« ΠροηγούμενηΣυνέχεια »
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,
and then we say
if AB contain EF three times, the measure of AB is 3,
Next suppose we wish to measure two straight lines AB, CD by the same standard EF.
AB contain EF m times
CD. 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 II. 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 1. 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 .... altitude the measure of the area of the will be denoted by in That is, when the base and altitude are commensurable, measure of area measure of base into measure of altitude.
Area of a Triangle.
If from one of the angular points A of a triangle ABC, a perpendicular AD be drawn to BC, Fig. 1, or to BC produced, Fig. 2,
Now if the measure of BC be b,
.AD... h, measure of area of
and if, in both cases, a parallelogram ABCE be completed of which AB, BC are adjacent sides,
area of ▲ ABC=half of area of ABCE.
.. measure of area of ▲ ABC is
Area of a Rhombus.
ABCE is bh ;
Let ABCD be the given rhombus.
Draw the diagonals AC and BD, cutting one another in O.
It is easy to prove that AC and BD bisect each other at right angles.
Then if the measure of AC be x,
twice measure of ▲ ACD. =twice xy
Area of a Trapezium.
Let ABCD be the given trapezium, having the sides AB, CD parallel.
Draw AE at right angles to CD.
Produce DC to F, making CF=AB.
Join AF, cutting BC in O.
Then in As AOB, COF,
:: BAO= 4 CFO, and ▲ AOB= ▲ FOC, and AB=CF; .. A COF▲ AOB. I. 26.
Hence trapezium ABCD= ▲ ADF.
Now suppose the measures of AB, CD, AE to be m, n, p respectively;
.. measure of DF=m+n, :: CF=AB.
Then measure of area of trapezium
=(measure of DFX measure of AE)
That is, the measure of the area of a trapezium is found by multiplying half the measure of the sum of the parallel sides by the measure of the perpendicular distance between the parallel sides.