PREFACE. This volume contains the Higher Branches of Synthetical Geometry, It consists of Treatises on Solid Geometry, Spherical Geometry, Spherical Trigonometry, the Projections of the Sphere, Perpendicular Projection, Linear Perspective, and Conic Sections. The first three Treatises are those contained in Playfair's Edition of Euclid's Elements, with some alterations. Several useful Definitions, Scholia, Corollaries, and Propositions, have been added. Instead of the first four propositions on Spherical Geometry, other four, from the excellent System of Mathematics by West, have been substituted, as they contain a more full exposition of principles. From the same work another important proposition, the eighteenth, has been added. The Spherical Trigonometry has been improved by inserting the twelfth and thirteenth, for other propositions which are somewhat intricate. These two are also taken from West's system, but the expressions in the demonstrations are considerably altered. The propositions which these have displaced, rather injure the symmetry of the system, as they require the aid of Analytical Trigonometry. A second demonstration has been given of the fifth proposition of the second book of Solid Geometry, depending on the principle established in the twenty-seventh proposition of the additional fifth book of the former volume. The two Treatises on the Projections of the Sphere are also adopted from West's system, except the problems on the Stereographic Projection, which, with the Treatises on Perpendicular Projection and Perspective, have been composed expressly for this work, The Treatise on Conic Sections is taken from the same work; and as the Corollaries are numerous, and most of them important, but undemonstrated in the above work, except a few, demonstrations have been added to those that required them, in order to remove unnecessary obstacles to the progress of the student, who will find a sufficient field for exercise in the undemonstrated theorems and problems annexed for this purpose to this Treatise. The Treatises on Projections have been added on account of their utility in some branches of practical science and of art; the Projections of the Sphere being necessary in Spherical Trigonometry, and in Nautical and Practical Astronomy; and Perpendicular Projection and Perspective being indispensable in constructing the diagrams in Geometry of three dimensions, and figures of objects in various branches of the arts and of philosophy. In the Treatise on Solid Geometry are found several examples of the use of the Method of Exhaustions employed by the ancient geometricians. For some remarks on this subject, and on a particular rule observed by Euclid in the composition of his geometry, the following quotation from the preface to Professor Playfair’s treatise is subjoined: “ With respect to the Geometry of Solids, I have departed from Euclid altogether, with a view of rendering it both shorter and more comprehensive. This, however, is not attempted by introducing a mode of reasoning looser or less rigorous than that of the Greek geometer; for this would be to pay too dear even for the time that might thereby be saved; but it is done chiefly by laying aside a certain rule, which, though it be not essential to the accuracy of demonstration, Euclid has thought it proper, as much as possible, to observe. · The rule referred to is one which regulates the arrangement of Euclid's propositions through the whole of the Elements, namely, that in the demonstration of a theorem he never supposes any thing to be done, as any line to be drawn, or any figure to be constructed, the manner of doing which he has not previously explained. In the two Books on the Properties of Solids that I now offer to the public, though I have followed Euclid very closely in the simpler parts, I have nowhere sought to subject the demonstrations to such a law as the foregoing, and have never hesitated to admit the existence of such solids, or such lines as are evidently possible, though the manner of actually describing them may not have been explained. In this way, also, I have been enabled to offer that very refined artifice in geometrical reasoning, to which we give the name of the Method of Exhaustions, under a much simpler form than it appears in the twelfth book of Euclid; and the spirit of it may, I think, be best learned when it is disengaged from every thing not essential to it. That this method may be the better understood, and because the demonstrations that require it are, no doubt, the most difficult in the Elements, they are all condueted as nearly as possible in the same way through the different Solids, from the pyramid to the sphere. The comparison of this last Solid with the cylinder concludes the eighth book, and is a proposition that may not improperly be considered as terminating the elementary part of Geometry.” This volume, with the preceding (the ELEMENTS of PLANE GEOMETRY), forms a sufficiently extended Elementary Course of Synthetical Geometry. The higher principles of Trigonometry, and the more abstruse properties of Curves, are fully and clearly investigated by the only adéquate method, which is founded on Algebraical Analysis, the application of which to these subjects constitutes the branches of Analytical Trigonometry and Analytical Geometry. ELEMENTS OF SOLID GEOMETRY. FIRST BOOK. Solid Geometry treats of the properties of geometrical figures existing in space. Hence, these figures possess extension in the three dimensions of length, breadth, and thickness ; they do not therefore exist in the same plane, but they may be represented by means of diagrams drawn on a plane. DEFINITIONS. 1. A straight line is perpendicular, or at right angles toa plane, when it makes right angles with every straight line meeting it in that plane. 2. A plane is perpendicular to a plane, when the straight lines drawn in one of the planes perpendicularly to the common section of the two planes, are perpendicular to the other plane. 3. The inclination of a straight line to a plane is the acute angle contained by that straight line, and another drawn from the point in which the first line meets the plane, to the point in which a perpendicular to the plane drawn from any point of the first line, meets the same plane. 4. The inclination of a plane to a plane is the acute angle contained by two straight lines drawn from any the same point of their common section at right angles to it, one upon one plane, and the other upon the other plane. 5. Two planes are said to have the same, or a like inclination to one another, which two other planes have, when their angles of inclination are equal to one another. 6. Parallel planes are such as do not meet one another though produced. 7. A straight line and plane are parallel, if they do not meet when produced. 8. The angle formed by two intersecting planes is called a dihedral angle. 9. Any two angles are said to be of the same affection, when they are either both greater or both not greater than a right angle. The same term is applied to arcs of the same or equal circles, when they are either both greater or both not greater than a quadrant. А B D PROPOSITION I. THEOREM. One part of a straight line cannot be in a plane, and another part above it. If it be possible, let AB, part of the straight line ABC, be in the plane, and the part BC above it; and since the straight line AB is in the plane, it can be produced in that plane ; let it be produced to D. Then ABC and ABD are two straight lines, and they have the common segment AB; which is impossible (Pl. Ge. I. Def. 3, Cor.)* Therefore ABC is not a straight line. PROPOSITION II. THEOREM. Any three straight lines which meet one another, not in the same point, are in one plane. Let the three straight lines AB, CD, CB, meet one another in the points B, C, and E; AB, CD, CB, are in one plane. Let any plane pass through the straight line EB, and let the plane be turned about EB, produced, if necessary, until it pass through the point C. Then, because the points E, 0, are in this plane, the straight line EC is in it (Pl. Ge. I. Def . 8); B for the same reason, the straight line u BC is in the same; and, by the hypothesis, EB is in it; * Pl, Ge, refers to the volume on Plane Geometry. |