# An introduction to linear drawing/Chapter 4

​﻿(1) FOURTH CLASS.
1. Draw a right line tangent to two circles. (fig. 1.)
Take either of the right lines in fig. 1. The circles
may be placed more or less distant from each other, and may even intersect or cut each other. A radius drawn from the centre to the point of contact will be perpen- dicular to the tangent.
2. Draw four tangents to two circles. (fig. 1.)
There may be two interior and two exterior tan- gents. The right line which joins the centres is also the point where the tangents must intersect each other.
3. Add two squares. (fig. 2.)
This figure and figure 3, present two rectangular tri- angles, on whose sides three squares are constructed. The two small squares have this peculiarity, that one of their sides is exactly one side of the triangle, and anoth- er is merely a prolongation of the other side of the right angle. If a semicircle be drawn on the greatest side of the triangle, it must touch the apex of the triangle.
It is a fact in geometry, that the . greatest of these three squares, contains a surface equal to the other two added together. ​﻿(2)
4. Add two given squares.
Make a right angle, and on its sides place the sides of the squares. These lengths will be the small sides of your right angled triangle; then draw the longest side, and you have the side of a square equal to the other two.
5. Double a square. (fig. 3.)
The triangle must be a rectangular isoceles, and the two small squares will then be equal to each other, or united they will be equal to the large square. ​﻿6. Cut off a square, (fig. 2.)
If the great square and one of the small ones be given, it is easy to find the size of the other.
First draw the great side, then the semicircle, then draw from either end of the great side, (which you will notice is the diameter of the semicircle) a cord of the semicircle, which is equal to a side of the given small square. The other cord which will finish the right angled triangle, is the side of the square required.
7. Take the half of a square. (fig. 3.)
A perpendicular to the middle of the long side, will strike the semi-circle, and a cord from this point of intersection to either end of the diameter or long side, will give the side of the square, half as large as the great square.
8. Make a graduated semicircle, usually called a Protracter. (fig. 4.)
By general consent a circle is divided into 360 equal parts, called degrees. A semicircle of course contains 180 degrees, that is, half of 360.
After having drawn a semicircle and its diameter, draw a perpendicular radius. This radius forms a right angle with the diameter, and cutting the semicircle in two equal parts or quarters of circles, leaves 90 degrees for each of them. If 90 degrees of a circle make a right angle, 45 degrees will make half a right angle, &c.
​﻿Or by another method. A radius, if made a cord of the semicircle, will allow three cords, each of which will contain 60 degrees; halve these arcs, and you have arcs of 30 degrees; halve the arcs of 30 degrees, and you have 15 degrees; cut these into three equal parts, and you have 5 degrees; then divide the arcs of five degrees into five parts, and you have the 180 degrees of the semicircle.
Whether the circle be large or small, it is divided into the same number of degrees; for if the radii of a small circle be lengthened out, and a larger circle drawn from the same centre, the radii will form the same part of the large as of the small circle, and the angle between any two radii will be unchanged.
9. Make an angle of 30 degrees on the graduated semicircle. (fig. 4.)
A radius drawn from the centre to the number 30 on the graduated semicircle, will form an angle of 30 degrees with the diameter of the semicircle. And so for any other number of degrees. It will be seen that any number of degrees less than 90 will make an acute angle, and more than 90 degrees will form an obtuse angle thus, in fig. 4, 30 degrees form an acute angle, and the remaining 150 degrees of the half circle form an obtuse angle.
Angles, therefore, are measured by their openings. Place the point of angle on the centre of the semi-circle, or the centre of the semicircle on the point of the angle, and then by seeing how many degrees the opening of the angle measures on the graduated edge of the semicircle, you will find the size of the angle. If the sides of the angle do not extend to the circumference, you may extend them till they do. If they extend beyond the circumference, measure the angle where the graduated circle cuts the sides.
​﻿10. After the pupil has drawn the semicircle, the monitor must require him to draw angles of various si- zes, from 1 to 180 degrees. Then, laying aside the semicircle, let him draw angles of various degrees, which the monitor will test by his brass semicircle, or by an- gles of pasteboard previously prepared : the latter art; the handiest if well cut.
