(22) From the definition of a circle, it follows immediately, that a line drawn from the centre to any point within the circle is less than the radius; and a line from the centre to any point without the circle is greater than the radius. Also, every point, whose distance from the centre is less than the radius, must be within the circle; every point whose distance from the centre is equal to the radius must be on the circle; and every point, whose distance from the centre is greater than the radius, is without the circle.

The word 'semicircle,' in Def. XVIII., assumes, that a diameter divides the circle into two equal parts. This may be easily proved by supposing the two parts, into which the circle is thus divided, placed one upon the other, so that they shall lie at the same side of their common diameter: then if the arcs of the circle which bound them do not coincide, let a radius be supposed to be drawn, intersecting them. Thus, the radius of the one will be a part of the radius of the other; and therefore, two radii of the same circle are unequal, which is contrary to the definition of a circle. (17.)

(23)

XIX. A segment of a circle is a figure contained by a right line, and the part of the circumference which it cuts off.

(24)

XX.

A figure contained by right lines only, is called a rectilinear figure.

The lines which include the figure are called its sides.

(25)

XXI. A triangle is a rectilinear figure included by three sides.

A triangle is the most simple of all rectilinear figures, since less than three right lines cannot form any figure. All other rectilinear figures may be resolved into triangles by drawing right lines from any point within them to their several vertices. The triangle is therefore, in effect, the element of all rectilinear figures; and on its properties, the properties of all other rectilinear figures depend. Accordingly, the greater part of the first book is devoted to the developement of the properties of this figure.

(26)

(27)

XXII. A quadrilateral figure is one

which is bounded by four sides.
The right lines, AB, CD, con-

necting the vertices of the

B

opposite angles of a quadrilateral figure, are called its diagonals.

XXIII. A polygon is a rectilinear figure, bounded by

more than four sides.

Polygons are called pentagons, hexagons, heptagons, &c., according as they are bounded by five, six, seven, or more sides. A line joining the vertices of any two angles which are not adjacent is called a diagonal of the polygon.

(28) XXIV. A triangle, whose three sides are equal, is said to be equilateral.

In general, all rectilinear figures whose sides are equal, may be said to be equilateral.

A

Two rectilinear figures, whose sides are respectively equal each to each, are said to be mutually equilateral. Thus, if two triangles have each sides of three, four, and five feet in length, they are mutually equilateral, although neither of them is an equilateral triangle.

In the same way a rectilinear figure having all its angles equal, is said to be equiangular, and two rectilinear figures whose several angles are equal each to each, are said to be mutually equiangular.

(29)

XXV. A triangle which has only two sides
equal is called an isosceles triangle.

A

The equal sides are generally called the sides, to distinguish them from the third side, which is called the base.

(30). XXVI. A scalene triangle is one which has no two sides equal.

(31)

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

That side of a right-angled triangle which is opposite to the right angle is called the hypotenuse.

(32)

(33)

XXVIII. An obtuse-angled triangle is that

which has an obtuse angle.

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

It will appear hereafter, that a triangle cannot have more than one angle right or obtuse, but may have all its angles acute.

(34) XXX. An equilateral quadrilateral figure is called a lozenge.

(35)

XXXI. An equiangular lozenge is called a
square.

We have ventured to change the definition of a square as given in the text. A lozenge, called by Euclid a rhombus, when equiangular, must have all its angles right, as will appear hereafter. Euclid's de finition, which is a lozenge all whose angles are right,' therefore,

contains more than sufficient for a definition, inasmuch as, had the angles been merely defined to be equal, they might be proved to be right. To effect this change in the definition of a square, we have transposed the order of the last two definitions.

(35)

XXXII.

An oblong is a quadrilateral, whose
angles are all right, but whose sides
are not equal.

This term is not used in the Elements, and therefore the definition might have been omitted. The same figure is defined in the second book, and called a rectangle. It would appear that this circumstance of defining the same figure twice must be an oversight.

(36)

XXXIII. A rhomboid is a quadrilateral,
whose opposite sides are equal.

0

This definition and the term rhomboid are superseded by the term parallelogram, which is a quadrilateral, whose opposite sides are parallel. It will be proved hereafter, that if the opposite sides of a quadrilateral be equal, it must be a parallelogram. Hence, a distinct denomination for such a figure is useless.

(37)

XXXIV. All other quadrilateral figures are called trapeziums.

As quadrilateral figure is a sufficiently concise and distinct denomination, we shall restrict the application of the term trapezium to those quadrilaterals which have two sides parallel.

