayon as the regulation tools, and in general they should use them. The figures may not be as striking, but they are more quickly made and they are more practical.
The definition of "plane" has already been discussed inChapter XII, and the other definitions of Book VI are not of enough interest to call for special remark. The axioms are the same as in plane geometry, but there is
at least one postulate that needs to be added, although it would be possible to state various analogues of the postulates of plane geometry if we cared unnecessarily to enlarge the number.
The most important postulate of solid geometry is as follows:One plane, and only one, can be passed through two intersecting straight lines.This is easily illustrated, as in most textbooks, as also are three important corollaries derived from it:
1.A straight line and a point not in the line determine a plane.Of course this may be made the postulate, as may also the next one, the postulate being placed among the corollaries, but the arrangement here adopted is probably the most satisfactory for educational purposes.
2.Three points not in a straight line determine a plane.The common question as to why a three-legged stool stands firmly, while a four-legged table often does not, will add some interest at this point.
3.Two parallel lines determine a plane.This requires a slight but informal proof to show that it properly follows as a corollary from the postulate, but a single sentence suffices.
While studying this book questions of the following nature may arise with an advanced class, or may be suggested to those who have had higher algebra:
How many straight lines are in general (that is, at the most) determined bynpoints in space? Two points determine 1 line, a third point adds (in general, in all these cases) 2 more, a fourth point adds 3 more, and annth pointn- 1 more. Hence the maximum is 1 + 2 + 3 + ... + (n- 1), orn(n-1)/2, which the pupil will understand if he has studied arithmetical progression.The maximum number of intersection points ofnstraight lines in the same plane is alson(n- 1)/2.
How many straight lines are in general determined bynplanes? The answer is the same,n(n- 1)/2.
How many planes are in general determined bynpoints in space? Here the answer is 1 + 3 + 6 + 10 + ... + (n- 2)(n- 1)/2, orn(n- 1)(n- 2)/(1 × 2 × 3). The same number of points is determined bynplanes.
Theorem.If two planes cut each other, their intersection is a straight line.
Among the simple illustrations are the back edges of the pages of a book, the corners of the room, and the simple test as to whether the edge of a card is straight by testing it on a plane. It is well to call attention to the fact that if two intersecting straight lines move parallel to their original position, and so that their intersection rests on a straight line not in the plane of those lines, the figure generated will be that of this proposition. In general, if we cut through any figure of solid geometry in some particular way, we are liable to get the figure of a proposition in plane geometry, as will frequently be seen.
Theorem.If a straight line is perpendicular to each of two other straight lines at their point of intersection, it is perpendicular to the plane of the two lines.
If students have trouble in visualizing the figure in three dimensions, some knitting needles through a piece of cardboard will make it clear. Teachers should call attention to the simple device for determining if a rod is perpendicular to a board (or a pipe to a floor, ceiling, orwall), by testing it twice, only, with a carpenter's square. Similarly, it may be asked of a class, How shall we test to see if the corner (line) of a room is perpendicular to the floor, or if the edge of a box is perpendicular to one of the sides?
In some elementary and in most higher geometries the perpendicular is called anormalto the plane.
Theorem.All the perpendiculars that can be drawn to a straight line at a given point lie in a plane which is perpendicular to the line at the given point.
Thus the hands of a clock pass through a plane as the hands revolve, if they are, as is usual, perpendicular to the axis; and the same is true of the spokes of a wheel, and of a string with a stone attached, swung as rapidly as possible about a boy's arm as an axis. A clock pendulum too swings in a plane, as does the lever in a pair of scales.
Theorem.Through a given point within or without a plane there can be one perpendicular to a given plane, and only one.
This theorem is better stated to a class as two theorems.
Thus a plumb line hanging from a point in the ceiling, without swinging, determines one definite point in the floor; and, conversely, if it touches a given point in the floor, it must hang from one definite point in the ceiling. It should be noticed that if we cut through this figure, on the perpendicular line, we shall have the figure of the corresponding proposition in plane geometry, namely, that there can be, under similar circumstances, only one perpendicular to a line.
