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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 straight lines A B, CB, is named the angle ABC, or CBA; that which is contained by AB, BD, 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 angles is called a right angle; and the straight line which stands on the other is called a per
pendicular 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
XIII.“ A term or boundary is the extremity of
anything." XIV. A figure is that which is inclosed by one or
more boundaries. XV. A circle is a plane figure con
tained 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 centre, 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 the diameter. XIX. “A segment of a circle is the figure con
tained by a straight line, and the circumference it
cuts off.” XX. Rectilineal figures are those which are con
tained 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 tri
angle is that which has three equal sides. XXV. An isosceles triangle is that which has only
two sides equal.
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.
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. , rhombus, is that which has all its sides
equal, but its angles are not right angles. XXXIII. A rhomboid, is that which has its op
posite 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 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.
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 pro
duced to any length in a straight line. III. And that a circle may be described from any
centre, at any distance from that centre.
I. 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.
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."
PROPOSITION I. PROBLEM. To describe an equilateral triangle upon a given finite straight line.
Let A B be the given straight line; it is required to describe an equilateral triangle upon it.
From the centre A, at the distance A B, describe (Postulate 3.) the circle BCD, and from the centre B, at the distance BA, describe the circle ACE; and from the point c, in which the circles cut one another, draw the straight lines (Post. 1.) CA, CB, to the points A, B; ABC shall be an equilateral triangle.
Because the point A is the centre of the circle BCD, AC is equal (Definition 15.) to AB; and because the point B is the centre of the circle ACE, BC is equal to BA: But it has been proved that ca is equal to AB; therefore CA, CB are each of them equal to AB; but things which are equal the same are equal to one another (Axiom 1.); therefore ca is equal to CB; wherefore ca, AB, BC are equal to one another ; and the triangle ABC is therefore equilateral, and it is described upon the given straight line A B. Which was required to be done.