9. Show that in a right triangle the difference between the sum of the perpendicular sides and the hypothenuse will be the diameter of the inscribed circle.

FIG. 86.

10. Show that the bisector of an interior angle and the bisectors of the non-adjacent exterior angles will pass through one point, and that point will be the centre of an escribed circle.

11. Show that if a regular hexagon be inscribed within a circle, each side will equal the radius of the circle.

12. Show that if any quadrangle be inscribed within a circle, the opposite angles will be supplementary.

13. Prove the converse.

14. Construct a segment of a given circle that shall be capable of containing a given angle.

15. With a given line as a chord construct a segment of a

circle that shall contain a given angle.

CHAPTER V.

56. Definitions. If a point move along a line, as AB, from any point, as A, toward B, it will in its course occupy the position of every point of the line as far as we may conceive it as moving.

If we fix our attention upon B, and speak of the distance of B from A, it is a definite thing, and is perfectly understood.

For purposes of description and comparison we frequently take some convenient unit of measure, and apply it to the distance. If the distance be a day's journey, we use the mile or the kilometre. If it be a distance, as in the figure, between points on the page of this book, we use inches or centimetres.

Remembering that the point in moving along the line from A to B occupies an infinite number of positions, one sees that the chances that the extremity of the measuring unit will not fall at B are as infinity (∞) to

one.

In the above figure, we would, "roughly speaking," say that B was 3 centimetres from A. If we desired, for any reason, more accurately to describe the distance, we should descend to fractional parts of this unit; the fractional part being less than the distance by which we failed to reach B when using the entire unit. Again, the chances are, as infinity to one, that the new unit of measure will not fall on B. The subdividing of the unit may be carried to any extent, depending entirely upon the required accuracy of description.

The most convenient subdivision is the decimal one.

If the applied unit or any of its subdivisions have their extremities at B, the distance AB and the unit are said to be commensurable. In general, however, if a point, as B, is taken at random on the line AB, its distance from A will not be commensurable in terms of any established unit. The distance is then said to be incommensurable with the unit.

It might be commensurable with respect to one unit and incommensurable with respect to another unit.*

A point may be so assumed that its distance from A shall be commensurable in terms of any unit that may be selected.

*NOTE IN ILLUSTRATION. - If we undertake to express decimally the distance of a point from A that is distant therefrom of the unit distance, we can only approximate to it. The first approximation would be .6, a nearer one would be .66, a still nearer .666. We might continue annexing decimal places as long as we please and we should never, in that way, reach the point that is of a unit's distance from A; although at each step we should come nearer the point.

is said to be the LIMIT toward which we approach as we increase the number of decimal places in .6666.....

57. We know from § 19 that parallel lines are everywhere equally distant from each other. Let AB and CD in the figure represent two parallel lines, and AC a line perpendicular to AB and CD. Represent the segment AC by (h).

If we cause the line AC to move parallel to itself a distance (b), the extremities of (h) remaining in AB and CD, the surface swept over by the segment (h) is described as "the area bh."

A

B

h

bh

C

D

FIG. 89.

AC and BD are parallel, and if the line CD, perpendicular to them and remaining always parallel to its initial position, should move to the position AB, the segment (b) would sweep over the area (hb). The areas swept over being the same, we have bh

=

hb.

Either side of the rectangle ACDB may be called the base; ordinarily it would be the lower one in the figure. The perpendicular distance between the base and its parallel is called the altitude. Sometimes this parallel is called the upper base.

The rectangle is accurately described by bh or hb.

In general, b and h are incommensurable with any assumed unit of length, and the area bh is incommensurable with the square, having that unit for its side. But for convenience of numerical description or comparison some unit square is taken and applied to the rectangle. If it be large tracts of land that are being considered, we use the acre or the hectare. If it be the areas of rectangles on a page of this book, the square inch would be appropriate.

h

A

H

B IN

If we should assume CM as the side of a unit square, and should lay it off from C towards A as many times as possible, an extremity would fall within a unit's length of A, as at H. If we lay off the same unit of length from C toward D, as many times as possible, an extremity would fall within a unit's length of D, as at K. If the line CH move parallel to itself and so that each point moves on a perpendicular to CA, until it occupy the position KN, it will have swept over an area expressed by CK × KN. It will be commensurable with the assumed unit area, and the number of times it will contain that unit is expressed by the product of the number of units of length in CH by the number of units of length in CK.

M

C

b

FIG. 90.

KD

If a nearer approximation is desired, the former unit of area is subdivided, so that a side of the new comparison square will be less than the distance by which in the preceding instance we failed of reaching either AB or BD.

We shall thus have an increased commensurable area, which will be a nearer approximation to the area bh.

This subdivision of the unit after the manner above indicated may be carried as far as we please, and a commensurable area be expressed in the terms of some measuring unit, which shall approximate as nearly as we may please to the incommensurable area bh.

bh is not necessarily incommensurable, but is generally so.