Simson; those who would unite modern improvements with the rigid method of the ancients, must study Playfair; and those who would have a complete view of geometry as it now is, without particular regard to the ancient method, must study Legendre.

As the student may desire to know in what respects the ancient and modern methods differ, we shall briefly state their general characteristics. Both agree in this, that certain principles or truths are taken for granted to begin with. They are taken for granted, because they cannot be proved; being self-evident the moment they are stated. These are called axioms, and are to geometry, what the foundations are to a building. Euclid's axioms are the following:

1. Things which are equal to the same thing are equal to one another.

2. If equals be added to equals the wholes are equal. 3. If equals be taken from equals, the remainders are equal.

4. If equals be added to unequals, the wholes are unequal.

5. If equals be taken from unequals, the remainders are unequal.

6. Things which are double of the same, are equal to one another.

7. Things which are halves of the same, are equal to one another.

8. Magnitudes which coincide with one another, that is, which exactly fill the same space, are equal to one another.

9. The whole is greater than its part.

10.

Two straight lines cannot inclose a space. 11. All right angles are equal.

12. If a straight line meets two straight lines so as to make the two interior angles on the same side of it taken together, less than two right angles, these straight lines being continually produced, shall at length meet upon that side upon which are the angles which are less than two right angles.

The last of these has been added by Euclid's Com

mentators.

The two methods differ in this. Euclid never supposes a line to be drawn, until he has first demonstrated the possibility and pointed out the manner of drawing

it. But in three cases the possibility cannot be demonstrated, because it is self-evident. These cases called postulates, and are the following:

are

1. Let it be granted that a straight line may be drawn from any one point to any other point.

2. Let it be granted that a terminated straight line may be produced to any length in a straight line.

3. Let it be granted that a circle may be described from any centre, at any distance from that centre.

The moderns, as Legendre, for example, are not thus scrupulous; but constantly suppose lines to be drawn, without demonstrating the possibility or explaining the

manner.

Lastly, the two methods differ in this. The moderns avail themselves of all the aid which Algebra can afford them. The ancients were unacquainted with Algebra. Accordingly Euclid was obliged to demonstrate the laws of proportion geometrically. Whereas in modern systems, these laws are supposed to have been previously demonstrated by the help of Algebra. The moderns also derive great advantage, in every part of geometry, from the use of Algebraic signs and symbols. The ancient reasonings, for want of these, were rendered exceedingly cumbrous and circuitous.

These are some of the general distinctions. But the student who would be able to estimate the comparative merits of the two systems, must examine both for himself.

ELEMENTS OF GEOMETRY.

SECTION FIRST.

OF LINES AND THEIR RELATIONS.

1. DEFINITION.-A point is position merely without any magnitude. The study of geometry properly begins with the consideration of a point, this being the first and simplest geometrical idea. If you were required to make a point with a pencil upon paper, you would merely place the sharpened end upon the paper, without moving it in any direction. If the pencil be as sharp as possible, this is the nearest approach you can make to a geometrical point as above defined. But as you cannot represent to the eye that which has absolutely no extension, it is sufficiently near the truth to call a point that which has an infinitely small extension. By infinitely small, we mean for the present, the smallest that can possibly be conceived.

2. DEF.-A line is the path described by the motion of a point. A point is the beginning and end of a line; for if you were required to make a line you would begin by placing the point of your pencil upon the paper; you would proceed to move it along the surface of the paper; and you would end by ceasing to move it. Here you make one point by placing the pencil; you make a line by moving it; and you make another point where you cease to move it. These points are the boundaries of lines. It is evident that if the describing point had no extension, the line would only have that which it acquires from the motion, namely length, without any breadth or thickness. But as such a line could not be represented to the eye, it is sufficiently near the truth to say -a line

F1

F1

has length with only an infinitely small breadth and thickness.

3. DEF.-A straight line is the path described by a point moving only in one direction. This will be readily understood if you consider how you would proceed to make a straight line. Your single endeavour would be to move the pencil throughout in one and the same direction. Thus if the pencil be placed at A (fig. 1) and if it move only in one single direction till it reaches B, the line A B is a straight line.

4. AXIOM.-A straight line is the shortest way from one point to another. By axiom is meant a proposition the truth of which is self-evident without reasoning. The above is one of this kind. If you were standing at a point A (fig. 1), and were required to run to the point B in the shortest possible time, would you keep always in the straight line A B, or would you deviate from it? You answer without a moment of hesitation, that you would keep in the straight line between the two points. Why? Because if you were to depart from it you would be obliged to return to it before you could reach B, since B is situated in it; and you would thus lose time. This is the only reason you could give; for if you were further asked why you would lose time by departing and returning, you could give no other reason for your belief than that the thing is self-evident, or no one can doubt it.

We

5. SCHOLIUM.-How to make a straight line. By scholium is meant any explanatory remark relating to what has gone before. If you ask how you can be sure, when attempting to make a straight line, that the describing point does not change its direction? answer that in practice this assurance is obtained by moving the pencil along the edge of an instrument called a rule, which is already ascertained to be straight. The rule is ascertained to be straight, by taking sight, as it is called, upon its edge, it being a fundamental principle in optics that the rays of light move in straight lines.

6. AXIOM.-Two points determine the position of a F 10 straight line. If a single point be givena's A (fig. 10), it is obvious that any number of straight lines may be drawn through it as in the figure, for the rule may be placed so as to have the point A coincide with its