Let ABCD, EFGH, be two planes cutting E one another in the line MN. Then, this line MN is a right one.

DEM. Suppose the line MN, which is common to the two planes, not to be a right line. In this case the points м and N might be joined by a right line MIN in the plane ABCD; and, as this right line is supposedH different from MN, it is not in the plane EFGH, and there might be another right D line, MON, drawn in the plane EFGH, be

M

F

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tween the same points м and N. Consequently, on this supposition, the two right lines MIN, MON, would enclose a space, which is impossible. Hence, the supposition is false; that is, the line of intersection MN is a right one. This, &c.

In common speech we say that a line is perpendicular to a plane when it leans no more to one side than another: this notion, stated a little more accurately, becomes the geometrical definition of a perpendicular to a plane.

DEF. I. A right line is said to be perpendicular to a plane when it is perpendicular to all the right lines in that plane which pass through the point where the line meets the plane.

Thus, if AB be a right line drawn to the plane CGFDHE, So that all the right lines in that plane which pass through the C point B, such as CD, EF, GH, are perpendicular to AB, then AB is said to be perpendicular to the plane CGFDHE.

ART. 4. If at the point where two right lines intersect, a right D line stand perpendicular to both, it shall also be perpendicular to the plane in which they lie.

Let AB be a right line perpendicular to both the right lines GH, IK, at their point of inter

F

L

E

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F

D

H

A

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M

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section, B. Then, AB is perpendicular to the plane CDEF, in which these lines, GH, IK, lie..

DEM. Through the point в draw any right line LM, in the plane CDEF. Take the portions BG, BI, BH, BK, equal to each other; and draw the right lines AG, AI, AH, AK. In the four triangles thus formed, ABG, ABI, Abh, abk, the four angles ABG, ABI, ABH, ABK, are granted equal, being all right angles. Likewise the side AB is common, and therefore as the sides BG, BI, BH, BK, have been taken equal, the four bases AG, AI, AH, AK, are equal, by ART. 1, GEOMETRY.

By ARTS. 1 and 2, GEOм., as the sides BI, BG, are equal respectively to the sides BH, BK, and the contained angles IBG, HBK, being vertically opposite, are also equal, therefore the angles BIL, BKM, are equal; and also the bases IG, HK. Consequently, the three sides AG, AI, IG, of the triangle AIG having been proved respectively equal to the three sides AH, AK, HK, of the triangle AHK, the angle AIL opposite AG is equal to the corresponding angle AKM opposite AH. But in the triangles IBL, KBM, the vertically opposite angles IBL, KBM, are equal; as also the angles. BIL, BKM; and the sides BI, BK: therefore, by ART. 46, GEOM., IL KM, and BL = BM. Thus, in the triangles AIL, AKM, we have shown that AI= AK, IL = KM, and the angle AIL = AKM; wherefore, by ART. 1, GEOM., al =AM. Hence, as in the triangles ABL, ABM, it has been proved that AL = AM, and BL = BM, the side AB being common, we have the angle Abl = abm, by Art. 6, GEOM.; that is, AB is perpendicular to LM. In the same manner it may be demonstrated that AB is perpendicular to every other right line in the plane CDEF; whence, by DEF. preceding, AB is perpendicular to the plane itself. This, &c.

Again: in common practice the mutual inclination of two planes, which meet or cut each other, would be considered as rightly measured by cutting them with a third plane on which both would stand upright, and then taking the angle between the lines where the latter plane cuts the two former.

If we now consider that the line of intersection, os, must also be upright on the plane IKLM, that is, perpendicular to the lines NO, OP, or they perpendicular to it, the following definition will appear to spring very naturally from the above popular method of estimating the inclination of two planes to each other.

DEF. II. When two planes cut each other, the angle between them, or their mutual inclination, is measured by the angle between two right lines drawn in those planes from the same point of their line of intersection, and perpendicular to it.

make with each other is measured by the angle between these right lines DC and DE. Any one such angle may be taken to measure the angle between the planes, inasmuch as their mutual inclination is evidently the same throughout their whole common length, AB. [NOTE C.]

DEF. III. Two planes are perpendicular to each other when the angle between them is measured by a right angle. Thus, in the above figure, if the angle between the perpen

diculars DC, DE, be a right angle, the planes MKLN, FGHI, are said to be perpendicular to each other.

LESSON I.

OF THE SPHERE.

DEFINITION IV. Any figure made where a solid is cut by a plane surface is called a Section of that solid.

Thus, if a die be cut by a plane parallel to any side of the die, each of the figures made where the plane passes is a section of the die; namely, a square.

Obs. A diameter of a circle remaining fixed, if the circle be made to revolve about this diameter, it is evident that the circumference will describe a surface which is everywhere equally distant from the centre of the circle; and that if the circle revolve progressively, the surface described will at length return into itself, so as totally to include a solid.

DEF. V. A sphere is a solid figure bounded by one surface, such, that all right lines drawn from it to one and the same point within the figure are equal to one another.

DEF. VI. In a sphere, the point from which the surface is everywhere equally distant is called the centre of the sphere.*

* Compare the above Observation and Definition with Observation and Def. 2, PROBLEMS, of our GEOMETRY. The Sphere is to Solids

Obs. It is plain that the centre of the sphere is also the centre of the revolving circle.

DEF. VII. A right line drawn from the centre of a sphere, and terminated in the surface, is called a Radius of the sphere.

DEF. VIII. A right line drawn through the centre of a sphere, and terminated both ways in the surface, is called a Diameter of the sphere.

ART. 5. Every section of a sphere, made by a plane, is a circle.

centre to the surface of the sphere, they are all equal, by DEF. VII. Hence, by DEF. II. GEOM., the bounding line of the section ADBEA is a circle.

Secondly: If the plane do not pass through the centre, c, of the sphere, as in figure 2. Draw the right line cr perpendicular to the plane of the section ADBEA; and also a right line, CD, CE, to any points, D, E, &c., of the boundary of the section. Join FD and FE By DEF. II. CF is perpendicular to FD and FE; therefore, by

2

what the Circle is to Plane figures. Also, it is sufficiently evident that the centre of a sphere must be within the figure, and that a sphere has but one centre; which principles we therefore assume; or they may be proved in the same manner as ARTS. 48 and 49, GEOMETRY.