THE THEORY OF LINEAL OR LONG MEASURE.

189. DEF. An Unit of lineal or long measure, is a straight line of a certain length, arbitrarily fixed upon; and by the determination of the ratios which other lines bear to it, we are enabled to compare with each other, all magnitudes of this description.

Thus, if the straight line ab be considered the lineal unit, the numerical magnitude of the straight line AB will manifestly be

the magnitude of ab: the magnitude of AB :: a lineal unit the lineal units in AB; that is,

:

the numerical magnitude of AB will be

= the magnitude of ab x

the lineal units in AB

a lineal unit

the magnitude of ab × the number of lineal units in AB: whence, representing the magnitude of ab by unity or 1, we shall have the numerical magnitude of AB represented by the number of lineal units contained in it; that is, if the lengths of two straight lines A B and CD be respectively 4 times and 6 times as great as the length of the lineal unit ab, the corresponding magnitudes of the lines AB and CD will be 4 and 6 respectively, which are expressed by the equalities

and a similar method of proceeding will shew that, if any straight line whatever be a multiple of the lineal unit, the numerical representative of its magnitude must be a whole number.

Next, if the proposed line AB be not an exact multiple of the lineal unit ab, but have a common measure with it, so that, when they are both divided by it, the common measure is contained 7 times and 3 times in them respectively; then, we shall evidently have

the magnitude of AB : the common measure :: 7 : 1; and the common measure: the magnitude of ab :: 1 : 3 that is, by Articles (122) and (123), we have

the magnitude of AB

=

7x the common measure;

x

1

the magnitude of AB = 7 × × the magnitude of ab

3

× the magnitude of ab

the magnitude of ab being represented by 1, as before; and thus we infer generally, that whenever a line is not a multiple of the lineal unit, but has a common measure with it, its magnitude may be represented by means of a fraction.

If, however, the proposed line AB be neither a multiple of the lineal unit ab, nor have any common measure with it, as, for instance, if AB = √2, then it is manifest that only an approximate arithmetical representation of it can be had, where the approximation may easily be carried far enough to answer every practical purpose, as appears from Article (167).

It need scarcely be observed here, that if the lineal unit be an inch, a foot, a yard, &c., the corresponding magnitudes of the proposed lines will be expressed in inches, feet, yards, &c., and their parts, respectively.

THE THEORY OF SUPERFICIAL OR SQUARE
MEASURE.

190. DEF. An Unit of superficial or square measure is a square surface or area, whereof the length of each side is equal to that of the lineal unit: thus, if ab represent the lineal unit,

the square abcd described upon it will be the superficial or square unit, having the two dimensions ab and ad,

:

which may be regarded as its length and breadth and the magnitude of any proposed surface or area, will manifestly be obtained by finding what multiple, part, or parts, the surface or area is of this unit.

191. The numerical representative of the Area of a rectangular parallelogram is equal to the product of those of two of its adjacent sides.

Let ABCD be a rectangular parallelogram, whereof the adjacent sides AB and AD contain 7 and 5 lineal units respectively; take AH = AK = the lineal unit, and draw KÈ and HF parallel to AB and AD, intersecting in G, so that the square AG, being equal to the square

abcd, may represent the superficial unit: then by EUCLID, VI. 1, we have

the area of the parallelogram ABEK: the area of the superficial unit AHĞK :: AB : AH :: 7 : 1;

whence, by articles (122) and (123),

the area of the parallelogram ABEK = 7 × the area of the superficial unit AHGK;

again, by the same proposition, we have

the area of the parallelogram ABCD : the area of the parallelogram ABEK : AD : AK :: 5 : 1; or, the area of the parallelogram ABCD = 5 × the area of the parallelogram ABEK;

and therefore, from the preceding equality, we obtain the area of the parallelogram ABCD = 5 × 7 × the area of the superficial unit AHGK;

whence, if the area of the superficial unit be represented by 1, the area of the parallelogram ABCD will, on the same scale, be represented by

5 x 7 or 35,

which is the product of two of its adjacent sides;

or, the area of the parallelogram ABCD = AB × AD

Also, from the general principle of the demonstration just given, it is evident that the same conclusion must hold good, if the sides be represented by fractions or irrational quantities, inasmuch as the proposition of geometry here made use of, has reference to quantity, and not to number only.

Ex. 1. Let the two sides of the rectangular parallelogram be equal to one another, and to 12 inches or 1 foot, so that ABCD becomes a square: then the area of the square ABCD

that is, if the side of a square contain 12 lineal inches, its area will comprise 144 superficial or square inches: or, in other words, 144 square inches are equal to 1 square foot.

Similarly, 9 square feet are equal to 1 square yard, and 30 or 30.25 square yards, to 1 square pole.

Ex. 2. Let the base AC, and the perpendicular altitude BC, of the triangle ABC right angled at C, be repre

sented

B

according to the principles above explained, by 4 inches and by 3 inches respectively: then it follows, from EUCLID, 1. 47, that

whence, by extracting the square roots of both sides of the equality, we have

AB 5 inches;

also, if the sides AC and BC were expressed in feet,

yards, &c., the corresponding value of AB would be found in those terms likewise.

same proposition,

3 feet and BC = 2 feet, we shall have, by the

=

9+ 4 = 13:

AB AC2 + BC2 = 32 + 22

=

and thence, by the extraction of the square root,

which is only an approximation to the true value, but may be continued to as much nicety as we please.

If we had AC = BC

=

1 yard, then would

AB2 AC2+ BC2 = 12 + 12 = 2:

=

and therefore, by performing the same operation, we have

AB = √2 1.4142135 &c. yards:

=

and this, in fact, proves the hypothenuse of a rightangled isosceles triangle, or the diagonal of a square, to be incommensurable with either of the sides.

From the last two instances, it appears that a quadratic surd may be expressed accurately in Geometry, though not so in Arithmetic; and it is also clear, from the mode of proceeding adopted, that any other geometrical proposition may be translated into the symbols of Arithmetic, and any part determined, when the number of the data is sufficient for the purpose.

192. From these principles, if the base and perpendicular altitude of a plane triangle be represented by numerical magnitudes, its area will be numerically represented by half their product.

For, let the base AB be equal to 8 feet, and the perpendicular altitude CD to 3 feet:

B

then, by EUCLID, I. 41, the area of the triangle ABC