will be easy if you consider how you would proceed to make one. Your single endeavour would be to move the pencil throughout in one and the same direction. Accordingly we define a straight line to be-the path described by a point moving only in one direction. Thus if the pencil F1 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. F1 4. 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, or let any one make the trial and he will find it so. Here then we have a proposition-a straight line is the shortest distance between two points--the truth of which every one believes instantaneously, and which no reasoning can render more evident than the mere statement or enunciation of it. Such a proposition is called an axiom, which is defined to be a proposition the truth of which is self evident—.

F10

5. The question now arises how you can be sure, when attempting to make a straight line, that the describing point does not change its direction? We 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. If a single point be given as 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 edge, and may then be turned round so as to have ever so many different positions, point A still coinciding with its edge. Hence we say-one point does not determine the position of a straight line-.But F 1 if there be two points given as A and B (fig. 1) it is obvi

the

ous that only one straight line can be drawn between or through them. Why? We might say because there can be but one shortest distance between two points. Or we might say because if the rule were so placed as to have the two points coincide with its edge, it could not be moved from this position without leaving one or both the points out of its edge. But neither of these reasons adds any force to our first belief. Hence it is received as an axiom, arising from the nature of a straight line, that-only one straight line can be drawn between or through two points-or in other words-two points determine the position of a straight line

7. We shall now explain the method of measuring and comparing straight lines. They are measured, like all other quantities, by taking some known quantity of the same kind as a standard, and seeking how often it is contained in them. Thus the standard by which we measure a straight line, must be a straight line of a known length, as an inch, a foot, a yard, etc. This standard, whatever it be, is called a linear unit, and we have the measure of a straight line when we know the number of linear units it contains. Thus if we take an ineh for the linear unit, and if we find it is contained 9 times in a given line as A B (fig. 1), we say the measure of A B is 9 inches. Since F 1 then the value of straight lines can be expressed in abstract numbers, and since abstract numbers are the object of arithmetic, it is obvious that the fundamental operations of arithmetic may be performed upon lines. This is called the application of arithmetic to geometry. Moreover since algebra is nothing more than general arithmetic, it follows that algebra as well as arithmetic may be applied to geometry.

8. It often becomes desirable to compare two straight lines, for the purpose of ascertaining how many times one is greater than the other. This is called finding their ratio. In order to do this, we must take for a common measure, a linear unit which is contained an exact number of times in each of the lines. When no such linear unit is known, as is frequently the case, the process for finding it is the same as that in arithmetic, for finding the greatest common measure of two numbers. Suppose the two lines to be compared are A B and C D (fig. 2). We F 2 propose to find their greatest common measure, and then

express their ratio in numbers. A proposition of this kind
is called a problem, which may be defined to be—an ope-
ration proposed to be performed. The performance of the
operation is called the solution of the problem. In solving
the problem before us, we first seek whether C D is con-
tained an exact number of times in A B. If it were con-
tained exactly 3 times for example, we should have their
ratio at once, namely 3 to 1. That is A B would be 3
times as great as C D. But we find upon trial that C D
is contained in A B twice and E B over. Therefore CD
is not a common measure. We next apply E B to C D.
and find that it is contained once and FD over. Therefore
E B is not a common measure. We next take F D and
apply it to E B. It is contained once and G B over.
Therefore F D is not the common measure.
This process
of applying the last remainder to the preceding must be
continued as long as there is a remainder. If no such
limit is attainable, the lines are said to be incommensurable.
If this limit can be attained, the line last applied is the
greatest common measure. Thus if G B. is contained
exactly twice in F D, G B is the common measure sought.
The ratio is then expressed as follows. G B, the linear
unit, is 1. Then, since it is contained twice in F D, F
D=2. But E B E G+G B=F D+G B=2+1=3.
Again C D C F+F D=E B+F D=3+2=5. Lastly
A B A E+E B=2 C D+E B=10+3=13. Accord-
ingly the ratio of A. B. to C. D. is that of 13 to 5; that
is A B is 13 of C D, or C D is of A B.

