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LIBRARY OF CONGRESS, 
 
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THE PRACTICE 
 
 OF 
 
 MECHANICAL DRAWING 
 
 FOR 
 
 SELF-INSTRUCTION, 
 
 BY 
 
 y 
 
 WILLIAMS WELCH, 
 
 Instructor of Drawing at Clemson College. 
 
 NEWBERRY, S. C. 
 
 Elbert H. Aull, Puewsher and Printer, 
 
 1895. 
 
 
<<.^ 
 
 Copyright, 1895, by Wms. Welch. 
 All Rights Reserved. 
 
PREFACE. 
 
 This book is intended to enable mechanics and others to learn to make me- 
 chanical drawings when thej^ cannot have the assistance of a teacher; and it is 
 also intended as a text-book for beginners in mechanical drawing. 
 
 Geometrical terms and problems, which are not needed in ordinary work,, 
 have been avoided. 
 
 WMS. WEI.CH. 
 C1.EMS0N C0L1.EGE, s. c, 
 Febrtiary 21st, 18 g^. 
 
CHAPTER IL 
 
 CONSTKUCTION OF GEOMETEICAL PEOBLEMS. 
 
 A draftsman should be familiar with the problems in this chapter, and should 
 be able to construct them very accurately when necessary. In practice, they are 
 sdldom used: but unless he knows how to draw them with precision, he will not 
 be apt to get them so nearly correct when drawing them approximately. 
 
 Problems of this kind are about the best exercises for beginners in mechanical 
 drawing. 
 
 The desired results will not be obtained unless extremely fine, hair- like 
 lines are drawn, and all centres and other points taken exactly on these lines. 
 Therefore it is important to use very hard pencils sharpened to a needle or chisel 
 point, and to avoid making holes in the paper with the compasses. 
 
 Given and required lines should be drawn fine in ink, and construction lines 
 drawn in pencil only ; but, to distinguish them in these problems, given lines are 
 heavy; required lines fine; and construction lines broken. 
 
 An ARC of a circle is drawn by stickmg one point of the compasses, Fig. 15» in 
 the paper and moving the other point around on the paper a short distance. The 
 distance between the points of the compasses is the radius of the arc. An arc 
 can be drawn when its centre and radius are given. 
 
 PROB. 1 . Draw a line which will divide a given line into two equal parte 
 and be perpendicular to it. (Take any line.) 
 
 With the ends of the given line as centres, and with equal radii, draw arcs which 
 will intersect on both sides of the line. The line joining the points of intersection 
 will be the one required. 
 
 PROB. 2. Draw a line perpendicular to a given line through a given point 
 in the line. (Take any point in a line.) 
 
 With the given point es a centre, and the same radius, draw arcs intersecting 
 the line. With the two jDoints of intersection as centres and equal radii, draw- 
 arcs intersecting on both sides of the line. A line joining these two last points 
 of intersection will be the perpendicular required. 
 
 PROB. 3. Draw a line perpendicular to a given line from any given point. 
 (Take point 2 ins. from line.) 
 
 With the given point as a centre, draw an arc which intersects the 
 line at two points. With these two points as centres, and equal radii, 
 draw intersecting arcs. The line joining the given point and the last point of 
 mtersection will be the perpendicular required. 
 
 An ANGLE is formed by two straight lines meeting in a point. Shcrttningor 
 lengthening the lines does Eot change the angle. Fcr the purycse of mrasuriug 
 angles and arcs, the whole circumference of anv circle is arbilr; rily divided into 
 
6 MECHANICAI. DRAWING. 
 
 360 equal arcs, called degrees. If the vertex of an angle (point where the sides 
 meet) is placed at the centre of the circle, the arc between the sides will contain 
 the same number of degrees as the angle. A right angle contains 90 degrees. 
 The radius of a circle will step around on the circumference exactly six times, 
 dividing it into arcs of 60 degrees. 
 
 PROB. 4, Draw a line which will divide a given angle, or arc, into two 
 ■equal parts. (Take any angle.) 
 
 With the vertex of the angle as a centre, draw an arc intersecting 
 the sides of the angle. With the points of intersection as centres, and 
 equal radii, draw intersecting arcs. The line joining the last point of inter- 
 section with the vertex, will bisect the angle and the arc contained between its sides. 
 
 Any point in this line will be equally distant from the sides of the angle. 
 
 PROB. 5. Divide a right angle into three equal parts, thereby constructing 
 angles of BO degrees. 
 
 With the vertex of the right angle as a centre, draw an arc intersecting the 
 sides. With the points of intersection as centres, and the same radius used in 
 drawing the arc, draw arcs intersecting the first arc. Lines drawn from the last 
 points of intersection to the vertex will trisect the right angle, forming angles of 
 ■30 degrees each. 
 
 PROB. 6. Construct an angle at a given point, equal to a given angle. (Take 
 any convenient angle and point.) 
 
 With the vertex of the given angle as a centre, draw an arc intersecting 
 the sides; and, with the vertex of the required angle as a centre, draw 
 an arc with the same radius. Make the two arcs between the sides equal 
 and the angles will be equal. 
 
 Lines are PARALLEL when they lie in the same plane and never meet if 
 produced indefinitely in both directions. The surface of the paper is the plane 
 in which lines are drawn. 
 
 PROB. 7. Draw a line parallel to a given line, at a given distance from 
 it. (Take 2 ins.) 
 
 With a radius equal to the given distance, and two points in the line — 
 one near each end — as centres, draw arcs. A line just touching these arcs will 
 be the parallel line required. 
 
 PROB. 8. Draw a line parallel to a given line, through a given point. (Take 
 point about dh ins. from line.) 
 
 Draw a line through the point and crossing the given line. Construct angles 
 around the point equal to the corresponding angles formed by the intersection of 
 the two lines. A side of these angles will be the parallel line required 
 
 PROB. 9. Divide a given line into any number of equal parts. (Take 7 parts.) 
 
 Draw a line through one end of the given line, and from that end measure 
 off the required number of equal divisions on this second line. Draw a line 
 through the last point of division and the other end of the given line, and draw 
 
MECHANICAL DRAWING. 7 
 
 liaes through all the other points of division parallel with this last line. These 
 parallel liaes will divide the given line into the required number of equal parts. 
 
 A TRIANGLE is a plain figure bounded by three straight sides. When one 
 angle is a right angle the figure is a right triangle; the longest side of which is 
 the hypotenuse, and the hypotenuse squared is equal to the sum of the squares of 
 the other two sides. When two of the sides are equal, the two angles opposite 
 them are equal and the triangle is isosceles. The angles of any triangle added 
 make two right angles The area of a triangle is equal to half its base multiplied 
 by its height. 
 
 PROB. 10. Form a right angle at the end of a given line by constructing a 
 right triangle on the line. 
 
 Measure off four equal divisions from the end of the line. With a radiu s 
 equal to 3 of the divisions, and the end of the line as a centre, draw an 
 arc. With a radius equal to 5 of the divisions, and the fourth point on the line 
 as a centre, draw an arc intersecting the first arc. Draw a line from the point of 
 intersection to the end of the given line and it will form a right angle with the line. 
 
 Peoof. — The hypotenuse squared is 25; 3 squared is 9; 4 squared is 10; 9 and 16 
 make 25. 
 
