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Examine in detail the cases :
(a) When D is infinitely remote, or CC' parallel to AB';
(d) When the vertices are in the same line perpendicular to the base;
(e) When the triangles are on different sides of the line in which their bases are situated. 18. Take the figure to 1. 47. Let FG, HK be produced to meet in R; and let P, Q be the intersections of FC, KB, with the sides AB, AC respectively, and N that of AL, BC. Then, it is required to
(a) The lines FG, HK intersect in AN produced :
The segments AP, AQ are equal; and if lines be drawn from P and Q respectively parallel to AC and AB, they will meet in AL:
(c) The points K, A, F are in a straight line.
(d) Let AD, AÉ intersect BC in S, T; and let Z be the intersection of CF, BK in AN. Then
BS : ST :: ST : TC; or BS.TC = ST :
BS :AZ :: AZ: CT; or BS.CT AZ. Properties of this figure appear to be inexhaustible. Upwards of fifty are given by Mr. Bransby in Vol. III. of the “Math. Repos. ;" and as geometrical exercises in earlier life, I once deduced about ten times that number. They appeared at the end to be more plentiful than at the beginning. 19. Turn to the figure in iv, 10, and show that
(a) AC-CB* = AC.CB:
GEOMETRY OF PLANES AND SOLIDS.
Two or three cautions must be given at the outset.
1. This subject is not to be considered difficult or abstruse, from its being so long delayed in its formal introduction into a geometrical course. The delay has arisen from the system of keeping the subjects of plane and solid geometry separate, and of discussing the one fully before the other is entered upon. There would, indeed, be no impropriety in the actual study of the two branches of the subject being carried on pari passû; although such a practice is not customary, and may have, if adopted, no especial advantage.
The only real difficulty that is attached to this part of the subject arises from the use of pictures of the figures instead of models of them in books. The picture in a plane proposition has a similitude in its general features to the perfect one which the mind contemplates in its investigations ; but this is lost when we use pictorial representations of figures in space, and the imagination is taxed to consider things as equal which are incapable in the picture of approximating to equality, and other such relations. However, if we substitute a model (no matter how rudely formed) for the picture, the difficulty is at once and entirely removed.
Beyond this there exists not the least difficulty in the conception or complexity in the reasoning, that can render the discussion of the line and plane, and figures formed by them, more intricate to the student than the First Book of Euclid was: indeed, from the power created by his previous geometrical experience, it ought to be less so to his mind. The propositions of plane geometry which are brought into play in the solid geometry are, moreover, of the most simple and elementary kind mostly the First Book of Euclid,—the others but rarely.
2. The student must take care to get clear views of the geometrical assumptions respecting the line and plane. A few remarks on this subject are subjoined.
(a.) We have a distinct conception of the line and plane antecedently to all geometrical argument, or even of verbal definition. No form of words can render one or the other more clear to the mind; but, on the contrary, all attempts at verbal definition of them tends to confuse the understanding. We do not inquire how the mind forms these conceptions, or whether they be innate or
not: it is sufficient that they are uniform and universal. Besides the conception of the figures, we have equally fixed in the mind certain relations of them. These are that two straight lines cannot coincide in two points without coinciding altogether to the utmost extent of their common prolongation ; that a straight line cannot coincide with a plane in two points without also coinciding with it at every point in that line; and that a plane cannot coincide with a plane in
A plane may
three points, not in a right line, without the coincidence being entire to the utmost common extension of those planes.
(6.) From these properties, which are properly called “axioms," other axioms (which partake of the nature of corollaries) are deducible. For instance, innumerable planes may pass through the same straight line, or through the same two points. Two planes which meet one another intersect in a straight line. be conceived to revolve about a given straight line till it passes through any given point without that line.*
(c.) Let it be clearly understood, that an axiom declares a property of which no logical proof is offered. The ground upon which it rests is antecedent to the logical part of geometry; its character and evidence is a question of metaphysics, and the axiom must be admitted before reasoning in a logical sense can be commenced. Let the student above all things keep clear of the frequent delusion of its being possible to build up a system of "geometry without axioms," whether the basis be definitions or
whatever else. 3. Let it also be understood what is the proper and true character of a definition. It is always a description of the thing meant by a word. The mode of description might be varied; but a definition is always, as has been just stated, nothing more and nothing less. The statement always describes the conditions requisite for making the figure, as Euclid's definitions of a triangle or a circle. They are often given in those absolute forms; but almost as often by describing the operations which will produce the figure. Thus a circle may be defined : “ If one extremity of a given straight line be fixed, and the line revolve about it in a plane, the other extremity will describe a circle.” Euclid's definition of a sphere is :-"A sphere is a solid figure described by the revolution of a semicircle about its diameter which remains fixed.” This might have been given thus : “ A sphere is a figure contained by one surface, and is such that every point in that surface is equally distant from a point within it.” It thus appears to be immaterial in some cases whether the figure be defined as previously existing or by means of its genesis. On the whole, it conduces to simplicity and brevity to define figures bounded by straight lines and planes in an absolute manner, and those bounded (or partially bounded) by curves and curve surfaces by means of their genesis. We are not, however, imperatively tied down to this arrangement; but whatever method we employ, the description must be adequate, complete, and devoid of superfluous conditions.
