29. Suppose the greatest number of diagonals which can be drawn in a polygon to be 35, how many sides has the polygon?

Answer. This number 35 is the half of a product found by multiplying together two numbers, whose difference is 3. We must therefore seek for two numbers whose difference is 3, and whose product is 70. These numbers are 10 and 7. The figure has 10 sides.

30. We will now calculate the whole number of ExTENTS which may lie between the points of intersection of straight lines intersecting one another at a given number of points.

Suppose the lines intersect at the greatest number of points. After finding the greatest number of points (22) in which each line is intersected, we must find the number of extents in each line, (27,) and multiply this product by the number of lines; we shall thus get the whole number of extents.

Example. If 6 straight lines intersect at the greatest number of points, then each line will intersect the other 5 lines. In each line will be 5 points of intersection, and

5×4
2

between 5 points lie = 10 extents; thus between

the points of intersection of the 6 lines lie 6 × 10 = 60 extents.

In 7 straight lines there will be 7-1=6 points in each;

(7-1) × (7-2)

2

extents.

In the same manner,

; in all 7 lines 7×(7-1)×(7-2)

3 lines give 3 × (3-1) × (3-2)_3×2×1

2

2

=

2

4×(4-1) × (4-2)_4×3×2

2

=

3

12

6× (6-1) × (6-2)_6×5×4

5×(5-1) × (5-2)_5×4×3

= 30

2

2

10×(10-1)×(10-2)_10×9×8

2

20 ،، 20×(20-1)×(20-2)_20×19×18

2

=3420

31. We will now compare lines together in regard to their length.

2 Lines are either equal or unequal. Call one A, the other B. Then A is either equal or not equal to B, and in the latter case A may be either longer or shorter than B.

3 Lines may all be of equal length.

2 may be equal, 1 not equal.

All may be unequal. 3 cases.

4 Lines. All equal.

3 equal-1 not equal.

2 equal-2 not equal, but equal to one an

other.

2 equal-2 not equal, and not equal to each

other.

All unequal. 5 cases.

6 Lines. All equal.

5 equal-1 unequal.

4 equal-2 unequal, but equal to each other.

4 equal-2 unequal, and unequal to each

other.

3 equal-3 unequal, but equal one to another.

3 equal-3 unequal, but 2 equal to each other. 3 equal-3 unequal, and unequal one to an

other.

2 equal-4 unequal, but equal one to another.

2 equal-4 unequal, but 3 equal one to
another.

2 equal-4 unequal, but equal by pairs.
2 equal-4 unequal, and of these 2 are
equal to one another, the other 2 un-
equal.

2 equal-4 unequal, and unequal one to an-
other.

All unequal. 13 cases.

Of these cases some are alike. The 8th is the same with the 3d, and the 9th with the 6th. Thus there are in all 11 different cases.

We can arrive at the same result by numbers without reference to forms. It is only necessary to seek into how many sets 6 can be divided, so that the sum of the numbers of each set shall be 6.

III. ANGLES.

32. We will now seek the number of angles which can be made by a given number of straight lines. Two straight lines may make 1, 2, or 4 angles, (fig. 16.) Three straight lines, if they meet at 1 point, may make at least 2, at most 6 angles, (fig. 17.)

If they meet at 2 points, they may make at least 2, at most 8 angles; they cannot make 7 angles, because 3 angles cannot be formed at one point by 2 straight lines.

If they meet at 3 points, they make at least 3, at most 12 angles; but never 11 angles.

In these calculations only the simple angles are considered. When three or more lines meet at one point beside the simple angles, others are formed which are the sum of two or more angles, and which may be called compound angles. If three lines meet at one point, beside the two simple angles, a third is formed of these two added together.

33. We will now consider the kind, as well as the number, of the angles which may be made by a given number of straight lines.

I. Two straight lines may make, (fig. 18,)

II. Three straight lines meeting at 1 point.

1. No line passing beyond the point of meeting.

1 R. and 1 ac.

(fig. 19,) or 1 ob. and 1 ac.

3 ob.

1 R. 2 ob. (fig. 20.)
2 ob. 1 ac.

2. One line passing beyond the point of meeting,

3. Two lines passing beyond the point of meeting,

(fig. 22.)

1 R. 1 ob. 3 ac.

3 R. 2 ac.

2 ob. 3 ac.

1 ob. 4 ac.

4. All three lines passing beyond the point of meeting, (fig. 23.)

2 R. 4 ac.

2 ob. 4 ac.

6 ac.

III. Three straight lines meeting at 2 points, (fig. 24.)

1. No line passing beyond the points of meeting.

2. When 1 line passes 1 point, (fig. 25.)

3. When 1 line passes 2 points, (fig. 26.)