XI.

A figure is that which is enclosed by one or more bounda

ries.

XII.

A circle is a plane figure contained 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.

XIII.

And this point is called the centre of the circle

There is therefore an essential difference between the circumference and the circle, the former being the boundary merely, and the latter the surface enclosed by it. Euclid however frequently employs the term circle, when the circumference only is meant. This is no doubt a violation of the very proper distinction which he has himself laid down, yet as it may be sufficient to have apprized the student here of this two-fold meaning of the term circle, there will be no necessity to alter the phraseology of the work in this respect.

XIV.

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

XV.

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

XVI.

Rectilineal figures are those which are contained by straight lines only.

The word "contained" which is preserved in this definition by all

editors, had better be changed for enclosed, the word employed in the definition of figure. There is a precision in the term enclosed that contained wants. An angle is said to be contained by its sides or the lines which form it, although these lines do not enclose any thing. A like change might be made in the definition immediately preceding, as also in definition XII.

XVII.

Trilateral figures, or triangles, by three straight lines.

XVIII.

Quadrilateral figures, by four straight lines.

XIX.

Multilateral figures, or polygons, by more than four straight

lines.

XX.

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

XXI.

Δ
Δ

An isosceles triangle is that which has two sides equal.

XXII.

A scalene triangle is that which has three unequal sides.

XXIII.

A right-angled triangle is that which has a right angle.

XXIV.

An obtuse-angled triangle is that which has an obtuse angle.

XXV.

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

If a triangle have neither a right angle, nor an obtuse angle, all its angles must be acute.

XXVI.

Of four-sided figures, a square is that which has all its sides equal, and all its angles right angles.

This definition has been long known to be objectionable on the ground of its redundancy. More is said of a square than is absolutely necessary to characterize it, as will be seen in the 46th proposition of the first book.

XXVII.

An oblong or rectangle is that which has all its angles right angles, but has not all its sides equal.

XXVIII.

A rhombus is that which has all its sides equal, but its angles are not right angles.

XXIX.

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.

XXX.

0

All other four-sided figures besides these, are called Trape

ziums.

XXXI.

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

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.

AXIOMS.

I.

THINGS which are equal to the same thing, are equal to one another.

II.

If equals be added to equals, the wholes will be equal.

III.

If equals be taken from equals, the remainders will be equal.

IV.

If equals be added to unequals, the wholes will be unequal.

V.

If equals be taken from unequals, the remainders will be 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, or which may be conceived to coincide with one another, that is, which exactly fill the same space, are equal to one another.

This axiom, referring to the most obvious test of geometrical equality, is scarcely comprehensive enough. Probably it would be an improvement to omit the words "that is, which exactly fill the same space;" as equality among ungles which do not fill space seems, by this condition, to be excluded.

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.

See NOTES at the end.