 PROB. 11. Construct a right triangle containing two angles of 45 degrees each. 
 
 Construct a right angle, (Prob. 1, 2 or 10.) With the vertex as a 
 centre, draw an arc intersecting the sides. Draw a line joining the points of inter- 
 section and it will form angles of 45 degrees with the sides. 
 
 PROB. 12. Construct a right triangle containing one angle of 30 degrees 
 and another of 60. 
 
 Construct a right angle. With the vertex as a centre and any radius, 
 draw an arc intersecting one side. With the point of intersection as a 
 centre, and twice the first radius as a radius, draw an arc intersecting the other 
 side. Draw a line through the two points of intersection, and it will form the re- 
 quired angles with the sides. 
 
 A. POLYGON is a figure bounded by three or more straight lines in the 
 same plane. When all the sides are equal, and all the angles are equal, it is a 
 regular polygon; and a circle can be drawn around it, touching all the angles, and 
 another can be drawn within it, touching all the sides at their middle points. 
 
 PROB. 13. Draw a regular polygon of 5 sides in a circle of a given diameter, 
 (Take diameter of circle 2 ins.) 
 
 Draw a diameter of the circle, and a radius perpendicular to it. Take 
 a point on the diameter, half way between the centre and the circum- 
 ference, as a centre; and, with a radius equal to the distance 
 from this point to the extremity of the radius, draw an arc intersecting the diam- 
 eter. The distance between this point of intersection and the extremity of the 
 radius will be a side of the required pentagon, and it can be completed by step- 
 ping the side around on the circumference and drawing lines joining these points. 
 
'S MECHANICAL DRAWIN©. 
 
 PROB. 14. Draw a regular polygon of 6 sides in a circle. (Take diameter of 
 circle 2 ins.) 
 
 The radius of the circle will be the sides of the inscribed hexagon, and 
 it can be completed by stepping the radius around oa the circumference and 
 drawing lines joining these points. 
 
 PROB. 15. Divide the circumference of a circle into 24 equal parts; thereby 
 making arcs and angles of 15 degrees each. (Take diameter of circle 2 ins.) 
 
 Draw two diameters perpendicular to each other. (Prob. 1.) With the radius of 
 the circle as a radius and the extremities of the diameters as centres, draw arcs 
 intersecting the circumference at 8 points. Bisect the arcs between all these 
 points (Prob. 4) and the required number of 'divisions will be made. 
 
 PROB. 16. Draw a regular pentagon with sides of a given length. (Take 
 sides 14 ins.) 
 
 Draw a side and erect a perpendicular to this side at one end (Prob. 2 
 or 10) equal in length to half the side. (Prob. 1.) From the other end of 
 the side draw a line passing through the extremity of the perpendicular, and ex- 
 tend this line a distance beyond the perpendicular equal to half the side. The 
 distance between the end of this line and the nearer end of the side will be the 
 radius of the circumscribed circle. It may be drawn and the polygon completed 
 by stepping the given side around on it 5 times. 
 
 PROB. 17. Draw a regular G sided polygon, the distance between the oppo- 
 site sides being given. (Take distance 1-| ins.) 
 
 Draw parallel lines the given distance apart (Prob. 7), and draw a 
 line making an angle of GO degrees with them. (Prob. 5.) This line between 
 the parallel lines will be the diameter of the circumscribed circle, and half of 
 it will be equal to a side of the required hexagon. It may be quickly completed 
 with a G0° triangle. Fig. 5. 
 
 PROB. 18. Draw a regular 8 sided polygon in a given square. (Take a 
 square 2x2 ins.) 
 
 Find the centre of the square by drawing diagonal lines through 
 the vertices (corners). With the vertices as centres, draw arcs passing 
 through the centre of the square and intersecting the sides. These points of 
 intersection will be the corners of the required octagon. 
 
 The circumference of a CIRCLE is drawn by a point moving around another 
 point (the centre) and remaining the same distance from it in the same plane. 
 A line drawn from the centre to the curve is the radius, and a line drawn across 
 the circle through the centre is the diameter. A line drawn across elsewhere is a 
 chord, and any part of the curve is an arc. The whole surface within the circum- 
 ference is the area, one-half is a semicircle, and one-fourth is a quadrant. The 
 surface enclosed by two radii and an arc is a sector, and that enclosed by a chord 
 and an arc is a segment. The circumference is equal to about 3 1-7 times the 
 diameter, or more exactly, 3.14I592G53590 times. The area of a sector is equal 
 

 Prob.7. 
 
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 Prcb.8. 
 
 
 Frob.^. 
 
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 \ 
 
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 VrobJO. 
 
 N 
 
 N 
 
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 Prob.//, 
 
 - 
 
 ProbJZ. 
 
 
 
 
 
 
 
MECHANICAL DRAWING. 9 
 
 to half its arc multiplied by the raiias, and the area of the whole circle is equal 
 to half the circumference multiplied by the radius. 
 
 PROB. 19. Draw a straight line, equal in length to the circumference of a. 
 given circle. (Take a circle with diam. 2 ins.) 
 
 Draw two diameters perpendicular to each other, and draw a chord from the 
 extremity of one to the extremity of the other. Draw a perpendicular to this 
 chord through its centre. (Prob. 4.) The part of this perpendicular between 
 the chord and the arc, added to three times the diameter, will be very nearly 
 equal to the circumference. It will be about 1-208 of the diameter too great. 
 
 Proof. — Take thechord=2; the radius will be\/2 = 1.414, and the part between 
 the chord and arc will be .414-=- 2. 828 = .1464 which is slightly greater than .1416. 
 
 The chord is equal to the radius of a circle which has twice the area of the- 
 given circle. 
 
 PROB. 20. Draw a square with an area equal to the area of a given circle. 
 (Take circle with diam. 2 ins.) 
 
 Draw a line equal in length to half the circumference Ol the given 
 circles (Prob. 19) and extend the line a distance equal to the radius. With 
 the middle point of the whole line as a centre, draw a semicircle passing through 
 its ends. At the point where the two lines meet, erect a perpendicular. (Prob. 2.) 
 The part of it between the line and semicircle will be a side of the required square. 
 
 Proof. — The two parts of the diameter multiplied together equal the perpen- 
 dicular squared; as it is a mean proportional between them. (See Wentworth's. 
 Geometry, page 157.) 
 
 A circle with an area two, three, four, five or more times as great as the area of 
 a given circle may be drawn by making the shorter part of a diameter equal to 
 the radius of the given circle and the longer part equal respectively to two, three, 
 four, five or more times that radius. The perpendicular to the diameter, from the 
 point where the two lines meet to the circumference, will be equal to the radius of 
 the required circle. 
 
 Proof. — Let r=: given radius; p= perpendicular, and ar the longer part of the 
 diameter. Then ar^ = p^-, r23.1416 = given area; p23.1416 = required area= 
 ar^d.14:lQ^^a times the given area. 
 
 PROB. 21. Draw a circle equal in area to a given equare. (Take side of 
 square ]f inches.) 
 
 Draw a line from a vertex of the square to the centre of oae 
 side. This line will be very nearly equal to the diameter of the required circle. 
 It will be about 1-100 of the side too small. 
 
 Proof. — Take side of square=l ; the diameter of circle will be 1.118, but it should 
 be 1.128 to make the areas exactly the same. 
 
