like manner, if a pen be drawn over this paper an effect is produced, which, in common language, would be called a line, right or curved, as the case may be. This, however, cannot, in the strict geometrical sense of the term, be a line at all, since it has breadth as well as length; for if it had not it could not be made evident to the senses. But having first obtained this rude and incorrect notion of a line, we can imagine that, while its length remain unaltered, it may be infinitely attenuated until it ceases altogether to have breadth, and thus we obtain the exact conception of a mathematical line.

The different modes of magnitude are ideas so extremely uncompounded that their names do not admit of definition properly so called at all.* We may, however, assist the student to form correct notions of the true meaning of these terms, although we may not give rigorous logical definitions of them.

A notion being obtained by the senses of the smallest magnitude distinctly perceptible, this is called a physical point. If this point were indivisible even in idea, it would be strictly what is called a mathematical point. But this is not the case. No material substance

can assume a magnitude so small that a smaller may not be imagined. The mind, however, having obtained the notion of an extremely minute magnitude, may proceed without limit in a mental diminution of it; and that state at which it would arrive if this diminution were infinitely continued, is a mathematical point.†

The introduction of the idea of motion into geometry has been objected to as being foreign to that science. Nevertheless, it seems very doubtful whether we may not derive from motion the most distinct ideas of the modes of magnitude. If a mathematical point be conceived to move in space, and to mark its course by a trace or track, that trace or track will be a mathematical line. As the moving point has no magnitude, so it is evident that its track can have no breadth or thickness. The places of the point at the beginning and end of its motion, are the extremities of the line, which are therefore points. The third of the preceding definitions is not properly a definition, but is a proposition, the truth of which may be inferred from the first two definitions.

As a mathematical line may be conceived to proceed from the motion of a mathematical point, so a physical line may be conceived to be generated by the motion of a physical point.

In the same manner as the motion of a point determines the idea of a line, the motion of a line may give the idea of a surface. If a mathematical line be conceived to move, and to leave in the space through which it passes a trace or track, this trace or track will be a surface; and since the line has no breadth, the surface can have no thickness. The initial and final position of the moving line are two boundaries or extremities of the surface, and the other extremities are the lines traced by the extreme points of the line whose motion produced the surface.

* The name of a simple idea cannot be defined, because the several terms which compose a definition signifying several different ideas can by no means express an idea which has no mauner of composition.-LOCKE.

The Pythagorean definition of a point, is 'a monad kaving position.'

The sixth definition is therefore liable to the same objection as the third. It is not properly a definition, but a principle, the truth of which may be derived from the fifth and preceding definitions.

It is scarcely necessary to observe, that the validity of the objection against introducing motion as a principle into the Elements of Geometry, is not here disputed, nor is it introduced as such. The preceding observations are designed merely as illustrations to assist the student in forming correct notions of the true mathematical significations of the different modes of magnitude. With the same view we shall continue to refer to the same mechanical idea of motion, and desire our observations always to be understood in the same sense.

The fourth definition, that of a right or straight line, is objectionable, as being unintelligible; and the same may be said of the definition (seventh) of a plane surface. Those who do not know what the words 'straight line' and 'plane surface' mean, will never collect their meaning from these definitions; and to those who do know the meaning of those terms, definitions are useless. The meaning of the terms right line' and 'plane surface' are only to be made known by an appeal to experience, and the evidence of the senses, assisted, as was before observed, by the power of the mind called abstraction. If a perfectly flexible string be pulled by its extremities in opposite directions, it will assume, between the two points of tension, a certain position. Were we to speak without the rigorous exactitude of geometry, we should say that it formed a straight line. But upon consideration, it is plain that the string has weight, and that its weight produces a flexure in it, the convexity of which will be turned towards the surface of the earth. If we conceive the weight of the string to be extremely small, that flexure will be proportionably small, and if, by the process of abstraction, we conceive the string to have no weight, the flexure will altogether disappear, and the string will be accurately a straight line.

A straight line is also sometimes defined 'to be the shortest way between two points.' This is the definition given by Archimedes, and after him by Legendre in his Geometry; but Euclid considers this as a property to be proved. In this sense, a straight line may be conceived to be that which is traced by one point moving towards another, which is quiescent.

Plato defines a straight line to be that whose extremity hides all the rest, the eye being placed in the continuation of the line.

Probably the best definition of a plane surface is, that it is such a surface that the right line, which joins every two points which can be assumed upon it, lies entirely in the surface. This definition, originally given by Hero, is substituted for Euclid's by R. Simson and Legendre.

Plato defined a plane surface to be one whose extremities hide all the intermediate parts, the eye being placed in its continuation.

It has been also defined as the smallest surface which can be contained between given extremities.'

Every line which is not a straight line, or composed of straight lines, is called a curve. Every surface which is not a plane, or composed of planes, is called a curved surface.

(9) VIII. A plane angle is the inclination of two lines to one another, in a plane, which meet together, but are not in the same direction.

This definition, which is designed to include the inclination of curves as well as right lines, is omitted in some editions of the Elements, as being useless.

(10)

(11)

IX. A plane rectilinear angle is the inclination
of two right lines to one another, which
meet together, but are not in the same
right line.

X. When a right line standing on another
right line makes the adjacent angles
equal, each of these angles is called a
right angle, and each of these lines is
said to be perpendicular to the other.

