THE

ELEMENTS OF EUCLID.

BOOK I.

DEFINITIONS.

1. A point is that which hath no parts, or which hath no mag- Book I.

nitude.

See Notes.

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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 ex-

treme points.
V. A superficies is that which hath only length and breadth.
VI. The extremities of a superficies are lines.
VII. A plain superficies is that in which any two points being See N.

taken, the straight line between them lies wholly in that su

perficies. VIII. “ A plane angle is the inclination of two lines to one See N.

" another in a plane, which meet together, but are not in

66 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.

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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 somewhere upon one of those straight lines, and the other upon

the other line: Thus the angle which is contained by the 6 straight lines AB, CB is named the angle ABC, or CBA; 6 that which is contained by AB, BD is named the angle ABD, or DBA ; and that which is contained by BD, 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 adja-
cent 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.
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 bound-

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

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 centre, and terminated both ways by the circumference. XVII. 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 sides. XXV. An isosceles triangle is that which has (only) two sides

equal.

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Book I. 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 ob-

tuse 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

sides equal, and all its angles right angles.

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.

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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 angles.
XXXIV. All other four sided figures besides these are call-

ed 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.

Book I.

POSTULATES.

1. 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 distance from that centre.

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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 un-

equal.
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.
IX. The whole is greater than its part.
X. Two straight lines cannot enclose a space.
XI. All right angles are equal to one another.
XII. “ If a straight line meets two straight lines, so as to

“ make the two interior angles on the same side of it taken
“ together less than two right angles, these straight lines
“ being continually produced, shall at length meet upon that
“ side on which are the angles which are less than two right
- angles. See the notes on Prop. 29. of Book I.”

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