 PROB. 22. Draw a circle passing through 3 given points. (Take points 1, 1|, 
 and 2 ins. apart.) 
 
 With two of the given points as centres, and with equal radii, 
 draw arcs which will intersect at two points. Draw a line through these two points 
 of intersection. With the third given point and either one of the others as centres. 
 
10 MECHANICAI, DRAWING. 
 
 draw arcs and another line as before. The point where these two lines 
 cross will be the centre of the required circle. 
 
 By taking any three points in the circumference of a given circle or arc, its 
 centre may be found in the same way. 
 
 PROS. 23. Draw, mechanically, an arc passing through 3 given points, with- 
 out using a centre. (Take points 1^, Ig and 2 |- ins. apart.) 
 
 Locate the three points on a piece of firm card-board, and cut straight lines passing 
 from the intermediate point through the two extreme points. Stirkfine pins in the two 
 extreme points on the drawing paper, aod hold a pencil in the vertex of the angle formed 
 in the card-board while it is moved back and forth against the pins. The pencil 
 will draw the required arc, and it can be inked with a curved ruler, Fig. 16. 
 
 PROB. 24-. Draw an arc with a given radius without using a centre. (Take 
 radius G inches.) 
 
 Part of a regular polygon of twelve sides can first be drawn, and the 
 arc drawn through three or more of the corners. (Prob. 23.) The angles at the 
 corners will be 150 degrees (Prob. 5); and the sides can be found by constructing 
 a triangle with the base equal to half the given radius, and the angles at the base 
 90 and 15 degrees. (Prob. 12 and 4.) The hypotenuse will be the required side. 
 If the radius is quite large, any regular polygon may be taken. The angles can 
 bo easily computed and the sides found by a table of chords. 
 
 A line is TANGENT io a circle when, however far produced, it passes through 
 hut one jyoint in the circumference of the circle. It will be perpendicular to a 
 radius of the circle drawn to the point of tangency. 
 
 PROB. 25. Draw a line tangent to an arc at a given point in the arc. (Take 
 radius of arc 5 ins.) 
 
 With the given point as a centre, draw arcs intersecting the 
 given arc at two points equally distant from the given point. "With these two 
 points as centres, draw intersecting arcs, and draw lines through the points of 
 intersection. (Prob. 22.) A line drawn perpendicular to this line through the given 
 point (Prob. 2) will be the tangent required. 
 
 PROB. 26. Draw a line tangent to a circle from a given point outside of 
 the citcle. (Take circle 2 ins. in diam.. point 3 ins. from centre.) 
 With the given point as a centre, draw an arc which will pass through the centre of the 
 circle. With the centre of the circle as a centre, and with a radius equal to the diameter 
 of the circle, draw an arc intersecting the first arc. Draw a line from the last point 
 of intersection to the centre of the circle. This line will cross the circumference 
 at the point of tangency; and a line drawn through it from the given point will be 
 the tangent required. Two tangents can be drawn. 
 
 PROB. 2T. Draw a line tangent to two circles of different diameters. (Take 
 circles 1 and 2 ins. in diam., centres 2 ins. apart.) 
 
 Draw a line through the centres, and extend it beyond the 
 smaller circle. Draw parallel lines through the centres of the circles. 
 
ProbJ3 
 
 ProbJ4 
 
 Prob./e 
 
 Proh.l? 
 
 
 J>TobJ8 
 
 A 
 
 
 •:>-] 
 
 
 .X 
 
 
 
 
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 FTob. /£) 
 
 JProh^O 
 
 JProb.2/ 
 
 Prob.£^ 
 
 Prob.£J 
 
 Prob.£' 
 
MKCHANICAL DRAWING. 11 
 
 Through the points where these parallel lines intersect the circumferences 
 draw a line, and extend it until it crosses the line passing through 
 the centres. From this last point of intersection draw a tangent to either circle. 
 (Prob. 26.) It will be the tangent required. Four such tangents can be drawn; 
 two will be interior, and two exterior tangents. 
 
 PROB. 28. Draw a circle tangent to a line at a given point, and passing 
 through another given point. (Take one point 1 in. from the line and 1| ins. 
 from the point of tangency.) 
 
 Draw a perpendicular to the line at the given point in it. 
 (Prob. 2.) With the points as centres and with equal radius, draw arcs inter- 
 secting at two places, and draw a line through the points of intersection. 
 (Prob. 22.) It will cross the perpendicular at the centre of the required circle. 
 
 PROB. 29. Draw a circle tangent to two lines and passing through a given 
 point equally distant from them. (Take lines making an angle of 60 degrees, and 
 a point 1 in. from vertex.) 
 
 Extend the lines until they meet, and draw a line 
 bisecting the angle between them (Prob. 4) Draw a perpendicular to this line 
 through the given point (Prob. 2), and bisect the angle which the perpendicular 
 makes with one of the sides. (Prob. 4,) The two bisecting lines will cross at the 
 centre of the required circle. 
 
 Note. — If the lines cannot be made to meet on the paper, draw lines parallel to and 
 equally distant from them (Prob. 7) and bisect the angle formed by these lines. 
 
 PROB. 30. Draw a circle tangent to three given lines. (Take one line 3 ins. 
 long, and the others making angles of 45 and 60 degrees at its ends.) 
 
 Extend the lines until they meet, and draw lines bisecting any two of the angles 
 formed. (Prob. 4.) These bisecting lines will cross at the centre of the required circle. 
 Its radius can be found by drawing a perpendicular to one of the given lines. 
 (Prob. 3.) 
 
 Two CIRCLES ARE TANGENT when the circumference of one passes 
 through but one point in the circumference of the other. A line drawn through 
 the centre of two tangent circles will pass through the point of tangency. 
 
 PROB. 31. Draw a circle with a given radius, tangent to a line and a given 
 circle. (Take radius of given circle 1 in., with centre 1^ ins. from line; radius 
 of required circle, | ins.) 
 
 Draw a line parallel with the given line at a distance from it 
 equal to the given radius (Prob. 7); and, with the centre of the given circle 
 as a centre, and a radius equal to the radius of the given cii'cle added to the 
 radius of the required circle, strike an arc. It will cross the parallel line at the 
 centre of the required circle. 
 
 PROB. 32. Draw a circle tangent to a circle and a line at a given point in 
 the line. (Take line and circle, as in Prob 31, and point in line 2 ins. from centre 
 of given circle.) 
 
 Draw a p3rp3niicular to the line through the given point 
 
12 MECHANICAI, DRAWING. 
 
 (Prob. 2) and extend it on either side of the line a distance equal to the radius of 
 the given circle. Draw a line from the extremity of the perpendicular to the 
 centre of the circle, and draw a perpendicular to this line through its middle 
 point. (Prob. 1.) It will cross the first perpendicular at the centre of the required 
 circle. Two such circles can be drawn; the given circle will be tangent to the ex- 
 terior of one and to the interior of the other. 
 
 PROB. 33. Draw a circle tangent to a line and to a circle at a given point 
 in the circle. (Take line and circle, as in Prob. 31, and point in circle 1 in. from 
 the line.) 
 
 Draw a line from the centre of the circle through the given point, and 
 draw another through the point tangent to the circle. (Prob. 25.) Extend the 
 tangent till it meets the given line, and draw a line bisecting the angle between 
 them. (Prob. 4.) The bisecting line will cross the first line at the centre of the 
 required circle. Two such circles can be drawn; the given circle will be tangent 
 to the exterior of one and to the interior of the other. 
 