Theorem.Oblique lines drawn from a point to a plane, meeting the plane at equal distances from the foot of the perpendicular, are equal, etc.
There is no objection to speaking of a right circular cone in connection with this proposition, and saying that the slant height is thus proved to be constant. The usual corollary, that if the obliques are equal they meet the plane in a circle, offers a new plan of drawing a circle. A plumb line that is a little too long to reach the floor will, if swung so as just to touch the floor, describe a circle. A 10-foot pole standing in a 9-foot room will, if it moves so as to touch constantly a fixed point on either the floor or the ceiling, describe a circle on the ceiling or floor respectively.
One of the corollaries states that the locus of points in space equidistant from the extremities of a straight line is the plane perpendicular to this line at its middle point. This has been taken by some writers as the definition of a plane, but it is too abstract to be usable. It is advisable to cut through the figure along the given straight line, and see that we come back to the corresponding proposition in plane geometry.
A good many ships have been saved from being wrecked by the principle involved in this proposition.
If a dangerous shoalAis near a headlandH, the angleHAXis measured and is put down upon the charts as the "vertical danger angle." Ships coming near the headland are careful to keep far enough away, say atS, so that the angleHSXshall be less than this danger angle. They are then sure that they will avoid the dangerous shoal.
If a dangerous shoalAis near a headlandH, the angleHAXis measured and is put down upon the charts as the "vertical danger angle." Ships coming near the headland are careful to keep far enough away, say atS, so that the angleHSXshall be less than this danger angle. They are then sure that they will avoid the dangerous shoal.
Related to this proposition is the problem of supporting a tall iron smokestack by wire stays. Evidentlythree stays are needed, and they are preferably placed at the vertices of an equilateral triangle, the smokestack being in the center. The practical problem may be given of locating the vertices of the triangle and of finding the length of each stay.
Theorem.Two straight lines perpendicular to the same plane are parallel.
Here again we may cut through the figure by the plane of the two parallels, and we get the figure of plane geometry relating to lines that are perpendicular to the same line. The proposition shows that the opposite corners of a room are parallel, and that therefore they lie in the same plane, or arecoplanar, as is said in higher geometry.
It is interesting to a class to have attention called to the corollary that if two straight lines are parallel to a third straight line, they are parallel to each other; and to have the question asked why it is necessary to prove this when the same thing was proved in plane geometry. In case the reason is not clear, let some student try to apply the proof used in plane geometry.
Theorem.Two planes perpendicular to the same straight line are parallel.
Besides calling attention to the corresponding proposition of plane geometry, it is well now to speak of the fact that in propositions involving planes and lines we may often interchange these words. For example, using "line" for "straight line," for brevity, we have:
Theorem.The intersections of two parallel planes by a third plane are parallel lines.
Thus one of the edges of a box is parallel to the next succeeding edge if the opposite faces are parallel, and in sawing diagonally through an ordinary board (with rectangular cross section) the section is a parallelogram.
Theorem.A straight line perpendicular to one of two parallel planes is perpendicular to the other also.
Notice (1) the corresponding proposition in plane geometry; (2) the proposition that results from interchanging "plane" and (straight) "line."
Theorem.If two intersecting straight lines are each parallel to a plane, the plane of these lines is parallel to that plane.
Interchanging "plane" and (straight) "line," we have: If two intersectingplanesare each parallel to aline, thelineof (intersection of) theseplanesis parallel to thatline. Is this true?
Theorem.If two angles not in the same plane have their sides respectively parallel and lying on the same side of the straight line joining their vertices, they are equal and their planes are parallel.
Questions like the following may be asked in connection with the proposition: What is the corresponding proposition in plane geometry? Why do we need another proof here? Try the plane-geometry proof here.
