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THE

ELEMENTS

O F

U C L I D.

EU

A

воок 1.

DEFINITION S.

I.

Point is that which hath no parts, or which hath no mag- See Notes,

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

V.

A fuperficies is that which hath only length and breadth.

VI.

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

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"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 fame "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 ftraight line.

N. B.

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N. B. When feveral angles are at one point B, any one of them is expreffed by three letters, of which the letter that is at the vertex of the angle, that is, at the point in which the ftraight lines that contain the angle meet one another, is put between the other two letters, and one of these two is fomewhere upon one of thofe ftraight lines, and the other upon the other line: Thus the angle which is contained by the ftraight 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 expreffed by a letter placed at that point; as the Angle at E.'

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X.

When a ftraight line' ftanding on ano-
ther ftraight line makes the adjacent
angles equal to one another, each of
the angles is called a right angle;
and the ftraight line which stands
on the other is called a perpendicular
to it.

XI.

An obtufe 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 inclofed by one or more boundaries.

XV.

XV.

A circle is a plane figure contained by one line, which is called the circumference, and is fuch that all ftraight lines drawn from a certain point within the figure to the circumference, are equal to one another :

Book I.

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A diameter of a circle is a ftraight line drawn through the see N. centre, and terminated both ways by the circumference.

XVIII.

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

XIX.

"A fegment 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.

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Multilateral figures, or polygons, by more than four straight lines.

XXIV.

Of three fided figures, an equilateral triangle is that which has three equal fides.

XXV.

An ifofceles triangle, is that which has only two fides equal

Book I.

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XXVI.

A scalene triangle, is that which has three unequal fides.

XXVII.

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

An obtufe angled triangle, is that which has an obtufe angle.

XXIX.

A

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

XXX.

Of four fided figures, a fquare 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 equal, but its angles are, not right angles.

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XXXIII.

See N. A rhomboid, is that which has its oppofite fides equal to one another, but all its fides are not equal, nor its angles right angles.

XXXIV.

All other four fided figures befides these, are called Trapeziums.
XXXV.

Parallel ftraight lines, are fuch as are in the fame plane, and
which, being produced ever so far both ways, do not meet.

Book I.

POSTULATES.

I.

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

LE

II.

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

III.

And that a cirele may be defcribed from any centre, at any distance from that centre.

AXIOM S.

I.

HINGS which are equal to the fame are equal to one an
other.

TH

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 exactly fill the fame space, are equal to one another.

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