Prop. 13. Cor.1. A plane surface may be prolonged to any extent in a plane.
Prop.13. Cor.2.

If a straight line in a plane be prolonged, its prolongation lies

wholly in that same plane. Prop. 13. Nom.

If a given straight line in a plane be turned in
that plane about one of its extremities which re-
mains at rest, till the straight line is returned to

the situation from which it set out, the plane
figure described by such straight line is called a circle, and its
boundary the circumference. The point in which the extremity of
the straight line remains at rest, is called the centre of the circle.
Any straight line drawn from the centre of a circle to the cir.
cumference, is called a radius of the circle; and any straight line
drawn through the centre and terminated both ways by the
circumference, is called a diameter of the circle.

When a circle is said to be described about the centre A with the radius

AB, the meaning is, that it is described by the revolution of the

given straight line AB about the extremity A. Prop.13. Cor.3.

A circle may be described about any centre and with any

radius. Prop.13. Cor. 4.

All the radië of the same circle are equal. And circles that

have equal radii, are equal. Prop.13. Cor.5.

A straight line from the centre of a circle to a point outside,

coincides with the circumference only in a point. Prop. 14.

Any three points are in the same plane. [That is to say,

one plane may be made to pass through them all.] Prop. 14. Cor.1.

Any three points which are not in the same straight line being joined, the straight lines which are the sides of the three-sided

figure that is formed lie all in one plane. "Prop. 14. Cor.2.

Any two straight lines which proceed from the same point, lie

wholly in one plane. Prop. 14. Cor.3. If three points in one plane (which are not in the same straight

line) are made to coincide with three points in another plane; the planes shall coincide throughout, to any extent to which they may be prolonged.

(The above Recapitulation contains the principal matters

likely to be referred to. But should reference be made to any thing that is not found in it, recourse is to be had to the Intercalary Book.)

FIRST BOOK continued from page 6.

XXXIII. Lines formed by the junction of several straight lines

which are not in one and the same straight line, and surfaces
formed by the junction of several plane surfaces which are not in
one and the same plane, are called compound. Lines other than
straight lines, and surfaces other than plane surfaces, [that is
to say, which are neither such, nor compounded of such,] are

called curved. XXXIV. Straight lines which proceed from the same point but

do not afterwards coincide, are said to be divergent.

XXXV. If through two divergent straight lines of unlimited * Interc,14. length a plane be* made or supposed to pass, and another straight Cor. 2.

line of unlimited length be turned about the point from which the two divergent straight lines proceed, continuing ever in the same plane with them, and so travel from the place of one to the place of the other; such travelling straight line is called the

radius vectus. XXXVI. The plane surface (of unlimited extent in some direc

tions but limited in others) passed over by the radius vectus in

travelling from one of the divergent straight lines to the other, See Note.

is called the angle between them.
Hence angles are compared together by
their extension sideways only; without

reference to the greater or smaller length
of the straight lines between which they

B' -C E4
lie. Thus the angle between the straight lines BC and BA, is
greater than that between BC and BD, or that between BD and
BA; and is in fact equal to their sum. Also if the radius vectus in-
stead of moving by the shortest road from BC to BA, should go round
by the contrary way, the plane surface so passed over is likewise an
angle. Such an angle may be called circuitous; and the other where

the radius vectus goes by the nearest road, direct. When several direct angles are at one point B, any one of them is ex

pressed by three letters, of which the letter that is at the vertex of the angle, [that is, at the point from which the straight lines that make the angle, proceed], is put in the middle, and one of the remaining letters is somewhere upon one of those straight lines, and the other upon the other. Thus the angle between the straight lines BA and BC, is named the angle ABC, or CBA ; that between BA and BD, is na ed the angle ABD, or DBA; and that between

BD and BC, is named the angle DBC, or CBD. But if there be only one such angle at a point, it may be named from a letter placed at that point; as the angle at E, or more briefly still, the

angle E. If the angle intended is the circuitous one, it must be expressed by the

use of the term, or something equivalent. But whenever the contrary

is not expressed, it is always the direct angle that is meant. XXXVII. 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 perpendicular to it;

and is also said to be at right angles to it. XXXVIII. An angle greater than a right angle,

is called obtuse. XXXIX. An angle less than a right angle, is called

acute. XL. Angles greater or less than right angles, are called by the

common title of oblique. XLI. A straight line joining the ex

tremities of any portion of the circum-
ference of a circle, is called a chord.


