also, the figure on B E is equivalent to the triangle C B D, that is, to the given figure.

EXERCISES.

(1.) Construct a regular hexagon equivalent to a square of 2 inches side.

(2.) The sides of a triangle measure 4 inches, 3 inches, and 2 inches respectively; construct an equivalent trapezoid, having the longest parallel side three times the length of the shorter, and the other side inclined at angles of 45°.

(3.) The sides of a trapezium measure 1 inches, 2 inches, 3 inches, and 4 inches, and the angle between the two longest sides is 30°; construct a regular octagon equivalent to the trapezium.

(4.) Construct a regular pentagon equal to the same trapezium. (5.) Construct a triangle equivalent to an equilateral triangle of 3 inches side, and having two of its angles 75° and 45° respectively.

ADDITION AND SUBTRACTION OF SIMILAR FIGURES.

238. PROBLEM 123.-To construct a figure which shall be equivalent (1) to the sum, (2) to the difference of two similar and similarly described figures.

(1.) Construct the right-angled triangle B A C, Fig. 190, having its sides, A B, A C, equal to homologous or corresponding sides of the two given figures; then shall the similar and similarly described figure on the hypothenuse B C be equivalent to their sum.

(2.) Construct the right-angled triangle B A C, Fig. 191, having its hypothenuse B C equal to a side of the largest of the

two given figures, and A C, one of its sides, equal to the homologous or corresponding side of the smaller; then shall the similar and similarly described figure on the other side, A B, be equivalent to the difference of the two figures.

Reason. The rectilineal figure described upon B C is (Theor. 33) equal to the sum of the rectilineal figures similar and similarly described upon A B and A C; and, consequently, the rectilineal figure on A B is equal to the excess of the similar and similarly described figure upon B C, over that described upon A C. Again, since circles are to another as the squares of their diameters (Theor. 34), and the square upon B C is equal to the

sum of the squares upon A B and A C; therefore, the circle having B C for its diameter, is equal to the sum of the two circles having A B and A C for their diameters.

Lastly, the same relation will also hold good for semicircles, or any equal portions of circles, such as similar segments of similar sectors.

EXERCISES.

(1.) Construct an equilateral triangle equivalent to the sum of two equilateral triangles, one inscribed in a circle of 2 inches d.ameter and the other described on the diameter.

(2.) Construct an equilateral triangle equivalent to the difference between the triangle just constructed and the equilateral triangle described about the circle.

(3.) The angles of a triangle are 45°, 60°, and 75°, and the perpendicular from the largest angle upon the opposite side measures 1 inch; construct (1) a square equivalent to the squares described upon the two shortest sides of the triangle; (2) a square equivalent to the difference between this square and that described on the largest side of the triangle.

(4.) A square is inscribed in an equilateral triangle whose sides are each 24 inches long; construct circles equivalent to the sum and difference of the circles described upon a side of the triangle, and a side of the inscribed square as diameters.

MULTIPLICATION AND DIVISION OF FIGURES.

239. PROBLEM 124.-To construct a similar figure having a given ratio to a given figure.

Take A C, Fig. 192, in the given ratio to A B, a side of the given figure; find A D, a mean proportional between A B and A C; on A D describe a figure similar and similarly situated to the given figure on A B; then the figure thus described shall have to the given figure the given ratio.

Reason. By the construction, A B is to A D as A D to A C; therefore (Theor. 32), the figure on A B is, to the similar and similarly situated figure on A D as A B to A C, or the figure on A D has to the figure on A B the ratio of A C to A B, that is, the given ratio.

EXERCISES.

(1.) The equal sides of an isosceles triangle measuring each 1.8 inches, and including an angle of 50° construct (1) a similar

H

triangle three times as large; (2) a similar triangle of one-third the size.

(2.) A regular hexagon being described about a circle of 1

inches diameter, construct (1) a regular hexagon of five times the size; (2) a regular hexagon of one-fifth the size.

(3.) A circle being inscribed in an equilateral triangle, each sidde of which measures 2 inches, construct (1) a circle to contain seven times the area, (2) a circle to contain one-seventh of the

area.

NUMERICAL REPRESENTATION OF AREAS.

240. Hitherto figures have been constructed equal in area to other figures, but differing in form, or figures similar in form to others but differing in area. By combining these two classes of problems, figures may readily be constructed, differing, in a given manner, both in form and area, from other given figures. The modes of determining the numerical representative of the area of a given figure, and of constructing a figure whose area is given by its numerical representative, remain to be considered. The number representing an area deperds upon the unit of

area assumed as the standard of comparison. The standards which are employed for this purpose vary with the standards which are used for the units of linear measurements, but are universally the squares described upon the linear units. Thus, the English units of length being the inch, the foot, the yard, &c., the units of area are the squares upon an inch, a foot, a yard, &c.; the French units of area are the squares upon the metre and its multiples and submultiples; and so with other nations, if we know the standard units of linear measure, the squares upon these units will be their units of surface measurement.

241. To find the number representing the area of a rectangle. Multiply together the numbers representing the linear dimensions of two adjacent sides, Fig. 193.

242. Conversely, to construct a rectangle of a given area.

It is only necessary to split into two factors the number representing the area, and construct the rectangles whose adjacent sides contain the numbers of linear units represented by the factors; thus, the area of a rectangle being stated to be 12 square inches, the sides of the rectangle will be 12 inches and 1 inch, 6 inches and 2 inches, or 4 inches and 3 inches, or 5 inches and 2 inches, or any two numbers of inches whose

product is 12.

243. PROBLEM 125.-To construct a square to contain an area represented by a given numerical quantity.

(1.) If the given numerical quantity be a perfect square, either whole or fractional, its square root will be a side of the required square. If the given numerical quantity be not an