BOOK I. but that an isosceles triangle may or may not be equilateral.]

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.

[Observe that no triangle can have more than one right angle, nor more than one obtuse angle.

Also a right angled triangle may be isosceles or not.

Figs. 1 and 2. are both right angled triangles. Fig. 1. is isosceles; fig. 2. is not isosceles.

Also an obtuse angled triangle may be isosceles or not.

Also an acute angled triangle may have all its three sides BOOK I. unequal, in which case it is scalene; or two of them equal, in which case it is isosceles; or all three sides equal, in which case it is equilateral.

a

Fig. 1. is a scalene acute angled triangle.
Fig. 2. is an isosceles acute angled triangle.
Fig. 3. is an equilateral acute angled triangle.]

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 its sides equal, but its angles

are not right angles.

DO

BOOK I.

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 called Trape-

ziums.

XXXV.
Parallel straight lines are such as are in the same plane, and
which, being produced ever so far both ways, do not meet.
Fig. 1.

Fig. 2.

[Fig. 1. lines parallel ; fig. 2. lines not parallel.
Observe particularly the words, in the same plane.]

a

A.
A parallelogram is a four-sided figure, of which the opposite

sides are parallel; and the diameter is the straight line
joining two of its opposite angles.
Fig. 1.

Fig. 2.

Fig. 3.

[All the above figures are parallelograms; those below are not parallelograms, both being four-sided figures. Fig. 4. bas one pair of opposite sides parallel.

Fig. 4.

Fig. 5.

Fig. 5. has no pair of parallel sides. ]

BOOK I.

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

distance from that centre.

[These three postulates are equivalent in practice to allowing a pen, a ruler, and a pair of compasses. Having these, the reader will be able to draw any of the figures in this book.]

AXIOMS.

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.

BOOK I.

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.

[In fig. 1. no single additional straight line could be made to inclose a space; in fig. 2. a third straight line could be drawn across the other two so as to inclose 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.”

[These are truths which no person can possibly doubt, who understands the terms in which they are expressed. There