Theorem.If two straight lines are cut by three parallel planes, their corresponding segments are proportional.
Here, again, it is desirable to ask for the corresponding proposition of plane geometry, and to ask why the proof of that proposition will not suffice for this one. The usual figure may be varied in an interesting manner by having the two lines meet on one of the planes, or outside the planes, or by having them parallel, in which cases the proof of the plane-geometry proposition holds here. This proposition is not of great importance from the practical standpoint, and it is omitted from some of the standard syllabi at present, although included in certain others. It is easy, however, to frame some interesting questions depending upon it for their answers, such as the following: In a gymnasium swimming tank the water is 4 feet deep and the ceiling is 8 feet above the surface of the water. A pole 15 feet long touches the ceiling and the bottom of the tank. Required to know what length of the pole is in the water.
At this point in Book VI it is customary to introduce the dihedral angle. The word "dihedral" is from the Greek,di-meaning "two," andhedrameaning "seat." We have the roothedraalso in "trihedral" (three-seated), "polyhedral" (many-seated), and "cathedral" (a church having a bishop's seat). The word is also, but less properly, spelled without theh, "diedral," a spelling not favored by modern usage. It is not necessary to dwell at length upon the dihedral angle, except to show the analogy between it and the plane angle. A few illustrations, as of an open book, the wall and floor of a room, and a swinging door, serve to make the concept clear, while a plane at right angles to the edge shows the measuring plane angle. So manifest is thisrelationship between the dihedral angle and its measuring plane angle that some teachers omit the proposition that two dihedral angles have the same ratio as their plane angles.
Theorem.If two planes are perpendicular to each other, a straight line drawn in one of them perpendicular to their intersection is perpendicular to the other.
This and the related propositions allow of numerous illustrations taken from the schoolroom, as of door edges being perpendicular to the floor. The pretended applications of these propositions are usually fictitious, and the propositions are of value chiefly for their own interest and because they are needed in subsequent proofs.
Theorem.The locus of a point equidistant from the faces of a dihedral angle is the plane bisecting the angle.
By changing "plane" to "line," and by making other obvious changes to correspond, this reduces to the analogous proposition of plane geometry. The figure formed by the plane perpendicular to the edge is also the figure of that analogous proposition. This at once suggests that there are two planes in the locus, provided the planes of the dihedral angle are taken as indefinite in extent, and that these planes are perpendicular to each other. It may interest some of the pupils to draw this general figure, analogous to the one in plane geometry.
Theorem.The projection of a straight line not perpendicular to a plane upon that plane is a straight line.
In higher mathematics it would simply be said that the projection is a straight line, the special case of the projection of a perpendicular being considered as a line-segment of zero length. There is no advantage, however, of bringing in zero and infinity in the course in elementary geometry. The legitimate reason for themodern use of these terms is seldom understood by beginners.
This subject of projection (Latinpro-, "forth," andjacere, "to throw") is extensively used in modern mathematics and also in the elementary work of the draftsman, and it will be referred to a little later. At this time, however, it is well to call attention to the fact that the projection of a straight line on a plane is a straight line or a point; the projection of a curve may be a curve or it may be straight; the projection of a point is a point; and the projection of a plane (which is easily understood without defining it) may be a surface or it may be a straight line. An artisan represents a solid by drawing its projection upon two planes at right angles to each other, and a map maker (cartographer) represents the surface of the earth by projecting it upon a plane. A photograph of the class is merely the projection of the class upon a photographic plate (plane), and when we draw a figure in solid geometry, we merely project the solid upon the plane of the paper.
There are other projections than those formed by lines that are perpendicular to the plane. The lines may be oblique to the plane, and this is the case with most projections. A photograph, for example, is not formed by lines perpendicular to a plane, for they all converge in the camera. If the lines of projection are all perpendicular to the plane, the projection is said to be orthographic, from the Greekortho-(straight) andgraphein(to draw). A good example of orthographic projection may be seen in the shadow cast by an object upon a piece of paper that is held perpendicular to the sun's rays. A good example of oblique projection is a shadow on the floor of the schoolroom.