(12) XI. An obtuse angle is an angle greater than a right angle.

(13) XII. An acute angle is an angle less than a right angle.

کے

(14) Angles might not improperly be considered as a fourth species of magnitude. Angular magnitude evidently consists of parts, and must therefore be admitted to be a species of quantity. The student must not suppose that the magnitude of an angle is affected by the length of the right lines which include it, and of whose mutual divergence it is the measure. These lines, which are

called the sides or the legs of the angle, are supposed to be of indefinite length. To illustrate the nature of angular magnitude, we shall again recur to motion. Let C be supposed to be the extremity of a right line C A, extending indefinitely in the direction C A. Through the same point C, let another indefinite right line C A, be conceived to be drawn; and suppose this right line to revolve in the same plane round its extremity C, it being supposed at the beginning of its motion to coincide with C A. As it revolves from C A, to CA,, C A,,

C

CA,, &c., its divergence from C A, or, what is the same, the angle it makes with C A, continually increases. The line continuing to revolve, and successively assuming the positions C A,, C A,, CA,, CA, &c., will at length coincide with the continuation CA, of the line CA, on the opposite side of the point C. When it assumes this position, it is considered by Euclid to have no inclination to CA,

and to form no angle with it.

Nevertheless, when the student advances further in mathematical science, he will find, that not only the line C A, is considered to form an angle with C A。, but even when the revolving line continues its motion past C A,; as for instance, to C A it is still considered as forming an angle with CA。; and this angle is measured in the direction A,, Ã ̧ Ã ̧, &c. to A。.

The point where the sides of an angle meet is called the vertex of the angle.

Superposition is the process by which one magnitude may be conceived to be placed upon another, so as exactly to cover it, or so that every part of each shall exactly coincide with every part of the other.

It is evident that any magnitudes which admit of superposition must be equal, or rather this may be considered as the definition of equality. Two angles are therefore equal when they admit of superposition. This may be determined thus; if the angles A B C and A'B'C' are those whose equality is to be ascertained, let the vertex B' be conceived to be placed upon the vertex B, and the side B'A' on the side B A, and let the remaining side B'C' be placed at the same side of B A with B C. If under these circumstances B'C' lie upon, or coincide with B C, the angles admit of superposition, and are equal, but otherwise not. If the side B'C' fall between B C and B A, the angle B' is said to be less than the angle B, and if the side B C fall between B'C' and B A, the angle B' is said to be greater than B.

B

C

As soon as the revolving line assumes such a position C A, that the angle A CA, is equal to the angle A, C As, each of those angles is called a right angle.

An angle is sometimes expressed simply by the letter placed at its vertex, as we have done in comparing the angles B and B'. But when the same point, as C, is the vertex of more angles than one, it is necessary to use the three letters expressing the sides as A C A ̧, A, C A 5, the letter at the vertex being always placed in the middle.

When a line is extended, prolonged, or has its length increased, it is said to be produced, and the increase of length which it receives is called its produced part, or its production. Thus,

if the right line AB be prolonged to A B', it is A said to be produced through the extremity B, and B B' is called its production or produced part.

B

Two lines which meet and cross each other are said to intersect, aud the point or points where they meet are called points of intersection. It is assumed as a self-evident truth, that two right lines can only intersect in one point. Curves, however, may intersect each other, or right lines, in several points.

Two right lines which intersect, or whose productions intersect, are said to be inclined to each other, and their inclination is measured by the angle which they include. The angle included by two right lines is sometimes called the angle under those lines; and right lines which include equal angles are said to be equally inclined to each other. . It may be observed, that in general when right lines or plane surfaces are spoken of in Geometry, they are considered as extended or

produced indefinitely. Whenever a determinate portion of a right line is spoken of, it is generally called a finite right line. When a right line is said to be given, it is generally meant that its position or direction on a plane is given. But when a finite right line is given, it is understood, that not only its position, but its length is given. These distinctions are not always rigorously observed, but it never happens that any difficulty arises, as the meaning of the words is always sufficiently plain from the context.

When the direction alone of a line is given, the line is sometimes said to be given in position, and when the length alone is given, it is said to be given in magnitude.

By the inclination of two finite right lines which do not meet, is meant the angle which would be contained under these lines if produced until they intersect.

(15)

XIII. A term or boundary is the extremity of any thing.

This definition might be omitted as useless.

(16)

XIV. A figure is a surface, inclosed on all sides by a line or lines.

The entire length of the line or lines, which inclose a figure, is called its perimeter.

A figure whose surface is a plane is called a plane figure. The first six books of the Elements treat of plane figures only.

(17) XV. A circle is a plane figure, bounded

by one continued line, called its
circumference, or periphery; and
having a certain point within it,
from which all right lines drawn
to its circumference, are equal.

If a right line of a given length revolve in the same plane round one of its extremities as a fixed point, the other extremity will describe the circumference of a circle, of which the centre is the fixed extremity.

(18)

XVI. This point (from which the equal lines are drawn) is called the centre of the circle.

(19) A line drawn from the centre of a circle to its circumference is called a radius.

(20)

XVII. A diameter of a circle is a right line drawn through the centre, terminated both ways in the circumference.

(21) XVIII. A semicircle is the figure contained by the diameter, and the part of the circle cut off by the diameter.