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A plane surface may be prolonged to any extent in a plane.

If a straight line in a plane be prolonged, its prolongation lies wholly in that same plane.

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 Jigure 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 circumference, 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.

A circle may be described about any centre and with any radius. All the radii of the same circle are equal. And circles that have equal radii, are equal. A straight line from the centre of a circle to a point outside, coincides with the circumference only in a point. Any three points are in the same plane. [That is to say, one plane may be made to pass through them all.]

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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.

Any two straight lines which proceed from the same point, lie wholly in one plane.

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 anything that is not found in it, recourse is to be had

to the Intercalary Book.)

Cor. 2.

See Note,


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, nhich are neither such, nor compounded of such, I 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
length a plane be” made or supposed to pass, and another straight
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,
is called the angle between them.
Hence angles are compared together by
their extension sideways only; without A ,D
reference to the greater or smaller length / o
of the straight lines between which they B C E
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 s acute.

XL. Angles greater or less than right angles, are called by the common title of oblique.

XLI. A straight line joining the ex- VTy U

tremities of any portion of the circumference of a circle, is called a chord. The same name is applied to a straight line

A B A B joining the extremities of any series of v=7 U

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 quadrilateral.

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.

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 thc 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.

ScholruM.—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. C *INTERC. 13. About the centre A, with the radius AB, de- /N Cor. 3. scribe” the circle BCD ; and about the centre B, D 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 one. 9 instance C) drawf the straight lines CA, CB, to the points A and Or. - e B. ABC shall be an equilateral triangle. Because the point A is the centre of the circle BCD, and C t|NTERc.13. and B are points in the circumference, AC is: equal to AB. And ** 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 "INTEno. 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 ABReferred back to, in the Scholium at the end of Prop. PII of the First Book.


See Note. PROBLEM.–From a point assigned, to dran, 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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