Miscellaneous Exercises on Book II.

1. In a triangle, whose vertical angle is a right angle, a straight line is drawn from the vertex perpendicular to the base; shew that the square on either of the sides adjacent to the right angle is equal to the rectangle contained by the base and the segment of it adjacent to that side.

2. The squares on the diagonals of a parallelogram are together equal to the squares on the four sides.

3. If ABCD be any rectangle, and O any point either within or without the rectangle, shew that the sum of the squares on OA, OC is equal to the sum of the squares on OB, OD.

4. If either diagonal of a parallelogram be equal to one of the sides about the opposite angle of the figure, the square on it shall be less than the square on the other diameter, by twice the square on the other side about that opposite angle.

5. Produce a given straight line AB to C, so that the rectangle, contained by the sum and difference of AB and AC, may be equal to a given square.

6. Shew that the sum of the squares on the diagonals of any quadrilateral is less than the sum of the squares on the four sides, by four times the square on the line joining the middle points of the diagonals.

7. If the square on one perpendicular from the vertex of a triangle is equal to the rectangle, contained by the segments of the base, the vertical angle is a right angle.

8. Produce a given straight line so that the rectangle contained by the whole line thus produced and another given straight line may be equal to the square on the produced part.

9. ABC is a triangle right-angled at A; in the hypotenuse two points D, E are taken such that BD=BA and CE=CA; shew that the square on DE is equal to twice the rectangle contained by BE, CD.

10. In any quadrilateral the squares on the diagonals are together equal to twice the sum of the squares on the straight lines joining the middle points of opposite sides.

11. If straight lines be drawn from each angle of a triangle to bisect the opposite sides, four times the sum of the squares on these lines is equal to three times the sum of the squares on the sides of the triangle.

12. CD is drawn perpendicular to AB, a side of the triangle ABC, in which AC=AB. Shew that the square on CD is equal to the square on BD together with twice the rectangle AD, DB.

13. If in any triangle BAC a line AD be drawn bisecting BC in D, shew that the sum of the squares on AB, AC is equal to twice the sum of the squares on AD, BD. ·

14. If ABC be an equilateral triangle, and AD, BE be perpendiculars to the opposite sides intersecting in F; shew that the square on AB is equal to three times the square on AF.

15. Divide a given straight line into two parts, so that the rectangle contained by them shall be equal to the square described upon a straight line, which is less than half the line divided.

NOTE 6.—On the Measurement of Areas.

To measure a Magnitude, we fix upon some magnitude of the same kind to serve as a standard or unit; and then any magnitude of that kind is measured by the number of times it contains this unit, and this number is called the MEASURE of the quantity.

Suppose, for instance, we wish to measure a straight line AB. We take another straight line EF for our standard,

EF

and then we say

if AB contain EF three times, the measure of AB is 3,

if

if

four......

х .......

..4,

...x.

Next suppose we wish to measure two straight lines AB, CD by the same standard EF.

where m and n stand for numbers, whole or fractional, we say that AB and CD are commensurable.

But it may happen that we may be able to find a standard line EF, such that it is contained an exact number of times in AB; and yet there is no number, whole or fractional, which will express the number of times EF is contained in CD.

In such a case, where no unit-line can be found, such that it is contained an exact number of times in each of two lines AB, CD, these two lines are called incommensurable.

In the processes of Geometry we constantly meet with incommensurable magnitudes. Thus the side and diagonal of a square are incommensurables; and so are the diameter and circumference of a circle.

Next, suppose two lines AB, AC to be at right angles to each other and to be commensurable, so that AB contains four times a certain unit of linear measurement, which is contained by AC three times.

Divide AB, AC into four and three equal parts respectively, and draw lines through the points of division parallel to AC, AB respectively; then the rectangle ACDB is divided into a number of equal squares, each constructed on a line equal to the unit of linear measurement.

If one of these squares be taken as the unit of area, the measure of the area of the rectangle ACDB will be the number of these squares.

Now this number will evidently be the same as that obtained by multiplying the measure of AB by the measure of AC; that is, the measure of AB being 4 and the measure of AC 3, the measure of ACDB is 4 × 3 or 12. (Algebra, Art. 38.)

And generally, if the measures of two adjacent sides of a rectangle, supposed to be commensurable, be a and b, then the measure of the rectangle will be ab. (Algebra, Art. 39.)

If all lines were commensurable, then, whatever might be the length of two adjacent sides of a rectangle, we might select the unit of length, so that the measures of the two sides should be whole numbers; and then we might apply the processes of Algebra to establish many Propositions in Geometry by simpler methods than those adopted by Euclid.

Take, for example, the theorem in Book 11. Prop. IV.

If all lines were commensurable we might proceed thus .— Let the measure of AC be x,

of CB... y,

Then the measure of AB is x+y.
(x + y)2 = x2 + y2 + 2xy,

Now

which proves the theorem.

But, inasmuch as all lines are not commensurable, we have in Geometry to treat of magnitudes and not of measures: that is, when we use the symbol A to represent a line (as in I. 22), A stands for the line itself and not, as in Algebra, for the number of units of length contained by the line.

The method, adopted by Euclid in Book II. to explain the relations between the rectangles contained by certain lines, is more exact than any method founded upon Algebraical principles can be; because his method applies not merely to the case in which the sides of a rectangle are commensurable, but also to the case in which they are incommensurable.

The student is now in a position to understand the practical application of the theory of Equivalence of Areas, of which the foundation is the 35th Proposition of Book I. We shall give a few examples of the use made of this theory in Mensuration.

Area of a Parallelogram.

The area of a parallelogram ABCD is equal to the area of the rectangle ABEF on the same base AB and between the same parallels AB, FC.

Now BE is the altitude of the parallelogram ABCD if AB be taken as the base.

Hence area of

ABCD=rect. AB, BE.

If then the measure of the base be denoted by b,

and

h,

altitude

will be denoted by bi

the measure of the area of the

That is, when the base and altitude are commensurable,

measure of area measure of base into measure of altitude.

S. E.