to measure the surface of any rectangle, we have only to compare the length of its base and altitude, respectively, with that of the base and altitude, respectively, of a rectangle which we take for the unit of surface. But in order that our measurement should be generally understood, it will not be sufficient to state how many times our superficial unit is contained in the rectangle we measure, unless we at the same time specify the size of that superficial unit we employ. This may be done, either by specifying the lengths of its base and altitude, or by a common agreement to employ the same unit of surface in all measurements. The latter expedient is evidently the best; and thus in England a square-inch, that is, the square of the linear unit, is generally employed as the superficial unit.

B

C

Example. Let ABCD be a rectangle, one of whose sides AD is two feet long, the adjacent side AB one foot and a half. What is the superficial magnitude of this rectangle? In order to answer this question, I suppose both AD and AB divided, the former into 24 inches, the latter into 18. I then multiply these numbers, which gives 432, the number of square inches, by ART. 89, in the surface ABCD. Hence, the

A

answer is, 432 square inches; the truth of which may be practically established, if the student pleases, by reckoning them. In each of the upright belts there are 18 squares, and as there are 24 of these belts, adding 18 twenty-four times, the result is 432; the whole number of squares in ABCD.

We e may now perceive the convenience of adopting for our superficial unit the square of the linear unit. If we specify the sides of a rectangle in inches, we have only to multiply the number of those in the two adjacent sides, and we obtain the surface of the rectangle immediately in square inches. If we specify the sides in feet, yards, or any other linear unit, we obtain the surface immediately in square-feet, square-yards, or the square of that linear unit employed. So that, if we chose any other superficial

unit than the square of the linear, we should have to find by another process how many of the former were in the surface measured immediately by the latter; which would be obviously superfluous.

By ART. 28, a parallelogram is equal to a rectangle on an equal base, and whose adjacent side is equal to the altitude of the parallelogram. By ART. 129, a triangle is equal to half a rectangle on an equal base, and whose altitude is equal to the altitude of a triangle. Hence, the surface of any parallelogram is found by multiplying the number of linear units in its base by the number of those in its altitude; the result will be the number of squares of that linear unit contained in the given parallelogram. Likewise: the surface of any triangle is found by multiplying the number of linear units in its base by the number of those in its altitude; half the result will be the number of squares of that linear unit in the given triangle *.

Example 1. How many square-inches is a square-foot ? LEARNER. 144.

Because a square-foot is the square of a line 12 inches long; that is, the rectangle under two lines, each 12 inches long. But, according to the rule just given, the surface of this rectangle is equal to 12 times 12, or 144, squareinches.

TEACHER. Hence you perceive that if we have any surface expressed by a number of square-inches, it is only necessary to divide that number by 144, and we obtain the number of square-feet in that surface.

Example 2. How many square-feet is a square-yard? LEARNER. 9.

Because a square-yard is the square of a line 3 feet in length; that is, the rectangle under two lines, each 3 feet long. Consequently, a square-yard is equal to 3 times 3, or 9 square-feet.

TEACHER. And consequently to 9 times 144 squareinches, or 1296.

* Instead of taking half this result, it will come to the same thing if we multiply the number of linear units in the base or altitude by half the number of those in the altitude or base.

Hence you perceive that, if we have any surface expressed by a number of square-feet, it is only necessary to divide that number by 9, and we obtain the number of square-yards in that surface. Or, if we have any surface expressed by a number of square inches, it is only necessary to divide that number by 1296, and we obtain the number of square-yards in that surface.

Conversely: If we have any surface expressed by a number of square-yards, we have only to multiply that number by 9, and we obtain the number of square-feet; or by 1296, to obtain the number of square-inches in that surface. And if we have any surface expressed by a number of square-feet, we have only to multiply that number by 144, and we obtain the number of square-inches in that surface.

Example 3. If the base of a parallelogram be 3 feet, and its altitude 5 feet, what is its superficial magnitude in square feet and square inches respectively?

Answer. 15 square-feet, 2160 square-inches. Here we take a square-foot and a square-inch, successively, as superficial unit.

