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Τ Η Ε

Ε Ι Ε E M E E N T S

OF

E U CL

L I

I D.

BOOK I.

D E F IN I TI ON S.

Pointi

I.
Point is that which hath no parts, or which hath no magnitude. See Not:s,

II.
A line is length without breadth.

IU.
The extremities of a line are points.

IV.
A straight line is that which lies evenly between its extreme points.

V.
A superficies is that which hath only length and breadth.

VI.
The extremities of a superficies are lines.

VII. A plane superficies is that in which any two points being taken, See N. the straight line between them lies wholly in that superficies.

VIII. " A plane angle is the inclination of two lines to one another in a See N. plane, which meet together, but are not in the same direction.”

IX.
A plane rectilineal angle is the inclination of two straight lines to one

another, which meet together, but are not in the fame straight
line.

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BK C E * N.B. When several angles are at one point B, anyone of them is expressed by three letters, of which the letter that is at ihe ves. tex of the angle, that is at the point in which the straight lines that contain the angle meet one another, is put between the other two letters, and one of these two is somewhere upon one of those straight lines, and the other upon the other line. thus the angle which is contained by the straight lines AB, CB is

named the angle ABC, or CBA ; that which is contained by * AB, DB is named the angle ABD, or DBA ; and that which is

contained by DB, CB is called the angle DBC, or CBD. but • if there be only one angle at a point, it may be expressed by a • letter placed at that point; as the angle at E.'

X.
When a straight line standing on another

straight line makes the adjacent angles
equal to one another, each of the an.
gles is called a right angle; and the
straight line which stands on the other
is called a perpendicular to it.

XI.
An obtuse angle is that which is greater than a right angle.

XII.
An acute angle is that which is less than a right angle.

XIII.
“ A term or boundary, is the extremity of any thing."

XIV.
A figure is that which is inclosed by one or more boundaries.

Book 1,

XV.
A circle is a plane figure contained by one line, which is called

the circumference, and is such that all straight lines drawn from
a certain point within the figure to the circumference, are equal
to one another,

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XVI.
And this point is called the center of the circle.

XVII.
A diameter of a circle is a straight line drawn thro' the center, and See N
terininated both ways by the circumference.

XVIII. A semicircle is the figure contained by a diameter and the part of the circumference cut off by the diameter.

XIX. « A segment of a circle is the figure contained by a straight line " and the circumference it cuts off.”

XX.
Rectilineal figures are those which are contained by straight
lines.

XXI.
Trilateral figures, or triangles, by three straight lines.

XXII.
Quadrilateral, by four straight lines.

XXIII.
Multilateral figures, or Polygons, by more than four straight lines.

XXIV.
of three sided figures, an equilateral triangle is that which has
three equal fides.

XXV.
An isosceles triangle, is that which has only two sides equal,

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Book 1.

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XXVI.
A scalene triangle, is that which has three unequal sides.

XXVII.
A right angled triangle, is that which has a right angle.

XXVII.
An obtuse angled triangle, is that which has an obtuse angle,

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XXIX.
An acute angled triangle, is that which has three acute angles.

XXX.
Of four sided figures, a square is that which has all its fides

equal, and all its angles right angles.

XXXI.
An oblong is that which has all its angles right angles, but has
not all its fides equal.

XXXII.
A rhombus is that which has all its fides cqual, but its angles,

are not right angles.

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See N.

XXXIII.
A rhomboid is that which has its opposite sides equal to one ano-

ther, but all its fides are not equal, nor its angles right angles.

Book 1.

XXXIV.
All other four sided figures besides these, are called Trapeziums.

XXXV.
Parallel straight lines, are such as are in the same plane, and

which being produced ever so far both ways, do not meet.

POSTULAT E S.

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I.
E T it be granted that a straight line may be drawn from
any one point to any other point.

II.
That a terminated straight line may be produced to any length
in a straight line.

III.
And that a circle may be described from any center, at any

distance from that center.

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A X I 0 MS.

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I.
HINGS which are equal to the same are equal to one
another.

II.
If equals be added to equals, the wholes are equal.

III.
If equals be taken from equals, the remainders are equal.

IV.
If equals be added to unequals, the wholes are unequal.

V.
If equals be taken from unequals, the remainders are unequal.

VI.
Things which are double of the fame, are equal to one another.

VII.
Things which are halves of the fame, are equal to one another.

VIII.
Magnitudes which coincide with one another, that is which exacto
ly fill the same space, are equal to one another.

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