11. Make a sphere and Us meridians. (fig. 5.)
Describe a circle and draw two diameters perpendic- ular to $ach other ; one for the axis, and the other for the equator, (a circle which goes round the earth at an equal distance from the ends of the axis, which ends are called poles.) Then draw arcs of a circle, all pass- ing through the poles, and whose centres are conse- quently on the perpendicular to the axis (that is, the equator) prolonged to the right or left hand. See Class III, problem 35. fig. 21.
These arcs have their centre as much farther off as they are nearer the axis. Their number is not import- ant, but if five be made on each side of the axis, as in the figure, each of file spaces between them will be just 15 degrees, or one twenty-fourth part of the whole sphere. These arcs in geography are called meridians..
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(6) ​﻿12. Make a sphere and the little circles which run parallel to the equator, (fig. 7.)
After having described a circle, and its two perpen- dicular diameters, as in the preceding problem, divide the circle by dots into arcs of, say, 15 degrees ; there will then be five dots and six arcs between the equator and each pole ; then divide the axis into the same number of parts. The next object is to draw an arc through the three points nearest the equator, then through the three next, and so on till all are drawn.
These arcs on a solid globe would be parallel to the equator, but do not appear so on a plane or flat surface. In geography, they are called Parallels of Latitude.
If an apple be taken and sliced from side to side, it will exactly represent the circles, which are planes cutting a sphere perpendicular to its axis.
13. Draw a sphere which shall unite the two pre- ceding problems. ("fig. 7.)
14. Draw an ellipse, (fig. 8.)
An ellipse is an oval, which may be more or less lengthened, as in figures 9 to 13. To make an ellipse : first cross two perpendicular right lines; the upper and lower halves to be of equal length, and the right and left hand to be equal also. You thus obtain the longest and shortest diameter of the ellipse, called its great and small axis. The next thing is to draw the curved lines as in ​﻿the figure. The length of the diameters may be varied at pleasure by the monitor.
There are various geometrical rules for drawing el- lipses, but it is not within the scope of our work to no- tice more than one of the simplest forms of ovals. Draw a circle and mark its centre and diameter. Then on one end of the diameter, draw another circle of the same size intersecting the former. Then opening the divi- ders the length of the diameter, place one foot on the lower point of intersection, and connect the two circles at top, and then do the same by the other point of inter- section and the bottom part of the oval.
A simple and amusing method is, to stick two pins into a piece of paper firmly, at any distance from each other ; tie the ends of a piece of string together, and put the string round both pins. Hold a pencil then at any part of the string, and move it round ; an ellipse will be formed, of which the two pins will be the two foci or centres. By lengthening or shortening the string, the figure may be made more or less elliptical.
15, Draw an oblique cone. (fig. 9.) ​﻿of the circle, and you have a cone. A cone is, in fact, a pyramid whose base is a circle, and not a pol- ygon. Sugar loaves are canes.
The height of a cone is a perpendicular let fall from the top or apex to the base. If this perpendicular fall exactly upon the centre of the base, the cone is upright.
The perspective, by changing the apparent dimensions of bodies, gives to the base of a cone the form of an ellipse. The cone presents no other difficulty than the ellipse.
16. Draw an upright cone. (fig. 10.)
parallel to the base. (fig. 9.)
18. Draw an oblique cylinder. (fig. 11.)
Draw two horizontal lines parallel to each other. Draw two equal circles, of which these shall be diame- ters. Let a right line go from centre to centre, and it will be the axis. Then draw lines from circumference to circumference, and you have a cylinder. A piece of the funnel of a stove is a cylinder. A cylinder is, in fact, a prism whose bases are circles instead of polygons* ​﻿The height of a cylinder is the length of the axis, or the distance from one base to the other. If the axis be perpendicular, the cylinder is said to be upright.
Here, as in the cone, the laws of perspective change the circles into ellipses. The axes of the ellipses, and that of the cylinder, may be given in inches by the monitor.
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19. Draw an upright cylinder. (fig. 12.)
20. Cut a cylinder by a section parallel to its base. (figs. 11 and 12v)
21. Make a cylinder whose axis shall be horizon- tal. (fig. 13.)