EUCL I D's

I ELEMENTS

BOOK I.

DEFINITIONS.

1.

POINT, is that which hath no Parts,

or Magnitude. А II. A Line is Length, without Breadth:

III. The Ends (or Bounds) of a Line,

are Points. IV. A Right Line, is that which lieth evenly between its

Points. V. A Superficies, is that wbich bath only Length and

Breadth: VI. The Bounds of a Superficies are Lires. VII. A Plain Superficies, is that which, lieth evenly be

tween its Lines. VIII. A Plain Angle, is the Inclination of two Lines to

one another in the same Plane, which touch each other, but do not both lie in the same Right Line. IX. If the Lines containing the Angle be Right ones, then the Angle is called a Right-lind Angle.

B

X. When

T

a

one.

X. When a RightLine, standing on another Right Line,

makes Angles on either Side thereof, equal between themselves, each of these equal Angles is a Right one; and that Right Line which stands upon the other, is.

called a Perpendicular to that whereon it stands.
XI. An Obtuse Angle, is that which is greater than a

Right one.
XII. An Acute Angle, is that which is less than a Right
XIII. A Term (or Bound) is that which is the Extreme

of any Thing.
XIV. A Figure, is that which is contained under one,

or more Terms.
XV. A Circle, is a plain Figure, contain'd under one

Line, called the Circumference; to which all Right
Lines, drawn from a certain Point within the Figure,
are equal.
XVI. And that Point is called the Center of the Circle.
XVII. A Diameter of a Circle, is a Right Line drawn

through the Center, and terminated on both Sides by
the Circumference, and divides the Circle into two
equal Parts.
XVIII. A Semicircle, is a Figure contain'd under a Di-

ameter, and that part of the Circumference of a Circle

cut off by that Diameter. XIX. A Segment of a Circle, is a Figure contain d un

der a Right Line, and Part of the Circumference of

the Circle [which is cut off by that Right Line.]
XX. Right-lin'd Figures, are such as are contain' d under

Right Lines.
XXI. Three-sided Figures, are such as are contained un-

der three Lines.
XXII. Four-sided Figures, are such as are contain’d un-

der four. XXIII. Many sided Figures, are those that are contain'd

under more than four Right Lines. XXIV. Of three-sided Figures, that is an Equilateral

Triangle, which hath three equal Sides. XXV. That an Isosceles, or Equicrural one, which hath only

two Sides equal. XXVI. And a Scalene one, is that which hath three un

' equal Sides, XXVII. Also of Three-sided Figures, that is, a Rightangled Triangle, which bath a Right Angle.

XXVIII.

a

XXVIII. That an Obtuse-angled one, which hath an

Obtuse Angle. XXIX. And that an Acute-angled one, which hath three

Acute Angles. XXX. Of Four-sided Figures, that is a Square, whose

four Sides are equal, and its Angles all Right ones, XXXI. That an Oblong, or Rectangle, a Figure which

is longer on one side than the other, which is Rightangled, but not equal fided. XXXII. That a Rhombus, which hath four equal Sides,

but not Right Angles. XXXIII. That a Rhomboides, whose opposite Sides and

Angles only are equal. XXXIV. All Quadrilateral Figures, besides these, are

called Trapezia. XXXV. Parallels are such Right Lines in the same

Plane, which if infinitely produc'd both Ways, would

a

never meet.

POST U L A T E S.

COM

1.

RANT that a Right-Line may be drawn

from any one point to another.
G
II. That a finite Right Line

may

be tinued directly forwards. III. And that a Circle may be describ's about any

Center, with any Distance.

1. HINGS equal to one and the same 17

Thing, are equal to one another.
T

II. If to equal Things, are added equal

Things, the Wholes will be equal. III. If from equal Things, equal Things be taken away,

the Remainders will be equal. IV. If equal Things be added to unequal Things, the

Wholes will be unequal. V. If equal Things be taken from unequal Things, the

Remainders will be ungenal. VI. Things which are double to one and the same Tbing,

are equal between themselves. VII. Things, which are' half one, and the same Thing,

are equal between themselves. VIII, Things which mutually agree together, are equal

to one another. IX. The Wbole is greater than its-Part

.. X. Two Right Lines do not contain a Space. XI. All Right Angles are equal between themselves. XII. If a Right Line, falling upon two other Right

Lines, makes the inward Angles on the same Side thereof, both together, less than two Right Angles, those two Right Lines; infinitely produc’d, will meet each other on that Side where the Angles, are less than Right ones.

Note, When there are several Angles at one Point,

any one of them is express’d by three Letters, of which that at the Vertex of the Angle is plac'd in the Middle. For Example ; In the Figure of Prop. XII. Lib. I. the Angle contain’d under the Right Lines AB, BC, is called the Angle ABC; and the Angle contain’d under the Right Lines AB, BE, is calựd the Angle A BE.

PRO

PROBLEM.
To describe an Equilateral Triangle upon a given finite

Right Line.

ET AB be the given finite Right Line,

upon which it is required to describe an L Equilateral Triangle.

About the Center A, with the Distance AB, describe the Circle BCD*;* 3 Polie

and about the Center B, with the same Distance BA, describe the Circle ACE; and from the Point C, where the two Circles cut each other, draw the Right Lines CA, CBt.

+ Poft. Then because A is the Center of the Circle DBC, AC shall be equal to A Bt. And because B is the $ 15 Def. Center of the Circle CAE, BC shall be equal to BA: but CA hath been proved to be equal to AB; there- . fore both CA and CB are each equal to AB. But things equal to one and the same thing, are equal between themselves, and consequently

CA is equal to CB; therefore the thțee Sides CA, A B, BC, are equal between themselves.

And so the Triangle BAC is an Equilateral one, and is described upon the given finite Right Line AB; which was to be done.

PROPOSITION II.

PROBLEM.

At a given Point, to put a Right Line equal to a

Right Line given.
LE
ET the Point given be A, and the given Right Line
BC; it is required

to put a Right Line at the Point A, equal to the given Right Line BC.

B 3

Draw