Example 4. If the base of a triangle be 9 feet, and its altitude 3 feet; what is its superficial magnitude in squareyards, square-feet, and square-inches, respectively?

Answer. 9 square-yards, 27 square-feet, 3888 square-inches.

Here we take a square-yard, a square-foot, and a squareinch, successively, as superficial unit.

Finally: Every rectilineal figure being divisible into triangles, the surface of each may be computed as above, and by adding these together, we obtain the whole surface of the given rectilineal figure.

It is for this reason that we choose a rectangular figure in preference to any other for a superficial unit. All rectilineal figures may be instantly reduced to triangles *, and the surfaces of these may be instantly computed by

Nay, curvilinear figures may be also included, by looking upon them as polygons of an infinite number of small rectilinear sides.

considering them as the halves of rectangles with equal bases and altitudes.

We have been minute in our explanation of this subject, chiefly because of the veil of obscurity, and indeed, absurdity, thrown over it by almost every geometrical treatise to be met with. This has arisen from the authors negligently applying arithmetical terms to geometrical quantities: they talk of " multiplying the line AB by the line BC"-of " the product of two lines"-of" a line being the square-root of a number." It would, perhaps, be better to avoid such modes of expression altogether, as they tend to confuse and mislead; but if they must be introduced, it is at least incumbent on the author to explain the sense in which they are employed, and are to be understood. When, therefore, a reader meets such expressions as these, he is always to recollect that the lines are considered, each as a certain number of linear units, and that it is the multiplication, product, &c. of these numbers, and not of the. lines, which is spoken of.

The incalculable utility of the Doctrine of Rectangles must now be apparent. By its means we are enabled to measure the superficial magnitude or area of all flat, straight-sided objects, as plains, walls, floors, &c. &c. as in the following examples.

Let there be a plane mirror ABCD of the usual form, that is to say, rectangular, and3 let its adjacent sides AB, AD, be respectively 5 feet 6 inches, and 4 feet 3 inches in length: what is the area of the mirror? LEARNER. 3366 square-inches.

TEACHER. Prove this.

LEARNER. As the sides AB and AD are A

respectively 66 and 51 inches long, multiplying these numbers, the result is 3366, for the number of square inches in

the area?

TEACHER. How many square-feet in the same area? LEARNER. 23 square-feet, with a remainder of 54 square-inches.

TEACHER. CD represents the wall hand-rail up a flight of stairs; CB and DE are its continuations on the landings below and above. A Housepainter wishes to measure the work done beneath this rail BCDE to the level AHGF of the floor: how is he to proceed?

B

A H

G

LEARNER. BC, CD, and DE, are supposed to be respectively equal and parallel to AH, HG, and GF: consequently, ABCH, HCDG, and GDEF, are each of them parallelograms, the first and last being rectangles. Wherefore, taking the lengths of AH and AB (suppose 2 and 3 feet respectively), we get twice three, or 6 square feet, for the area of ABCH.-Again, taking the length of HG, and the altitude of HCDG (suppose 6 feet and 2 feet 6 inches respectively), we get 72 multiplied by 30 inches, or 2160 square inches, that is, 15 square-feet, for the area of HCDG. Finally, taking the lengths of GF (suppose 1 foot), we get 3 square feet, as GD is equal to AB, for the area of GDEF. Hence, the area of the whole piece of work, ABCDEFGH, must be 24 square-feet.

TEACHER. It is on the principle set forth in ART. 89, that almost all Land-Surveying and plane Mensuration depend. Corrections and modifications must be used where the surfaces to be measured are not exactly flat, or exactly rectilineal; but the foundation principle is this theorem. Suppose the triangular piece of ground ABC is to be measured, let us see how the surveyor may find its area on the above principle *.

A

B

D

Let him measure any of its sides, as AC, and the perpendicular BD on that side from the vertex of the opposite angle. Let these lines be respectively 40 and 20 yards

* It is not exactly in the following way that lands or surfaces on the Earth are usually measured; because, however simple in theory, it is not always so in practice. But in very many cases it is quite sufficient, and in all measurements of surfaces it is indispensable as a preliminary

element.