The postulate simply predicates that the elementary figures are constructible, leaving out all implication of the means of making them, as in the three in Euclid's First Book. The cause appears to be,—not that “the process is mechanical,” or that “ the method is self-evident," or any one of the many conjectures that have been made; but that if the modus operandi were predicated, no means exist for demonstrating the truth of the construction. The form of the problem could hardly be
* To these may be added the inferences :--the three sides of a triangle are in one plane; if two straight lines meet one another, they are in one plane; and some others of like kind, which are occasionally required, but which do not require to be dwelt upon at length here.
given to them under these circumstances, as their being left without demonstration would in initio violate the general rule upon which the system was developed.
4. The construction of the problems in space that are requisite as subordinate to demonstration are always given by Euclid; and, indeed, an adherence to this practice is in some respects desirable. Those which occur are however so simple, and the processes little more than corol· laries from theorems actually given, that I have yielded to the wishes of the officer to whom the Board has committed the control of these volumes, and omitted them altogether. A few of them will, however, inevitably make their appearance in the “ Descriptive Geometry” hereafter.
DEFINITIONS. 1. Parallel planes are such as, however far produced in all possible directions, they will never meet.
2. A line and plane are parallel when, however far produced in all possible directions, they will never meet.
3. A dihedral angle is formed by two planes which intersect. The line of intersection is called the edge of the angle, and the planes themselves the faces of the angle.
4. If from any point in the edge of a dihedral angle two straight lines be drawn, one in each of the faces perpendicular to the edge, they form the profile angle of the planes. Their inclination to one another is called the inclination of the planes. If the profile angle be a right angle, the planes are said to be perpendicular to one another.
5. If a straight line be perpendicular to each of two straight lines at their point of intersection, it is said to be perpendicular to the plane passing through those two lines. Conversely, the plane is said to be perpendicular to the line.
6. A line which is neither parallel nor perpendicular to a plane is said to be oblique to the plane. The inclination of an oblique line to a plane is the acute angle contained by that line and another drawn from the point in which it meets the plane, to the point in which a perpendicular from any point in the line meets the plane.
7. The trace of a line or plane (or any surface, indeed) is the point or line in which that line or plane meets some specified plane.
8. A solid angle (or more descriptively, a polyhedral angle), is formed by three or more planes which meet in a point, and each plane limited in its expansion by the two adjacent ones.
It is called trihedral, tetrahedral, pentahedral, etc., according as there are three, four, five, etc., planes.
The planes are called faces; their linear intersections, edges; and the point in which they all meet, the vertex of the solid angle.
9. If any number of lines be parallel and intercepted between two parallel planes, and planes join these two and two consecutively, the figure produced is a prism.
The prism is, for geometrical purposes, considered as capable of indefinite prolongation both ways from the parallel planes, or in the direction of the lines.
The lines are called the edges of the prism; the parallel planes, the ends or bases of the prism; and the planes joining the parallel lines, the faces.
10. If planes passing through a point without a plane, pass also through the sides of a polygon in that plane, and be limited by their adjacent intersections, they form a pyramid.
The pyramid takes the name of triangular, tetrangular, pentangular, etc., according to the figure of the polygon.
That polygon is the base, the point is the vertex, the planes are the faces, and their intersections the edges of the pyramid.
The pyramid is, geometrically, capable of indefinite extension, both below the base and beyond the vertex.
11. If a line making any angle with the plane of a circle move continually over the circumference, and always keep parallel to its first position, it generates a cylindrical surface.
If this surface be cut by two planes parallel to the circle, the intercepted portion is a cylinder.
These two planes are called the ends or buses of the cylinder ; the original circle the directrix; the moving line the generatrix; and a line through the centre parallel to the generatrix, the axis of the cylinder.
12. If about a point taken without the plane of a circle a straight line move always touching the circumference of the circle, it generates a conical surface. The portion of it intercepted between the point and circle is called a cone. The point is the vertex of the cone, the circle is its base, and the generatrix in any position is called an edge of the
13. If a circle revolve about any diameter till its plane takes a reversed position, it will generate a spherical surface or a sphere.
The terms centre, diameter, etc., are the same as in a circle.
14. Polyhedron is a figure bounded by plane faces, and takes its name from the number of those faces.
Thus the tetrahedron has four faces, the hexahedron six, the octohedron eight, the dodecahedron twelve, and the icosahedron twenty. These names, however, are only applied when all the faces of the figure are equal to one another, forming subjects of much interest to the ancients; but, except from their being amongst the most frequent forms assumed by crystals (both artificially and naturally produced), they would be mere curiosities in modern science.
In modern science the polyhedron is always described, but seldom named, with one or two exceptions.
15. When three pairs of parallel planes mutually intersect, they enclose a figure called the parallelopiped.
It is oblique or rectangular, according as the containing pairs of planes are oblique or rectangular. The cube is the most confined instance of it.
16. Similar polyhedrons are such as have their angular points or vertices similarly situated, each with respect to the others.
* The terms cone and cylinder are generally employed to signify not only the parts here defined by them, but the surfaces generally extended as far as may be necessary
This definition is complete; all others yet given involve superfluous conditions, just as the definition of similar plane polygons does.