 PROB. 34. Draw a circle with a given radius, tangent to two given circles. 
 (Take given circles 1 and 2 ins. in diameter, and centres 2 ins. apart ; radius of 
 required circle | ins.) 
 
 With the centre of one circle as a centre, and with a radius 
 equal to its radius added to the radius of the required circle, draw an arc. With 
 the centre of the other circle as a centre, and with a radius equal to its radius 
 added to the radius of the i-equired circle, draw another ai-c. The two arcs will 
 cross at the centre of the required circle. 
 
 PROB. 35. Draw a circle tangent to two given circles at a given point in one 
 circle. (Take given circles as in Prob. 34, and point in larger circle 30 degrees from 
 aline between their centres.) 
 
 Draw a line through the given point and the centre of that cii'cle. 
 With the centre of that circle as a centre, draw three or four arcs 
 crossing this line near the centre of the required circle. With the centre of the 
 other circle as a centre, and a radius equal to its radius added to the distance 
 from the given point to the first arc, draw an arc intersecting the first arc; then 
 with the same centre and the same radius increased by the distance from the first 
 arc to the second, draw an arc intersecting the second arc; then in the same way 
 draw one intersecting the third arc and so on. With a curved ruler, Fig IG, 
 draw a curved line through these points of intersection. It will cross the first line at 
 the coutre of the required circle. Tae curve is part of a hyperbola (Prob. 44) and 
 curves towards the smaller circle. 
 
 PROB. 3G. Draw a circle tangent to three given circles of different diam-ters. 
 (Take centres of circles all 2 ins. apart; with radii of f, | and 1 inch.) 
 
 Draw two curves as in Prob. 35, and they will cross each other at the centre of the 
 required circle. 
 
 An ELLIPSE is drawn by a point moving around iwo other points in the same 
 
Prck25. 
 
 Proh26. 
 
 ProbZa, 
 
 Pmh.29. ^ 
 
 ProhSO. 
 
 ProhJl 
 
 ProKJ2. 
 
 Proh.33, 
 
 PwhJ4. 
 
 Prch3§. 
 
 Proh36. 
 
MECHANICAL DRAWING. IS 
 
 plane, so that the distance between it and one point, added to the distance between 
 it and the other point, remains the same. The two fixed points are the focii. A 
 line drawn through the focii is the longer axis of the ellipse, and a perpendicular 
 to it through its middle point is the shorter axif^. 
 
 PROB. 37. Draw an ellipse with pins and a string; the axes being given. 
 (Take axes 1^ and 2| ins.) 
 
 Draw the shorter axis. With its ends as a centres, and a radius 
 equal to half the longer axis, draw arcs intersectiDg at two points. These 
 points will be the focii. Stick pins in the focii and in one end of the shorter axis, 
 and tie a linen thread around the three pins. Remove the pin from the shorter 
 axis. The point of a pencil held tight against the string will draw the required 
 ellipse. 
 
 The position of the focii and the length of the string can be computed: 
 The distance from the point where the axes cross to the focii will be equal to the 
 base of a right triangle; the altitude of which will be half the shorter axis, and 
 the hypotenuse half the longer axis. The string will be equal in length to the 
 longer axis added to the distance between the focii- 
 
 PROB. 38. Draw an ellipse by taking 3 points on a straight edge; the axes 
 being given. (Take axis 1^ and 2|^ ins.) 
 
 Draw the axes perpendicular to each other through their middle points. Take 
 3 points on the straight edge of a piece of paper or a scale, Fig. 17, and make the 
 distance between the first and second points equal to half the shorter axis; and, 
 between the first and third, equal to half the longer axis. Keep the second point 
 on the longer axis, and the third point on the shorter axis. The first point will 
 draw the required ellipse. 
 
 The straight edges of a square may be held on part of the axes, while a thin 
 piece of wood, with pins in it for guides, is used for drawing the curve; or an 
 instrument called a trammel may be used for drawing large ellipses, and an 
 ellipsograph used for drawing small ones. 
 
 PROB. 39. Draw an ellipse approximately with 4 arcs; the axes being given, 
 (li and 2| ins.) 
 
 Draw the axes as in Prob. 38 and from one end of the longer axis measure a 
 distance equal to halt the shorter axis. Construct an angle of 45 degrees at this 
 point meeting the shorter axes (Prob. 11). With this point as a centre, and half 
 the hypotenuse of the triangle formed, as a radius, strike an arc intersecting the 
 longer axis. With the point where the axes cross as a centre, and a radius equal 
 to the distance from it to the further point of incersection, draw arcs crossing the 
 axes at 4 points. Draw lines from the two points in the shorter axis through 
 the two in the longer axis and extend them. These four points will be the centres, 
 and the arcs will be drawn through the extremities of the axis, between the ex- 
 tended lines. 
 
 PROB. 40. Draw an ellipse approximately with 8 arcs; the axes being 
 given. (1^ and 2 J- ins.) 
 
14 MECHANICAL DRAWING. 
 
 Eight centres are used; two on each axis, and one in each angle between 
 them. Draw the two axes as in Prob. 38. From an extremity of each axis 
 draw a line, 1 and 2, parallel with the other axis and forming a rectangle 
 on half the axes. Draw line 3 from the extremity of one axis to that of the 
 other, diagonally across the rectangle. From the outside corner of the rectangle 
 draw line 4 perpendicular to the diagonal and it will cross the axes at two of the 
 required centers, a and b. With the intersection of the axes as a centre, and half 
 the shorter axis as a radius, strike an arc intersecting the longer axis at 5. With 
 a point half way between this point and the farther extremity of the longer axis, 
 as a centre, draw a semicircle 6 passing through these two points. With the radius 
 of this semicircle as a radius and the extremity of the shorter axis as a centre, 
 strike an arc intersecting the shorter axis. With the centre b on the shorter axis 
 extended, as a centre, draw arc 7 passing through the point of intersection oq the 
 shorter axis. With the extremity of the longer axis as a centre, and a radius equal 
 to the length of the shorter axis between the centre and semicircle, as a radius, 
 draw arc 8. It will intersect the last arc drawn at another one of the required 
 centres c. Draw line 9 through the centres b and c, and line 10 through c and a. 
 Three of the arcs will be drawn between these lines extended for one-fourth of the 
 ellipse. The other centres and lines can be easily found from these. 
 
 PROB. 41. Draw a perpendicular and tangent to an ellipse at a given point 
 on the curve. (Take axes 1^ and 2 J ins. and a point about equally distant from the 
 ends of the axes.) 
 
 Draw a line from both focii through the given point, and bisect the angles they 
 form (Prob. 4). One of the bisect'ng lines will be the required perpendicular, 
 and the other the required tangent. 
 
 PROB. 42. Find the two axes of an ellipse; the curve only being given. 
 (Take same sized ellipse.) 
 
 Draw any two parallel lines across the ellipse, and draw a line through the 
 middle points of these lines. With the middle point of this line as a centre, draw 
 a circle intersecting the curve at four points. With these four points as centres 
 and equal radii, draw intersecting arcs and draw lines through the points of inter- 
 section. These lines will be the required axes. 
 
