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XXIV.

Of three-sided figures, an equilateral triangle is that which has three equal sides.

XXV.

An isosceles triangle is that which has 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 quadrilateral or four-sided figures, a square 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 has all its sides equal, but its angles are not right angles.

XXXIII.

A rhomboid has its opposite sides equal to each other, 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.

A.

A parallelogram is a four-sided figure, of which the opposite sides are parallel: and the diameter, or the diagonal is the straight line joining two of its opposite angles.

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.

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AXIOMS.

I.

THINGS which are equal to the same thing 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.

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.

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To describe an equilateral triangle upon a given finite straight line.
Let AB be the given straight line.

It is required to describe an equilateral triangle upon AB.

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From the center A, at the distance AB, describe the circle BCD; (post. 3.)

from the center B, at the distance BA, describe the circle ACE; and from C, one of the points in which the circles cut one another, draw the straight lines CA, CB to the points A, B. (post. 1.) Then ABC shall be an equilateral triangle.

Because the point A is the center of the circle BCD,
therefore AC is equal to AB; (def. 15.)

and because the point B is the center of the circle ACE,
therefore BC is equal to AB;

but it has been proved that AC is equal to AB;
therefore AC, BC are each of them equal to AB;

but things which are equal to the same thing are equal to one another; therefore AC is equal to BC; (ax. 1.)

wherefore AB, BC, CA are equal to one another:
and the triangle ABC is therefore equilateral,
and it is described upon the given straight line AB.
Which was required to be done.

PROPOSITION II. PROBLEM.

From a given point, to draw a straight line equal to a given straight line.
Let A be the given point, and BC the given straight line.
It is required to draw from the point A, a straight line equal to BC.

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From the point A to B draw the straight line AB;
upon AB describe the equilateral triangle ABD, “(1. 1.)

(post. 1.)

(post. 2.)

from the center B, at the distance BC, describe the circle CGH,

and produce the straight lines DA, DB to E and F;

(post. 3.) cutting DF in the point G:

and from the center D, at the distance DG, describe the circle GKL, cutting AE in the point L.

Then the straight line AL shall be equal to BC.
Because the point B is the center of the circle CGH,
therefore BC is equal to BG; (def. 15.)
and because D is the center of the circle GKL,
therefore DL is equal to DG,

and DA, DB parts of them are equal; (1. 1.) therefore the remainder ÂL is equal to the remainder BG; (ax. 3.) but it has been shewn that BC is equal to BG,

wherefore AL and BC are each of them equal to BG;

and things that are equal to the same thing are equal to one another; therefore the straight line AL is equal to BC. (ax. 1.) Wherefore from the given point A, a straight line AL has been drawn equal to the given straight line BC. Which was to be done.

PROPOSITION III. PROBLEM.

From the greater of two given straight lines to cut off a part equal to the less.

Let AB and C be the two given straight lines, of which AB is the greater.

It is required to cut off from AB the greater, a part equal to C, the less.

D

EB

From the point A draw the straight line AD equal to C; (1. 2.) and from the center A, at the distance AD, describe the circle DEF (post. 3.) cutting AB in the point E.

Then AE shall be equal to C.

Because A is the center of the circle DEF,
therefore AE is equal to AD; (def. 15.)
but the straight line Cis equal to AD; (constr.)
whence AE and Care each of them equal to AD;
wherefore the straight line AE is equal to C. (ax. 1.)

And therefore from AB the greater of two straight lines, a part AB has been cut off equal to C, the less. Which was to be done.

PROPOSITION IV. THEOREM.

If two triangles have two sides of the one equal to two sides of the other, each to each, and have likewise the angles contained by those sides equal to each other; they shall likewise have their bases or third sides equal, and the two triangles shall be equal, and their other angles shall be equal, each to each, viz. those to which the equal sides are opposite.

Let ABC, DEF be two triangles, which have the two sides AB, AC equal to the two sides DE, DF, each to each, viz. AB to DE, and AC to DF, and the included angle BAC equal to the included angle EDF.

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