3. Draw a rectangle of four by five inches. What is its area? Ans. 20 square inches. 4. Draw a rectangle of six by eight inches. What is its area? Ans. Let the Pupil draw the foregoing Problems on the Blackboard. No. 4 has twice the length of sides of No. 3. How many times larger than No. 3 is it? (Four times larger; because No. 3 contains one square inch, and No. 4 contains four square inches.) No. 5 has four times the length of sides of No. 3. How much larger than No. 3 is it? (Sixteen times larger.) No. 5 has twice the length of sides of No. 4. How much larger than No. 4 is it? (Four times larger.) No. 7 has twice the length of sides of No. 6. How much larger is No. 7 than No. 6? (Four times larger.) From the foregoing it appears that, by increasing the lengths of the sides of a square or a rectangle to two times their length, we form a similar figure four times as large; by increasing to three times, we form a similar figure nine times as large; by increasing to four times, we form one sixteen times as large; by increasing to five times, we form one twenty-five times as large, etc. The same principle holds true with regard to a figure of any number of sides. ELEMENTARY PRINCIPLE.-The areas of similar plane figures are as the squares of their similar sides. If, therefore, we have a plane figure of any number of sides, and wish to make another similar to it, but four times as large, we double the lengths of the sides; because 2 times 2 are four: if we wish to make one nine times as large, we treble the lengths of the sides; because 3 times 3 are nine: if we wish to make one sixteen times as large, we quadruple the lengths of the sides; because 4 times 4 are sixteen: and so on to the square of any given number. Let the teacher explain more fully, if necessary, what is meant by the square of a number, and especially when that number represents the length of a given line. PROBLEMS FOR PRACTICE. 1. Draw a square similar to No. 3, but nine times as large. A polygon is a plane figure having many sides and many angles. The term is generally applied to a plane figure of more than four angles and four sides. Free-hand exercises on the blackboard.-Let the pupil follow out, on the blackboard, a course of exercises similar to those prescribed for the tint-lined drawing-paper. LESSON IV. Diagonals.-Diagonals are lines drawn in the direction of a diagonal of a primary erect square. A primary diagonal is a line drawn diagonally from one corner to another of a primary erect square. No. 1 is made up of primary diagonals in two directions. No. 2 is a primary diagonal square. What is its area equal to? (Two square inches; inasmuch as it includes four halves of the small primary erect squares.) What is the area of No. 3 ?* What is the area of No. 4? No. 5? No. 6? If No. 2 have its sides doubled in length, how much larger will the figure be? If No. 2 have its sides trebled in length, how much larger will the figure be? PROBLEMS FOR PRACTICE. 1. Draw a diagonal square similar to No. 2, but sixteen times as large; that is, containing sixteen times the area of No. 2. sides be, compared with the sides of No. 2 ? How long must the 2. Draw a diagonal square similar to No. 2, but twenty-five times as large. How long must the sides be, compared with the sides of No. 2 ? 3. Draw a diagonal square similar to No. 3, but nine times as large. 4. Draw a diagonal rectangle similar to No. 5, but four times as large. *The halves of square inches included within the figures in this lesson might be marked with dots, for greater facility in counting them. In drawing these problems let the pupils arrange them in such a manner as to economize the space on the drawing-paper. To find the area of any diagonal square, or other diagonal rectangle: RULE A.-Multiply the length in primary diagonals by the breadth in primary diagonals, and TWICE the product will be the area, in measures of the primary eréct squares. Rule A is only a special application of Rule I. (Reason for the rule.-The length in primary diagonals multiplied by the breadth in primary diagonals will give the number of primary diagonal squares; and we then multiply by 2, because there are two primary erect squares in each primary diagonal square.) Thus, in No. 3, multiply 2, the length in primary diagonals of one side, by 2, the length in primary diagonals of another side, and the product will be 4; and twice four will be the area in primary erect squares, or square inches. What is the area of a diagonal square of 7 diagonals to a side? (Ans. 98 square inches.) What is the area of a diagonal rectangle of 5 by 7 diag onals? (Ans. 70 square inches.) Let the pupil carry out the same system on the blackboard. LESSON V.