 A PARABOLA is drawn by a point moving in a plane, so that its distance from a 
 given point remains equal to its distance from a given line. The fixed line is the 
 focus and the line is the directrix. 
 
 PROB. 43. Draw a parabola by finding points in the curve. (Take focus 
 ^ in. from directrix.) 
 
 Draw lines parallel with the directrix (use T-square, Fig. 4.) With the 
 distance from the directrix to the first parallel line as a radius, and the 
 focus as a centre, draw an arc intersecting the first parallel line at two points. 
 With the distance from the directrix to the second parallel line as a radius, and 
 the focus as a centre, draw an arc intersecting the second parallel line. In the 
 
MECHANICAI, DRAWING. 15 
 
 same way draw arcs intersecting the other parallel lines. These points of inter- 
 section will be points in the required curve. It may be drawn in iak, free hand or 
 with a curved ruler, Fig. 10, or with the compasses by finding centres and 
 radii by trial, which will draw arcs through three of the points at a time. 
 
 A HYPERBOLA is drawn by a point moving in a plane, so that its distance 
 from a given point remains equal to its distance from a given circle. The fixed 
 point and the centre of the circle are the focii. 
 
 PROB. 44, Draw a hyperbola by finding points in the curve. (Take radius 
 of circle 1 in. ; focii 1^ ins. apart.) 
 
 With the centre of the circle as a centre, draw a number of arcs 
 where the required curve is to be drawn. With the distance from the 
 circle to the first arc as a radius, and the other focus as a centre, 
 draw an arc intersecting the first arc at two points. With the distance from the 
 circle to the second arc as a radius, and the same focus as a centre, draw an arc 
 intersecting the second arc. In the same way draw arcs intersecting the other 
 arcs. These points of intersection will be points in the required curve, and it 
 may be drawn in ink in the same way as the parabola. 
 
 A HELIX is generated by a point moving uniformly around a given line, and 
 also moving in the direction of the line at a fixed distance from it. The given 
 line is the axis. A corkscrew, a wire spring, and screw-threads are illustrations 
 of a helix. Two views are necessary in a drawing to show a helix. The bottom 
 view will be a circle and the side view will be reversed curves. 
 
 PROB. 45. Draw a tielix with a given diameter and a given rise per revo- 
 lution. (Take diameter 2 ins. and rise 1 in. per revolution.) 
 
 Draw a circle for the bottom view. From the centre of this circle draw the 
 axis for the side view. Measure the rise per revolution on this axis. Divide the 
 rise into 24 equal parts (Prob. 9 or with Scale, Fig. 17), and draw lines through 
 these points of division perpendicular to the axis (use T-square). Divide half the 
 circle unto 12 equal parts (Prob. 15). From the first point of division on the 
 circle, draw a line parallel with the axis (use a triangle. Fig. 5) and intersecting 
 the first line which is perpendicular to the axis; from the second point of division 
 on the circle draw another parallel line intersecting the second perpendicular line; 
 from the third draw one intersecting the third, and so on. These points of inter- 
 section will be points in the required curve. The curve may be traced on a piece 
 of firm card-board or thin wood, and trimmed out smoothly with a keen pen-knife. 
 With this curve a great many revolutions of the helix may be made neatly with 
 ink in the drawing. 
 
 A SPIRAL is drawn by a point moving uniformly around a given point in the 
 same plane, and moving away from it at the same time. A watch spring is a^ 
 illustration of a spiral. This curve is used for drawing cams. 
 
 PROB. 46. Draw a spiral moving uniformly from the centre at given rate. 
 (Take 1 in. per revolution.) 
 
16 MECHANICAL DRAWING. 
 
 Draw lines through the centre, making angles of 15 degrees (Prob. 15). Meas- 
 ure the distance per revolution on one of these lines, and divide the distance into 
 24 equal parts. With the point as a centre and a radius equal to the distance to 
 the ^rs^ point of division, draw an arc intersecting one of the lines; with a radius 
 equal to the distance to the second point of division, draw an arc intersecting the 
 n<ext line; from the third point of division draw an arc intersecting the third line, 
 and so on. These points of division will be points in the required spiral. It may- 
 be drawn in ink with the compasses by finding centres and radii by trial, which will 
 draw arcs through three of the points at a time. 
 
 The INVOLUTE of a circle is drawn by a point in a straight line which rolls 
 on the circle. A pencil fastened to a string, which is kept stretched as it is un- 
 wound from a spool, -vrill draw an involute of a circle. This curve is used for 
 drawing the teeth of gear-wheels. 
 
 PROB. 47. Draw the involute of a given circle. (Take circle 1 in. in 
 diameter.) 
 
 Divide the circle into 24 equal parts. (Prob. 15), and draw lines tangent to the 
 circle at these points of division (with triangles. Fig. 5). With one point of divis- 
 ion as a centre, and with a radius equal to the length of the arc between the points 
 of division, draw an arc from the circle to the first tangent line; with the next point 
 of division as a centre, draw an arc from the end of this arc to the next tangent 
 line; with the third point as a centre, continue the curve to the third tangent line, 
 and so on. When 24 divisions are taken the curve will be more accurate, if these 
 ■centres are taken on a circle whose diameter is about 1-94 greater than the diam- 
 eter of the given circle. 
 
 A CYCLOID is drawn by a point in the circumference of a circle which rolls 
 on a line. If the generating circle rolls on the outside of a circle, the curve is an 
 epicycloid ; and, if it rolls on the inside, it is a hypocycloid. The circle on which 
 it rolls is the pitch circle. These curves are used for drawing the teeth of gear- 
 wheels. 
 
 PROB. 43. Draw epi- and hypocj^cloids; the diameters of the circles being 
 given. (Take diameter of pitch circle 4 ins., and diameter of generating circles 1 
 in., and let both curves start from the same point.) 
 
 Draw part of the pitch circle, and draw a generating circle tangent to it on the 
 ■outside, and draw another tangent on the inside. With the centre of the pitch 
 circle as a centre, draw arcs passing through the centres of the generating circles. 
 Make a number of equal divisions on the pitch circle, and draw lines through 
 them from the centre and intersecting the two arcs. With these points of inter- 
 section on the arcs as centres, draw parts of the generating circle where the curve 
 is to be drawn. With a point of division on the pitch circle as a centre, and the 
 distance between the points of division as a radius, draw an arc from the pitch 
 circle to the nearest pari of the generating circle; with the next point of division 
 as a centre, draw an arc which will continue the curve to the next part of the gen- 
 
ProhJZ 
 
 £Ciip6 
 
 ProhSS. 
 
 PrchM 
 
 Proh40. 
 
 ProA41. 
 
 Proh4Z. 
 
MKCHANICAI. DRAWING. 17 
 
 erating circle; with the third point as a centre, continue the curve to the third part 
 of the generating circle, and so on. Unless the points of division are quite small, 
 the radii of the arcs will be slightly too great. 
 
 I 
 
CHAPTER III. 
 
 PROJECTIOIS^ AND DEVELOPMENT OF SOLIDS. 
 
 The problems in this chapter are given to show how the different 
 views of an object are arranged and how patterns for curved surfaces are 
 cut. Two views are usually sufficient to represent the dimensions of a 
 
 solid. The top views, when drawn, are placed exactly above the prin- 
 
 cipal views; the bottom views, exactly below them; and the side and end 
 views, near the sides and ends which they represent. 
 