5

5

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9. If a line be not straight, it must be because the describing point has changed its direction once or more. When these changes of direction do not take place so often as to prevent your perceiving the intervals A B, BC, C F3 D (fig. 3) between any two successive changes, the line, which is made up of straight lines, is called a broken or polygonal line.

10. When the direction changes so often that you cannot perceive the intervals between the successive changes, F 4 as in A B (fig. 4), the line thus described is called a curved line. In this case though you cannot actually perceive the intervals between which there is no change, yet this does not hinder your conceiving that there are such intervals. Indeed there must be such intervals from the

very nature of the motion, but the changes are so frequent
that they are infinitely small. We shall therefore define
a curved line to be-a line made up of infinitely small straight
lines-. This is the best idea you can form of a curved
line, for
you thus make a straight line the unit or element
of all lines, a principle which will be found to be of great
utility hereafter, when we come to compare curved and
straight lines.

The Circle and its Parts.

11. There is one curved line, which, both on account of its simplicity and importance, is more remarkable than any other, namely the circumference of a circle. Suppose the straight line A B (fig. 5), having the point A fixed, to F 5 turn as upon a pivot about this point, till, having performed a complete rotation, it returns to its first position. We must here remark that the surface of the paper represents a plane which is defined to be a surface in which any two points being taken, the straight line joining these points, lies wholly in that surface. Now in the above construction the describing line A B is supposed to remain always in the same plane represented by the surface of the paper, understanding for the present by the word surface-that which has length and breadth without thickness. We shall treat more particularly of surfaces in the next section. These things being premised, we call the path described by the point B the circumference of a circle, which we define to be a curved line all the points of which are equally distant from a point within called the centre. The whole space enclosed is called a circle, the moving line A B a radius, and the fixed point A the centre. The radius is the same in every position, or as it is commonly expressed—all radii of the same circle are equal. A line drawn through the centre till it meets the circumference each way, is called a diameter. Therefore-a diameter is equal to twice the radius, and all diameters of the same circle are equal- When we wish to speak of any portion of the circumference as G B, we call it an arc, and the straight line G B joining its extremities is called a chord. The portion of space comprehended between an arc G B and its chord is called a segment. The portion of space E A

B comprehended between the two radii E A, A B, and the arc E B is called a sector.

12. Every diameter bisects the circle and its circumference-. To bisect is to divide into two equal parts. A proposition like the above is called a theorem, which is defined to be a proposition the truth of which is to be demonstrated by a process of reasoning. To enunciate a proposition is to state it in words. We proceed to demonstrate the proposition above enunciated. Let the two portions of the circle D E B, D G B above and below the diameter D B, be folded one upon the other, so that the folded edge shall coincide with the diameter. The two portions of the circumference will exactly coincide with each other; for if they did not, there would be points in them unequally distant from the centre, which would be contrary to the definition of a circle. If the two portions coincide or fill the same space, they are equal. This is an axiom. Therefore the diameter bisects the circle and its circumference, which was to be demonstrated. Each portion of the circumference cut off by the diameter is called a semicircumference, and each portion of the circle is called a semicircle.

13. In the same circle or in equal circles, if two arcs are equal, their chords will be equal, and conversely if two chords are equal their arcs will be equal. This proposition, which is one of great consequence, is demonstrated in a manner similar to the preceding, that is by superposition as it is called. Let the arc D F be supposed equal to D G F5 (fig. 5). Then if the lower portion of the figure be folded upon the upper as before, the arcs D G and D F coinciding, the point G will fall upon the point F, and the chords D G and D F having two points D and F common must coincide throughout, since only one straight line can be drawn between two points (6). Therefore if the two arcs are equal, their chords are equal. We are now to prove the converse, namely if two chords as D G and D F are equal, their arcs are equal. If the chord D G be applied to D F, as they are by hypothesis or supposition equal, the point G must fall upon F. Then the arcs D G and D F, belonging to the same circle and having two points common, must have all their points common, since they must all be equally distant from the centre by the definition. Therefore if two chords are equal their arcs are equal.