edge, and may then be turned round so as to have ever so many different positions, the point A still coinciding with its edge. Therefore one point does not determine the position of a straight line. But if there be two points given as A and B (fig. 1) it is obvious 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 the above is received as an axiom arising from the nature of a straight line.

7. SCHOLIUM.-How to measure straight lines and express them by numbers-Linear Units. Lines 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 inch 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 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.

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8. PROBLEM.-To find the greatest common measure of two straight lines in order to express their ratio in numbers. By problem, is meant an operation proposed to be performed. The performance of the operation is called the solution of the problem. proceed to solve the above, taking A B and C D (fig. 2,) F 2 for the two lines to be compared. The process is similar to

We

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measure.

that in arithmetic for finding the greatest common
divisor of two numbers. If the learner is not al-
ready acquainted with the use of signs, he must
observe that the sign (=) signifies equal to; and
the sign (+) which is called plus, signifies added to.
SOLUTION. First seek how often the smaller line
CD is contained in the larger 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
CD is contained in A B twice and E B over. There-
fore C D is not a common measure. We next apply
the remainder E B to C D and find that it is contained
once and F D over. Therefore E B is not a common
We next take the remainder F D and apply
it to E B. It is contained once and G B over. There-
fore 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 incom-
mensurable. 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 com-
mon 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=F D÷
G B=2+1=3. Again C D E B÷F D=3+2=5.
Lastly A B 2 C D E B-10+3=13. According-
ly the ratio of A B to C D is that of 13 to 5; that is
A B is of C D, or C D is of A B.

9. DEF.-A polygonal line is the path described by a point which changes its direction at intervals so large that they can be perceived. 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, CD (fig 3) between any two successive changes, the line, which is made up of straight lines, is called a broken or polygonal line.

10. DEF.-A curved line or curve is the path described by a point which changes its direction at intervals so small that they cannot be perceived. When the direction changes so often that you cannot perceive the in

tervals between the successive changes, 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. COROLLARY I.-A curved line may be considered as made up of infinitely small straight lines. By corollary is meant a proposition which follows as a consequence from a preceding proposition or definition. The above corollary follows directly from the definition of a curve, and furnishes the best idea you can form of it, 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. COR. II.-A straight line and a curve can only coincide for an infinitely small extent. To coincide is to lie one upon the other and exactly fill the same space. The above corollary follows directly rom the two definitions of a straight line and a curve.

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THE CIRCLE AND ITS PARTS.

DEF.-A circle is a figure bounded by a curved line, all the points of which are equally distant from one point called the centre. Thus if all the points in the curved line B C G DFE (fig. 5) are equally distant from the centre A, the figure is a circle.

If the

straight line A B, having the point A fixed, be suppósed to turn about this point, till having performed a complete rotation, it returns to its first position, this motion would describe the circle. The moving line A B is called the radius, and the bounding line B C GDFE the circumference. Any straight line as D B drawn through the centre to meet the circumference each way is called a diameter. Any portion of the circumference as B C G is called an arc. Any straight line as G B joining the extremities of an arc is called a chord. Any portion of a circle comprehended between an arc and its chord is called a segment, as the

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segment B G C. Any portion of the circle comprehended between two radii and an arc is called a sector, as the sector E A B. COR.-In the same circle all radii are equal – each diameter is double the radiusall diameters are equal — every chord is less than its arc. All these follow directly from the definitions.

12. THEOREM.-Every diameter bisects the circle and its circumference. By theorem is meant a proposition the truth of which is to be demonstrated by a process of reasoning. To bisect is to divide into two equal parts. To enunciate a proposition is to state it in words. We proceed to demonstrate the theorem above enunciated, by the method called superposition. DEMONSTRA

TION. Let the two portions of the circle D E B, D G B (fig. 5) 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. 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. THEOREM.-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 is demonstrated in a manner similar to the preceding, that is by superposition. DEM. 1.-Let the arc DF be supposed equal to D G (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. 2.-We are now to prove the converse, namely if two chords as DG and DF 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 DG 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.

14. PROB.-Having any arc given, to make another equal to it. It is proper here to remark that the instrument used in making arcs and measuring them. is called a compass or more generally compasses. Being very common we shall not describe it. SOL.-If the arc B C (fig. 6) be given and you wish to make another as D F equal to it, you first describe an indefinite arc DF with the same radius as that of B C, because they must belong to the same circle. Then take the chord B C in the compasses, and placing one foot in D move the other round till it cuts D F in F. The arc D F will be equal to BC because their chords are equal (13.)

15. SCHOLIUM.-How the circumference is divided. It often becomes necessary to compare an arc with an entire circumference or with another arc of the same circumference. For this purpose every circumference is supposed to be divided into 360 equal arcs called degrees and marked thus (.) For instance 60° is read 60 degrees. As all circumferences whether great or small, are divided into the same number of parts, it follows that a degree, which is thus made the unit of arcs, is not a fixed value, but varies for every different circle. It merely expresses the ratio of an arc, namely, to the whole circumference of which it is a part, and not to any other. As we sometimes have occasion for an unit less than a degree, each degree is divided into 60 equal parts called minutes and marked thus ('). Again each minute is divided into 60 equal parts called seconds and marked thus ("). When extreme minuteness is required the division is sometimes continued to thirds and fourths, maked thus (''), (''''). As a quarter of a circumference, or, as it is generally called, a quadrant contains 90°, and as small numbers are more convenient than larger ones, it is usual in practice to refer all arcs to a quadrant, instead of an entire circumference. Thus considering a quadrant as unity, we say that a degree is, a minute, and a second of the quadrant in which it is taken.

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