THK

ELEMENTS OF EUCLID.

BOOK I.

DEFINITIONS.

I. A POINT is that which hath no parts, or which hath no mag. See Notes. nitude.

II. A line is length without breadth.

III. 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 See N. " in a 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 same
straight line.

B

N. B. " When several angles are at one point B, any one of ? them is expressed by three letters, of which the letter that is at the vertex 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 some

where 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 o is contained by AB, DB is named the angle ABD, or DBA; 6 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
expressed by the letter placed at that point; as the angle at E."

X.
When a straight line standing on ano-

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

be

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 enclosed by one or more boundaries.

XV.

Book I. 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:

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

XVII.
A diameter of a circle is a straight line drawn through the cen-
tre, and terminated both ways by the circumference.

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

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

Book L

ΔΔ Δ

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

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

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

A

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 sides

equal, and all its angles right angles.

1

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

XXXII.
A rhombus, is that which has all its sides equal, but its angles

are not right angles.

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

another, but all its sides are not equal, nor its angles right
apgles.

Book I.

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.

POSTULATES.

I: Let 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 centre, at any dig.

tance from that centre.

AXIOMS.

1.
THINGS 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 same, are equal to one another.

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

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

The whole is greater than its part.