 In the figures, the outlines are drawn heavy on the bottom and right 
 to shade them; and surfaces, which are cut by a plane, are distinguished 
 by having oblique parallel lines drawn across them, called section-lines or 
 hatching. 
 
 A good drawing board, T-square, triangle, and dividers or measuring 
 scale, are essential for making these and similar drawings. The student 
 
 should copy the developments on heavy paper and cut them out and bend 
 them into shape to test the accuracy of his drawings. 
 
 A point is PROJEGTED on a line or a plane when a straight 
 line is drawn from it to the line or plane. The line drawn is the pro- 
 
 jecting line, and the point where the projecting line intersects the other 
 line or pierces the plane is the projection of the given point on that line 
 or plane. 
 
 A surface is DEVELOPED when it is removed from a solid 
 and spead out on a plane. 
 
 A PRISM is a solid with two equal parallel bases which are poly- 
 gons, and with faces (sides) which are all parallel to the same line. 
 When the bases are regular polygons, the prism is regular; and when the 
 faces are perpendicular to the bases, it is a right prism. 
 
 PRO-B. A 9. Draw three views and the development of the 
 
 faces of a regular 6-sided prism with its top cut away by a plane which 
 makes an angle of 45 degrees with its base. (Take prism i inch across 
 corners and 2 ins. high.) 
 
 Draw a regular hexagon for the bottom view. Project vertical lines 
 
 upwards from the corners of the bottom view, as indicated in the figure 
 by dotted lines with arrow points. These projecting lines determine the 
 
 edges in the view which is above. Horizontal lines projected to the 
 
 right from the corners determine the edges in the view which is at the 
 right of the bottom view. At any convenient distance from the bottom 
 
 view, draw straight lines across the projecting lines perpendicular to them, 
 for the base in the other two views. Draw a line across the upper view 
 
6 MECHANICAL DRAWING. 
 
 at the required distance from the base and making the required angle with 
 it. This' last line represents the cutting plane. It crosses all the 
 
 edges of the prism, and the length of each edge may be measured, and 
 the corresponding edges in the view at the right made the same length. 
 
 The development of the faces may be drawn by drawing parallel 
 lines, as far apart as the edges really are on the prism itself, and making 
 each line equal in length to the corresponding edge of the prism. The 
 
 distance between the parallel lines is equal to the distance between the 
 corners in the bottom view. The length of each one may be projected 
 
 from the upper view, as indicated in the figure by dotted lines. 
 
 PROB. SO. Draw three views and the development of the 
 
 faces of a regular 8-sided prism with its bottom cut away by a plane 
 which makes an angle of 30 degrees with its base. (Take prism i inch 
 across corners and 2 ins. high.) 
 
 Draw a regular octagon for the top view. Project lines downwards 
 
 from the corners and they will determine the edges in the view which is 
 below it. Draw a line across this view making the required angle with 
 
 the base. It will cut all the edges, and the length of each edge may 
 
 be measured and the view at the right and the development of the faces 
 drawn as in Prob. 49. 
 
 PROB. 51. Draw three views and the development of the 
 
 faces of a regular 24-sided prism with its top cut away by a plane which 
 makes an angle of 60 degrees with its base. (Take prism i inch across 
 corners and 2 ins. high.) 
 
 Draw a regular polygon of 24 sides for the bottom view. Project 
 
 lines from its corners for the edges in the other two views. Draw a 
 
 line across the upper view, making the required angle with its base, and 
 complete the other view and the development as in Prob. 49. 
 
 A CYLINDER is a solid with two equal, parallel bases which 
 are circles or other plane curves, and with a curved lateral surface which 
 is generated by a straight line moving parallel with itself and constantly 
 touching the curves. Any position of the generating line is an element 
 
 of the cylinder. The line joining the centres of the bases of a regular 
 
 prism or a cylinder is its axis. 
 
 The volume of a prism or cylinder is equal to its base multiplied by 
 its altitude (height). 
 
 PROB. 52. Draw three views and the development of the 
 
 whole surface of a circular cylinder with its bottom cut away by a plane 
 which makes an angle of 45 degrees with its axis. (Take cylinder i 
 
 inch in diam. and 2 ins. high.) 
 
 Draw a circle for the top view and divide it into 24, 48, or any num- 
 ber of equal parts. Project lines drawn from these points of division 
 
MECHANICAL, DRAWING. 7 
 
 and they will be elements of the cylinder in the lower view, which are 
 equally distant from each other. Draw the base and draw a line across 
 
 the lower view making the required angle with its axis. It will cut all 
 
 the elements, and the length of each one may be measured, and the oth- 
 er view and the development of the surface drawn as in Prob. 51, with 
 the exception that the width across the development is equal to the cir- 
 cumference of the base, and the outline is a curve traced through the ex- 
 tremities of the elements. The outline of the surface which is cut by 
 the plane is known to be an ellipse. Its shorter axis is equal to the 
 diameter of the cylinder and its longer axis is equal to the line drawn 
 across the lower view. 
 
 A PYRAMID is a solid with one base, and triangular /«,f(?j- which 
 meet at a point called the apex or vertex. When the base is a regular 
 
 polygon and the faces are equal, the pyramid is 7-egular. 
 
 PROB. 53. Draw three views and the development of the 
 
 faces of a regular 6-sided pyramid with its top cut away by a plane 
 which makes an angle of 45 degrees with its base. (Take pyramid i^ 
 
 ins. across corners of base and 2 ins. high with y^ inch of top cut off.) 
 
 Draw a regular hexagon for the outline of the base in the top view, 
 and draw a horizontal line at a convenient distance below it for the base 
 in the view below it. Project the corners of the base from the top 
 
 view down to the base in the lower view. Locate the apex in the low- 
 
 er view at the required distance above the base and exactly below the 
 centre of the top view. Draw lines from the corners of the base to the 
 
 apex in both views. These lines are the edges of the pyramid. Draw 
 
 a line across the lower view making the required angle with the base. 
 It will cut all the edges. Project the points where they are cut up to 
 
 the corresponding edges in the top view, as indicated in the figure by dot- 
 ted lines. Draw lines joining these points on the edges in the top view, 
 and its outline will be completed. The other view may be projected to 
 
 the right of the lower view. The distance across its base is equal to 
 
 the vertical distance across the top view; and the base, apex, and points 
 where the edges are cut, can be projected from the lower view, as indi- 
 cated in the figure by dotted lines. 
 
 The development of the faces of the complete pyramid is 6 equal 
 isosceles triangles. Their bases are equal to the distance between the 
 
 corners of the base of the pyramid in the top view, and their sides are 
 equal to the true length of the edges of the pyramid. The development 
 of the faces which are partly cut away may be drawn by finding the 
 length of the remaining part of each edge. The edges on the right and 
 
 left, in the lower view to the left in the figure, are parallel with the sur- 
 face of the paper and are therefore drawn in their true lengths; but the 
 
8 MECHANICAL DRAWING. 
 
 other edges come towards the front at one end and are therefore shorter 
 on the drawing than they really are on the pyramid. Their true lengths 
 may be found by projecting their extremities to the line at the right or 
 left, as shown in the figure by the dotted lines with arrow-points. The 
 
 true length of the remaining part of each edge may be measured off on 
 the corresponding edge of the development of the complete pyramid, and 
 the required development completed as in Prob. 49. 
 
