Let A B C D be any parallelogram; the sides A B and D c are equal to each other, and so are the sides A D and в ç. Also,

the opposite angles at a and c, and those at в and D, are equal to each other. And if a diameter is drawn from a to c, or from B to D, that diameter will divide the surface of the parallelogram into two equal parts; namely, into two triangles, having all their sides and angles equal, only placed in reversed positions upon the diameter.

Because D C and a в are parallel, the angles at D and a are together equal to two right angles, and because a D and B c are parallel, the angles at A and B are also together equal to two right angles. Leave out the angle a, which is common to those equals, and the remaining angle D is equal to the remaining angle B. But, because of the parallels, the angles в and c are together equal to two right angles, or to the two angles at D and A. From these equals, take away D and в which are also equal, and the remainders, which are a and c, must be equal. But again: the alternate angles D C A and C A B on the parallels D C, A B, are equal, and so are the alternate angles D A C, AC B, on the parallels D A, C B. Therefore, the two triangles A B C and C A D, have the side a c common, and all their angles équal; therefore, the sides opposite the equal angles are equal, that is, A B in the one is equal to c n in the other, and B c in the one equal to D A in the other. Consequently these triangles are equal to each other in every respect; and as the two together make up the whole parallelogram, each one singly must be equal to the half of it. Therefore, the opposite sides

and angles are equal, and the diameter bisects the parallelogram.

The converse of this, that straight lines which join the extremities of equal and parallel straight lines towards the same parts, are themselves equal and parallel, follows, as a matter of course, from this equality of the opposite sides of a parallelogram.

2. If a parallelogram has one right angle, all its angles are right angles.

This follows immediately from what was shown in article 1. The opposite angles are equal, so that if one is a right angle the one opposite to it must also be a right angle; but the four angles are altogether equal to four right angles, and these two opposite ones are equal to two, so that there remain other two right angles for the remaining two opposite angles of the figure; but they, too, must be equal to each other, or each of them must be a right angle. Consequently, if one angle of a parallelogram is a right angle, each of the four must be a right angle.

3. A parallelogram, which has its angles right angles, is called a rectangular parallelogram, or simply a RECTANGLE; and a figure of this kind is said to be contained by the two sides which are about or include any of its angles. This last expression has reference to the area or surface of the rectangle ; and the one of the containing sides is called the base, and the other the altitude or height, and the one may always be considered as representing length, and the other as representing breadth. Thus, when we speak of the rectangle A B C D as a surface, we call it the rectangle A B, B C, which means that the expression for the surface is the product of a в multiplied by в c, that is, A B XB C, both being understood to express the same kind of measure as lines, and the surface as many squares of the lineal measure, as the product of the numbers. This is the original connexion between Geometry and Arithmetic; and the princi

ples of it have been explained in a former part of this volume, so that we need not dwell upon it at present, nor adduce any example in illustration, farther than the following simple figure:

DF

B

A B C D is any rectangle whatever, of which в c is the length or base, and A в the breadth or altitude, The value, that is, the area of the rectangle, must remain the same, while the length and breadth remain the same, or, which is the same thing, equal; for the area is the product of those dimensions or factors, unaffected by any thing except their individual lengths and the fact of their standing to each other in the relation of length and of breadth; and while they remain equal and preserve this relation, there is no circumstance which can in any way alter the area or value. Thus, for instance, if a D is produced towards E, still parallel to в c, but to any distance whatever, and c E joined as by the dotted line, and B F drawn through в parallel to c E, then the oblique parallelogram F B CE must also have the same area or surface as the rectangle A B C D. Also, if a B is produced indefinitely, and two parallels drawn from D and c, in any direction, but both in the same, till they meet A B produced in the points H and G, the parallelogram HDCG thus formed must be equal in surface to the rectangle A B C D, in what place or in what position soever it hap

pens to be situated; for we have already shown that place and position, are not elements which can affect the value of any magnitude or quantity.

This, which is a very useful property, is usually cited in the words, "Parallelograms upon the same or equal bases, and between the same parallels, are equal to each other." But there is some objection to the words "between the same parallels," inasmuch as, to a beginner, they tie down the equal parallelograms to one locality; and, therefore, the words "equal bases and altitudes," or " equal lengths and breadths," are preferable, inasmuch as they are not tied down to any particular locality, but can be understood and applied whether the parallelograms are between the same parallels or not.

the

From mere inspection of the equal parallelograms in the above figure, it will be seen that while the area or surface remains of the same value, the lengths of the sides may undergo changes. Thus, for instance, in the oblique parallelogram B C E F, parallel sides which are dotted, namely в F and c E, are each greater than B A or C D ; for they are opposite right angles at a and D, while B A and C D are opposite angles at F and E in the same triangles, each of which is less than a right angle. But A B might have been produced any length to G, so that the continuation is still parallel to c D. CG might be joined and D H drawn from D parallel to c G, and meeting a G in н; and the parallelogram G C D H must still be equal in area to the right angle A B C D; for the two dimensions, of which its area or surface is the product, remain unchanged, both in their lengths and in their relation of base and altitude, or length and breadth ; while the sides which bound the oblique parallelogram, and make angles less than right angles with the parallels, are always the greater the less the angles that they make. Hence we have this general conclusion, that if in any figure whatever we can get two dimensions which stand to each other in the relation of

432

PARALLELOGRAM AND TRIANGLE.

length and breadth, and correctly represent the length and breadth of that figure as a rectangle, then the product of those two dimensions will always be a correct expression for the area or surface of the figure, whatever may be the shape of that figure, provided only that it is bounded by straight lines.

It follows from this, that any straight lined figure whatever may be reduced to a rectangle of exactly the same value, and its surface as a whole expressed in square measures, that is, measures which are the squares of some known measures of length. This is the principle which applies to the measuring of land, and of all other surfaces whatever, whatever may be the form and dimensions, if the surface is a plane, or regarded as such.

4. Upon looking back at the diagram in article 1, page 428, it will be understood that while A B is the length or base of the parallelogram, a perpendicular let fall from D upon a B, or from c upon A в produced, must express the altitude or breadth of the parallogram A B C D; and if we call this perpendicular p, then the area or surface of the parallelogram will be expressed by the product A B × P.

Now the triangle A c B, which is on the same base with the parallelogram and has the same altitude, was shown to be equal in surface to half the parallelogram; and if the base and altitude had been equal, the surface would have been the same, as the mere situation, by not affecting either the value or the relation of either of the factors, cannot possibly affect the value of the product of those factors. Hence we have the general conclusion if a triangle and parallelogram have equal bases and altitudes, the area of the triangle is half that of the parallelogram.

From this we can immediately derive a method of finding the area either of a parallelogram or triangle, when the base and altitude are known. It is: for the parallelogram, multiply the base and altitude; and for the triangle, multiply the same, and take half