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some point (say B) there must be an angle of less than 180° with the line AB-viz. ABP, and another angle PBC with the line in which it would have moved had it gone on (1.) But if P continue to travel along BP which has one point (B) in common with BA and therefore cannot have another, by what means is it to return to its first position A, which it must do if space is to be enclosed? The answer is at hand:-If P is to return to its former path AB, it must make such inclinations with the lines BP and BA as will make up the deflection PBC from the line BC (2.)
That is, the angle BPA + the angle PAB must equal the angle PBC, And since ABP + PBC
-angle in a straight line
.. ABP + BPA + BAP=180°.
In other words: "The three angles of a triangle are together equal to 180°." And this is one condition which must be fulfilled if space is enclosed. Thus the angles of a triangle appear to be only the angle of a straight line broken up. As a simple example of the above theorem, let the student cut off the three corners of a triangular piece of paper, and place them together. He will find that they form a straight line.
It must be remarked that the above condition is sufficient only to bring back the point P to its original path AB, and not to any particular point in it such as A. Thus if there were any point D in AB to which P returned, the same condition would be fulfilled, and space would be enclosed (3.) to bring it back to the particular point A will be considered in Part II.
The further condition necessary
THUS we have determined one condition that space may be enclosed by the smallest possible number of straight lines: namely, that there be three points not in the same straight line, the angles at which are together equal to the angle in a straight line. Hence it follows immediately that "two straight lines cannot enclose space," inasmuch as unless they coincide they can have but one point in common, and the angle at that point must be less than 180°.
In will be easy from this to deduce the condition for of boundary points-a boundary point being one at which the angle is less than 180°. For suppose that there be a fourth point
D to which P has to travel before its return
dition will be involved as before. For PD makes an angle APD, with AP therefore for P to return, it must make the angles APD+PDA+DAP=180° (1.) And therefore the angles at A, B, P and D altogether are equal to or "contain" 360°. Thus in fact every new boundary point after the second adds 180° to the sum of the angles of the figure, and therefore one condition that space may be enclosed by straight lines, is that the angles of the enclosing figure equal twice as many right angles as the figure has
more than two sides (which is the same proposition as Euclid I. 32, Cor. 1.) "
We have seen then the least number of straight lines which can contain space, namely, three. Is there any limit on the other side also? Now since the figure is formed of straight lines, the number can only be limited by the size of the straight lines themselves; and in any given space as the number of boundary points increases, the length of the straight lines joining them decreases: so that the greatest limit to the number of boundary points will be reached when the
length of the straight lines is as small as possible—that is, when each line consists of two points only. In other words, the limit to the number is attained when the boundary points lie immediately
a This theorem may be simply expressed in an algebraic form, thus:
If n be the number of sides of any polygon, then the sum of its angles (n-2) 180°. And therefore, if the polygon be regular, the
And so on. And it may be observed that though the number of degrees in the angle increases every time, yet as the denominator of the fraction must be always larger than the numerator, the number of degrees can never (unless indeed n become infinite) attain 180°-that is, the number in a straight line. We shall see how this corresponds with a subsequent chapter.
together, each two consecutive points forming a fresh straight line of inappreciable length. And in this case the Polygon (that is, the figure enclosed by any number of straight lines) becomes a Circle, and its sides, when produced, are Tangents. Hence, a circle is the limiting form of a polygon when the number of sides becomes immeasurably great and the length of the sides immeasurably small. Thus we have obtained a curious relation between the polygon and the circle, and we proceed to that of the circle and straight line, by the examination of which we shall be able to determine the other condition necessary for the limitation of space -viz., that none of the enclosing points shall be at an immeasurably great distance. It will be well, however, first to enumerate the various forms of the polygon, to which Euclid alludes. They are as follows:
That which has 3 boundary points is called a Triangle.
Of Trapezia and Triangles again, Euclid defines the following kinds. Of Trapezia
That which has its opposite sides equal and its angles not right-angles is called a Rhomboid.
That which has all its sides equal and its angles right-angles is called a Rhombus.
That which has its opposite sides equal and its angles right
angles is called an Oblong.
That which has all its sides equal and its angles right-angles is called a Square.
That which has three unequal sides is called Scalene.
In this list we must observe that "Trapezium" has been used as a general name for all four-sided figures; but Euclid only uses it to denote those to which he does not give any more distinctive name such as 66 Square," "Rhombus," &c. And it must be further remarked that although these figures may in general have all their sides of different lengths, yet (except in the case of Triangles and of Parallelograms, of which we shall speak presently) Euclid confines his attention to those which have all their sides equal and by consequence, though he does not prove it, their angles equal also. Such Polygons are called "Regular."
IF I am the possessor of £5, it is evident that until I have spent that sum I cannot be in need of borrowing of a friend. If I have a glass of water by my side, I need not suffer thirst until after that water be drunk up. If I walk ten miles towards a place, I cannot until I have walked more than ten miles in the opposite direction be further from the place than I was before I started. Or to put this into mathematical language-If a quantity gradually decrease from being positive until it become negative, it must pass through zero. Hence in Geometry, when we find an angle gradually decreasing and then increasing in the opposite direction, we know that there must have been some point at which it vanished altogether. Such a point is called a "critical" point, i.e., one at