Part 30 (1/2)

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 raph, for example, is not fore in the camera If the lines of projection are all perpendicular to the plane, the projection is said to be orthographic, froood exaraphic 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 schoolrooht 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 the 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 thement

There are many simple illustrations of this case For exae of the ceiling and the various edges of the floor of the schoolrooalleries 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 Greek _polys_ (rasping the le 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 wethe 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 wethe ratio of the spherical polygon to the whole spherical surface It should also be observed that just as we ons in a plane, so we led, 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 syloves, all their parts being ed in opposite order Our hands, feet, and ears afford other illustrations of syles 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 vertex _V_ to approach the plane that cuts the edges in _A_, _B_, _C_, _D_, , the edges continuing to pass through these as fixed points The sules about _V_ approaches what limit? On the other hand, suppose _V_ recedes indefinitely; then the sum approaches what limit? Then what are the two lile were concave, ould the proof not hold?

FOOTNOTES:

[87] These er Lehrues to intending buyers

[88] An excellent set of stereoscopic views of the figures of solid geoland, is published by Underwood & Underwood, New York Such a set may properly have place in a school library or in a classrooeous

CHAPTER XX

THE LEADING PROPOSITIONS OF BOOK VII

Book VII relates to polyhedrons, cylinders, and cones It opens with the necessary definitions relating to polyhedrons, the ety and valuable when brought into the work incidentally by the teacher ”Polyhedron” is from the Greek _polys_ (many) and _hedra_ (seat) The Greek plural, _polyhedra_, is used in early English works, but ”polyhedrons” is the form now more commonly seen in A sawed, like a piece of wood sawed from a beam) ”Lateral” is from the Latin _latus_ (side) ”Parallelepiped” is from the Greek _parallelos_ (parallel) and _epipedon_ (a plane surface), froy to ”parallelogram” the word is often spelled ”parallelopiped,” but the best iven ”Truncate” is from the Latin _truncare_ (to cut off)

A few of the leading propositions are now considered

THEOREM _The lateral area of a prise by the periht section_

It should be noted that although soive the proposition that parallel sections are congruent, this is necessary for this proposition, because it shows that the right sections are all congruent and hence that any one of them may be taken

It is, of course, possible to construct a prisht section, that is, a section cutting all the lateral edges at right angles, is impossible In this case the lateral faceswhat is called a _prismatic space_ This ter 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 prisested in connection with beams, corridors, and priss Most geometries supply sufficient material in this line, however

THEOREM _An oblique prisht section of the oblique prise of the oblique pris up to the mensuration of the prisous proposition in plane geole, 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 propositionproposition, 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_