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X. Equal and fimilar folid Figures, are thofe that are contained under equal Numbers of fimilar and equal Planes.

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XỈ. A folid Angle, is the Inclination of more than two Right Lines that touch one another, and are not in the fame Superficies Or, a folid Angle is that which is contained under more than two plane Angles which are not in the fame Superficies, but being all at one Point.

XII. A Pyramid is a folid Figure contain❜d under Planes fet upon one Plane, and put together at one

Point.

XIII. A Prism is a folid Figure contained under Planes, whereof the two oppofite are equal, fimilar, and parallel, and the others Parallelograms.

XIV. A Sphere is a folid Figure, made when the Diameter of a Semicircle, remaining at reft, thé Semicircle is turned about till it returns to the famé Place from whence it began to move.

XV. The Axis of a Sphere is that fixed Line about which the Semicircle is turned.

* XVI. The Centre of a Sphere is the fame with that of the Semicircle.

XVII. The Diameter of a Sphere, is a Right Line drawn thro' the Center, and terminated on either Side by the Superficies of the Sphere.

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XVIII. A Cone is a Figure defcribed when one of the Sides of a Right-angled Triangle, containing the Right Angle, remaining fixed, the Triangle is turned about till it returns to the Place from whence it first began to move. And if the fixed Right Line be equal to the other that contains the Right Angle, then the Cone is a rectangular Cone; but if it be lefs, it is an obtuse-angled Cone; if greater, an acute-angled Cone:

XIX. The Axis of a Cone is that fixed Right Line about which the Triangle is moved.

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XX. The Bafe of a Cone is the Circle defcribed by the Right Line moved about.

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XXI. A Cylinder is a Figure defcribed by the Motion of a Right-angled Parallelogram, one of the Sides containing the Right Angle, remaining fixed, while the Parallelogram is turned about to the fame Place from whence it began to be moved.

XXII. The Axis of a Cylinder is that fixed Right Line about which the Parallelogram is turned. XXIII. And the Bafes of a Cylinder are the Circles that be defcribed by the Motion of the two oppofite. Sides of the Parallelogram.

XXIV. Similar Cones and Cylinders are fuch, whofe Axes and Diameters of their Bafes are proportional.

XXV. A Cube is a folid Figure contained under fix equal Squares.

XXVI. A Parallelepipedon is a Figure contained under fix quadrilateral Figures, whereof those which are oppofite are parallel.

XXVII. A Polyhedron is a Solid of many Sides

or Faces.

PROP. I.

BT

D

A

E

One part AC of a Right Line cannot be in a Plane, and another part CB without the fame.

Continue out AC in the Plane to F; then if CB be in the fame ftreight Line with AC,

two right Lines AB, AF will have a common

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abfurd.

PROP

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DEB to be in one Plane,

For fuppofe the Part EGF of the Triangle and the part GDFB

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to be in another; then the part EF of the Right Line EB is in one Plane, and the part FB in another, which is abfurd. Therefore a 1. 11. the Triangle EDB is in one Plane. And fo likewife are the Right Lines ED, EB; confequently the Wholes AB, DC are a in one Plane. Q. E. D.

A

a

PROP. III.

If two Planes AB,
CD mutually cut each
other, their common
Section EF is a Right
Line.

E

H

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tion EF be not a D Right Line,

b` draw

the Right Line EGF in the Plane AB, and the Right Line EHF in the Plane CD. Therefore two Right Lines EGF, EHF, include a Space, which is abfurd.

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If the Right Line EF ftands at Right Angles to two Right Lines AB, CD, mutually cutting one another, in the common Section E; it shall also be at Right Angles to the Plane ACBD drawn thro' them.

M 3

.b

Take

1. poft. 1.

14 ax. I.

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a fchol.

34. 1.

e

29. I.

conft. 3.26.1.

h

11 4. I.

i 8. 1.

с

parallel to CB, and AC parallel
To the Ang. GAE = EBH.
AGE=ĚHB. But alfo AE

GE 8

=

h

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с

to DB; and and the Ang. EB. whence GE EH, and AG BH. Wherefore because of the Right Angles (from the Hyp.) and fo confequently equal ones at E, the Bases FA, FC, FB, FD are equal. Therefore the Triangles ADF, FBC, are mutually Equilateral to each other; and fo the Ang. DAF =CBF. Therefore in the Triangles AGF, FBH, the Sides FG, FH are equal; and confequently likewise the Triangles FEG, FEH, are mutually equilateral. Therefore the Angles FEG, FEH, are equal; and accordingly" right ones. 10def.1. In like manner FE is at right Angles to all Lines drawn thro' E in the Plane ADBC. and 3 def.11, therefore it is "at right Angles to that fame Plane. QE. D.

4. I.

18. I.

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If a Right Line AB ftands at Right Angles to three Right Lines AC, AD, AE, mutually cutting one another, on their common Section; thofe three Right Lines are in one Plane.

For

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Planes. Now because BA, by the Hypoth. is perpendicular to the Right Lines AC, AD, it

is

b

C

perpendicular to the fame Plane FC, and 4. 11. confequently to the Right Line AD. There-3 def.xx. fore (fince AB is alfo in the fame Plane AE,

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as AG is) the Angles BAG, BAE are right ones, and fo equal, the Part to the Whole, which is abfurd.

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If two Right Lines AB, DC, be at Right Angles to the fame Plane EF, thofe Right Lines, AB, DC, are Parallel.

Draw AD, to which let DG AB be perpendicular in the Plane EF; and join BD, BG, AG. In the Triangles BAD,

e

e

f

DG; and AD is com- conft.

h

=

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ADG, because the Angles DAB, ADG are hyp. right ones; and AB mon; therefore fhall BD = AG; wherefore in the Triangles AGB, BGD mutually equilateral to each other, the Ang. BAG BDG; S. I. whereof BAG being a right one, BDG fhall be fo alfo. But the Ang. GDC is fuppofed to be a right one; therefore the right Line GD is perpendicular to three right Lines, DA, DB, CD; which therefore are in one Plane, in i 5. 11.

M 4

i

which

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