Theorem.Between two straight lines not in the same plane there can be one common perpendicular, and only one.
The usual corollary states that this perpendicular is the shortest line joining them. It is interesting to compare this with the case of two lines in the same plane. If they are parallel, there may be any number of common perpendiculars. If they intersect, there is still a common perpendicular, but this can hardly be said to be between them, except for its zero segment.
There are many simple illustrations of this case. For example, what is the shortest line between any given edge of the ceiling and the various edges of the floor of the schoolroom? If two galleries in a mine are to be connected by an air shaft, how shall it be planned so as to save labor? Make a drawing of the plan.
At this point the polyhedral angle is introduced. The word is from the Greekpolys(many) andhedra(seat). Students have more difficulty in grasping the meaning of the size of a polyhedral angle than is the case with dihedral and plane angles. For this reason it is not good policy to dwell much upon this subject unless the question arises, since it is better understood when the relation of the polyhedral angle and the spherical polygon is met. Teachers will naturally see that just as we may measure the plane angle by taking the ratio of an arc to the whole circle, and of a dihedral angle by taking the ratio of that part of the cylindric surface that is cut out by the planes to the whole surface, so we may measure a polyhedral angle by taking the ratio of the spherical polygon to the whole spherical surface. It should also be observed that just as we may have cross polygons in a plane, so we may have spherical polygonsthat are similarly tangled, and that to these will correspond polyhedral angles that are also cross, their representation by drawings being too complicated for class use.
The idea of symmetric solids may be illustrated by a pair of gloves, all their parts being mutually equal but arranged in opposite order. Our hands, feet, and ears afford other illustrations of symmetric solids.
Theorem.The sum of the face angles of any convex polyhedral angle is less than four right angles.
There are several interesting points of discussion in connection with this proposition. For example, suppose the vertexVto approach the plane that cuts the edges inA,B,C,D, ..., the edges continuing to pass through these as fixed points. The sum of the angles aboutVapproaches what limit? On the other hand, supposeVrecedes indefinitely; then the sum approaches what limit? Then what are the two limits of this sum? Suppose the polyhedral angle were concave, why would the proof not hold?
Book VII relates to polyhedrons, cylinders, and cones. It opens with the necessary definitions relating to polyhedrons, the etymology of the terms often proving interesting and valuable when brought into the work incidentally by the teacher. "Polyhedron" is from the Greekpolys(many) andhedra(seat). The Greek plural,polyhedra, is used in early English works, but "polyhedrons" is the form now more commonly seen in America. "Prism" is from the Greekprisma(something sawed, like a piece of wood sawed from a beam). "Lateral" is from the Latinlatus(side). "Parallelepiped" is from the Greekparallelos(parallel) andepipedon(a plane surface), fromepi(on) andpedon(ground). By analogy to "parallelogram" the word is often spelled "parallelopiped," but the best mathematical works now adopt the etymological spelling above given. "Truncate" is from the Latintruncare(to cut off).
A few of the leading propositions are now considered.
Theorem.The lateral area of a prism is equal to the product of a lateral edge by the perimeter of the right section.
It should be noted that although some syllabi do not give the proposition that parallel sections are congruent, this is necessary for this proposition, because it showsthat the right sections are all congruent and hence that any one of them may be taken.
It is, of course, possible to construct a prism so oblique and so low that a right section, that is, a section cutting all the lateral edges at right angles, is impossible. In this case the lateral faces must be extended, thus forming what is called aprismatic space. This term may or may not be introduced, depending upon the nature of the class.
This proposition is one of the most important in Book VII, because it is the basis of the mensuration of the cylinder as well as the prism. Practical applications are easily suggested in connection with beams, corridors, and prismatic columns, such as are often seen in school buildings. Most geometries supply sufficient material in this line, however.
