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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 conducted 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.
EDINBURGH, September 1, 1837.
ELEMENTS OF SOLID GEOMETRY.
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.
1. A straight line is perpendicular, or at right angles to a 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.
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, C, are in this plane, the straight line EC is in it (Pl. Ge. I. Def. 8); A for the same reason, the straight line
BC is in the same; and, by the hypothesis, EB is in it;
* Pl. Ge, refers to the volume on Plane Geometry.
therefore the three straight lines EC, CB, BE, are in one plane; but the whole of the lines DC, AB, and BC, produced, are in the same plane with the parts of them EC, EB, BC (I. 1.) Therefore AB, CD, CB, are all in one plane.
COROLLARY 1.-It is manifest that any two straight lines which cut one another are in one plane.
COR. 2.-Only one plane can pass through three points, or through a straight line and a point; and these conditions therefore are sufficient to determine a plane.
PROPOSITION III. THEOREM.
If two planes cut one another, their common section is a straight line.
Let two planes AB, BC, cut one another, and let B and D be two points in the line of their common section. From B to D draw the straight line BD; and because the points B and D are in the plane AB, the straight line BD is in that plane (Pl. Ge. I. Def. 8);
for the same reason, it is in the plane CB; the straight line BD is therefore common to the planes AB and BC, or it is the common section of these planes.
PROPOSITION IV. THEOREM.
If a straight line stand at right angles to each of two straight lines in the point of their intersection, it will also be at right angles to the plane in which these lines are.
Let PO be perpendicular to the lines AB, CD, at their point of intersection O, it is perpendicular to their plane. For, draw through O any straight line EF in their plane. In OD take any point G, and make GDOG, and through G draw GF parallel to OB, to meet OF in F; join DF, and produce DF to meet AB in B, and join PD, PF, and PB.
Because OG GD, therefore (Pl. Ge. VI. 2.) BF=FD; and because, in triangle DOB, the side DB is bisected in F, therefore (Pl. Ge. II. a.) DO2 + OB2 = 2 OF2 + 2 FB3.
And for a similar reason, in triangle DPB, DP2 + PB2 = 2 PF2+2 FB2. But since POD is given a right angle, therefore (Pl. Ge. I. 47) PD2 = PO2 + OD2; and for a similar reason PB2 = PO2 + OB2. Therefore PD2 + PB2 =2PO2+OD2+OB2-2 PO2+2 OF2+2 FB2, for it was shown that OD2+OB2=20F2+2 FB2. But it was also proved that PD2 + PB2 = 2 PF2+2 FB2; and therefore 2 PF2+2 FB2=2 PO2+2 OF2+2 FB2; or, taking 2 FB2 from both, 2 PF2 = 2 PO2 + 2 OF2, or PF2 = PO2 + OF2. Therefore (Pl. Ge. I. 48) POF is a right angle. In a similar manner it may be shown that PO is perpendicular to any other line through O in the plane ACBD; therefore it is perpendicular to that plane (I. Def. 1.)
COR. 1.-If a plane be horizontal in any two directions, it is so in every direction.
COR. 2.-The perpendicular PO is less than any oblique line as PB, and therefore it measures the shortest distance from the point P to the plane.
PROPOSITION V. THEOREM.
If three straight lines meet all in one point, and a straight line stand at right angles to each of them in that point, these three straight lines are in one and the same plane.
Let the straight line AB stand at right angles to each of the straight lines BC, BD, BE, in B, the point where they meet; BC, BD, BE, are in one and the same plane.
If not, let, if it be possible, BD and BE be in one plane, and BC be above it; and let a plane pass through AB, BC, the common section of which with the
plane, in which BD and BE are, shall be
right angles to the plane passing through them (I. 4); and therefore makes right angles (I. Def. 1) with every straight line meeting it in that plane; but BF, which is in that plane, meets it; therefore the angle ABF is a right angle; but the