 PROB. 54. Draw three views and the development of the 
 
 faces of a regular 8-sided pyramid with its bottom cut away by a plane 
 which makes an angle of 30 degrees with its base. (Take pyramid ij^ 
 
 ins. across corners of base and 2 ins. high. ) 
 
 Draw a regular octagon for the outline of the base in the top view 
 of the complete pyramid, and draw a horizontal line below it for the base 
 in the side view below it. Draw the corners, as in Prob. 53, and draw 
 
 a line across the side view making the required angle with the base. 
 This line will cut all the edges, and the points where they are cut may 
 be projected up to the corresponding edges in the top view. The view, 
 
 at the right and the development of the faces may be drawn as in Prob. 
 
 53- 
 
 A CON EI is a solid with one base which is a circle or other plane 
 curve, and with a curved lateral surface generated by a straight 
 line' passing through a fixed point and constantly touching the curve. 
 Any position of the generating line is an element of the cone. The line 
 
 joining the centre of the base with theHapex of a regular prism or a cone 
 is its axis. 
 
 The volume of a pyramid or cone is equal to its base multiplied by 
 one-third of its altitude. 
 
 PROB. 55. Draw two views and the development of the sur- 
 
 face of a right circular cone, which has its top cut away by a plane par- 
 allel with its base. (Take cone 2^ ins. across base and i in. high 
 with Yz in. of top cut off. ) 
 
 Draw a circle for the outline of the base in the top view, and draw 
 a horizontal line below it equal in length to the diameter of the circle, 
 for the base in the lower view. Locate the apex at the required dis- 
 
 tance above the base in the lower view, and draw elements to it from the 
 extremities of the base. Draw a line across the lower view parallel 
 
 with the base and cutting away the required amount of the top. The 
 
 outline of the cut surface in the top view is a circle with its diameter 
 equal to the line drawn across the lower view. 
 
 The development of the surface of a cone is a sector of a circle Avhose 
 radius is equal to an element of the cone, and the arc is equal in length 
 to the circumference of the base of the cone. The part cut away from 
 
Trob.49 
 
 O 
 
 Prism. 
 
 Proh.SO. 
 
 Pr6b.J2. 
 
 cylinder 
 
MECHANICAL DRAWING. 9 
 
 the development is drawn with the same centre and with a radius equal to 
 an element of the part which is cut away from the cone. 
 
 When the elements of a cone form angles of 60 degrees at its base 
 and apex, the development is a semicircle. 
 
 Note.— This problem is used in cutting out flanges and such tin utensils as funnels, dippers, 
 strainers, pails, coffee-pots, etc. 
 
 PROB. 56. Draw two views and the development of the 
 
 whole surface of a cone which has its top cut away by a plane parallel 
 with its base, and its bottom cut away by a plane which makes an angle 
 of 30 degrees with its base. (Take cone 2 ins. across base and 2 ins. 
 
 high with ^ in. of top cut off.) 
 
 Draw two views and the development of the whole cone as in Prob. 
 55. Divide the circle in the top view into 24, 48, or any number of 
 
 equal parts and project these points of division down to the base in the 
 lower view. From these points draw lines to the apex in both views. 
 
 These lines are equally distant elements of the cone. Draw a horizon- 
 
 tal line cutting off the top and an oblique line cutting off the bottom in 
 the lower view. The two. lines cut all the elements and the length of 
 
 each one may be found and the development completed, as in Probs. 54 
 and ^S, by tracing a curve through the extremities of the elements. 
 
 The surface of the cone which is cut by the oblique plane is known 
 to be an ellipse. Its longer axis is equal to the oblique line which is 
 
 drawn across this cone, and its shorter axis is equal to the diameter of 
 the cone at the middle of the oblique line, as indicated in the figure by 
 the broken horizontal line. 
 
 Solids INXERSEGX when one pierces the other, or when they 
 are cut so as to fit each other or become united. If they only touch 
 
 they are tangent but not intersecting. 
 
 PROB. 57. Draw three views and the development of the 
 
 whole surface of a cylinder intersected by a larger cylinder which is per- 
 pendicular to it. (Take diam. of one cylinder i in. and the other i^ 
 ins.) 
 
 Draw three views of the smaller cylinder as in Prob. 52. In the 
 
 lower view, draw a circle for the end view of the larger cylinder. Diraw 
 equally distant elements on the smaller cylinder, as in Prob. 52. These 
 
 elements are all intersected by the larger circle, and the length of each 
 one may be measured and the other two views and the development of 
 the lateral surface completed as shown in the figure. 
 
 The development of the surface, which fits against the larger cylinder, 
 is eliptical, and points in its outline may be found thus: Draw a straight 
 line through the centre of the top view of the smaller cylinder and extend 
 it. Take any convenient part of this extended line arid make it equal 
 
10 MECHANICAL DRAWING. 
 
 in length to the part of the larger circle, in the lower view, against which 
 the smaller cylinder fits. Make divisions on it equal to the correspond- 
 
 ing divisions which are made on that part of the larger circle by the ele- 
 ments of the smaller cylinder. Draw perpendicular lines through these 
 points of division. Project the nearest points of division, which are on 
 the top view of the smaller cylinder, to the nearest perpendicular line; 
 the next-nearest to the next line, and so on. The outline may be traced 
 through these points of intersection. 
 
 PROB. 58. Draw three views and the development of the 
 
 lateral surface of a cylinder intersected obliquely by a larger cylinder. 
 (Take diam. of one cylinder i in. and the other i}i ins. with their axes 
 making an angle of 60 degrees with each other.) 
 
 Draw three views of the smaller cylinder and draw equally distant el- 
 ements on it as in Prob. 52. Draw the larger cylinder making the re- 
 quired angle with the smaller one in the lower view. The elements 
 drawn on the smaller cylinder are intersected by elements of the larger 
 cylinder, and these elements and the points of intersection may be found 
 thus: Draw arcs, in the upper and lower views, with radii equal to the 
 radius of the larger cylinder, and their centres on the axis of the larger 
 cylinder. In the top view, draw elements of the larger cylinder through 
 the points of division on the top view of the smaller cylinder, and extend 
 them to the arc. Measure off these points of division, which are made 
 on the arc in the upper view, on the corresponding part of the arc in the 
 lower view; and draw elements of the larger cylinder through them. 
 These elements will intersect the elements drawn on the smaller cylinder. 
 The same elements, which intersect in the top view, intersect in the low- 
 er one, and these points of intersection determine the length of each ele- 
 ment of the smaller cylinder. The third view and the development may 
 be drawn as in Prob. 57. 
 
 PROB. S9. Draw three views and the development of the 
 
 surface of a regular 6-sided prism intersected by a cone whose axis coin- 
 cides with the axis of the prism. (Take prism i in. across corners, and 
 elements of cone forming an angle of 60 degrees at its apex. ) 
 
 Draw three views of a regular 6-sided prism, as in Prob. 49. Lo- 
 
 cate the apex of the cone on the axis of the prism at the same distance 
 from the base in both the side views. From these points draw elements 
 of the cone forming equal angles with the axis of the prism. They in- 
 
 tersect the edges of the prism in one view, and the middle of the faces in 
 the other view. As all the edges are the same length, and all the faces 
 
 go up the same distance in the middle, their extremities may be located 
 and curves drawn through them, as shown in the figure. These curves 
 
MECHANICAL DRAWING. 11 
 
 are known to be hyperbolas (Prob. 44) but they are usually drawn as 
 arcs of circles. 
 Note.— This problem is given to show how the chamfer is drawn on nuts and bolt-heads. 
 
