Book XI.

PROP. XV. THEOR.

F two straight lines meeting one another, be parallel Sce N.

to two straight lines which meet one another, but are not in the same plane with the first two; the plane which passes through these is parallel to the plane passing through the others.

Let AB, BC two straight lines meeting one another, be parallel to DE, EF that meet one another, but are not in the same plane with AB, BC. the planes through AB, BC, and DE, EF Thall not meet though produced.

From the point B draw BG perpendicular to the plane which 2. 11. 11.
passes through DE, EF, and let it meet that plane in G; and
through G draw GH parallel b to ED, and GK parallel to EF. b. 31. s.
and because BG is perpendicular to the plane through DE, EF, it
shall make right angles with e-

E
very straight line meeting it in
that plane 'but the straight B

C. 3. Def.11, lines GH, GK in that plane

K
meet it. therefore each of the
angles BGH, BGK is a right
angle. and because BA is paral- A
lel 4 to GH (for each of them is

H

d. 9. Ir. Parallel to DE, and they are not both in the same plane with it) the angles GBA, BGH are together equal to two right angles. and BGH is a right ande, e. 29. 5. therefore allo GBA is a right angle, and GB perpendicular to l... for the same reason, GB is perpendicular to BC. since therefore the straight line GB stands at right angles to the two stra 2bo lines BA, BC, that cut one another in B; GB is perpendicou: [ 4.15. to the plane through BA, BC. and it is perpendicular to the plane through DE, EF; therefore BG is perpendicular to each of its planes through AB, BC and DE, EF. but planes to which the fame straight line is perpendicular, are parallel to one anot'er. & 14, 17. therefore the plane thro' AB, BC is parallel to the plane te DE, EF. Wherefore if two straight lines, &c. Q: E. D.

Book XI.

PROP. XVI. THEOR.

If two parallel planes be cut by another plane, their
IF

common sections with it are parallels.

Let the parallel planes AB, CD be cut by the plane EFHG, and let their common sections with it be EF, GH. EF is paralici to GH.

For, if it is not, EF, GH shall meet, if produced, either on the side of FH, or LG. first, let them be produced on the side of TH, and meet in the point K. therefore since EFK is in the piane Ab, every point in EFK is in that plane; and K is a point in EFK; therefore K is in the plane AB. for the same renfon K is also in the plane CD. wherefore the

H planes AB, CD produced meet

D one another; but they do not meet, since they are parallel by the Hypothcfis. therefore the A straight lines EF, GH do not E

G meet when produced on the side of TH. in the fame manner it may be proved that EF, GH do not meet when produced on the side of EG. but straight lines which are in the same plane and do not meet, though produced either way, are parallel. therefore EF is parallel to GH. Wherefore if two parallel planes, &c. Q. E. D,

PRO P. XVII. THEOR.
IF

F two straight lines be cut by parallel planes, they
Thall be cut in the same ratio.

Let the straight lines AB, CD be cut by the parallel planes GH, KL, MN, in the points A, E, B; C, F, D. as AE is to EB, fo is CF to FD.

Join AC, BD, AD, and let AD meet the plane KL in the point X; and join EX, XF. because the two parallel planes KL, MN are cut by the plane EBDX, the common sections EX, BD are paral,

del'. for the same reason, because the two parallel planes GH, KL Book XI.
are cut by the plane AXFC,
the common sections AC, XF

a. 16. 11.

10
are parallel. and because EX
is parallel to BD, a side of the

G
triangle ASD, as AE to EB,
so is 6 AX to XD. again, be-

b. 2. 6. cause XF is parallel to AC, a

I
side of the triangle ADC, as E
AX to XD, fo is CF to id. K
and it was proved that AX is
to XD, as AE to EB. therefore

N
e as AE to EB, fo is CT to M

B
FD. Wherefore if two straight

C. II. So lines, &c. Q. E. D.

PRO P. XVIII. THEOR.
I
TF a straight line be at right angles to a plane, every

plane which passes thro' it shall be at right angles
to that plane.

