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BOOK III.

AREA OF FIGURES.

MANY geometrical questions are most simply treated by the ordinary methods of arithmetic.

Of this kind are all questions of ratio and proportion, and most questions as to the area of figures.

It is therefore most important to acquire clear ideas as to the way in which the different geometrical magnitudes are measured.

A straight line is measured by ascertaining how many yards, feet, inches, or other units of length it contains.

An angle is measured by ascertaining how many degrees, minutes, seconds, or other units of angular space it contains.

A surface is measured by ascertaining how many acres, square yards, square feet, or other units of surface it contains. The measurement of straight lines is familiar to everybody.

The measurement of angles we have already explained. The measurement of the area or surface of figures is less but the measurement of a rectangle is a simple operation, and we shall shew that the measurement of any recti

lineal figure may eventually be reduced to the measurement of a rectangle.

In all that follows, whenever the product of two magnitudes is spoken of, it must be understood of the product of the numbers representing these magnitudes.

A line cannot be multiplied by a line, nor a surface by a surface, but the numbers representing a line or a surface may be combined in any way, in accordance with the ordinary laws of arithmetic.

We shall often have occasion to employ the simpler laws of numerical proportion, and it may be convenient to state the more important of them here. They are these.

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In applying arithmetic to geometry we are met by the preliminary difficulty that certain magnitudes are incommensurable with regard to one another, that is, cannot be exactly

measured with the same unit. This difficulty is less serious than it appears, and we shall pass it by, giving proofs which are directly applicable only to commensurable magnitudes. The more advanced student will easily convince himself that whatever is true of these is true also of magnitudes which are not commensurable.

EQUAL FIGURES.

=

The word equal and the sign when used of figures denote equality of area, and nothing more.

Congruent figures are equal, since they can be made to coincide, but equal figures are not necessarily congruent. Two fields may be of the same size, though their boundaries differ in every respect.

THEOREM I

A parallelogram ABCD is bisected by its diagonal. For the triangles ABD, BDC have been shewn to be congruent (1. 16), and are therefore equal.

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COR. The complements about the diameter, that is, the figures AK, KC are equal.

also

For the triangle ABD = the triangle CDB;

the triangle EBK = the triangle FKB, and the triangle HKD the triangle GDK; Therefore the remaining figure the remaining

figure KC.

=

THEOREM II.

Parallelograms upon the same base and between the same parallels are equal.

Let ABCD, EB CF be parallelograms upon the same base BC and between the same parallels BC and AF, they shall be equal one to another.

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Whence in the triangles EAB, FDC

EA=FD, AB=DC_and_ BE = CF.

Therefore the triangles are congruent and equal.

[I. 16.

From the whole fig. ABCF take away the triangle FDC, there is left the parallelogram ABCD.

Again, from the whole fig. ABCF take away the triangle EAB.

There is left the parallelogram EBCF. Therefore the parallelogram ABCD the parallelogram EB CF.

=

COR. Parallelograms upon equal bases and between the same parallels are equal.

For they may always be placed so that their equal bases coincide, which will be equivalent to their being on the same base.

THEOREM III.

Any triangle ABC is half of a parallelogram DBCE on the same base and between the same parallels.

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But ABC is half of ACBF; therefore of DBCE.

THEOREM IV.

Triangles upon the same base and between the same parallels are equal.

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For each is half of any parallelogram on that base and between those parallels.

COR. 1. Triangles upon equal bases and between the same parallels are equal.

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