to each. For the faces in the two supplemental trihedral angles being the supplements of the measures of the dihedral angles about the original trihedral angles (Prop. 46) will be equal, each to each; and therefore by this proposition the dihedral angles contained by these faces will be equal, each to each; consequently the faces of the original trihedral angles, which are the supplements of the measures of these dihedral angles, will be equal, each to each.

If two trihedral angles have two faces or plane angles of the one, equal to two faces of the other, each to each, and the dihedral angle contained by the two faces in the one equal to the dihedral angle contained by the two faces equal to them of the other; they shall have their third faces equal, and their remaining dihedral angles shall be equal, each to each, namely those to which the equal faces are opposite.

Let the trihedral angles at S and V (fig. 53) have the two faces ASB, ASC equal to the faces DVE, DVF, each to each, and the dihedral angle contained by the faces ASB, ASC equal to the dihedral angle contained by the faces DVE, DVF; they shall have their third faces BSC, EVF equal; the dihedral angle contained by the faces ASB, BSC shall be equal to that contained by the faces DVE, EVF; and the dihedral angle contained by ASC, BSC equal to that contained by DVF, EVF.

The same construction being made as in the last proposition, it may be shown as before, that GH is equal to KL, and the angle SGH equal to the angle VKL; that GI is equal to KM, and the angle SGI equal to the angle VKM; that GP and NP are equal to KT and QT, each to each; and that GO and NO are equal to KR and QR, each to each. And because the dihedral angle contained by the faces ASB, ASC is equal to that contained by the faces DVE, DVF, the inclination of the former planes, the angle PNO, is equal to the inclination of the latter planes, the angle TQR (Prop. 38); therefore in the two triangles PNO, TQR, the sides PN, NO are equal to TQ, QR, each to each, and the angle PNO is equal to the angle TQR; therefore the base PO is equal to the base TR (I. 4). Hence in the two triangles PGO, TKR the three sides in the one are equal to the three sides in the other, each to each, and therefore the angle PGO is equal to the angle TKR (I. 8); conse. quently, in the two triangles HGI, LKM, the two sides HG, GI are equal to the two LK, KM, each to each, and the angle HGI is equal to the angle LKM, and therefore the base HI is equal to the base LM (I. 4). Since then, in the two triangles HSI, LVM, the two sides HS, SI are equal to the two LV, VM, each to each, and the base HI has been shown equal to the base LM, the angle HSI is

equal to the angle LVM; that is, the third face of the trihedral angle at S is equal to the third face of the trihedral angle at V.

Since the three faces of the trihedral angle at S are equal to the three faces of that at V, their remaining dihedral angles are equal (Prop. 48).

Wherefore, if two trihedral angles, &c.: which was to be proved. Cor. From this and Prop. 46, it follows that, if two trihedral angles have a face, or plane angle, in the one equal to a face in the other, and the dihedral angles adjacent to these faces equal, each to each, their other faces will be equal, each to each, and their remaining dihedral angles also equal. For in this case, the two supplemental trihedral angles will have two faces in the one equal to two faces in the other, each to each, and the dihedral angles contained by these faces equal (Prop. 46); therefore, by this proposition, they will have their third faces equal, and their remaining dihedral angles equal, each to each; and consequently the faces in the original trihedral angles being the supplements of these dihedral angles (Prop. 46), will be equal, each to each, and the third dihedral angles in the original trihedral angles, which are the supplements of the third faces in the supplemental angles, are also equal.

THEOREMS TO BE DEMONSTRATED.

1. If through a point in a given straight line several equal straight lines be drawn, making equal angles with the given line, the extremities of these straight lines will be in the circumference of a circle of which the plane is perpendicular to the given straight line, and of which the centre is in this line.

2. When a straight line meeting three straight lines in a plane makes equal angles with them, these angles are right angles, and the straight line is perpendicular to the plane.

