EUCLID's

ELEMENTS.

BOOK I.

DEFINITION S.

A

POINT, is that which hath no Parts, or Magnitude.

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 which hath only Length and Breadth:

VI. The Bounds of a Superficies are Lines.

VII. A Plain Superficies, is that which lieth evenly between its Lines.

VIII. APlain Angle, is the Inclination of two Lines to one another in the fame Plane, which touch each other, but do not both lie in the fame Right Line.

IX. If the Lines containing the Angle be Right ones, then the Angle is called a Right-lin❜d Angle. X. When

B

X. When a RightLine, standing on another Right Line,
makes Angles on either Side thereof, equal between
themfelves, each of these equal Angles is a Right one;
and that Right Line which ftands upon the other, is
called a Perpendicular to that whereon it stands.
XI. An Obtufe Angle, is that which is greater than a
Right one.

XII. An Acute Angle, is that which is less than a Right

one.

XIII. ATerm (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'dFigures, are fuch as are contain'd under
Right Lines.

XXI. Three-fided Figures, are fuch as are contained un-
der three Lines.

XXII. Four-fided Figures, are fuch as are contain❜dunder four.

XXIII. Many fided Figures, are thofe that are contain'd
under more than four Right Lines.

XXIV. Of three-fided Figures, that is an Equilateral
Triangle, which hath three equal Sides.

XXV. That an Ifofceles, or Equicrural one, which hath
only two Sides equal.

XXVI. And a Scalene one, is that which hath three unequal Sides.

XXVII. Also of Three-fided Figures, that is, a Rightangled Triangle, which hath a Right Angle.

XXVIII.

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

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

XXX. Of Four-fided Figures, that is a Square, whofe four Sides are equal, and its Angles all Right ones. XXXI. That an Oblong, or Rectangle, a Figure which is longer on one fide 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, whofe oppofite Sides and Angles only are equal.

XXXIV. All Quadrilateral Figures, befides these, are called Trapezia.

XXXV. Parallels are fuch Right Lines in the fame Plane, which if infinitely produc'd both Ways, would

never meet.

I.

G

POSTULATE S.

RANT that a Right-Line may be drawn from any one Point to another.

II. That a finite Right Line may be continued directly forwards.

III. And that a Circle may be defcrib'd about any Center, with any Distance.

I.

HINGS equal to one and the fame
Thing, are equal to one another.

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 ungeual.

VI. Things which are double to one and the fame Thing, are equal between themselves.

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

VIII. Things which mutually agree together, are equal to one another.

IX. The Whole is greater than its Part.0

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 fame Side thereof, both together, less than two Right Angles, thofe two Right Lines, infinitely produc'd, will meet each other on that Side where the Angles, are lefs than Right ones.

NOTE, When there are feveral Angles at one Point, any one of them is exprefs'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 call'd the Angle A BE.

PRO

PROPOSITION I.

PROBLEM.

To defcribe an Equilateral Triangle upon a given finite

L

Right Line.

ET AB be the given finite Right Line,
upon which it is required to describe an
Equilateral Triangle.

About the Center A, with the Distance AB, defcribe the Circle BCD *; * 3 Post, and about the Center B, with the fame Distance B A, defcribe the Circle ACE; and from the Point C, where the two Circles cut each other, draw the Right Lines CA, CBt.

Then because A is the Center of the Circle DBC, AC fhall be equal to AB. And because B is the Center of the Circle CAE, BC fhall be equal to BA': but CA hath been proved to be equal to AB; therefore both CA and CB are each equal to AB. But things equal to one and the fame thing, are equal between themselves, and confequently CA is equal to CB; therefore the three Sides CA, AB, BC, are equal

between themselves.

And fo 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

+ Poft.

15 Def.