The same name is applied to a straight line
joining the extremities of any series of
straight lines that approaches to the like
form. The points A, B, where the
chord meets the ends of the series, are called the cusps; and the

angles A and B, the angles at the cusps. XLII. Figures which are bounded by straight lines, are called rectilinear. Linear figures of all kinds are understood to lie wholly in one plane,

when the contrary is not expressed. XLIII. Of rectilinear figures, such as are contained by three straight lines, are called triangles. XLIV. Those contained by four straight lines, are called quad

rilateral. XLV. Those contained by more than four, are called polygons.

Figures in which a number of sides is specified or intimated, are

always understood to be rectilinear, when the contrary is not expressed.


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XLVI. Of triangles, such as have two sides equal,

are called isoskeles. XLVII. A triangle which has all its three sides equal, is called equilateral. Hence all equilateral triangles are at the same time isoskeles ; but

isoskeles triangles are not all equilateral. XLVIII. A triangle which has a right angle, is called right-angled. The side opposite to the right angle is called the hypotenuse.

The other two sides are sometimes called the base and perpendicular. XLIX. A triangle which has an obtuse angle, is

called obtuse-angled. L. A triangle which has all its angles acute, is called

acule-angled. LI. A triangle which has all its angles oblique, is called

oblique-angled. LII. The nomenclature of the various kinds of quadrilateral

figures cannot with propriety be given, till these figures have been shown to be capable of possessing certain properties from which their distinctions are derived. It is therefore to be found in the places where such properties are demonstrated. (See the Nomenclature at the end of Propositions XXVIII A, XXXIII,

and XXXIV bis, of the First Book. LIII. In any quadrilateral figure, a straight line joining two of the opposite angular points is called a diagonal. For brevity, quadrilateral figures may be named by the letters at two

of their opposite angles, when no obscurity arises therefrom. LIV. Of polygons, such as have five, six, seven, eight, nine,

ten, eleven, twelve, and fifteen sides respectively, are called a pentagon, hexagon, heptagon, oktagon, enneagon, dekagon, hendekagon, dodekagon, pendekagon. For polygons with other numbers of sides, names might probably

be found, or be framed from the Greek ; but they are not in

common use.

SCHOLIUM.—Henceforward all lines, angles, and figures linear or superficial, whether single or formed by the junction of many, will be understood to lie wholly in one plane, viz. the plane of the paper on which they are represented; when the contrary is not expressed.


See Note. PROBLEM.-To describe an equilateral triangle upon a given

straight line.

Let AB be the given straight line. It is required to describe an equilateral triangle upon it.

About the centre A, with the radius AB, de-
Cor. 3.

scribe* the circle BCD; and about the centre B, D A B E
with the radius BA, describe the circle ACE.
Because the circles ACE and BCD pass through
each other's centres, each will cross the other in two places and be

crossed by it. From a point in which the circles meet (as for + INTERC. 9. instance C) drawt the straight lines CA, CB, to the points A and Cor.

B. ABC shall be an equilateral triangle.

Because the point A is the centre of the circle BCD, and C INTERC.13. and B are points in the circumference, AC ist equal to AB. And Cor. 4.

because the point B is the centre of the circle ACE, and C and A are points in the circumference, BC is equal to AB. But it has been shown that AC is equal to AB; therefore AC and BC

are each of them equal to AB. And things which are equal to the *INTERC. 1.

same, are* equal to one another; therefore AC is equal to BC. Wherefore AC, BC, AB are equal to one another, and the triangle ABC is equilateral ; and it is described upon the given straight line AB. Which was to be done.

And by parity of reasoning, the like may be done in every other instance.

SCHOLIUM. --It has not yet been proved that the place where the two circles cross one another is only a point. There might, therefore, for all that has yet been proved, be more equilateral triangles than one, describable on the same side of AB. Which if it were possible (though it will hereafter be shown that it is not), would in no way affect the accuracy of the assertion that it has been shown how to construct an equilateral triangle upon AB.Referred back to, in the Scholium at the end of Prop. VII of the First Book.


PROPOSITION II. See Note. PROBLEM.- From a point assigned, to draw a straight line equal

to a given straight line.

First Case. Let A be the point assigned, and BC the given

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