BG is perpendicular to both the Planes drawn thro' A B, BC, and DE, EF. But those Planes to which the fame Right Line is perpendicular, are * parallel; † 14 of this. therefore the Plane drawn thro' AB, B C, is parallel to the Plane drawn thro' DE, EF. Wherefore, if two Right Lines, touching one another, be parallel to two Right Lines touching one another, and not being in the fame Plane with them; the Planes drawn through thofe Right Lines are parallel to each other; which was to be demonftrated.

།::

PROPOSITION XVI.'

THEOREM.

If two parallel Planes are cut by another Plane, their common Sections will be parallel.

LET two parallel Planes A B, CD, be cut by any

"

Plane EFGH; and let their common Sections be EF, GH. I fay, E F is parallel to GH.,

For, if it is not parallel, EF, G H, being produced, will meet each other either on the Side FH, or EG. First, let them be produced on the Side FH, and meet in K; then, because E F K is in the Plane A B, all Points taken in E F K will be in the fame Plane. But K is one of the Points that is in EFK; therefore K is in the fame Plane A B. For the fame Reason, K is alfo in the Plane CD; wherefore the Planes A B, CD, will meet each other. But they do not meet, fince they are fuppofed parallel; therefore the Right Lines EF, GH, will not meet on the Side FH. After the fame manner it is proved, that they will not meet, if produced on the Side EG. But Right Lines, that will neither way meet each other, are parallel; therefore E F is parallel to GH. If, therefore, two parallel Planes are cut by any other Plane, their common Sections will be parallel; which was to be demonftrated.

*

XVII.

PROPOSITION

THEOREM.

If two Right Lines are cut by parallel Planes,
they shall be cut in the fame Proportion.

LET two Right Lines A B, CD, be cut by parallel
Planes GH, KL, MN, in the Points A, E, B,
C, F, D. I fay, as the Right Line A E is to the Right
Line E B, fo is C F to F D.

For, let A C, B D, A D, be joined; let A D meet the Plane KL in the Point X; and join EX, XF. Then, because two parallel Planes K L, MN, are cut by the Plane EBD X, their common Sections 16 of this. EX, B D, are parallel. For the fame Reason, because two parallel Planes G H, KL, are cut by the Plane AX FC, their common Sections A C, FX, are parallel; and because E X is drawn parallel to the Side B D of the Triangle ABD, it fhall be, as, AE is to EB, fo is + AX to XD. Again, becaufe X F is drawn parallel to the Side AC of the Triangle ADC, it fhall be +, as A X is to X D, fo is C F to FD. But it has been proved, as AX is to XD, fo is A E to E B. Therefore, as AE is to E B, fo is t CF to FD. Wherefore, if two Right Lines are cut by parallel Planes, they shall be cut in the fome Propor tion; which was to be demonftrated.

† 2.6.

11. 5.

If a Right Line be perpendicular to fome Plane, then all Planes paffing through that Line will be perpendicular to the fame Plane.....

ET the Right Line A B be perpendicular to the Plane CL. I fay, all Planes that pass through A B, are likewife perpendicular to the Plane CL.

For, let a Plane DE pafs thro' the Right Line A B, whofe common Section with the Plane CL, is the Right Line CE; and take fome Point F in CE; from which let F G be drawn in the Plane D E, perpendi

cular

cular to the Right Line CE: Then, because A B is perpendicular to the Plane C L, it fhall also be per- Def. 3. pendicular to all the Right Lines which touch it, and are in the fame Plane: Wherefore, it is perpendicular to CE; and, confequently, the Angle A B F is a Right Angle: But G F B is likewife a Right Angle; therefore A B is parallel to F G. But A B is at Right Angles to the Plane CL; therefore F G will be + at + 8 of this. Right Angles to that fame Plane. But one Plane is perpendicular to another, when the Right Lines drawn in one of the Planes, perpendicular to the common Section of the Planes, are I perpendicular to the other † Def. 4. of Plane. But FG is drawn in one Plane D E, per- bis. pendicular to the common Section C E of the Planes, and it has been proved to be perpendicular to the Plane CL: In like manner any other Line in the Plane D E, drawn perpendicular to C E, is proved to be perpendicular to the Plane CL. Therefore the Plane DE is at Right Angles to the Plane CL. After the fame manner we demonftrate, that all Planes paffing thro' the Right Line A B, are perpendicular to the Plane CL. Therefore, if a Right Line be perpendicular to fome Plané, then all Planes paffing through that Line, will be perpendicular to the fame Plane; which was to be démonstrated.

PROPOSITION XIX.

THEOREM.

If two Planes, cutting each other, be perpendicular to fome Plane, then their common Sellion will be perpendicular to that fame Plane.

LET

BC,

ET two Planes A B, B C, cutting each other, be perpendicular to fome third Plane, and let their common Section be B D. I fay, BD is perpendicular to the faid third Plane, which let be ADC.

For, if poffible, let B D not be perpendicular to the third Plane; and from the Point D, let DE be drawn in the Plane A B, perpendicular to AD; and let D F be drawn in the Plane BC, perpendicular to CD: Then, because the Plane A B is perpendicular to the third Plane, and DE is drawn in the Plane A B, perpendicular to their common Section AD; DE

fhall