-No. 1 is an erect cross, representing one thin piece, 2 inches by 8 inches, laid at right angles across another piece 2 inches by 6 inches. First draw the upper piece, marked 1, and shade it lightly. The lower piece might have the shading described in No. 4 of Lesson II. No. 2. Draw the pieces in the order in which they are numbered. The lower piece is first shaded with diagonal lines, the same as the upper piece, and the shading is finished by drawing lines between the diagonals first drawn. Nos. 3 and 4. In these, and in all similar figures, the upper pieces-supposing that the pieces are in a horizontal position should be drawn first. In most outline drawings, and in lightly shaded drawings, the outline is made heaviest on the side opposite to the direction from which the light is supposed to come. Thus, in No. 4, the light is supposed to come in the direction of the arrow a; and hence the outlines are made the heaviest where the shadows would naturally fall. No. 5. Observe the direction in which the light falls upon this figure, as indicated by the arrow b, and the consequent heavy outlines of those sides of the four pieces which would be in shadow. The shading in No. 7 should render each square distinct from the others. No. 8 is a pattern made up of only one figure, repeated continuously, and so arranged as to cover the entire surface. A very great variety of patterns, consisting wholly of repetitions of one figure to each pattern, may easily be designed, and drawn by the aid of the ruled paper. What is the area of each of the squares, as they are numbered, in No. 7. The area of the pattern figure in No. 8? PAGE TWO. LESSON VI. Two-space Diagonals.-By a two-space diagonal is meant the diagonal of a rectangle which is twice as long as it is broad. It is a diagonal which passes over two spaces on the ruled paper. No. 1. The lines in No. 1 are two-space diagonals. They should be copied, without the aid of a ruler, until they can be drawn with tolerable accuracy, and with facility. At b lines are first drawn as at a; and then lines are drawn intermediate between them; c is first drawn the same as b, and is then filled in with intermediates. In this manner great uniformity of shading may be attained. No. 2 is drawn in a manner similar to No. 1. First trace each line lightly, and continue to pass the pencil over it until it is drawn with accuracy. No. 3. As two square inches are represented in the dotted rectangle, and as the line a b divides the rectangle into two equal parts, therefore on each side of the line there is an area equal to one square inch, Ç 2 BRAR OF THE UNIVERSITY OF No. 4. What area is embraced within the dotted square? Then how much is embraced within the portion a? No. 5. What area is embraced within the dotted rectangle? Then what area is embraced within the portion a? The portion marked a in No. 4 is a triangle-a figure of three sides and three angles. It is an acute-angled triangle, because each angle is less than a right angle. (See Lesson III.) The portion marked a in No. 5 is called an obtuse-angled triangle, because one of the angles is greater than a right angle. No. 6 is a figure called a rhombus. A rhombus is a figure which has four equal sides, the opposite sides being parallel; but its angles are not right angles. What area is embraced in the upper half of No. 6? In the whole figure? No. 7. What area is embraced in each of the parts a of No. 7? In the central rectangle b? In the whole rhombus? (16 square inches.) No. 8. In the dotted figure No. 8 there are three of the small squares; hence the dotted figure contains an area of three square inches. But the part b (as shown in No. 3 and No. 5) contains an area of one square inch, and the part c an area of one square inch; hence the part a must contain an area of one square inch also. No. 9. What area is embraced in the rhombus No. 9? (Let the pupil prove that each part a embraces an area of one square inch, the same as a in No. 8.) No. 10. What area is embraced in No. 10? How is it shown that the upper part marked 1 contains an area equal to one square inch? No. 11. What area is embraced in the star figure No. 11? (Let the pupil prove that each of the points marked 1 contains an area of one square inch.) No. 12 is an octagonal or eight-sided figure. A regular octagon has eight equal sides and eight equal angles; but here, while the sides are equal, the angles are not all equal. What is the area of each of the parts a of the octagon? Of the whole octagon? The shading of the central square of No. 12 is produced by carrying the pencil from left to right with a running dotting motion. In industrial drawing it is desirable to designate the different sides or surfaces of objects |