3. All the parallels to a given straight line, drawn through different points of any straight line, are in the same plane.

4. If, on the same side of a plane, equal and parallel straight lines be drawn from different points of a straight line in the plane, the extremities of these parallels will be in a straight line parallel to the plane.

5. If a straight line be perpendicular to a plane, every plane parallel to the line will be perpendicular to that plane.

6. If two planes be perpendicular to each other, every straight line perpendicular to one of them will be either parallel to the other, or be wholly contained in it.

And, conversely, if a straight line be parallel to a plane, every plane perpendicular to this line will also be perpendicular to the former plane.

7. If a plane be parallel to two straight lines which cut one another, it will be parallel to the plane of these lines.

8. The triangles formed by joining the corresponding extremities of three equal and parallel straight lines, not all in the same plane, are equal, and their planes are parallel.

9. If, from different points of a plane, equal and parallel straight lines be drawn on the same side of it, their extremities will be in a plane parallel to the former.

10. If a straight line be parallel to a plane, every straight line parallel to that line will be parallel to the plane, or be wholly contained in it.

11. Two straight lines respectively parallel to two parallel straight lines are parallel to each other.

12. Two planes respectively parallel to two parallel planes are parallel to each other.

13. If two planes be respectively parallel to two planes which cut one another, the intersection of the former planes will be parallel to that of the latter.

14. A straight line which is parallel to one of two parallel planes is parallel to the other, or is wholly contained in it.

15. Planes respectively perpendicular to two straight lines which intersect, likewise intersect.

16. When a straight line meets a plane, any straight line perpendicular to the plane will meet any plane perpendicular to the former straight line.

17. Planes perpendicular to the sides of a triangle at their middle points intersect each other, in the same straight line.

18. Parallel straight lines which meet a plane make equal angles with it.

19. Parallel planes which meet a straight line make equal angles with it.

20. If the faces of two dihedral angles be parallel, they will be equal when their faces are both towards the same or towards contrary parts; and one will be the supplement of the other when two of their faces are towards the same parts, and the other two towards contrary parts. 21. The planes which bisect the three dihedral angles about a trihedral angle intersect in the same straight line.

22. If two straight lines not in the same plane be divided in the same ratio, three parallel planes may be drawn, whereof two pass through the corresponding extremities of the straight lines, and the third through the points of section.

THE

ELEMENTARY PRINCIPLES

OF

DESCRIPTIVE GEOMETRY.

1. DESCRIPTIVE GEOMETRY has been defined, "The science which teaches methods of representing accurately geometrical magnitudes, and how to perform graphically all possible operations upon these magnitudes*." By means of it questions which embrace the three dimensions of space are reduced to constructions which may be effected on a plane. Although solid geometry and descriptive geometry treat of the same magnitudes, they differ essentially from each other. In the former, magnitudes in space are in general represented rather arbitrarily, according to the appearance they may present as viewed on a plane, and although the conditions arrived at are not affected by want of accuracy in the representations, we have here no means of construction which will give the real dimensions of the magnitudes represented; the latter affords these means.

To describe a sphere that shall circumscribe a triangular pyramid, is the same problem with regard to space, which the description of a circle about a triangle is with regard to a plane. In the latter problem, the perpendiculars to two of the sides of the triangle at their middle points, will intersect in a point, and the perpendicular to the third side, at its middle point, must meet the two other perpendiculars in their intersection: in the former problem, the planes perpendicular to each pair of the edges of one of the solid angles, at their middle points, will meet in a line; the three lines of intersection will meet in a point; and in the same point must meet, the intersections of the planes drawn perpendicular to the other edges of the pyramid, at their middle points. In the case of the circle, all the operations can be effected, and the lines, both those given and those required in the construction, can be correctly represented in their true dimensions, because they are all in the same plane. In the sphere the case is very different: the lines given, not being

* Vallée, Traité de la Géométrie Déscriptive, Introduction, p. xvi.