Let the straight line AB be at right angles to the plane CK. every plane which pales through AB shall be at right angles to the plane CK.

Let any plane DE paf; through AB, and let CE be the common
fetion of the planes DE, CK; take any point F in CE, from
which draw FG in the plane

G A H
DE at right angles to CE. and
because AB is perpendicular to
the plane CK, therefore it is also

K
perpendicular to every straight
line in that plane meeting it'.

2.3. Def.adica
and consequently it is perpendi-
cular to CE, wherefore ABF is
a right angle; but GFB is like c
wife a right angle; therefore AB is parallel b to FG. and AB is at b 28.8.
right angles to the plane CK; therefore FG is also at right angles
to the same plane. but one plane is at right angles to another plane c. 8. 11.
when the straight lines drawn in one of the planes, at right angles

Bock XI. to their common section, are also at right angles to the other Wplane d; and any straight line FG in the plane DE, which is at d.4. Def.si. sight angles to CE the common fection of the planes, has been

proved to be perpendicular to the other plane CK; therefore the plane DE is at right angles to the plane CK.

In like manner, it may be proved that all the planes which pass through AB are at right angles to the plane CK. Therefore if a straight line, &c.

Q. E. D.

IF F two planes cutting one another be each of them

perpendicular to a third plane; their common section shall be perpendicular to the fame plane.

Let the two planes AB, BC be each of thein perpendicular to a third plane, and let BD be the common section of the first two. BD is perpendicular to the third plane.

If it be not, from the point D draw, in the plane AB, the
straight line DE at right angles to AD the common section of tre
plane AB with the third plane; and in the plane BC draw DF at
right angles to CD the common fection of
the plane BC with the third plane. and be-

B
cause the plane AB is perpendicular to the
third plane, and DE is drawn in the plane
AB at right angles to AD their common E F

section, DE is perpendicular to the third 2.4. Def.vs. plane ? in the same manner, it may be

proved that DF is perpendicular to the
third plane. wherefore from the point D
two straight lines stand at right angles to D
the third plane, upon the same side of it,

A

C b. 13.01. which is impossible b. therefore from the

point D there cannot be any straight line at right angles to the third plane, except BD the common section of the planes AB, BC. BD therefore is perpendicular to the third plane. Wherefore if two planes, &c. Q. E. D.

Book XI.

PROP. XX. THEOR.

IF a solid angle be contained by three plane angles, See N.

any two of them are greater than the third. Let the folid angle at A be contained by the three plane angles BAC, CAD, DAB. any two of them are greater than the third.

If the angles BAC, CAD, DAB be all equal, it is evident that any two of them are greater than the third. but if they are not, let BAC he that angle which is not less than either of the other two, and is greiter than one of them DAB; and at the point A in the fti aight line AB, make in the plane which passes through BA, AC, the angle BAE equal' to the angle DAB; and make AE 2. 2 3. equal to AD, and through E draw BEC

D cutting AB, AC in the points B, C, and jon DB, DC. and because DA is equal to Al, and AB is common, the two DA, AB are equal to the two EA, AB, and A the angle DAB is equal to the angle EAB. therefore the base DB is equal b

b. 4. t. to the base BE. and because BD, DCB E are greater than CB, and one of thern BD has been proved equal c. 20. 1. to Be a part of CB, therefore the other DC is greater than the remaining part EC. and because DA is equal to AE, and AC comino, but the base DC greater than the base EC; therefore the angle DAC is greater than the angle EAC; and, by the d. 25. 1. construction, the angle DAB is equal to the angle BAE; wherefore the angles DAB, DAC are together greater than the angle BAC. but BAC is not less than either of the angles DAB, DAC, therefore BAC with either of them is greater than the other. Wherefore if a folid angle, &c. Q. E. D.

PROP. XXI. THEOR.
EVERY folid angle is contained by plane angles

which together are less than four right angles. Tirft, Let the folid angle at A be contained by three plane angles BAC, CAD, DAB. these three together are less than four right angles.