EXERCISES ON THE PROPOSITIONS. PROP. I., BOOK I.-Describe an equilateral triangle upon the lower side of a given finite straight line. PROP. II., BOOK I.—From a given point to draw a straight line equal to a given finite straight line. Let A be the given point, and BC the given finite straight line: the equilateral figure employed in the construction is to be drawn on the lower side of the line A B. PROP. IV., BOOK I.-This is not really a proposi tion, though commonly so ranked; it is but a definition of coincidence, the triangles A B C, DEF being merely made to coincide. PROP. V., BOOK I.-Prove the proposition, letting the triangles AFC and AG B, and also FBC and GCB be separate figures. PROP. IX., BOOK I.-Draw the figure so that the equilateral triangle DFE may lie on the upper side of DE and be greater than DA E. Do the same with DFE less than DA E. PROP. XI., BOOK I.-Let the point c be at the end of the given line A B, and then work the proposition. PROP. XV., BOOK I.-Prove the proposition with a pair of scissors, a sheet of cardboard, and a lead pencil. PROP. XVI., Book I.-Bisect BC and show that the angle BCG is greater than A B C. PROP. XVII., Book I.-With a paper triangle show practically that any two angles of a triangle are together less than two right angles. PROP. XXII., Book I.-Why cannot a triangle be described of which the sides shall be respectively 3, 4, and 8 feet? D PROP. XXIV., Book I.-In what three ways can two triangles be proved equal to each other? PROP. XLVII., Book I.-This proposition is employed by the carpenter. To get a right angle for the inclination between two pieces of wood, as in making door and window frames, and in setting up a roof with the edge of the roof containing a right angle, he lays down his timber on the ground as nearly at right angles as he can determine. He then measures off a distance equal to three feet on one joist or beam, and four feet on the other from the extremity placed in apposition with the former. If the line joining these two points measures five feet exactly he has hit the right angle, since 52=32+42. 25=9+16. If this is not the case, he shifts the beams until he has secured this end. PROP. I., BOOK II.-Let B D=4, DE=2, E C=3, and A=7. Then AXBC=7X (4+2+3)=63. But AXBC=7×4+7×2+7x3=63 as before. PROP. II., BOOK II.-Let A C=a, and CB=b. Then Or AB=a+b. Q. E. D PROP. III., BOOK II.-Let A C=a, and с в=b. Then And AB=a+b. i.e., a+ab=a2+ab. Q. E. D. PROP. IV., BOOK II.-Let A C=a, and C B=b. Then Then AB=a+b. (a+b)2=a2+b2+2×axb. i.e., a2+2ab+b2=a2+2ab+b3. Q.E.D. PROP. VII., BOOK II-Let A C=a, and C B=b. i.e., AB=a+b. (a+b)2+a2=2x(a+b)×a+b3. 2a2+2ab+b2= 2a+2ab+b2. Q. E. D. MENSURATION. Mensuration is the art and science of measuring surfaces and solids. It requires an elementary knowledge of Arithmetic and Geometry. Surfaces may be divided into plane and curved; and each of these may be again sub-divided into regular, or geometrical (symmetrical), and irregular. SUPERFICIAL, OR SQUARE MEASURE. 144 Square Inches (sq. in.) make 9 Square Feet 30 Square Yards, or 2721 sq. ft. 40 Perches make 4 Roods, or 4840 Yards 10,000 Square Links 1 Square Foot sq. ft. 1 Square Rod, pole or perch. 1 Rood, 1 Acre, γ. a. 1 Square Chain. 10 Square Chains, or 100,000 Links 1 Acre. The Irish Acre is equal to 1a. 2r. 19p. English. The Welsh Acre contains commonly, 2 English ones. The Scotch Acre contains 4 Roods, of 40 Falls each, but the Fall is larger than the English Perch. 3 Roods and 6 Falls are equal to an English Acre. The Scotch Acre is therefore larger, as 160 to 126. 36 square yards, or the square of 18 feet of stone, or brick-work, are a rod; in some places, 100 superficial feet of flooring, one square. 4840 square yards one square acre; 640 square acres one square mile. The square foot contains 183-346 circular inches. A circle one foot in diameter contains 113 097 square inches. CUBIC, OR SOLID MEASURE. 1728 Cubic Inches make 27 Cubic Feet 1 Cubic Foot. 1 Cubic Yard.* 40 Feet of Round Timber, or 50 Feet of 1 Ton, or Load. Hewn Timber 42 Cubic Feet 1 Ton of Shipping. A Cubic Foot of Water weighs 1000 oz. Avoirdupois. A Cord of Wood is 4 feet broad, 4 feet deep, and 8 feet loug, eing 128 cubic feet. A Solid Yard of Earth is called a Load. LINEAL, OR LONG MEASURE. By this, all measures of length, breadth, height, depth, and distance are reckoned. 12 Inches (in.) make 3 Feet, or 36 inches 1 Foot, 1 Yard, ft. yd. fa. 5 Yards, or 16 feet 4 Poles, or 22 yards 40 Poles, or 10 chains, or 220 yards 8 Furlongs, or 1760 yards 3 Miles, 1 Pole, Rod, or perch, p. 1 Land-Chain,* ch. 1 League, fur. m. 1. Besides these are-a Line, one-twelfth of an inch; a Barleycorn, one-third of an inch; a Hand, 4 inches, used for the height of horses; Span, 9 inches; Cubit, 13 feet; Military pace, 2 feet; Geometrical pace, 5 feet. 60 Geographical or sea miles, or 69 oneninth Statute miles, make a degree of the Meridian. An Irish Pole is 7 yards, and Mile 2240 yards. A Scotch Pole is 6 yards 6 inches, and Mile 1984 yards. *The Chain consists of 100 Links, each Link being equal to 7.92 inches. MENSURATION OF PLANE SURFACES. I. The area of a square, oblong, or rectangle. RULE. Area = Length x Breadth. EXAMPLES. 1. Find the area of a square 13 feet in the side. Ans. 13 x 13 = 169 sq. ft. 2. Find the area of an oblong 12ft. 6in. by 6ft. 3in. 3. What is the area of a rectangle, of which the sides are 3 and 19 ft. respectively? Ans. 57 sq. ft. 4. One side of a rectangle is 13 feet, and the area 169 sq. feet; what is the length of the other side? Ans. 169 13. 13 II. To find the area of a triangle when the base and perpendicular height are given. A triangle is half the rectangle that can be described on the base, with a vertical height equal to the perpendicular height of the triangle. But the area of such a rectangle = triangle = Base x Height 2 EXAMPLES. 1. Find the area of a triangle of which the base is 14, and the perpendicular height 7 feet respectively. Area = 14 x 7 = 7x7 = 49 sq. ft. 2. Find the area of a triangle, when the base 12 feet, and the vertical height 72 feet, are given. Ans. 432 sq. ft. 3. What is the vertical height of a triangle containing 640 sq. yds., when the base is 8 feet? Base x Height 640 × Height. Area = Height = 4. What is the base of a triangle containing 320 sq. feet, when the vertical height is 4 feet? Ans. 160 feet. 5. Find the area of half an oblong field, measuring Ans. 768 sq. feet. 64 feet by 24. 6. What is the area of a triangular fieid A B C, of which the base AB is 1566 links, and the vertical height DC 564 links ? Ans, 4 ao. 1 rood 26+ perches. |