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OF THE MEASUREMENT OF SURFACES.
Preliminary Remarks. We determine the length of a line, by finding how many times another line, which we take for the measure, is contained in it. The line which we take for the measure, is chosen at pleasure ; it may be an inch, a foot, a fathom, a mile, &c. If we have a line upon which we can take the length of an inch 3 times, we say that line measures 3 inches, or is 3 inches long. In like manner, if we have a line upon which we can take the length of a fathom 3 times, we call that line 3 fathoms, &c. To find out which of two lines is the greater, we must measure them. If we take an inch for our measure, that line is the greater, which contains the greater number of inches. If we take a foot for our measure, that line is the greater, which con. tains the greater number of feet, &c.
To measure the extension of a surface, we make use of another surface, commonly a square (O), and see how many times it can be applied to it; or, in other words, how many of those squares it takes to cover the whole surface. The length of the square side is arbitrary. If it is an inch, the square of it is called a square, inch; if it is a foot, a square foot; if it is a mile, a square mile, &c. The extension of a surface, expressed in numbers of square miles, rods, feet, inches, &c., is called the area of the surface. Remark 2. If we take one of the
Fig. I. sides of a triangle for the basis; the perpendicular dropped from the vertex of the opposite angle to that side is called the altitude or height of the
Fig. II. If in the triangle ABC, (Fig. I.)
B for instance, we call AC the basis the perpendicular BD will be its height (altitude). If the perpendicular BD should fall without the trian
gle ABC, (as in Fig. II.) we need only extend the basis, and then drop the perpendicular BD upon the farther extension (CD) of the basis AC.
any perpendicular, MN, CO, PQ, &c. dropped from the opposite side BC, or its farther extension CR upon that basis, or its farther extension DS, will incasure the height of the rectangle. For in a parallelogram the opposite sides are parallel to each other (see Definitions), and all the perpendiculars, dropped from one of two parallel lincs to the other, are equal (Query 12. Sect. I.) What in this respect holds of a parallelogram is applied also to a square, a rhombus, and a rectangle; for these three figures are only modifications of a parallelogram.—(See Definitions.)
As in a rectangle ABCD, the adjacent sides c. AB, BD, are perpendicular to each other, it is evident that if AB is taken for the basis, the side BD itself will be the height of the rectangle.
Remark III. We call two geometrical figures equal * to one another, when they have equal areas, (see preliminary remark to sect. II.)—Thus a triangle is said to be equal to a rectangle when it contains the saine number of square miles, rods, feet, inches, &c. as that rectangle.
0 1 2 3 4 5 6 If the basis AB, of a rectangle ABCD, measures 6 inches, and the 2 height, the side BC, 4 1 inches, how many square inches are there in the rectangle?
* The term equivalent would undoubtedly be better; but as there is no generally adopted sign in Mathematicks to express, that two
A. If a rectangle is 4 inches high, I can divide it like the rectangle ABCD (see the figure) into four rectangles each of which shall be one inch high, and have its basis equal to the basis of the whole rectangle. And as the basis AB, of the rectangle measures 6 inches, by raising upon it, at the distance of an inch from each other, the perpendiculars 1, 2, 3, 4, 5, each of the four rectangles will be divided into 6 square inches; and therefore the whole rectangle ABCD into 24 square inches.
Q. How many square inch- 0 1 2 3 4 5D es are there in a rectangle, whose basis is 5, and height
1 3 inches?
B A. Fifteen.
Because in this case I can divide the rectangle into 3 rectangles of 5 square inches each.
Q. Supposing the measurements of the first rectangle (see the 1st figure) were given in feet, in rods, or in miles, instead of inches, how many square feet, rods or miles would there be in the rectangle?
A. If its measurements were given in feet, it would contain 24 square feet; if they were
1 2 3
things are equivalent without being exactly the same, we are obliged to use the sign equal.
given in rods, it would contain 24 square rods ; and if in miles, 24 square miles; for in these cases I need only imagine the lines, 1, 2, 3, 4, &c. to be drawn a foot, a rod, a 'mile apart ; the number of divisions will remain the same; nothing but their size will be altered.---And in the same manner ;
if the measurements of the second rectangle were given in feet, rods, or miles, it would contain 15 square feet, rods, or miles &c. *
Q. Can you now give a general rule for finding the area of a rectangle ?
A. Yes. Multiply the length of the basis, given in miles, rods, feet, inches, &c. by the height expressed in units of the same kind.
Q. Can you now tell me how to find the area of a square ?
A. The area of a square is found by multiplying one of its sides by itself.–For a square is a rectangle whose sides are all equal (see Definitions) and the area of a rectangle is found by multiplying the basis by one of the two adjacent sides.
* The teacher may also give his pupils a
14 78 rectangle, whose measurements are both given
1 1 12 in fractions; for instance, a rectangle of 3} inches in length and 24 inches high, and then 1 1 1 12 shew by the figure that this rectangle will measure 6 square inches, two half square inches, and f of a square inch, in the whole 77 square inches, which is the answer to the multiplication of 3} by 24.