(38) XXXV. Parallel right lines are such as

are in the same plane, and

which, being produced continually in both directions, would never meet.

It should be observed, that the circumstance of two right lines, which are produced indefinitely, never meeting, is not sufficient to establish their parallelism. For two right lines which are not in the same plane can never meet, and yet are not parallel. Two things are indispensably necessary to establish the parallelism of two right lines, 1°, that they be in the same plane, and 2°, that when indefinitely produced, they never meet. As in the first six books of the Elements all the lines which are considered are supposed to be in the same plane, it will be only necessary to attend to the latter criterion of parallelism.

POSTULATES.

(39) I. Let it be granted that a right line may be drawn any one point to any other point.

from

10

༤་ྒུ

ELEMENTS OF EUCLID.

(40) II. Let it be granted that a finite right line may be produced to any length in a right line.

(41) III. Let it be granted that a circle may be described with any centre at any distance from that centre.

(42) The object of the postulates is to declare, that the only instruments, the use of which is permitted in Geometry, are the rule and compass. The rule is an instrument which is used to direct the pen or pencil in drawing a right line; but it should be observed, that the geometrical rule is not supposed to be divided or graduated, and, consequently, it does not enable us to draw a right line of any proposed length. Neither is it permitted to place any permanent mark or marks on any part of the rule, or we should be able by it to solve the second proposition of the first book, which is to draw from a given point a right line equal to another given right line. This might be done by placing the rule on the given right line, and marking its extremities on the rule, then placing the mark corresponding to one extremity at the given point, and drawing the pen along the rule to the second mark. This, however, is not intended to be granted by the postulates.

The third postulate concedes the use of the compass, which is an instrument composed of two straight and equal legs united at one extremity by a joint, so constructed that the legs can be opened or closed so as to form any proposed angle. The other extremities are points, and when the legs have been opened to any degree of divergence, the extremity of one of them being fixed at a point, and the extremity of the other being moved around it in the same plane will describe a circle, since the distance between the points is supposed to remain unchanged. The fixed point is the centre; and the distance between the points, the radius of the circle.

It is not intended to be conceded by the third postulate that a circle can be described round a given centre with a radius of a given length; in other words, it is not granted that the legs of the compass can be opened until the distance between their points shall equal a given line.

AXIOMS.

(43) I. Magnitudes which are equal to the same are equal to each other.

(44) II. If equals be added to equals the sums will be equal.

(45)

(46)

III. If equals be taken away from equals the remainders will be equal.

IV. If equals be added to unequals the sums will be

unequal.

(47)

V. If equals be taken away from unequals the remainders will be unequal.

(48) VI. The doubles of the same or equal magnitudes are equal.

(49) VII. The halves of the same or equal magnitudes are equal.

VIII.

Magnitudes which coincide with one another, or exactly fill the same space, are equal.

(50)

(51) IX.

The whole is greater than its part.

(52)

X.

(53) XI.

(54) XII.

Two right lines cannot include a space.

All right angles are equal.

B

If two right lines (A B, C D) meet a third right
line (AC) so as to make the
two interior angles (B A C and
DCA) on the same side less
than two right angles, these two
right lines will meet if they be

D

produced on that side on which the angles are less than two right angles.

(55) The geometrical axioms are certain general propositions, the truth of which is taken to be self-evident, and incapable of being established by demonstration. According to the spirit of this science, the number of axioms should be as limited as possible. A proposition, however self-evident, has no title to be taken as an axiom, if its truth can be deduced from axioms already admitted. We have a remarkable instance of the rigid adherence to this principle in the twentieth proposition of the first book, where it is proved that two sides of a triangle taken together are greater than the third;' a proposition which is quite as self-evident as any of the received axioms, and much more self-evident than several of them.

On the other hand, if the truth of a proposition cannot be established by demonstration, we are compelled to take it as an axiom, even though it be not self-evident. Such is the case with the twelfth axiom. We shall postpone our observations on this axiom, however, for the present, and have to request that the student will omit it until he comes to read the commentary on the twenty-eighth proposition.

Two magnitudes are said to be equal when they are capable of exactly covering one another, or filling the same space. In the most ordinary practical cases we use this test for determining equality; we apply the two things to be compared one to the other, and immediately infer their equality from their coincidence.

By the aid of this definition of equality we conceive that the second and third axioms might easily be deduced from the first. We shall not however pursue the discussion here.