 A POLYHEDRON is a solid with four or more faces. When 
 the faces are equal, regular, polygons and the vertices are equal, the poly- 
 hedron is regular. There are but five regular polyhedrons. Three of 
 them have respectively four, eight, and twenty faces, which are triangles; 
 one (the cube) has six faces, which are squares; and one has twelve 
 faces which are pentagons. 
 
 PROB. 60. Draw three views and the development of the 
 
 faces of a regular polyhedron of 20 sides intersected by a square prism. 
 (Take edges of polyhedron y% ins. and prism ^ ins. square. ) 
 
 This problem is intended as an exercise which will give the student 
 a clearer understanding of how the different views of an object are pro- 
 jected and arranged. He should first make the polyhedron by drawing 
 the development, shown in the figure, on card-board, and cutting through 
 the full lines and half through the dotted lines. The edges may then 
 be brought together and held in place by pasting strips of paper over 
 them. He can then study the problem out for himself. 
 
 Any point in the top view must be exactly above the same point in 
 the view below it; and any point in one side view must be exactly to the 
 right or left of the same point in the other side view. A side view is 
 
 placed opposite and near the side it represents. 
 
 A SPHERE is a solid with a curved surface, every point of 
 which is equally distant from the ce?itre. The distance between the sur- 
 
 face and the centre is the radius of the sphere; and the distance through 
 the centre is the diameter. 
 
 A zone is part of the surface of a sphere between two circles which 
 lie in parallel planes. A lune is part of the surface between two semi- 
 
 circles which meet at the extremities of a diameter. 
 
 The surface of a sphere is equal to its diameter multiplied by its cir- 
 cumference; and the volume of a sphere is equal to its surface multiplied 
 by one-third its radius. 
 
 PROB. 61. Draw three views and the development of the 
 
 surface of a regular 6-sided prism intersected by a sphere. (Take 
 
 sphere i^ ins. in diam. and prism i inch across corners with its axis 
 passing through the centre of the sphere.) 
 
 The outline of the sphere in each view is a circle. At the centre 
 
 B|v~ of the top view draw a regular hexagon for the top view of the prism. 
 ^1^ Project lines from the corners of the hexagon to the other two views of 
 ^^Hthe sphere. Draw the top base of the prism on each of the side views 
 
12 MECHANICAL DRAWING. 
 
 sphere intersects the edges of the prism in one view, and the middle of 
 the faces in the other. These points of intersection may be measured 
 
 from the base and curves drawn through them as in Prob. 59. 
 
 PROB. 62. Draw three views and the development of the 
 
 surface of a cylinder intersected by a sphere. (Take sphere i^ ins. in 
 
 diam. and cylinder i in. with its axis passing 3-16 ins. from centre of 
 sphere. ) 
 
 Draw three views of the sphere as in Prob. 61. Draw a circle in 
 
 the top view, for the top view of the cylinder. Place the centre of this 
 
 circle at the required distance to the right or left of the centre of the 
 sphere. Draw the other two views of the cylinder and divide it into 
 
 equally distant elements, as in the previous problems. Draw horizontal 
 
 lines on the top view of the sphere through the points of division on the 
 top view of the cylinder. These lines will be semicircles in the view 
 
 below the top view, and will intersect the same elements which they in- 
 tersect in the top view. These last points of intersection determine the 
 length of each element, and the other view and the development may be 
 completed as in the previous problems. 
 
 PROB. 63. Draw the development of the surface of a 
 
 sphere, approximately, by dividing it into lunes. (Take sphere i^ ins. 
 
 in diam. and divide surface into 12 lunes.) 
 
 The length of a lune is equal to one-half the circumference of the 
 sphere, and the width of it at its centre is equal to one of the parts into 
 which the circumference is divided. 
 
 Draw a line equal in length to the circumference of the sphere and 
 divide it into the required number of equal parts. Draw a perpendicu- 
 
 lar line half way between two of these points of division and extend it 
 on both sides a distance equal to one-fourth the circumference of the 
 sphere. Draw arcs passing through the extremities of the perpendicular 
 
 line and the two points of division which are on each side of it. These 
 arcs will be very nearly the outlines of the lune. 
 
 The lunes may all be drawn and cut out and fitted over a solid 
 sphere and pressed or beaten into shape. In practice it is difficult to 
 
 fasten all the points of the lunes together and it is therefore better to cut 
 them off and fill out the place with a circular piece. 
 
 PROB. Q^. Draw the development of the surface of a 
 sphere approximately by dividing it into zones. (Take sphere i}4 ins. 
 
 in diam. and divide surface into zones 15 degrees wide.) 
 
 Each zone may be treated as part of a cone, and its development 
 drawn as in Prob. 55. The radius, with which the development of a 
 
 zone is drawn, is an element of the complete cone, which would be tan- 
 gent to the sphere at the centre of that zone. 
 
Proh.oZ 
 
 Intersecting Cylinders 
 
 Proh.58. 
 
 PrQb.59. 
 
 Prob.GO. 
 
 FolyheciroTly 
 
MECHANICAL DRAWING. 13 
 
 Draw a circle equal to the circumference of the sphere, and draw a 
 vertical line through its centre. From one of the points where this 
 
 line intersects the circle, divide one-fourth of the circle into 6 equal parts. 
 Draw tangents to the circle from these points of division to the vertical 
 line. These tangents form angles with the vertical line of 15, 30, 45, 
 
 60 and 75 degrees respectively. The width of each zone is equal to the 
 
 distance between the points of division. The development of the zone 
 
 at the centre is straight, and is equal in length to the circumference of 
 the sphere. The outside arcs of the developments of the two zones next 
 
 to it are drawn with a radius equal to the Jongest tangent added to half 
 the width of the zone. Their length is made equal to the circumfer- 
 
 ence of the sphere. The inside arcs and ends of the developments are 
 
 determined as in Prob. 55. The developments of the next two zones 
 
 are semicircles and their outside arcs are drawn with a radius equal to 
 the next longest tangent added to half the width of the zone. The out- 
 side arcs of the next two are equal in length to the inside arcs of the 
 semicircular ones. The outside arcs of the next smaller are equal in 
 
 length to the inside arcs of the greater ones next to them. The open- 
 
 ings in the smallest zones are drawn to their centres. 
 
 If the zones are all drawn touching each other, as in the figure, the 
 distance between the centres of the smallest zones is equal to one-half the 
 circumference of the sphere. 
 
 If the draftsman has a measuring scale by which he can measure dec- 
 imally, and a protractor by which he can lay off angles; he can make the 
 width of each zone equal to .262 times the radius of the sphere and the 
 tangent lines equal respectively to 3.732, 1.732, i.ooo, .577 and .268 times 
 the radius. The middle development is straight and equal in length to 
 6.283 times the radius. The two developments next to it contain 93^ 
 
 degrees; the next two 180 degrees (semicircles). The next two have 
 105 degrees cut out; the next two 48^, and the two smallest 12^^ cut 
 out. 
 

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