52. How do you draw a line which shall be tangent to the circumference of a circle at a given point? 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 53. How do you draw through a given point without a circle a line which shall be tangent to the circumference? 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, and they will be tangent to the circle BED at the points B and D. 54. What is an ellipse? It is an oval curve ACBD. E 55. What is the longest line which can be drawn within the curve called? What is the shortest line called? The longest line AB is called the transverse axis; and the shortest line DC is called the conjugate axis. The point E, at which they intersect, is called the centre of the ellipse. 56. What are the foci of an ellipse? They are two points F and H, determined by describing the arc of a circle with D as a centre, and a radius DF equal to AE, half of the transverse axis. B H 57. How will you describe an ellipse when you know the two axes AB and CD? First, find the foci F and H by describing an arc with D as a centre, and with a radius equal to AE. Secondly, take a string or thread equal in length to AB, and fasten the extremities at the foci F and H. Then place a pencil against the string and move it round, bearing it tight against the string, and the point will describe the ellipse ADBC. QUESTIONS TO BE PUT FROM FIGURES MADE BY THE TEACHER UPON THE BLACK-BOARD. For what is it used? How is it divided? How are angles measured? When one straight line meets another, what is the sum of the angles on the same side? How many degrees in one right an- If there are several angles, what is Describe the ruler and triangle, and How do you divide a line into any the manner of using them. How do you draw a perpendicular? What is a Scale of Equal Parts? What is a unit of the scale? Explain how you take from the scale a given number of parts. Explain the Diagonal Scale. What is a Scale of Chords? How will you lay off an angle? What is the Semicircular Protrac tor? number of equal parts? How do you describe a square? How do you inscribe an equilateral triangle in a given circle? How do you inscribe a hexagon in a circle? How do you inscribe a dodecagon? How do you lay off an angle with it? How do you inscribe in a circle a Describe Gunter's Scale. How do you bisect a line? How do you draw a perpendicular at a given point? polygon having any number of sides? How do you inscribe a square? an octagon ? How do you make an angle equal How do you circumscribe a square to a given angle? How do you bisect an arc? How do you draw a parallel to a given line? When two angles of a triangle are given, how do you find the third ? about a circle? How do you draw a line tangent to a circle at a point of the circumference? How from a point without the circumference? Note. After the teacher shall have made the above figures, or most of them, on the black-board, and the pupils copied them on their slates, let the students then be called to the black-board in turn, and practised in the drawing of them. BOOK III. SECTION I. OF DRAWING IN GENERAL. 1. What are drawings? DRAWINGS are representations to the eye of the forms, dimensions, positions, and appearance of objects. They form a written language, which is easily comprehended by every one. 2. What are the uses of drawing? Drawing, to the practical man, furnishes a simple means of describing and explaining a thing in a brief and striking manner. On this account, alone, its great advantages are everywhere apparent. Drawings, also, impress the mind with images approaching nearer to the reality, than any other means of description. The pen of the ablest historian presents but a feeble image, when compared with the pictured canvass of the painter, or the life-like forms of the sculptor. 3. When you look at a single object, what do you observe that distinguishes it from other objects? When we observe a single object, we discover that we are able to recognise it by means of three properties which distinguish it from other objects, viz.: its form, its light and shade, and its color. If we consider more than one |