An introduction to linear drawing/Chapter 3
A Plane is a level. Place one die upon another, and together they make a prism, cut by a plane where the separation between the dice is.
25. Make a six sided prim, and double it by
lengthening it. (fig. 19.)
When one die is put upon another, the first die is doubled in length.
26. Make a jive sided prism, and cut it by three planes parallel to its base.
A long prism may be drawn and cut as in Prop. 24, or a short prism be first made and lengthened as many times as you please.
27. Draw a Jive sided prism in a horizontal position. (fig. 20.) Cut it by a plane parallel to its base.
THIRD CLASS.
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1. Describe a circle and mark its centre, (fig. 1.)
By constant practice the pupil will be able to draw a circle, and mark its centre with great exactness. The monitor with a pair of dividers will prove it. The pu- pils in the Monitorial School have various methods of marking circles, without the aid of dividers, the most (8)
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14, Describe a circle and draw a tangent to it.
Tangent comes from a latin word, which means to touch. A tangent is a right line which touches a cir- cle, but does not cut any of it off. If a right line be drawn from the centre of the circle or arc to the point of contact (that is, the point where the tangent touches the circle) the two right lines, (that is, the radius and the tangent) will be perpendicular to each other, (fig.9.)
The monitor may test this with his quadrant of pasteboard ; or, marking two places on the tangent at equal distances from the point (f contact, he may see with his dividers if these points are at equal distances from the centre of the circle.
15. Draw four tangents to a circle, forming a quadrilateral or four sided figure.
This figure need not form a perfect square, as in figure 10.
16. Circumscribe or surround a circle with a square.
(fig- 8.)
(fig. 10.) When the four tangents make right angles with each other, the figure is a square. In other cases, any di- rection may be given to two of the tangents.
In figure 10 we say the circle is circumscribed by the square, or the circle inscribed in a square*
17. Inscribe a square in a circle. (fig. 11.)
When a polygon has all the points of its angles touch- ing a circle, it is said to be inscribed in a circle, and the circle circumscribes the polygon.
18. Double an arc of a circle. (fig. 6.)
This is more difficult than Prop. 11. First draw an arc and mark the centre of its circle, then prolong the arc to two, three, Stc. times its former size.
19. Draw a tangent to a circle from a given point outside. (fig. 8.)
20. Draw two tangents to a circle from a given point. (fig. 12.)
Observe that in drawing a tangent to a circle in problem 14, any part of the circle may be taken, but when a tangent is drawn from a given point, it can hit but two points of the circle, as in fig. 12.
21. Cut a circle into six equal parts, or, in other words, inscribe a regular hexagon in a circle.) (fig. 13.)
The radius of any circle is equal to one side of the hexagon to be inscribed in it. The monitor, therefore,
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22. Cut a circle into three equal parts, and inscribe an equilateral triangle, (fig. 13.)
After the hexagon is correctly drawn by problem 21st, it is easy to inscribe the triangle required in this, by drawing a cord between two points of the hexagon.
If cords be then drawn between the three remaining points of the hexagon, another triangle will be formed, whose base will be opposite the base of the other trian- gle, forming a beautiful figure resembling a star.
23. Make two unequal circles tangent outside.
nequal circles are circles of unequal size only.
24. Make two unequal circles tangent inside, (fig.15.)
25. The centres and the point of contact being giv- en, perform problems 23 and 24.
The monitor will mark the centres, he. When the circles touch either within or without, the point of con- tact and the two centres will be in a right line, and these may be tested with a rule.
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14.) A regular polygon has all its sides equal, and all its angles of an equal opening. When such a polygon is inscribed in a circle, the sides are cords of equal arcs, and the points cut the circle into equal parts.
26. Inscribe a regular octagon in a circle. (fig. 16.)
Draw two diameters perpendicular to each other, then divide each quarter of the circle into halves by other diameters ; then draw arcs from diameter to di- ameter.
27. Inscribe a regular pentagon in a circle. (fig. 17.)
It is difficult by the eye alone to divide the circum- ference into five equal parts, and the object of this problem is to exercise the pupils.
28. Make a triangle, and circumscribe a circle.
First make a triangle, and then the object is to de- scribe a circle which shall cut each of its three points. To do this, raise a perpendicular on the middle of one
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(fig. 18.) of the sides, and then on the middle of another side. These perpendiculars will cross each other, and the point of section (that is, the point where they cut each other) will be the centre of the circle required.
In the figure, the dotted lines show the perpendiculars and centre.
29. Make a circle, and draw a tangent triangle. (fig. 19.)
Three tangents to a circle are easily made, but the monitor may increase the difficulty by giving directions to the tangent sides. Thus, let two sides be at right angles, obtuse or acute; let the triangle be equilateral, &c.
30. Draw a regular pentagon, and circumscribe it with a circle. (fig. 17.)
31. Draw a regular hexagon, and circumscribe it with a circle. (fig. 13.)
32. Draw a regular octagon, and circumscribe it with a circle. (fig. 16.)
In the former problems, the circle was made first, now the polygon.
33. Inscribe a circle in a triangle. (fig. 19.)
To find the centre of the circle, draw a line from the middle of either side of the triangle to the apex opposite, then do the same by another side and its opposite apex; the place where these two lines cross each other, will be the centre of the circle to be inscribed. See the dotted lines in fig. 19.
34. Make an arc which shall pass through two given points. (fig. 20.)
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After having marked two points, trace an arc of a circle which shall pass through both of them. The centre must be somewhere on a perpendicular to the middle of a cord which would join these two points.
35. Make several arcs pass through two given points•
Draw one arc as in problem 34, then a cord from point to point, then a perpendicular to the cord, and then you may make any number of arcs pass through the two points, all of whose centres must be on the per- pendicular.
This problem will assist the pupil in drawing the me- ridians on a map of the globe.
36. Describe a circle, and circumscribe it with a hexagon. (fig. 22.)
Cut the circumference into six equal arcs, as if you wished to inscribe the hexagon. Then draw a radius to each point, and six tangents perpendicular to the
- radii will form the regular hexagon required. Or, the
tangents may be drawn as in fig. 22, where they meet and form an angle on the prolonged radii.
(fig. 21.)
(22) 37. Inscribe and circumscribe a circle with regular and parallel hexagons. (fig. 23.)
38. Inscribe a circle in a regular hexagon. (fig. 22.) This problem is the inverse of the 36th. First draw
the hexagon, then describe the circle, touching it on all sides. The centre of the circle may be found by rais- ing perpendiculars on the middle of any two sides until they cross each other. The point where they cross, is the centre.
39. Make a triangle whose three sides are given•
Trace three right lines for sides. Take one of them, the longest if you please, for the base, and then make a point where you think the other two sides will reach. The difficulty is to ascertain exactly where this point should be. With dividers it may be easily found in the following manner :—After you have drawn the base, open the dividers the length of the next side to be drawn, and placing one foot of the dividers on one end of the base, draw an arc with the other foot. Then taking the length of the third side, place one foot on the other end of the base, and draw an arc which shall cross the former arc : the point where the arcs cross each other is the summit or apex required.
If the two arcs cannot cross, the problem is said to be absurd : for no triangle can be made of the given sides. Each of the three sides must be shorter than the two others would be if united.
(fig. 24.)
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