Theorem.An oblique prism is equivalent to a right prism whose base is equal to a right section of the oblique prism, and whose altitude is equal to a lateral edge of the oblique prism.
This is a fundamental theorem leading up to the mensuration of the prism. Attention should be called to the analogous proposition in plane geometry relating to the area of the parallelogram and rectangle, and to the fact that if we cut through the solid figure by a plane parallel to one of the lateral edges, the resulting figure will be that of the proposition mentioned. As in the preceding proposition, so in this case, there may be a question raised that will make it helpful to introduce the idea of prismatic space.
Theorem.The opposite lateral faces of a parallelepiped are congruent and parallel.
It is desirable to refer to the corresponding case in plane geometry, and to note again that the figure isobtained by passing a plane through the parallelepiped parallel to a lateral edge. The same may be said for the proposition about the diagonal plane of a parallelepiped. These two propositions are fundamental in the mensuration of the prism.
Theorem.Two rectangular parallelepipeds are to each other as the products of their three dimensions.
This leads at once to the corollary that the volume of a rectangular parallelepiped equals the product of its three dimensions, the fundamental law in the mensuration of all solids. It is preceded by the proposition asserting that rectangular parallelepipeds having congruent bases are proportional to their altitudes. This includes the incommensurable case, but this case may be omitted.
The number of simple applications of this proposition is practically unlimited. In all such cases it is advisable to take a considerable number of numerical exercises in order to fix in mind the real nature of the proposition. Any good geometry furnishes a certain number of these exercises.
The following is an interesting property of the rectangular parallelepiped, often called the rectangular solid:
If the edges area,b, andc, and the diagonal isd, then (a/d)2+ (b/d)2+ (c/d)2= 1. This property is easily proved by the Pythagorean Theorem, ford2=a2+b2+c2, whence (a2+b2+c2) /d2= 1.In casec= 0, this reduces to the Pythagorean Theorem. The property is the fundamental one of solid analytic geometry.
If the edges area,b, andc, and the diagonal isd, then (a/d)2+ (b/d)2+ (c/d)2= 1. This property is easily proved by the Pythagorean Theorem, ford2=a2+b2+c2, whence (a2+b2+c2) /d2= 1.
In casec= 0, this reduces to the Pythagorean Theorem. The property is the fundamental one of solid analytic geometry.
Theorem.The volume of any parallelepiped is equal to the product of its base by its altitude.
This is one of the few propositions in Book VII where a model is of any advantage. It is easy to make oneout of pasteboard, or to cut one from wood. If a wooden one is made, it is advisable to take an oblique parallelepiped and, by properly sawing it, to transform it into a rectangular one instead of using three different solids.
On account of its awkward form, this figure is sometimes called the Devil's Coffin, but it is a name that it would be well not to perpetuate.
Theorem.The volume of any prism is equal to the product of its base by its altitude.
This is also one of the basal propositions of solid geometry, and it has many applications in practical mensuration. A first-class textbook will give a sufficient list of problems involving numerical measurement, to fix the law in mind. For outdoor work, involving measurements near the school or within the knowledge of the pupils, the following problem is a type:
If this represents the cross section of a railway embankment that islfeet long,hfeet high,bfeet wide at the bottom, andb´feet wide at the top, find the number of cubic feet in the embankment. Find the volume ifl= 300,h= 8,b= 60, andb´= 28.
If this represents the cross section of a railway embankment that islfeet long,hfeet high,bfeet wide at the bottom, andb´feet wide at the top, find the number of cubic feet in the embankment. Find the volume ifl= 300,h= 8,b= 60, andb´= 28.
The mensuration of the volume of the prism, including the rectangular parallelepiped and cube, was known to the ancients. Euclid was not concerned with practical measurement, so that none of this part of geometry appears in his "Elements." We find, however, in the papyrus of Ahmes, directions for the measuring of bins, and the Egyptian builders, long before his time, must have known the mensuration of the rectangular parallelepiped. Among the Hindus, long before the Christian era, rules were known for the construction of altars, and among the Greeks the problem of constructing a cube with twice the volume of a givencube (the "duplication of the cube") was attacked by many mathematicians. The solution of this problem is impossible by elementary geometry.
