Practical Geometry.-Problems. PROBLEM XXVI. To inscribe a square in a given circle. 39. Let ABCD be the given circle. Draw two diameters DB and AC at right angles to each other, and through the points A, B, C and D, draw the lines AB, BC, CD and DA: then ABCD will be an inscrib ed square. B 40. By bisecting the arcs AB, BC, CD, and DA, and joining the points of bisection, we can form an octagon; and by bisecting the arcs which subtend the sides of the octagon, we can inscribe a polygon of sixteen sides. PROBLEM XXVII. To circumscribe a square about a circle. 41. Draw two diameters AB and CD at right angles to each other; and through their extremities A, B, C and D, draw A lines respectively parallel to the diameters CD and AB: a square will thus be formed circumscribing the circle. B QUEST.-39. Explain the manner of inscribing a square in a circle. 40. Also, an octagon. 41. Explain the manner of circumscribing a square about a circle. Practical Geometry.-Problems. PROBLEM XXVIII. To draw a line which shall be tangent to the circumference of a circle at a given point. 42. Let A be the given point. Through A draw the radius AC, and then draw DA perpendicular to the radius at the extremity A. The line DA will be tangent to the circumference at the point A. D PROBLEM XXIX. Through a given point without a circle to draw a line, which shall be tangent to the circumference. 43. Let A be the given point without the given circle BED. Join the centre C and the given point A, and bisect the line CA at O. With O as a centre, and OA as a radius, describe the circumference ABCD. Through B and D draw the lines AB and AD, B E A and they will be tangent to the circle BED at the points B and D. QUEST.-42. Explain the manner of drawing a tangent line to a circle at a given point of the circumference. 42. Explain the manner of drawing a tangent line to a circle through a given point without. Mensuration of Surfaces. PART III. SECTION I. MENSURATION OF SURFACES. 1. The arca of any figure, has already been defined to be the measure of its surface. (Part I. § V. Art. 7). This measure is merely the number of squares which the figure is equal to. A square whose side is one inch, one foot, or one yard, &c., is called the measuring unit; and the area or content of a figure is expressed by the number of such squares which the figure contains. 2. In the questions involving decimals, the decimals. are generally carried to four places, and then taken to the nearest figure. That is, if the fifth decimal figure is 5, or greater than 5, the fourth figure is increased by one. QUEST.-1. What is the area of a figure? What is the measure? What is a square whose side is 1 foot, 1 yard, &c. called? How is the area or content of a figure expressed? 2. In questions involving decimals, to how many places are the figures generally carried? What is meant by taking the nearest figure? Mensuration of Surfaces. 3. Surveyors, in measuring land, generally use a chain called Gunters' chain. This chain is four rods, or 66 feet in length, and is divided into 100 links. 4. An acre is a surface equal in extent to 10 square chains; that is, equal to a rectangle of which one side is ten chains, and the other side one chain. One-quarter of an acre, is called a rood. Since the chain is 4 rods in length, 1 square chain contains 16 square rods; and therefore, an acre, which is 10 square chains, contains 160 square rods, and a rood contains 40 square rods. The square rods are called perches. 5. Land is generally computed in acres, roods, and perches, which are respectively designated by the letters A. R. P. When the linear dimensions of a survey are chains or links, the area will be expressed in square chains or square links, and it is necessary to form a rule for reducing this area to acres, roods, and perches. For this purpose, let us form the following QUEST.-3. What chain is generally used by land surveyors? What is the length of this chain? How is it divided? 4. What is an acre of land? What is a quarter of an acre called? What are square rods called? 5. In what is land generally computed? signated? How is each denomination de |