Ifeequals the edge of the given cube, thene3is its volume and 2e3is the volume of the required cube. Therefore the edge of the required cube ise∛2. Now ifeis given, it is not possible with the straightedge and compasses to construct a line equal toe∛2, although it is easy to construct one equal toe√2.
Ifeequals the edge of the given cube, thene3is its volume and 2e3is the volume of the required cube. Therefore the edge of the required cube ise∛2. Now ifeis given, it is not possible with the straightedge and compasses to construct a line equal toe∛2, although it is easy to construct one equal toe√2.
The study of the pyramid begins at this point. In practical measurement we usually meet the regular pyramid. It is, however, a simple matter to consider the oblique pyramid as well, and in measuring volumes we sometimes find these forms.
Theorem.The lateral area of a regular pyramid is equal to half the product of its slant height by the perimeter of its base.
This leads to the corollary concerning the lateral area of the frustum of a regular pyramid. It should be noticed that the regular pyramid may be considered as a frustum with the upper base zero, and the proposition as a special case under the corollary. It is also possible, if we choose, to let the upper base of the frustum pass through the vertex and cut the lateral edges above that point, although this is too complicated for most pupils. If this case is considered, it is well to bring in the general idea ofpyramidal space, the infinite space bounded on several sides by the lateral faces, of the pyramid. This pyramidal space is double, extending on two sides of the vertex.
Theorem.If a pyramid is cut by a plane parallel to the base:
1.The edges and altitude are divided proportionally.2.The section is a polygon similar to the base.
To get the analogous proposition of plane geometry, pass a plane through the vertex so as to cut the base. We shall then have the sides and altitude of the triangle divided proportionally, and of course the section will merely be a line-segment, and therefore it is similar to the base line.
The cutting plane may pass through the vertex, or it may cut the pyramidal space above the vertex. In either case the proof is essentially the same.
Theorem.The volume of a triangular pyramid is equal to one third of the product of its base by its altitude, and this is also true of any pyramid.
This is stated as two theorems in all textbooks, and properly so. It is explained to children who are studying arithmetic by means of a hollow pyramid and a hollow prism of equal base and equal altitude. The pyramid is filled with sand or grain, and the contents is poured into the prism. This is repeated, and again repeated, showing that the volume of the prism is three times the volume of the pyramid. It sometimes varies the work to show this to a class in geometry.
This proposition was first proved, so Archimedes asserts, by Eudoxus of Cnidus, famous as an astronomer, geometer, physician, and lawgiver, born in humble circumstances about407 B.C.He studied at Athens and in Egypt, and founded a famous school of geometry at Cyzicus. His discovery also extended to the volume of the cone, and it was his work that gave the beginning to the science of stereometry, the mensuration part of solid geometry.
Theorem.The volume of the frustum of any pyramid is equal to the sum of the volumes of three pyramids whose common altitude is the altitude of the frustum, and whosebases are the lower base, the upper base, and the mean proportional between the bases of the frustum.
Attention should be called to the fact that this formulav= 1/3a(b+b'+ √(bb')) applies to the pyramid by lettingb'= 0, to the prism by lettingb=b', and also to the parallelepiped and cube, these being special forms of the prism. This formula is, therefore, a very general one, relating to all the polyhedrons that are commonly met in mensuration.
Theorem.There cannot be more than five regular convex polyhedrons.
Eudemus of Rhodes, one of the principal pupils of Aristotle, in his history of geometry of which Proclus preserves some fragments, tells us that Pythagoras discovered the construction of the "mundane figures," meaning the five regular polyhedrons. Iamblichus speaks of the discovery of the dodecahedron in these words: