LIBRARY OF CONGRESS.* ' ©1^. - ®op^ng|l %i UNITED STATES OF AMERICA. AN OUTLINE COT^RSE JVIECHA^IICAli) DRAWIJMG EVENING DRAWING CLASSES, Cflf- W. S. LOCKE, tl INSTRUCTOR IN MECHANICAL DRAWING, sf RHODE ISLAND SCHOOL OF DESIGN. ;"^Pij 23 u PROVIDENCE, R. I. Copyright 1SS9 and 1S91. \ ^ (, ^ \"' INDEX. INTRODUCTION. FREE-HAND DRAWING. PLANE GEOMETRY. Problems in Plane Geometry. SOLID GEOMETRY. Square Root. Cube Root. ORTHOGRAPHIC PROJECTIONS. Revolutions. True Length of a Line. Problems in Projections. CONIC SECTIONS. INTERSECTIONS and DEVELOPMENTS. ISOMETRIC PROJECTIONS. LINEAR PERSPECTIVE. WORKING DRAWINGS. DESIGN. CAMS. Face Cams. Square Cams. Edge Cams. Leaders. GEARING. Single Curve Gearing. Rack to Mesh with Single Ct^rve Gear. Single Curve Gears having less than 30 teeth. IV. INDEX. Bevel Gear Blanks. Epicycloidal Gears. Proportions of Gear Wheels. Teeth of Gear Wheels. Strength of a Tooth. Comparison of Teeth. Gear Designing. Sample Plate of Involute Teeth. TABLES AND DIAGRAMS. Strength of Materials. Factor of Safety. Horse Power of Shafting. Strength of Belting. Decimal Equivalents. Handle Table. Proportions of Bolts and Nuts. Alphabets. Areas of Circles. Circumferences of Circles. INTRODUCTION. This book is not a treatise on Mechanical Drawing, but an outline course on the subject. A mastery of its contents will not make the student an accomplished draughtsman, but it is hoped that such mastery will' help him a long way toward that desired result. The book is put out chiefly for that class of students^ who, from age or condition, are not able to take the time necessary to go through a complete course of study with the view of making themselves mechanical draughtsmen. The difficulty of gauging the needs of this body of students makes the success of any such effort problematical ; nevertheless, this trial is made with the hope of ultimate success. In attempting to meet the want referred to, the style of writing at- tempted, is forcible rather than elegant, direct rather than graceful. While much is lost by adopting this style, it is hoped that more is gained. *i FREE-HAND DRAWING. The student, like the child, must '' creep before he can walk,-' that is, do some preparatory work before he can stride along in his chosen path. In this pre- liminary work, the foundation, care should be taken. Study and work thoroughly, rather than fast. Make clear to your mind each subject as it is taken up, and by what is understood that which follows will be made easier. Before writing became common, the saying, *' The pen is mightier than the sword,'' was accepted by the world. In these days, when time is very valuable, drawings are generally used in all kinds of business. Drawing is the universal language, and the ability to make good drawings is of more value, to-day, than any mere physical power. More or less elaborate drawings are used to illustrate form, fancy, or thought, wherever man communicates with his fellow. The art of making pictures has grown from rude outlines made on bark of trees with charcoal to magnificent oil paintings in splen- did colors. It is not the object of this book to teach this art, but it will be well for the student of Mechanical Drawing to learn to make outline drawings, first, with- out instruments, other than pencil and paper. The ability to make these free-hand drawings is useful and valuable, always cultivating the eye and hand for more complicated work. Representations of simple forms MECHANICAL DRAWING. should be attempted at first, as even these may prove difficult to the beginner. IVe must learn to know what our eyes see and then to represent that correctly. Fio.l. R(?,^ . Figure i represents three '' views," top, side, and end of a simple block. The student should begin with some such sim- ple form, and learn to see and represent it correctly. To do this, place the block on the level of the eye, in such a position that only four lines, bounding one plane of the block, can be seen. These simple drawings having been made, the block should be placed lower and an attempt made to draw a " perspective" of it. Fig. 2 is an accurate perspective of the block, three of whose faces are shown in Fig. i . While the repre- sentation is correct, quickly recognized and simple, there are some details that will repay study. First, it is stand- ing on a level surface because certain lines are vertical. Second, the lines that form the bottom and top do not, in themselves, appear as level lines, while we know they must be. Third, the angles made at the meeting of the lines are not right angles — the corners do not appear as being square, though we know they are so. If the stu- dent sees and understands these three points, and also the fact that the drawing — Fig. 2 — is correct, he has taken a good step in the right direction. It will not be arrived at in an hour, and many times a day the attention MECHANICAL DRAWING. 9 should be fixed on a form of definite outlines and an attempt made to see and represent it. This training- should continue until the student can draw, with fair speed and accuracy, such simple inanimate objects as he handles in his business. It will often happen that the ability to do this will save time and money to employer or employed, or both. The draughtsman is lame with- out it. The fundamental principles of perspective being thus understood, the student may continue to practice free-hand drawing both to his pleasure and profit, but for the purpose of accomplishing the work laid out in this Course, we may leave free-hand drawing for the next step. This may well be ' Geometrical Drawing, for the science which treats of Position and Extension must help us to correct such inaccuracies as we have now noticed in work where the eye is the only judge of correctness. INSTRUMENTS. Before we take up Geometrical Drawing we should know something of instruments and their use. The skill of hand already acquired will be of great use in handling instruments, and the accuracy of eyesight^ necessary in making measurements. Accurate 77ieasiu'e- ments are absolutely necessary to good Geometrical and Mechanical Drawing. A complete outfit for Mechanical Drawing may be bought for $15.00, and half this sum will buy instruments sufficient for most plain work. The following is a satisfactory outfit : One drawing board, one T-square, two triangles, one scale, two rub- bers, pencils, and case of instruments. The following- are good sizes: Board, twenty-four by thirty-two inches, T-square with thirty-inch blade, a six-inch rub- ber 45° triangle, a ten-inch rubber 30-and-6o° triangle, lO MECHANICAL DRAWING. a twelve-inch steel scale, pencil and ink eraser, one each, four thumb tacks, four and six H. Faber's drawing pen- cils, one case instruments, containing one five-and-one- half-inch compasses, with pen, pencil and needle points, one five-inch spring dividers, two right line pens, three- inch spring dividers, three-inch spring bow pencil and three-inch spring bow pen. This may come well within $15.00, and be composed of goods that will wear for many years. In using the T-square it should be placed on the board with its head held against the left hand side. All hori- zontal lines should be drawn with the pencil held against the upper side of the blade of the square as it is thus held. Perpendicular lines should be drawn with the combined help of T-square and triangle. With the 45*^ and the 30-and-6o° triangles the following angles, with a horizontal, may be obtained : 15°, 30°, 45°, 60^, 75 and 90^. The student will profit by a facility in placing his triangles to make these angles. In the Geometry which follows, a method is given of dividing an angle in halves, so that the student should be able to make any desired angle. The most convenient scale is a steel scale, twelve inches long, graduated to sixteenths on one edge and thirty-seconds on the other. It is made by Brown & Sharpe, and sold everywhere for $1.25. The compasses are for making circles and arcs of cir- cles. In drawing a circle never take hold of a point of the dividers, but always hold the compasses by the head or handle. The joints. in the stocks are used to bend the points to nearly right angles with the paper when draw- ing large circles. In using the right-line pen always hold it in a plane at right angles to the paper. Always draw, never push the pen. This last is a good rule also in using the pencil, which should be sharpened to a chisel edge rather than to a point. This sort of '' point" MECHANICAL DRAWING. II is also best for the pencil used in the compasses. It will wear longer and make a better line. Much care should be taken in all these operations, for these are details of the art of drawing, which is composed of details. Ten times more time is lost erasing incorrect lines than is lost in careful decision as to whether or not they will be right. In drawing, it is very clearly seen that "haste makes waste." The first requisite of a drawing, is ACCURACY, the second, neatness. After these, in a sheet of drawings, comes, third, arrangement. With careful practice one may learn to draw fast, but no one can make good, reliable drawings in a hurry. PLANE GEOMETRY. DEFINITIONS. Geojnetry is the Science of Position and Extension, A Point has merely position j without extension. Extension has three dimensions : Le7igth^ Breadth and Thickness. A Line has only one dimension, namely, length. All lines are either straight or curved. Note. — In this book the word "line" is understood to mean a straight liyie^ unless otherwise specified. A Surface has two dimensions : Length and breadth. A Solid has the three dimensions of extension : length, breadth and thickness. The Position of a Point is determined by its Direction and Distance from any known point. A Plane is a surface in which any two points being taken, the straight line joining those points lies wholly in that surface. Axiom. A straight line is the shortest distance be- tween two points. THE ANGLE. An Angle is formed by two lines meeting or crossing each other. MECHANICAL DRAWING. I3 The V'ertex of the angle is the point where its sides meet. The magnitude of the angle depends solely upon the difference of direction of its sides at the vertex or amount they are spread apart. The- magnitude of the angle does not depend upon the length of its sides. When one straight line meets or crosses another, so as to make the two adjacent angles equal, each of these angles is called a Right angle, and the lines are said to be perpendicular to each other. Thus the angles ABC and ABD (Fig. 3) being equal, are right angles. An Acute angle is one less than a right angle, as K (Fig. 3)- An Obtuse angle is one greater than a right angle as -V(Fig. 3)._ Parallel Lines are straight lines which have the same Direction, as AB^ CD (Fig. 3). Parallel lines cannot meet, however far they are pro- duced. POLYGONS. A plane figure is a plane terminated on all sides by lines. If the lines are straight, the space which they contain is called a rectilineal figure, ox polygon {F Fig. 3). The polygon of three sides is the simplest of these figures, and is called a tiHangle; that of four sides is called a quadrilateral ; that of five sides, ?i pentagon; that of six, a hexagon^ &c. A triangle is denominated equilateral {E Fig. 3), when the three sides are equal, isosceles (/ Fig. 3) , when two only of its sides are equal, and scalene (^9 Fig. 3), when no two of its sides are equal. H MECHANICAL DRAWING. A right-triangle is one which has a right angle. The side opposite to the right angle is called the hypothcmise. Thus ABC (Fig. 3) is a triangle right-angled at A, and the side BC is the hypothenuse. Among quadrilateral figures, w^e distinguish : The square (Sq. Fig. 3), having its sides equal, and its angles right angles. The rectangle {^R Fig. 3), having its angles right angles, and its opposite sides equal. T\\Q parallelogram (/* Fig. 3), which has its opposite sides parallel. The rhombus or lozenge {Rh. Fig. 3), which has its sides equal, w^ithout having its angles right angles. Rq.S. The trapezoid ( 7' Fig. 3), which has two only ot its sides parallel. A diagonal is a line which joins the vertices oi two angles not adjacent, as AC in the figure of the paral- leloijrani. MECHANICAL DRAWING. 15 THE CIRCLE. Definitions, The circumfereitce of a circle is a curved line^ all the points of which are equally distant from a point within, called the centre. The circle is the space enclosed by this curved line. The radius of a circle is any straight line, as AB^ AC, AD (Fig. 3), drawn from the centre to the circumfer- ence. The diameter of a circle is a straight line, as \BD, drawn through the centre, and terminated each w^ay by the circumference. A semicircufiiference is one half of the circumference, and a semicircle is one half of the circle itself. An arc of a circle is any portion of its circumference, as BFE, The circle is supposed to be divided into 360 equal parts called degrees, marked ^ . Thus " an angle of 90*^" is, of course, one-quarter of a circle. 60^, is one- sixth, 45^, one-eighth, &c. The chord of an arc is the straight line, as BE, which joins its extremities. The segment of a circle, is a part of a circle compre- hended between an arc and its chord, as EFB, A tangent (Fig. 3) is a line, which has only one point in common with the circumference, as HD. A polygon is said to be circumscribed about a circle, when all its sides are tangents to the circumference ; (Fig. 3,) and, in this case, the circle is said to be in- scribed in the polygon. A polygon is inscribed in a circle when all its vertices are in the circumference of the circle, (Fig. 3.) Problems in Plane Geometry. Note. — Lay out a sheet of drawing paper with tifteen equal rectangles inside the three-quarter inch margin. Write a problem and make a graphical solution in each rectangle In each problem there are certain things given and certain things required. Carefully note and make use of the things given in finding the things required. I. Problem. To find the position of a point in a plane, having given its distances from two known points in that plane. Solution. Let the known points be A and B. From the point ^ as a centre, with a radius equal to the dis- tance of the required point from ^, describe an arc. Also, from the point ^ as a centre, with a radius equal to the distance of the required point from B, describe an arc cutting the former arc ; and the point of intersec- tion C is the required point. a. By the same process, another point D may also be found which is at the given distances from A and B, and either of these points therefore satisfies the condi- tions of the problem. b. If both the radii were taken of equal magnitudes, the points C and D thus found would be at equal dis- tances from A and B, c. The problem Is impossible, when the distance between the known points is greater than the sum of the given distances or less than their diflerence. MECHANICAL DRAWING. I 7 d. If the required point is to be at equal distances from the known points, its distance from either of them must be greater than half the distance between the known points. 2. Problem. To divide a given straight line AB into two equal parts ; that is, to bisect it. Solution. Find by § b^ a point C, above the line, at equal distances from the extremities A and B. Find also another point D.^ below the line, at equal distances from A and B. Through C and D draw the line CD^ which bisects AB at the point E. 3. Pi'oblem. At a given point A^ in the line BC^ to^ erect a perpendicular to this line. Solution. Take the points B and C at equal distances from A ; and find a point D equally distant from B and C. Join AD and it is the perpendicular required. 4. Problem. From a given point A., above the straight line BC, to let fall a perpendicular upon this line. Solution, From A rs a centre, with a radius suffi- ciently great, describe an arc cutting the line BC in two points B and C ; find a point B> below BC\ equally dis- tant from B and C, and the line AB> is the perpendicular required. 5. Problem. To draw a perpendicular to a line at one end. Solution. Let AB be a horizontal line* With A as centre and radius AB draw an arc. With same radius and centre B draw arc, cutting first one at D. With D as centre and same radius draw arc over A. Draw line through B and D^ meeting the last arc at /s. ^ Line EA will be perpendicular to line AB. iS MECHANICAL DRAWING. 6. Problem. To make an arc equal to a given arc AB^ the centre of which is at the given point C. Solution. Draw the chord AB. From any point D as a centre, with a radius equal to the given radius CA^ describe the indefinite arc FH. From 7^ as a centre^ with a radius equal to the chord AB, describe an arc cutting the arc FH \\\ H, and we have the arc FH^=l arc AB. 7. Problem. At a given point A., in the line AB^ to make an angle equal to a given angle K. Solution. From the vertex A", as a centre, with any radius, describe an arc IL meeting the sides of the angle ; and from the point ^ as a centre, by the preceding prob- lem, make an arc BC equal to IL. Draw AC, and we have angle BAC = angle K. 8. Problem. To bisect a given arc AB. Solution. Find a point D at equal distances from A and B. Through the point F> and the centre C draw the line CP>^ which bisects the arc AB at F. 9. Problem. To bisect a given angle A. Solution. From ^ as a centre, with any radius, de- scribe an arc BC^ and by the preceding problem, draw the line AF to bisect the arc BC^ and it also bisects the angle A. 10. Problem. Through a given point A, above a given straight line BFC^ to draw a straight line parallel to the line BFC. Solution. Join FA, and, by Problem 7, draw AD^ making the angle FAD — AFC, and AD is parallel to BFC MECHANICAL DRAWING. I9 1 1 . Problem. Two sides of a triangle and their in- cluded angle being given, to construct the triangle. Solution. Make the angle A equal to the given angle ^ take AB and AC equal to the given sides, join BC^ and ABC is the triangle required. 12. Problem, The three sides of a triangle being given, to construct the triangle. Solution. Draw AB equal to one of the given sides^ and, by § I, find the point C at the given distances AC and BC from the point C, join AC and BC, and ABC is the triangle required. Note. — The problem is impossible, when one of the given sides is. greater than the sum of the other two. 13. Problem. The adjacent sides of a parallelogram and their included angle being given, to construct the parallelogram . Sohttion. Make the angle A equal to the given angle ^ take AB and AC equal to the given sides, find the point Z^, by § I, at a distance from B equal to AC, and at a distance from C equal to AB. Join BD and DC^ and ABCD is the parallelogram required. Note. — If the given angle is a right angle, the figure is a rectangle; and, if the adjacent sides are also equal, the figure is a square. 14. Problem. To find the centre of a given circle or of a given arc. Solution. Take at pleasure three points. A, B, C, on the given circumference or arc ; join the chords AB and BC^ and bisect them by the perpendiculars DE and FG ; the point O in which these perpendiculars meet is the centre required. 20 MECHANICAL DRAWING. 15. Problem. Find by the same construction as in Problem 14 a circle, the circumference of which passes through three given points not in the same straight line. 16. Problem. Through a given point to draw a tangent to a given circle. Solution. If the given point A is in the circumference, draw the radius CA^ and through A draw ^Z) perpendic- ular to CA^ and AD is the tangent required. 17. Problejn. Through a given point to draw tan- gents to a given circle. - Solution. If the given point A is without the circle, join it to the centre by the line AC ; upon AC as a diam- eter describe a circumference cutting the given circum- ference in J/ and A/' ; join AM and AA^^ and they are the tangents required. 18. Problem. To inscribe a circle in a given triangle ABC. Solution. Bisect the angles A and B by the lines AO and BO, and their point of intersection O is the centre of the required circle, and a perpendicular let fall from O upon either side is its radius. Note. — The three lines AO, 7?0 and TO, which bisect the three angles of a triangle, meet at the same point. 19. Problem. To divide a given straight line AB into any number of equal parts. Solution. Suppose the number of parts, for example, is six. Draw the line AO, making an acute angle with line AB\ take AC of any convenient length and apply it six times to AO. Join B and the last point of divis- ion, Z>, by the line BD. Through the last point of division but one draw a line (see Problem 10) parallel to BD. This line will cut AB, at E, one-sixth of the distance from B toward A. Apply EB six times to AB, MECHANICAL DRAWING. 21 20. Problem. To find a mean proportional between two given lines. Solution. Draw the straight line ACB, making AC equal to one of the given lines, and EC equal to the other. Upon ACB as a diameter describe the semicircle ADB. At C erect the perpendicular CZ^, and CD is the required mean proportional. 21. Problem. To divide a given straight line ACB at the point C in extreme and 7nean ratio, that is, so that we may have the proportion : AB : AC=AC: CB. Solution. At end B erect the perpendicular BP> equal to half of ACB. Join AP>, take DP from D on AD equal to BD, and AC equal to AP, and C is the required point of division. REGULAR POLYGONS. Defi7titions. A regular polygon is one which is at the same time equiangular and equilateral. Hence the equilateral triangle is the regular polygon of three sides, and the square the one of four. An equilateral polygon is one which has all its sides equal ; an equiangular polygon is one which has all its angles equal. 22. Problem. To inscribe a square in a given circle ► Solution. Draw two diameters, AB and CD, perpen- dicular to each other ; join AD, DB, BC, CA ; and ADBC is the required square. 23. Problem. To inscribe a regular hexagon in a given circle. Solution. Take the side BC of the hexagon equal to the radius ^C of the circle, and, by applying it six 22 MECHANICAL DRAWING. times round the circumference, the required hexagon BCDEFG is obtained. 24. Pj'oblem. To describe a regular decagon in any circle. Sohctioii. Divide the radius of a (three-inch) circle in extreme and mean ratio. (Problem 21.) The longer part is equal to one side of the regular decagon required. Apply it ten times to the circum- ference, and join the points by straight lines, making the decagon. Aiake a pentagon by joining the alternate vertices of the decagon. 25. Problem. To circumscribe a circle about a given regular polygon ABCD^ &c. Solution. Find, by Problem 14, the circumference of a circle w^hich passes through three vertices, A^ B, C ; and this circle is circumscribed about the given polygon. 26. Pi^oblem. To inscribe a circle in a given regular polygon A BCD, &c. Solution. Bisect two sides of the polygon by perpen- diculars, the point of intersection is the centre of the required circle. The sides of the polygon become tangents to the circle. 27. Problem. To inscribe a pentagon in a circle. Solution. Draw a diameter AB and a radius CD per- pendicular to it. Bisect BC at E. With centre at E and radius ED draw arc DF^ — F being on AC. With Z> as a centre, and DF as a radius, draw arc EG, meet- ing the circumference at G. Draw line DG. It is one side of the required pentagon. 28. Problem. To construct a regular polygon of any number of sides in a circle (Approximate method). MECHANICAL DRAWING. 23 Solution. Draw a diameter AB and divide it into as many equal parts as there are sides in the required poly- gon—say eight. With A and B as centres, and radius AB draw arcs intersecting in C. Draw line from C through the second point of division of AB to meet the circumference at D. AD is one side of the required polygon. AREAS. Definitions. Equivalent figures are those which have the same surface. The area of a figure is the measure of its surface. The unit of surface is the square w^hose side is a linear unit ; such as a square inch or a square foot. The area of a square is the square of one of its sides. A parallelogram is equivalent to a rectangle of the same base and altitude. The area of a parallelogram is the product of its base by its altitude. Parallelograms of the same base are to each other as their altitudes ; and those of the same altitude are to each other as their bases. All triangles of the same base and altitude are equiva- lent. The area of a triangle is half the product of its base by its altitude. Every triangle is half of a parallelogram of the same base and altitude. The circumference of a circle is equal to 3. 141 6 multi- plied by its diameter., or r. D. The area of a circle is equal to 3. 14 16 multiplied by the square of its radius., or r: R\ The area of a trapezoid is half the product of its alti- tude by the sum of its parallel sides. 24 MECHANICAL DRAWING. 29. Problem. To construct a square equal to three times a given square. Solution. Extend one side of the given square, and lay off on it the length of its diagonal. Draw a line from the point at which this diagonal ends to the extreme angle of the square, and upon this line construct a square, w^hich will be the square required. 30. Problem. To make a square equivalent to the sum of two given squares. Solution. Construct a right angle C ; take CA equal to a side of one of the given squares ; take CB equal to a side of the other ; join AB, and AB is a side of the square sought. A square may be found equivalent to a given triangle, by taking for its side a mean proportional between the base and half the altitude of the triangle. A square may be found equivalent to a given circle, by taking for its side a mean proportional between the radius and half the circumference of the circle. PRACTICAL PROBLEMS IN PLANE GEOMETRY. Note. — Divide the sheet into four equal parts and put a problem in each. Make an angle of 75° and bisect it. 2. Draw six-ineh parallel lines, y<^//r inehes apart. 3. Divide a seven-ineh line into nine equal parts. '4. Make a simple belt pulley withy^z'^ spokes. 5. Make an isosceles triangle equivalent to a right- angled triangle whose sides are three., four ?iwdi five inches long. MECHANICAL DRAWING. 25 6. Make an isosceles triangle equivalent to a par- allelogram whose sides are three^ and foicr inches^ and whose angles are 6d^ and 120°. 7. Make a square equivalent to a right-angled trian- gle whose sides are three, four and five inches, 8. Make a square equivalent to a circle whose diame- ter \% four inches. SOLID GEOMETRY. DEFINITIONS. From the definitions of Plane Geometry we may recall at this time those of a Line', a Surface, a Plane, and a Solid. A Solid has the three dimensions of extension ; length, breadth and thickness. Every solid bounded by planes is called ?<. polyedron. The bounding planes are called the faces ; whereas the sides or edges are the lines of intersection of the faces. A polyedron of four faces is a tetraedroji^ one of six is a hexaedrou^ one of eight is an octaedron^ one of twelve a dodecaedron^ one of twenty an icosaedrou, &c. The tetraedron is the most simple of polyedrons ; for it requires at least three planes to form a solid angle, and these three planes leave an opening, which is to be closed by a fourth plane. A prism is a solid comprehended under several par- allelograms, terminated by two equal and parallel poly- gons. The bases of the prism are the equal and parallel polygons. The convex surface of the prism is the sum of its parallelograms. MECHANICAL DRAWING. 27 The altitude of a prison is the perpendicular distance between its bases. A right prism is one whose lateral faces or parallelograms are perpendicular to the bases. J^.. Fig. 3A. Note. — In this book the word prism is to be taken to mean a right prism ^ that is one whose lateral faces are rectangles. In this case each of the sides has an altitude equal to that of the prism. A prism is triangtdar ^ quadrangular.^ pentagonal^ hex- agonal., &c., according as its base is a triangle, a quad- rilateral, a pentagon, a hexagon, &c. The prism, whose bases are regular polygons of an infinite number of sides, that is, circles, is called a cylinder. The line which joins the centres of its bases is called the axis of the cylinder. In the right cylinder the axis is perpendicular to the bases, and equal to the altitude. The right cylinder may be considered as gen- erated by the revolution of a rectangle about one of its sides. The other side generates the con- vex surface, and the ends generate the bases of the cylinder. A prism whose base is parallelogram has all its faces parallelograms, and is called a parallelopiped. When all the faces of a parallelopiped are rectangles, it is called a right parallelopiped. ^ The cube is a right parallelopiped, com- prehended under six equal squares. The cube, each of whose faces is the unit of surface, is assumed as the ////// of solidity. Fig. 30. Fig. 3B. 28 MECHANICAL DRAWING. The volume^ solidity^ or solid cojitents of a solid, is the measure of its bulk, or is its ratio to the unit of solidity. The cubic inch is a good unit of solidity. The area of the convex surface of a prism or cylinder is the perimeter^ or the circumference of its base inulti- plied by its altitude. The formula for the convex surface of a cylinder is,. 6' = t: DH^ where S^^ the convex surface, 7rz=3.i4i6,. D = diameter, and If = height. Careful attention is called to this mode of expression,, for it is convenient and will be used hereafter. In the expression S = t: DH, the quantities written together are supposed to be multiplied together, thus S= t: Z>Zr becomes S= tt x jDxIf. Any prism or cylinder is equivalent to a right prisn> of the same base and altitude. The solidity or volume of any piHsm or cylinder is the product of its base by its altitude. Prisms or cylinders of equivalent bases and equal altitudes are equivalent. Let R be the radius, and A area of the base of a cylin^ der; and r. — 3.1416, we have A = 3.i4i6x^'. Denoting, also, the altitude by H and the solidity or volume of the cylinder by V^ we have V=AxJf = -X R'xII=r.R'H. A pyramid is a solid formed by several triangular planes proceeding from the same point and terminating in the sides of a polygon. This point is the vertex or apex of the pyramid. The altitude of the pyramid is the distance of its vertex from its base. A pyramid is regular when the base is a regular polygon, and the perpendicular let fall from the vertex upon the base, passes, through the centre of the base. This per- PiG^ pendicular line is the axis of the pyramid. MECHANICAL DRAWING. 29 When the base of a pyramid is a circle it is called a <:^;^^ -DH\ As before, it must be remembered that the quantities written together are multiplied together. The volume or solidity of a cone is one- third of the product of its base by its altitude or height. Let H represent the height. The area of the base is 3.1416X R^ ; hence the volume of a cone is F = 1/371 R^H. The volume of any pyramid is one -third of the product of its base by its height. Pyramids or cones of equivalent bases and equal alti- tudes are equivalent. Any pyramid or cone is a third part of a prism or cylinder of the same base and altitude. THE SPHERE. DEFINITIONS. A Sphere is a solid terminated by a curved surface, all the points of which are equally distant from a point within called the centre. *3 30 MECHANICAL DRAWING. The sphere may be conceived to be gen- erated by the revolution of a semicircle about its diameter. The radius of a sphere is a straight line Fig. E. draw^n from the centre to a point in the sur« face ; the diameter or axis is a line passing through the centre, and terminated each way by the surface. All the radii of a sphere are equal ; and all its diam- eters are also equal, and double the radius. Every section of a sphere made by a plane is a circle. The section made by a plane w^hich passes through the centre of the sphere is called a great circle. Any other section is called a small circle. The radius of a great circle is the same as that of the sphere, and therefore all the great circles of a sphere are equal to each other. The centre of a small circle and that of the sphere are in the same straight line perpendicular to the plane of the small circle. The area of the surface of a sphere is the product of its diameter by the circumference of a great circle. The surface of a sphere is equivalent to four great circles. The surfaces of spheres are to each other as the squares of their radii. From the foregoing we may deduce the following formulas : Surface of a sphere :=^ S ^ 4' R^ ^^ - Z>^, for since the area of one circle is -i?^ the surface of a sphere is 4 X -K'^4-R\ The solidity or volume of a sphere is one-third of the product of its surface by its radius. The surface — S= 4- R" multiplied by Yi R\'$> V — {4' R') X Yz R = 7: ^T.R\ or Volume of sphere = V = MECHANICi\L DRAWING. 3 1 For the convenience of the student, rules for extract- ing the Square and Cube Roots are here inserted. SQUARE AND CUBE ROOT. RULE. To extract the square root of any number : 1 . Beginning with the units figure, point off the ex- pression into periods of two figures each. 2. Find the greatest square in the number expressed by the left hand period, and write its square root as the first figure of the 7^oot. 3. Subtract this square from the part of the number used, and to the remainder unite the next two terms of the given number for a new dividend. 4. Double the part of the root already found for a trial divisor ; and by it divide the new dividend — less the last figure — and write the quotient as the next figure of the root. Also, write it at the right of the trial divisor^ the combined figures making the true divisor. 5. Multiply the true divisor by the last figure of the root and subtract the product from the new dividend. 6. If there are any more terms of the root to be found, unite with the remainder the next two terms of the given number, and take for a trial divisor, double the root already found, and proceed as before. RULE. To extract the cube root of a number : I. Beginning with the units figure, point off the ex- pression into periods of three figures each. 32 MECHANICAL DRAWING. 2. Find the greatest cube in the number expressed by the left hand period, and write its cube root as the first figure of the root. 3. Subtract the cube from the part of the number used, and with the remainder unite the next three figures of the given number for a new dividend. 4. Take three times the square of the part of the root already found, with two ciphers annexed, for a trial divisor, and by this divide the new dividend, and write the quotient as the next term of the root. 5. To the trial divisor add three times the first term of the root.^ with a cipher annexed, multiplied by the last term, also the square of the last term of the root, 6. Multiply this sum by the last term of the root^ and subtract the product from the new dividend. 7. If there are more terms of the root to be found, unite with the remainder the next three figures of the given number, take for a trial divisor three times the square of the part of the root now found, and proceed as before. ORTHOGRAPHIC PROJECTIONS. All Mechanical Drawing is founded on Mathematics — principally on Plane, Solid and Descriptive Geometry. We have now some knowledge of Plane and Solid Geometry. Descriptive Geometry is that branch of Mathematics which has for its object the explanation of the methods of representing, by drawings, all geometrical magni- tudes ; also, the solution of problems relating to these magnitudes in space. Drawings are so made as to present to the eye, situ- ated at a particular point, the same appearance as the magnitude or object itself, were it placed in the proper position. The representations thus made are the p7'o- jections of the object. The planes upon which these projections are usually made are the planes of projectio7i. The point at which the eye is situated is the point of sight. Definition. When the point of sight is in a perpen- dicular, drawn to the plane of projection, through any point of the drawing, and at an infinite distance from this plane, the projections are Orthographic. This result is reached, physically, if we suppose the eye to be as large as the object and placed in the perpen- dicular referred to, and at any convenient distance. Definition. When the point of sight is within a finite distance of the drawing, the projections are Sccnographic, commonly called the Perspective. 34 MECHANICAL DRAWING. The student should gain a sound knowledge of Ortho- graphic projection before attempting Perspective, hence our attention will be directed to the former for the present. In Orthographic project- ions, three planes of pro- jections ( sometimes two suffice) are used, at right angles to each other, one horizontal and the other two vertical, called respect- ively the horizontal and ver- tical planes of projection, and denoted by the letters H and V. Let a rectangular cross (Fig. 5) ,be imagined self- suspended near a lower corner of a room, or between three sheets of paper, placed in a similar position, name- ly, at right angles to each other ; the three principal dimensions, length, breadth and thickness, of the cross being each perpendicular to one of the sheets of paper— which serve as the three planes of projection. As indi- cated by the dotted lines, let perpendiculars be drawn from the principal points of the cross to each plane of projection. Let the two vertical sheets be now laid down on a table, keeping the top of the cross in line on both. Now (Fig. 4) we have the three projections of the cross on one plane in the manner in which it is proper to repre- sent them as Orthographic projections. It is easily seen that neither of these projections is a correct representation of the cross as we see it, and, also, that collectively the three projections truly represent the cross in length, breadth and thickness. Here, then, is the value of this method of representing objects. MECHANICAL DRAWING. 35 All that these projections need to make them working drawings^ are the dimensions in figures. The projection on H is called the Plan^ and the two on V and F, are called Elevations. ^ V !• 1 \ \- \^ / The representation (in Fig. 5) is in Perspective. Fig- ures I and 2 represent the whole principle in the same manner. GROUND LINE. It is to be observed that the line which (in Fig. 5) is made by the intersection of H and F, is preserved in Fig. 4. It is called the Ground Line. The representa- tion of the object above the Ground Line is called an « Elevation, and the one below is called the Plan, of the object. Returning now to Fig. 5, it will be seen that the Plan is drawn upon the plane you look down upon, and the elevations upon planes you look upon horizontally. After a little experience, the Ground Line becomes as 36 MECHANICAL DRAWING. imaginary as the Equator, but like the latter serves its purpose. ^. V. g. 9 IG, il Figure 6 represents in Plan and Elevation a triangular prism ; Fig. 7, a rectangular prism ; Fig. 8, a square pyramid ; Fig. 9, a hexagonal pyramid ; Fig. 10, a right cylinder, and Fig. 11, a cone. These names, objects and representations should be kept in mind, for they will be referred to many times. This subject of Orthographic projections is the most important of all subjects to the mechanical draughts- man. He uses it a thousand times to one of any other method of representation, and should be proportionally well acquainted with it. In all that follows in this book this acquaintance will be cultivated, until, it is hoped, any line in space may be comprehended and drawn. The subjects which immediately follow, are especially cho.sen to develop facility in making Orthographic pro- jections, as well as to gain accurate knowledge of various solids and their combinations. REVOLUTIONS. So far we have confined ourselves to projections of objects placed at right angles to the planes of projection, MECHANICAL DRAWING. 37 but it will be easily understood that in making drawings of machines or houses, we shall find many lines which are not so related to natural planes of projection already described. For an example, let us take the rectangular prism (Fig. 7) just used. To assist us in the right wa}^, we figure the corners of the front side (Fig. 12). In the plan the figures double, but make no confusion so long as we have the elevation to look at. Tip the prism, now, so that the base line 3, 4 will make an angle of 30"" with the ground line, kee^oing the plane of the face i, 2, 3, 4, parallel to the ground line. To make a second plan of the prism in this second position : As the different representations or views of an object are supposed always to be in positions perpen- dicular to each other, the corner i, for example, will be found in a perpendicular to ground line. As the prism was not inclined to the vertical plane, the desired corner will be found in a line through i of the plan, parallel to the ground line. The perpendicular from i in the second elevation, and the parallel from i in the first plan meet, making i of the second plan. In the same way seven other corners are found and the new plan finished. We have now a plan and elevation of the prism as it is inclined to the GL. Move this second plan to the right and incline it to the GL at an angle of 45 ^\ Now from the third plan draw a perpendicular, and from the second elevation draw a parallel, from corner i. The point in which these two lines meet is corner i in tlie third elevation. One by one seven other j^oints may be found, completing the elevation of the prism as it ap- pears inclined to both planes of projection. This work brings us to a point where we may attempt the problem : " To find the true length of a line." n 38 MECHANICAL DRAWING. TRUE LENGTH OF A LINE. RULE. Revolve one of the projections of the line until it is parallel to the Ground Line. Construct the other pro- jection of the line to agree with the second position. This construction will be the true length of the line. EXPLANATION. Let ;//, ;/, represent the horizontal, and ;// n the vertical, projections of -a line. To proceed according to the rule : Revolve ///, ;/, about ;;/ until // is at ;/" and ;//, ;/' is parallel to G L. In this revolution it is supposed that the angle of the line with the hori- p^Q p zontal plane is not changed. If this be so, ;/' has not changed its height above the horizontal plane, and its new position is to be found at the intersection of a parallel from n and a perpendicular from ;/", or at N. .;;/' not having moved, the true length of the liur must be ;;/' N. For practice, take the two projections of the line i, 4, in the third plan and elevation of the prism just drawn in " revolutions." Revolve the elevation of the line and construct the plan. The accuracy of the work may be tested by the first elevation of the line. PROBLEMS. 1. Find the true length of a line whose elevation ap- pears as a line 4" long, inclined 45^ to the Ground Line, and whose plan has an angle of 30^ to the G. L. 2. Draw two views of a rectangular prism 3 x 2 x t and give the true length of its diagonal. MECHANICAL DRAWING. 39 3. Draw two projections of a hexagonal pyramid, having a base inscribed in a circle whose diameter is three inches, and whose altitude is five inches. Give the true length of the centre line of one side. 4. Draw a hexagonal pyramid exactly like that of Problem 3, and give the true length of the line which joins the middle point of the middle line of a side, to the centre of the base. CONIC SECTIONS. The Conic Sections are so called because they are sec- tions of a cone. We have had a definition of a plane. Imagine two such surfaces passed through a solid, at a distance from each other of less than the thousandth part of an inch. The slice of the solid between the planes is termed a Section, It is also called a lamina^ or a slice section. Also, we often use the term Section when but one plane is passed. The Conic Sections are taken from a right cone and are, the Triangle, Circle, Ellipse, Parabola and Hyper- bola. The Triangle is a section cut from a cone by two planes passed through the apex cutting the base. A Circle is a section of a right cone cut at right angles to the axis. The Ellipse is a curved section cut at any angle to the axis, large enough to cut both sides of the cone. The Hyperbola is a curved section cut from the cone parallel to the axis and perpendicular to the base. The Parabola is a curved section cut from a right cone parallel to one of the sides, as it appears in elevation. Of these Sections, the triangle and circle are in con- stant use, in drawing ; the ellipse in frequent use ; the parabola occasionally, and the hyperbola but rarely. MECHANICAL DRAWING. 41 Fig. G. To learn, practically, what these curves are and how to get them from a cone, proceed as follows ; Make a plan and elevation of a cone, having a base four and a half inches in diameter and six and one- half inches high* In the elevation draw the elevations of the Sections, as represented. The plans of but two of the Sections are given, for more would make confusion. It will be easily understood that F G \^ the base of the hyperbola, and D E the base of the triangle. These are in their true lengths. The altitudes of all the Sections are given in the ele- vation of the cone ; that of the hyper- bola, for example, is H L. If, now, we erect a perpendicular, equal to H Z, to the middle of a base line equal to F G, the points corresponding to F, L and G will be three important points in the hyperbola. If, now, we want more points, upon which to construct the curve, we proceed as follows : Pass a plane through the cone parallel to the base, cutting the hyperbola. It will cut a circle from the cone, as N^ O. This circle is on the convex surface of the cone. As the hyperbola is also upon its surface they must intersect at A. Drawing, now, the plan of circle JV O from centre X, we find that the width of the circle, at A^ is B C. But the two curves intersect at A. Hence B C IS also the width of the hyperbola at a distance, jET A^ from the base. Set off H A on the axis of the hyperbola, from the base, and draw a line parallel to the base. On this line set off ^ S^ each side, frona the axis. These points will be points in the hyperbola. More 42 MECHANICAL DRAWING. planes must be passed, more circles drawn, and more points obtained to be accurate, especially near the top of the curve. The parabola and ellipse are obtained in the same w^ay. In the parabola care must be taken to set off R F on the axis and the perpendicular from F, that is, F Z as the width, of the parabola at height R F^ X E F> IS, the plan of the triangle. It is to be drawn " full size," but no directions are given, as the student is expected to work out for himself the major part of these Sections. A knowledge gained by zuork is retained and used, when frequently careful instruction is forgotten. As the ellipse occurs frequently in drawing and requires expensive instruments for its delineation, the following approximate method, by arcs of circles, is given : Draw major axis A B^ and half minor axis CD. Complete the rect- angle A B X a. Draw A D. Draw a b, per- pendicular to C D. Ex- tend C n to e b. With radius C D draw arc D /. With diameter A f draw semicircle A e f. Draw radius g h. T Lay off h /on c b to / F^^- H. Through j draw an arc with b as centre. From A as centre and radius C e draw an arc, cutting arc through/ at k. Through k draw bl, and through m draw ko : in is a centre for A o, k centres for o /, and b, for ID. A o I D \^ one-quarter of the ellipse. The parabola may be described as follows : Suppose the parabola to have a base CD and an alti- tude A B. Extend A B to E, making B E equal to BA. MECHANICAL DRAWING. 43 Draw E C and E D. Divide C E and E D into any number of equal parts, numbering one from the top and the other from the bottom. Join i, i?^ — 2, 2, — 3, 3, &c. These lines will all be tangents to the parabola. With a French curve the para- bola may be drawn. In drawing an irregular curve with a French curve in this way, be sure that the instrument touches three of the points through which the curve is to be drawn. Intersections and Developments. As the memory will easily recall, there are many lines seen on a manufactured article, or on a drawing of it, that are not lines of any individual part of the article, but lines that occur where two or more forms intersect or join each other. Such lines are called intersection lines. H Intersections is the name given to that part of geometrical drawing that treats of the in- tersection lines and their cor- rect delineation. It v^ill be seen at once that a thorough knowledge of geo- -metrical solids will be neces- sary to the student who desires to take a full course in inter- sections. On the other hand, we all have a fair understanding of many geometrical forms that meet our eyes in com- bination every day, and with these we will make a beginning. Let A B^ C D^ represent two projections of a hexagonal prism, and -fi", 3, 6, ZT, i, 4, two projections of a hex- agonal pyramid, passing through, or intersecting the prism. To learn what the intersection line will be, proceed as follows : MECHANICAL DRAWING. 45 First, number the angles of the base of the pyramid so that the position of a line in both projections maybe easily noted. The line i-E intersects the prism at Z, whose elevation is at K, on line i H. The line 6-E intersects the prism at J/, whose elevation is at JV j hence the in- tersection of the face Zr,-i,-6, of the pyramid, with the prism, is the line K JV. ^-E is directly under 6-E^ so its elevation will be, at O. By similar reasoning, the intersecting point on /\-H is at P, and K-N-O-P is the intersection li7te. It will easily be seen that the similar line on the other side of the prism is constructed in the same way. It will be good practice for the student to copy the plan of the combination, with the line \E at an angle of 15^ with the Ground Line, and then construct the elevation. We will suppose a triangular prism passed through a right cylinder (Fig. 13). First make a plan and two elevations of the cylinder* In the middle of the elevation which is projected from the plan, draw the end elevation of the prism. Now draw the plan, assuming that the prism projects from the cylinder at either side, and that centre line of prism and cylinder coincide ii;i Z>, E^ C. 46 MECHANICAL DRAWING. In the side elevation it is evident that there will be an intersection line ; that none appears in the plan and end elevation is evident because in tlie plan it coincides w^ith the outline of the cylinder, and in the end elevation it coincides with the outline of the prism. We can lay out the top and bottom lines and ends, on the side elevation, from the other views. The point where tlie lower line of the prism pierces the cylinder is found as follow^s : In the plan draw the line Z, 7V^, perpendicular to the diameter Z, O^ through the point J/, where the prism pierces the cylinder. Lay off the distance Z, O from X to F, the latter being the desired point. The top point of the intersection line is on the cir- cumference. To find other points in the intersection line, pass the planes i and 2 and proceed in the same w^ay as in the case of the bottom line. It is now required to develop the surface of the cylin- der. In Mechanical Drawing this means to draw an equivalent plane figure. This may be illustrated by fitting the surface of the cylinder with a covering of paper. When this paper is unrolled and spread on a table, we have a surface equivalent to that of the cylin- der. It is now required to outline, in this develop7nent of the cylinder, the hole that the prism makes. Let A^ H (Fig. 13) represent part of the development of the cylinder. Let a perpendicular at Z>, represent the axis of the prism. Lay ofi' the arc Z, J/, developed as a straight line on each side of the axis, making the line R^ P. The third corner is found on the axis at the alti- tude of the prism from R^ P. Intermediate points are laid oft^ from the axis on traces of the planes i and 2 in the same manner. MECHANICAL DRAWING. 47 We have had the intersection of plane surfaces with plane surfaces, plane surfaces with curved surfaces, and now we have curved with curved surfaces. Fig. 13 A illustrates the intersection of two cylinders. The method of finding the intersection line is as fol- lows : In the plan pass the plane i' through the cylin- ders parallel to both axes. It will cut a rectangle from each. ^/ is one end of the rectangle cut from the smaller cylinder, and c f o- // is the elevation of the rectangle in its true size. To find the rectangle cut from the larger cylinder, draw a semicircle ABC, show- ing one-half of the ii\\(\ of the c\ Under. The trace of 48 MECHANICAL DRAWING. the plane across it is seen in the line 3, 4, which, of course, is one-lialf the end of the rectangle. This half laid off each way from D^ in the elevation, at a a^ shows VIS that the rectangle \% a a a a. We see that these rect- angles intersect at X and three other points, which must be intersection points, that is, points which are in both solids, for they were cut by one plane, y and other points are found by the same process. To find point Z, pass a plane through A tangent to smaller cylinder. The rectangle in the smaller cylinder is reduced to a single line, and the rectangle of the larger cylinder in- tersects it at Z. The curved intersection line G xyz may now be drawn by free-hand, by a template whittled from thin wood, or by a French curve. Duplicated three times, it will complete the elevation as seen in the plate. It is now required to develop the cylindrical sur- face of the larger cylinder. The semicircle ABC^ Fig. 13 A, is the basis of our work. Set a pair of spac- ing dividers closely and step round the curve. Suppose that it is 20 steps. In Fig. 13 B these 20 steps will make the straight line T B^ which double.d gives the full development B B. TIV is, of course, the length of the cylinder. To find tlie hole made in this cylinder, lay off H G as T G^ ^7 A as 7^7? ^nd by as 7^8. Planes i, 2, in MECHANICAL DRAWING. 49 Fig. 13 B, are then in same position as in Fig. 13 A. With the dividers space o^ HJV and transfer it to V S at 9, H O 2X 10, and H P ?it z. Planes i' 2' through 9 and 10 will then have the same position on the surface as in Fig. 13 A. The points x and jf v^ill be at the intersec- tion of the traces of the planes, and G and z on the cen- tre lines, and Gxyz will be the developed curve, and may be duplicated to complete the figure. It will not be a true ellipse, departing from that curve as shown at Q. In all these intersections there is but one method used — that of passing a plane through the solids as they appear in one view and then constructing, in another view, the plane surfaces cut out of the solids, from vs^hence the intersections of the planes give points in the intersections of the solids. Once have a clear idea of what a plane is, and how it should be passed, and the subject of intersections becomes easy. A hint of the method of passing the plane : Always cut the solids by the planes so as to get the simplest pos- sible figures. In making developments the ability to get the true length of any line, shown by two or more projections, will save time and assist toward accuracy. In making working drawings, or other drawings of combined solids, these intersection lines constantly ap- pear, and facility in their representation is a necessity to the draughtsman. The workman also is greatly bene- fited by a knowledge of them, which, in short, are parts of the forms which he handles every day. ISOMETRIC PROJECTIONS. Prof. Farish, of Cambridge, England, in 1820, gave the term Isometrical Perspective to a particular projec- tion, which represents a cube from a position where J:hiee sides appear as equal rhombuses. The term Isometric means equal measure. Let three straight lines be drawn, intersecting in a common point and perpendicular to each other, two of them being horizontal aud the third vertical — like the three adjacent edges of a cube. Then let a fourth straight line be drawn through the same point, making equal angles with the first three, as the diagonal of a cube. If, now, a plane be passed per- pendicular to this fourth line, and the straight lines and other objects be orthographically projected upon it, the projections are called Isonetric. The three straight lines first drawn are the co-ordinate axes ; and the planes of these, taken two and two, are the co-ordinate planes. The common point is the origin. The fourth line is the Iso7netric Axis. Since the co-ordinate axes make equal angles with each other, and with the plane of projection, it is evi- dent that their projections will make equal angles with •each other, two and two, that is, angles of i3o°. Hence, MECHANICAL DRAWING. 5r (Fig. 14,) if any three straight lines, as Ax, Ay and Az^ be drawn through the point A, making with each other angles of 120°, these may be taken as the projections of the co-ordinate axes, and are the directrices of the drawing. It is further evident, that if any equal distances be taken on the co-ordinate axes, or on lines parallel to either of them, their projections will be equal to each other, since each projection will be equal to the distance itself into the cosine of the angle of inclination of the axes with the plane of projection. The angle which the diagonal of a cube makes with^ either adjacent edge is known to be 54^ 44'; therefore, the angle which either edge, or either of the co-ordinate axes, makes with the plane of projection will be the complement of this angle, viz., 35° 16'. If a scale of equal parts be constructed, the unit of the scale being the projection of any definite part of either co-ordinate axis, as one inch, or one foot, will be one inch multiplied by the natural cosine of 35^ 16'. We may from this scale determine the true length of the isometric projection of any given portion of either of" the co-ordinate axes, or of lines parallel to them, by taking from the scale the same number of units as the number of inches or feet in the given distance. Con- versely, the true length of any line in space may be found by applying its projection to the Isometric scale, and taking the same number of inches or feet, as the number of parts covered on the scale. Or: The isometrical length of a line, is the true length of a line multiplied by the natural cosine of 3^^ i6\ Now this cosine is .816; hence, if we multiply the true length of a line — say one inch long— by .Sr6, we will get the isometrical length of the line, that is, 52 MECHANICAL DRAWING. one inch multiplied by .8x6, which equals .8x6 (thou- sandths) of an inch, or about (yf) thirteen sixteenths of an inch. Now, if we have a 35^ 16' triangle, and a scale thir- teen-sixteenths full size, we have the special tools neces- sary to make an isometrical drawing. A much easier way is generally adopted. It is customary to use a 30^ triangle in place of a 35° 16' triangle, and a full size scale in place of a |f scale. Since in most of the frame work connected with ma- chinery, and in various kinds of buildings, the principal lines to be represented occupy a position similar to the co-ordinate axes, namely, perpendicular to each other^ one system being vertical, and two others horizontal, the Isometric projection is used to great advantage in their representation. A still greater advantage arises from the fact that in a drawing thus made, all lines parallel to the directrices are constructed on a full size scale. If the isometrical projection of a point be required^ the following operation is sufficient : Thus in Fig. 14, let A Z, A V and A X he the direc- trices, A being the projection of the origin. On A^ F, lay o?i A, F equal to the distance of the point from the co-ordinate plane X, Z. Through P draw P, M' parallel to A, Z, and make it equal to the distance of the point from the plane X Y. MECHANICAL DRAWING. 53 Through M^ draw M\ M parallel to AX, and make it equal to the third given distance, and M will be the required projection. The projection of any straight line parallel to either of the co-ordinate axes may be constructed by finding, as above, the projection of one of its points, and drawing through this, a line parallel to. the proper directrix. If the line is parallel to neither of the axes, the pro- jections of its ends may be found, as above, and joined by a straight line, which will be the projection required. The projection of curves may be constructed by find- ing a sufficient number of the projections of their points. PROBLEM. To construct the Isometric projection of a cube : Let the origin be taken at one of the upper corners of the cube, the base being horizontal, and let AX^ A Y and AZ (Fig. 14), being the directrices. From A, on the directrices, lay ofl^ the distances AX\ AZ and A F, each equal to the length of the edge of the cube. These lines will be the projections of the three edges of the cube which intersect at A, Through X, Y and Z draw Xe, Xg, Ye, Yc, Zc and Zg, parallel to the directrices, completing the three equal rhombuses A, X, e,Y, etc. These will be the projections of the three faces of the cube— which are seen — and the representation will be complete. It must be very carefully noted that only lines of 30^ with the horizon, and perpendicular lines, may be meas- ured on an isometrical drawing. All other lines arc more or less distorted. The following is a good method for drawing *5 54 MECHANICAL DRAWING. AN ISOMETRIC CIRCLE. Fig. K. Draw ABCD with 30^ triangle. Bisect AD at /. Erect perpendicular HX. With radius HO describe arc OX. With radius OX describe arc EX. OE is one- half longer axis. Bisect OE at G, Through G draw JN. With radius NJ, draw arc JK. Trisect OE at /andi^. With radius FE describe arc EL. Lay off FE on JG from / to M. With radius JH, centre AI, describe arc cutting EL at L. Through F draw LFFy with centre F describe arc LJ, completing one-quarter of the ellipse. Transfer centres and complete. Isometrical projection is especially valuable to the architectural draughtsman, as it explains many con- structions that could hardly be done by plans, elevations and sections, and it also unites with pictorial represen- tation, the applicability of a scale. For drawings for the Patent Office it is a convenient and easy method, combining the requisites of many projections ; but as a drawing of what could absolutely be seen by the eye, it is not truthful, and, therefore, when pictorial illustration o?ily is requisite, the drawing should be made in linear perspective. LINEAR PERSPECTIVE. From the definition of Orthographic projections which ^ve have had, we understand that they can never present to the eye of an observer a perfectly natural appearance, and hence this mode of representation is used only in drawings made for the purposes of mechanical or archi- tectural constructions. Whenever an accurate picture of an object is desired, the scenographic method must be used, and the point of sight chosen where it would naturally be placed in looking at the object repre- sented. That application of the principles of Descrip- tive Geometry which has for its object the accurate representation, upon a single plane, of the details of the form and the principal lines of a body, is called Linear Perspective. The art by which a proper coloring is given to all parts of the representation, is called Aerial Perspective. This, properly, forms no part of a Course like this, and so is left entirely to the taste and skill of the artist. The surface upon which the representation of a body is made, is called the plane of the picture j the plane of the picture is usually taken between the object to be represented and \\\q. point of sight, in order that the draw- ing may be of smaller dimensions than the object. It is also taken vertical, as in this position it will, generalh', be parallel to many of the important lines of the object. The Orthographic projection of the point of sight on 56 MECHANICAL DRAWING.* the plane of the picture, is called the principal point of tJic picture ; and a horizontal line through this point and in the plane of the picture, is the horizon of the picture. Observation has made it evident that the greatest angle under wdiich one or more objects can be distinctly seen is one of 90*^. If between the object and the eye there be interposed a transparent plane, (such as one of glass) the intersections of the visual rays (the visual rays are those reflected rays of light from the object to the eye which make it visible,) with this plane are termed per- spectives of the points from which the rays come. Referring now to Plate A, we use two horizontal lines in drawing the Perspective of an object. One is the intersection of the picture plane with the ground, and is called the Ground Line {G. L.) ; the other is the Horizon Line^ (already mentioned) and is located par- allel with, and five feet above the G. L. This relation of these two lines never changes as to direction, but always changes to conform with the scale of any par- ticular picture. It is supposed to be at the same height from the ground as the eye of the observer, and the Orthographic projection of the point of sight becomes the principal point of the picture {F, P.), The point of sight, or Point of the Observer {P. O.)^ is selected at will, in a perpendicular to the plane of the picture at the P. P. The point P. O. being selected, its distance from P. P. is laid oft' on the Horizon Li fie each side of P P. These points on the horizon line are termed van- ishing points of diagonals or Distance Points. If lines be drawn from these points to P. O., the included angle will be found to measure 90^, All lines drawn through the object, at an angle of 45^ with the picture plane, vanish at the distance points. All perpendiculars to the picture plane, through the object, vanish at the principal point. MECHANICAL DRAWING. 57 PROBLEM. To construct the perspective of a Square. Place the Orthographic plan of the square H E A F as far in front of' the G. L. as it is supposed to be behind it. First make a perspective of the diagonal F E. To do this, continue F E to O. The perspective of this diagonal is — as we have already learned — O E>. To get the per- spective of the perpendicular AE, extend it to B. Its perspective is B P. Consequently, the perspective of E^ which is at the intersection oi F O and A B, is at X, the intersection of the perspectives oi F O and AB. From the plate it is seen that the perspectives of the other cor- ners of the square are obtained in the same way. Hence, the RULE. The perspective of a point is found at the intersection of the perspectives of a diagonal and a perpetidicular through the point. It should be noticed that the perspective of the diagonal F E \?> the diagonal X Z. 58 MECHANICAL DRAWING. The application of this simple rule enables us to con- struct the perspective of any line lying on the ground, as, for example, the circle A B C D E F G H \^^ in per- spective, the ellipse i 2345678. The process of con- structing the perspective circle is simply that outlined in the rule, as will be easily understood from the plate. In making perspectives of solids, we use the rule given with Plate A. To find. the perspective of a point not on the ground, proceed as follows : The points of intersection with the G. L. of a diagonal and a perpendicular through the point, are transferred to a horizontal line located r:S far above the G. L. as the point is above the ground. From these last points, the perspective of the point is constructed as it would be from the G. L. In Plate B, the perspective of a point B is found seven MECHANICAL DRAWING. 59 feet from the ground, by transferring the points x^ y, on the G. L. to line O F, seven feet from the ground, and proceeding as in constructing point 2. The representa- tion is of a square prism seven feet high. This prism and the square in Plate A are represented in parallel perspective — one side being parallel to the G L. 6o MECHANICAL DRAWING. The second representation on Plate B — that of a prism two feet square and three feet long, is in angular per- spective. The method used is that given in the descrip- tion of Plate A. The prism is placed in an awkward position purposely, to show that the rule given is ample for all conditions. In Plate C, we find a new kind of line, namely, one inclined to both the picture plane and the ground. Its perspective is found by our one rule. It is necessary to make two elevations first, from which the heights of points are taken. Architects use this method. By using two distance points and placing the plan at an angle of 45° with the G. Z., much work is saved, e. g., w^hen A i;s located, half the work of finding other eave corners is done. Note the method of finding centres of v^all surfaces by diagonals. Plate C shows the proper method of procedure in con- structing the perspective of any object. By it, the great object of properly placing the thing to be represented, is gained, and the representation appears natural, though much more room is required to perform the operations. WORKING DRAWINGS. It is the opinion of the author that a fair understand- ing of the principles touched upon in previous chapters is necessary to enable an ordinary draughtsman to do his work ; and, conversely, having such understanding, it will be easy for him to learn to make good working drawings. A working drawing is a drawing made for workmen to use in making the thing drawn. A working drawing should represent clearly the form, material and dime?isions of the thing to be constructed. The design must be made perfectly correct, because from it are taken the dimensions for the working drawing. Dimensions should be taken several times before being put on a drawing, and the workman required to work from the dimensions. This requirement is necessary, because the paper on which a drawing is made will shrink and swell with the changes in the weather, making it un- trustworthy as a guide to correct figures. It will be noted from what has been said, that the most important part of a working drawing is the dimensions. A sheet of working drawings should always bear a title, if possible > in the lower right hand corner. The title should answer with utmost brevity the following questions: What is it? What scale? When and by whom ? Some manufacturers require, in addition, the name of the firm and the number of sheets of working drawings necessary to represent all parts of the machine. 6 62 MECHANICAL DRAWING. Figure 15 is a working drawing for a small jour- nal or box. A few points may be made from it. The figures are placed where they would nat- ^1"^ urally be looked for.— They are made much larger and clearer than figures would be on an ordinary engraving. A workman desires to see these figures at a glance, not to be obliged to hunt for them. They are placed in a way to obscure the outline of the box the least possi- ble. The lines drawn for the index points are usually in red and do not obscure the real drawing as much as the black ones in this drawing do. This matter of putting dimensions upon working drawings seems to be the chief stumbling block of inexperienced draughtsmen. A knowledge of how the thing is made is the very greatest assistance in making a working drawing. In running out the red dimension lines, care should be taken that they run perpendicularly to the surface or line whose dimension is to be given. Index points and figures are always in black ink. Small dimensions are generally clearer if indicated as in the dimension ^" in Fig. 15. The distances between finished surfaces are important and should be put on first. There are various ways of indicating a surface that is to be finished, one of the simplest being writing the italic /across the line. Dis- tances between or from centre lines are generally impor- tant though often forgotten. The matter of centre lines itself is fundamental in making a working drawing. MECHANICAL DRAWING. 63 The centre lines of each view should be drawn first, and the remainder of the drawing laid out from them. A tapped hole should be marked T, as well as the thread indicated. In making three views of a curved line, plot the third from the other two rather than guess at it, for incorrect drawings are pretty sure of being expensive — in reputation to the draughtsman, and in money to the manufacturer. In the detail drawing of the Head of a Screw Slotting Machine which accompanies this subject some of these points may be noted. -3H> o\i \ K^ HEAD SCREW SLOTTER l^z^lFt. /2-2G-90 WSL S iU S^ Fig. 16. The plan is turned improperly that the curves may be in full lines. It should, according to rule, be inverted- Points are taken on the curves in the two elevations and are transferred to the plan, where, by chance, they fall nearly into an arc of a circle. The student's drawing 64 MECHANICAL DRAWING. from which this illustration was taken had an o-gee curve in place of this quarter circle. The centres of the arcs, forming the curves of the sides of the head, should always be given. Almost any curve may be matched up with arcs of circles, and irregular or French curves should not be used for outlines, for the curves are difficult of duplication by the pattern maker, and so lose time and money, through the draughtsman, where it should be saved. On the contrary, where radii are given, the pattern maker may easily get his outlines, whatever the scale of the drawing. It will be noticed that some of the lines of the draw- ing are heavier than others. These heavier lines are supposed to cast shadows from the direction of 45^ over the left shoulder of the draughtsman. In the plan this light " goes off" from the solid above and to the right of it, and on the elevation, at the right and bottom. In details the plan is generally shaded as are the other views because the objects may be placed in one position as well as another. It is not well, how- ever, to always follow the rigid rule of shading right hand and bottom lines. For example, a cylinder, if standing on end, should have the bottom line shaded, but if it were lying down, the bottom line should not be shaded, for the darkest line on a cylinder is some dis- tance from the edge. Much more sound instruction might be written on the art of making working drawings, but this is only an "Outline Course" in drawing, and this subject is dis- missed with this parting warning : Do not think of the state of the drawing after it has been in the shop a week, but make it as neatly, clearly and distinctly as you can. MECHANICAL DRAWING. 65 »6 66 MECHANICAL DRAWING. MECHANICAL DRAWING. 67 DESIGN Having learned to make working drawings, there is a natural desire to take the next step, and learn to design the buildings or machines that require the explanatory working drawing. The designing of cams and gearing will be explained, and a finished design is shown of a machine designed to roll hot bar steel into finished forg- ings for the market. While it is not the custom to shade such designs, the present one is so finished to give the student some instruction in that art. In the subjects just mentioned much will be learned, incidentally, of designs, in a class of facts that soon be- come common-place to the draughtsman or designer, but are as necessary to him as his paper. A good working table of strength of materials is given, also diagrams for ascertaining strength of shaft- ing and belting, together with data useful in connection with these tables and diagrams ; but it is beyond the scope of this work to teach a subject that requires origi- nal thought and oftentimes original research. Prof. R. H. Thurston says : * " The work of design- ing metal parts of machinery involves the intelligent consideration of the cheapest and most satisfactory methods of moulding those which are to be cast, as * Materials of Engineering, Part II., Article 144. MECHANICAL DRAWING. 69 well as of forging parts made of wrought iron and steeL The pattern maker must also know how to prepare the pattern so as to avoid the difficulties frequently met with in moulding. The moulder is required to know how to mould the piece in order to secure sound castings ; and the founder must understand the mixing and melting of metals in such a manner as will give castings of the re- quired quality. The engineer should know what forms can be cheaply made in cast metal, and what cannot be cast without difficulty, or without liability to come from the mould unsound. He should be able to instruct the pattern maker in regard to the form to be given the pat- tern in order to make the moulder's work easy and satis- factory, to tell the moulder how to mould the pattern, with what to ffil his flask, how to introduce the molten metal, and to provide for the escape of air, gas, and vapor, and he should be able to specify to the founder the brands and mixtures of iron to be chosen." There is one suggestion to be made to the beginner in designing which is of supreme importance to him, namely : that the moving parts ^ not the frame, is the machine— in the same sense that the spirit of a man, not his body, is the man. It is well, generally, to- design the moving parts of the machine, and re-design them, until each part, the relation of each to each, and all as a unit, is satisfactory. After this, the frame may be designed to support and withstand the stresses or shocks of the moving parts. Oftentimes it will require skill to do this, but once done it is finished, whereas a moving part, shaped awkwardly or weakly, to accom- modate a frame, will be a continual source of annoyance^ CAMS The designing of Cams is taken up here because it is easy of mastery, interesting beyond nearly all other simple elements of machinery, and useful far beyond existing engineering practice. Definition. (Worcester.) *'Cam Wheel — Awheel formed so as to move eccentrically, and produce a reciprocating and interrupted motion in some other part of machinery connected with it." This definition, though fairly good, gives but a misty idea of the form or construction of a Cam. Generally speaking, cam motions produce the best irregular posi- tive motions known. They dispense w^ith expensive link or lever w^ork, and may easily be so designed, made and applied, as to produce required motions to the last degree of accuracy. Cams may be classified as follows : into Wipers and Frog Cams, Face Cams, Square Cams and Edge Cams. A Wiper is shown in Fig. i. It is used for operations where the power is grad- ually stored and suddenly expended. Trip hammers are often operated by such cams. Its object is gained by various proportions, but always with the same idea — a gradual rise, and sudden drop — the breadth of face and length of drop being proportioned to the work to be done. MECHANICAL DRAWING. 7I FACE CAMS. When a Face Cam is to be used in a machine its de- sign is usually left until the other moving parts and the points of support have been designed and located. If possible, the cam is placed on a shaft that is useful in the machine for some other operation. If this is not possible, then shafts are put in for the cam and cam lever, and the cam shaft given a positive revolution by gearing from the prime mover of the machine, and the cam designed as follows : First, about the centre F, (Fig. 2) draw a circle to represent the shaft and then one to show the cam hub. Outside these draw a third which shall pass through the centre of the cam-roll at its inside position. This position should be selected as near as possible to the cam hub, to save power and space. From the same centre draw a fourth circle through the centre of oscillation of the cam lever (X) . This last circle is divided into any convenient number of parts. That of Fig. 2 is divided into 33 but 32 would be better. From the centre selected for the lever, with a radius equal to its length — which is from centre of lever shaft to centre of cam roll — draw an arc from near cam centre to the circle through centre of lever shaft. Locate the cam roll at the intersection of this arc with the third circle. This radial arc, through the first position of the cam roll, is numbered i. It does not, necessarily, pass through the centre of the cam, though that is the best arrangement. Now, through each one of the remain- ing 32 dividing points, draw arcs like number i, with centres on circle through X. On these cd^cs the succes- sive positions of the eentre of the cam roll are platted. If it is desired to have a throw of two inches in one- quarter revolution, then, with 32 divisions, a position 72 MECHANICAL DRAWING. one-quarter inch farther from the centre of the cam, is taken on 8 successive arcs, when the required throw of two inches will have been accomplished. If, now, a ^' rest'* is required, the points on successive arcs will be at the same distance from the centre of the cam shaft. From each one of the points as centres, just mentioned, a circle is drawn of the size of the cam roll, and lines drawn tangent to all these circles will represent the path in which the roll is to move. The cam of Fig. 2 has one-half revolution rest and then does its work in one- eleventh of a revolution. The rest is represented by making the path concentric with the cam shaft. In MECHANICAL DRAWING. 73 turning from this part of the path — just after arc 15 — several extra positions of the cam roll are platted, and again at the end of the "throw" several more. The object of this is to ascertain whether or not we are try- ing to change the direction of the path too abruptly. As long as each of the circles representing the roll shows a part of its circumference beyond the others, on tlie in- side line of the path, the curve is practicable ; provided ^ also, that the radius of curvature of the inside side of the path is less than that of the roll. These conditions must be fulfilled or the roll will not move round the curve. There are no very sharp curves in this cam, but those at 15 and 19 will illustrate the rule. From 20 to 22 the " return " is rapid— to draw 7" from its work — and from 22 to I the return is made gradually and easily. A cam like this, with bell crank and " tucker" T^ may be used in a paper folding machine. Frtr^ rig.2B ^y///////y////.^///////A Fig. 2 B shows something further of the construction of the cam and its lever. It is a section of the cam and roll on a line perpendicular to Z X with T left out and the bell-crank turned round, to show full length of L X, The lever should be fastened to its shaft by two set- screws, and the hub B should be three times as long as the cam roll. Fig. 2 B shows the proper form of stud C for the cam roll. Hub A should be of same length as hub B. This cam is designed in a wheel, but it should be un- derstood that only so much of it as will serve to hold the 7 74 MECHAN-ICA.L DRAWING. roll in its place and the cam on the shaft is necessary. It may be in form of a wheel, to prevent entanglements with clothing or parts of machinery. On arc 19 the thickness of metal outside the path is supposed to be of just sufficient strength to do its work. This makes the outside boundary of the cam. As seen in Fig. 2 B the luheel is a simple disc and rim. A '^frog " ca7?i is designed in the same way as a face cam. If all of this cam, outside the inside line of the path were cut away, the remaining — inner — part would be a frog cam. The cam roll for a frog cam must be kept against the cam by a spring or weight. The principle of designing face and frog cams, 7i'//// the lever ^ is not generally understood, but from the fore- going explanations it may be seen : 1st. That if these cams were designed on radial lines, and the cam roll oscillated on a lever, an allow^- ance would have to be made for the difference of angu- lar position. To make this perfectly clear, design a cam of a quarter rest, a quarter throw, a quarter rest and a quarter return, by the method just described and also by radial lines. In the former the '' quarters" will not be equal ones— of 90^ — but in the latter the quarters will be equal. The conclusion must be that if the latter method of design is to be used the cam roll must move in a radial line, not in an arc of a circle. The advantage of the former method is that most cams do actuate a lever and the method plats accurately each successive position of its cam roll. 2d. That as long as L X remains of same length, and X at same distance from F, the cam will be the same — to accomplish a given motion — whether A' is at the right or left, above or below Y. MECHANICAL DRAWING. 75 3d. That if length of Z X is changed, the cam — to produce same results— will have to be. This is true whether point X is moved or not. 4th. That, retaining X in its relative position to F, lengthening L X increases the power and decreases the speed attained at T, ^nd shortening L X produces the contrary effects. In both these changes the shape of the cam — to produce a given motion— would be of different form. There is no better practice for the beginner than to take the same requirements and design the cam under differing dimensions and relations of shafts. He will then learn of the adaptability of the method just de- scribed to various trying conditions and requirements. SQUARE CAM. Figures 3 and 4 are side and section views of a "Square Cam" and its immediate attachment, the '^ Yoke." This cam has often been used in sewing ma- chines and boot and shoe machinery. In the position shown the top and bottom parts are arcs of circles hav- ing the centre of the shaft for a centre, and are, in canr language, " rests." This is an unfortunate term here, for this cam has no rest nor gives any to its attachments. The essentials in drawing this cam are but two. Firsty there is a point on e^ch of the diagonals E F^ I H^ which is a centre for two arcs of circles, each forming a part of the outline of the cam. Second, the sum of the radii of these two arcs is equal to the sum of the radii of the two arcs of '' rest." Great care must be taken in drawing this cam to get the outline perfectly. The cam K^ Fig. 3, will, by means of its yoke, pivoted at N^ de- scribe with the point the " square " figure H' E I' F. This result is not obtained so simply by any other means. To lay out the cam, begin as in the case of the Face 76 MECHANICAL DRAWING. Cam ; draw a circle to represent the cam shaft. Through its centre draw two lines at right angles to each other^ preferably at angles of 45^, which will make four divis- ions of the outline of the cam. In the lowest of these divisions draw a quarter circle for a quarter of the cam^ making its radius such as will give the cam sufficient strength. In the opposite quarter draw another quarter circle, the difference between the raclii of the two quarter circles being the ^'' throw'' of the cam. Two horizontal lines may now be drawn tangent to the quarter circles, and two perpendicular ones also, 7?iaking a perfect square divided into two equal rectangles by a perpendicular through the ceritre of the shaft. The remainder of the outline of the cam may now be completed, with the help of previous statements, bearing in mind that each of the remaining arcs, of the outline MECHANICAL DRAWING. 77 of the cam, must be tangent to the perpendicular sides of the square. The yoke of Fig. 3 is similar to one designed for a' nailing machine, the stress on it being chiefly in a ver- tical direction. If there were to be any considerable stress sidewise, the yoke should be strengthened about the fulcrum N. Fig. 4 is a section cut by a vertical plane through the centre of the shaft of Fig. 3. It will be easily under- stood, with the exception of the " sliding " block about the stud N^ which serves as a fulcrum. It is a square block bored to fit the stud. A better design for the stud would be one having a larger diameter through the block and washers, and then cut down for the threaded portion, at each end, and fitted with a common hex- agonal nut on the face. EDGE CAMS. Figures 5, 6 and 7 show an Edge Cam and methods used in its design. Fig. 5 shows it in connection with roll and lever. Fig. 6 shows the ordinary division of the circumference into 8 divisions of 45"' each. Fig. 7 shows the development of the circumference with the path traced upon it. The largest diameter of the roll is used in tracing the path. It is assumed that three throws and three rests are required. The first quarter, i to 3, is a rest in the middle of the lever angle. In the next 8th revolution, (3 to 4), the lever takes its extreme " back" position. The quarter, 4 to 6, is a rest in this extreme position. In the next quarter, 6 to 8, the work is supposed to be executed. The whole throw forward is eftected at an easy angle in one-quarter revolution. One sixty-fourth rest is given to hold the work, and seven sixty-fourths revolution re- turns the lever to its first position. 78 MECHANICAL DRAWING. It Will be noticed that the cam roll for this cam is conical. This is neces- sary, because that part of the roll near- est A must travel a much greater dis- tance than the end nearest the centre of the cam. Now that your attention is called to the fact, you will readily com- prehend this necessity ; but there are many cams of this kind running with rolls that are right cylinders simply be- cause their designers forgot this. In Fig. ^ the cam lever is shown at point I , with the centre line of the lever pass- ing through the centre of the throws of the cam. From 3 to 4, the roll drops seven-sixteenths of an inch, and from 4 to 6, the lever makes an angle with the centre line of the cam, as is shown by the dotted line drawn from A' to the correct position of the roll. In laying out the cui^ves of the cam path at the ends of the "throws," much care must be taken not to have sharp an- gles. The cen- tre line is laid out — that is — the centre line of movement. The changing of this line from a rest to a throw, or the reverse, is always made in the arc / / k V \ v6 \ n ^ ■i'tl > » »0 V 5^ \ \ i i_ii 1 MECHANICAL DRAWING. 79 of a circle. The radius of this arc must be at least one- eighth of an inch larger than the largest radius of the cam roll. This ensures a round corner for the path. It is better, where possible, to make this arc one-quarter of an inch larger than the radius of the roll. The change in direction of this centre line can never be over 45° for forward or working angles. 30^ angles should not be exceeded. From this description it will be seen that the develop- ment is the important feature in designing this cam. The larger the diameter of the cam the easier the angles. This and all other cams should be keyed or pinned to the shaft. The cam levers for all cams should be fasten- ed with set-screws, and the correct timing of the ma- chine effected by slipping the lever on its shaft until the proper position is reached. LEADERS. When cams are to be cut, it is best to make the for- mer — called the "leader" in this case — much larger than the cam, so as to avoid inaccuracies and unsteady lines in the cam. The cam blank is also marked. This is accomplished by duplicating the development on a piece of thin tin, which is then fastened to the blank and the outline prick-punched through. Sometimes the devel* opment is on strong manilla paper, which is wrapped on the cam and pricked through. Edge Cams are sometimes, and Face Cams generally^ cast with the path in them. Face Cam patterns and castings are sometimes made so nicely that a cold-chisel and file will in a very few minutes smooth the casting sufficient for the roll. Such cams wear a long time. Usually cams are cut from leaders of double their size. These leaders are disks of cast iron, half an inch thick, 8o MECHANICAL DRAWING. turned smoothly and emery-polished, so that the scriber lines may be seen easily. The leader for such a cam as No. 5, for example, would be a plain disk eight inches in diameter with the throw of seven-eighths of an inch and two throws of seven-sixteenths of an inch laid off on its edge. This, when ready for use, would resemble the cam but slightly, although having the same rises and falls on its periphery as the centre of the cam roll will make in traveling its path. The true cam in the cases of the Face Cam and the Edge Cam is the curve traced by the centre of the roll. GEARING. In this chapter the student may find : First, a clear indication of methods of drawing three forms of gear teeth. Second, a diagram and its description by which gear wheels may be proportioned ; and, third, a few practical hints. Several standard works have been con- sulted, especially the treatise published by the Brown & Sharpe Mfg. Co., from which are taken, by permission, methods for delineating single-curve and involute forms of teeth, and also useful formulas and explanations. While a larger portion of the space is used in describing methods used in drawing teeth than in explanation of the designing of gears, the student should direct his chief thought to the latter work, since there are several manufacturers who make better gear teeth than any in- experienced man can hope to. Let two cylinders, mounted on parallel axes, have their convex surfaces in contact. If now we turn one cylinder, the adhesion of its surface to the surface of the other will make that turn also. The surfaces touching each other, if there is no slip, will evidently move through the same distance in a given time. This sur- face speed is called linear velocity. Linear velocity is the distance a point moves in a given direction in a given time. These cylinders in turning about their axes also pass through angles whose vertices are at the axes 82 MECHANICAL DRAWING. of the cylinders. The angular distance passed through in a given time is called angiUar velocity. If one cylin- der is twice as large as the other, the smaller will make two turns while the larger makes one, but the linear velocity of the cylinders is equal. This combination would be a very useful one in mechanism if we could be sure that the cylinders would not slip on each other. Let grooves be cut on the circumferences of the cyl- inders of a size equal to the spaces between the grooves, and the material taken out in making the groove placed on the spaces between the grooves. The spaces are called lands^ and the parts placed upon them addenda. A land and its addendum is called a tooth. A toothed cylinder is called a gear. Two or more gears with teeth interlocking are called a train. The circumference of the cylinders, without teeth, is called the pitch circle. This circle exists geometrically in every gear and is called the pitch circle^ or the primitive circle. In the study of gear wheels it is the problem to shape the teeth that the pitch circles will just roll on each other without slipping. The groove between two teeth is called a space. In cut gears the width of space and thickness of tooth at pitch line are equal. The circular pitch is the distance measured on the pitch line, or pitch circle, which embraces a tooth and a space. In cast gears the tooth is from .46 to .48 of the circular pitch. CLASSIFICATION. If we conceive the pitch of a pair of gears to be the smallest possible, we finally reduce the teeth to mere lines on the original pitch surfaces. These lines are called elements of the teeth. Gears may be classified MECHANICAL DRAWING. 83 with relation to the elements of their teeth, and also with relation to the direction of their shafts. First. Spur Gears ; those gears connecting parallel shafts and whose tooth elements are straight. Second. Bevel Gears ; those gears connecting shafts whose axes meet when sufficiently prolonged, and the elements of whose teeth are straight lines. In bevel gears the surfaces that touch each other, without slip- ping, are upon cones or parts of cones, whose apexes are at the point where the centre of their shafts meet. Third. Worm Gears ; those whose axes neither meet nor are parallel, and the elements of whose teeth are helical, or screw-like. A modification of this form of tooth is the skew-bevel wheel, which is used in some cases with a smooth surface which is a zone or frustrum of an hyperboloid of revolution. A hyperboloid of rev- olution is a surface resembling a dice-box, generated by the revolution of a straight line around an axis from which it is at a constant distance, and to which it is inclined at a constant angle. Of modifications of the spur-gear we have the internal gear, the elliptical gear, the segment and the rack. We will consider Bevel Gears and Spur Gears. We will use the following abbreviations : Let D = the diameter of addendum, or ful/ size circle. \jei D' := the diameter of pitch circle. Let P' = the circular pitch. Let / = the thickness of tooth at pitch line. Let s = the addendum, or face of tooth. Let/ = the clearance. Lfet D' = 2 s, or working depth of tooth. Let Z>' -\- / = whole depth of space. Let JV = number of teeth in one gear. §4 MECHANICAL DRAWING. Let TT =z 3.1416, or circumference when diameter is i. If we multiply the diameter of any circle by tt, the product is the circumference of that circle. If we divide the circumference by tt, the quotient will be the diam- eter of that circle. The circular pitch and number of teeth in a wheel being given, the diameter of the wheel and size of tooth parts are found as follows : Dividing by 3.1416 = multiplying by g.^^^^ = .3183 ; hence, multiply the circumference of a circle by .3183 and the product is the diameter of the circle. Multiply the circulai' pitch by .3183 and the product w^ill be the same part of the diameter of the pitch circle that the circular pitch is of the circumference of pitch circle. This part, or modulus is called a diavieter pitch. The diameter pitch = addendum of tooth = s. Cir- cular pitch multiplied by .3183 1=1 s^ or .3183 P' = s. The number of teeth in a wheel multiplied by a diam- eter pitch equals diameter of pitch circle, JVs = £>'. Add two to the number of teeth, multiply the sum by s and the product will be the whole diameter, (7V^-|-2) s = D. One-tenth of thickness of tooth at pitch line, equals amount added to bottom of space for clearance, yV = /• ^^« = ^' = Radius of pitch circle. Distance between centres of two ndieels equals the sum of the two pitch circle radii. In making drawings of gears and in cutting racks, it is necessary to know the circular pitch in whole inches and the natural divisions of an inch, as one-half inch pitch, one-quarter inch pitch, etc., but since it is difficult to measure the circumference of the pitch circle and di- vide it into equal parts, it is much better that the diam- eter of a gear should be of a size conveniently measured. The same applies to the distance between centres. MECHANICAL DRAWING. 85 Hence it is generally more convenient to assume the pitch in terms of the diameter. A definition of a diam- eter pitch and the method of obtaining it from the cir- cular pitch has been given. If the circumference of the pitch circle is divided by the number of the teeth in the gear, the quotient v^ill be the circular pitch. If the diameter of the pitch circle is divided bv the number of the teeth, the quotient will be a diameter pitch. Thus, if a gear has forty-eight teeth and a pitch diameter of twelve inches, the diameter pitch is twelve inches divided by forty-eight, or one-quarter of an inch. Naturally, in deciding dimensions of teeth for a gear, a diameter pitch of some convenient part of an inch is taken. In speaking of diameter pitch, only the denominator of the fraction is named. One-third of an inch diame- ter pitch is called 3 diametrical pitch. Diametrical pitch is the number of teeth to one inch of diameter of pitch circle. Represent this by F. Thus, one-quarter inch diameter pitch becomes 4 diametrical pitch, or 4 P, be- cause there would be four teeth on the gear to every inch of diameter of its pitch circle. The circular pitch and different parts of the teeth are derived from the diametrical pitch as follows : (i) ^-^ = P\ or 3.1416 divided by the diametrical pitch is equal to the circular pitch. (2) 7 = i^, or one inch divided by the thickness of one tooth equals number of teeth to one inch. (3) p = /, or 1.57 divided by the diametrical pitch gives thickness of tooth at pitch line. (4) p r=: Z^', or number of teeth in a gear divided by the diametrical pitch equals diameter of the pitch circle. The diameter of the pitch circle of a wheel hav- ing 60 teeth, 12 P^ would be, consequently, five inches. S6 MECHANICAL DRAWING. (5) —p- = Z>, or, add 2 to the number of teeth in a wheel and divide the sum by the diametrical pitch, and the quotient will be the lohole diameter of the gear or the diameter of the addendum circle. The diameter of gear blank for a gear of sixty teeth, 12 /^, would be, conse- quently, 5 ^2 inches. (6) > = P^ or number of teeth divided by diameter of pitch circle, in inches, gives the diametrical pitch. (7) -^ = P^ or add 2 to the number of teeth, divide by whole diameter and quotient will be diametrical pitch. PD' = N^ or pitch circle diameter multiplied by dia- metrical pitch equals number of teeth in the gear. (8) Formula (i) may be transposed, -^ =/^. SINGLE CURVE GEARS. Single curve teeth are so called because their working surfaces have but one curve, which forms both face and flank of tooth sides. This curve is, approximately, an involute. In a gear of 30 teeth or more, this curve may be the single arc of a circle, whose radius is one-fourth the radius of the pitch circle. A fillet is added at the bottom of the tooth, to make it stronger, equal in radius to one-sixth the widest part of tooth space. A cutter made to leave this fillet has the advantage of wearing longer than it would if brought up to a corner. In gears having less than thirty teeth this fillet is made the same as just given, and the sides of teeth formed with more than one arc, as will be shown fartheron. Having calculated the data of a gear of 30 teeth, \" circular pitch we proceed as follows : 1. Draw pitch circle and point it ofl' into parts equal to one-half circular pitch. 2. From one of these points, as at B^ (see plate MECHANICAL DRAWING. 87 Single Curve Gear) draw radius to pitch circle, and upon this radius describe a semicircle ; the diameter of this semicircle being equal to radius of pitch circle. Draw addendum, working depth and whole depth cir- cles. 3. From the point B^ where semicircle, pitch circle, and outer end of radius to pitch circle meet, lay off a GEAR, 30 TEETH, ^"CIRCULAR PITCH, P'=f'or .75" N=30 P =4.1£ = .375" = .2387" D"= .4775" S^-/= .2762" D"+/- .5150" D' = 7.1610" D =7.7384" SINGLE GEAR. 88 MECHANICAL DRAWING. distance on semicircle equal to one-fourth of the radius of pitch circle, shown in the figure at B A, This is laid off as a chord. 4. Through this new point A, upon the semicircle, draw a circle concentric to pitch circle. This last is called the base circle^ and is the one for centres of tooth arcs. In a certain system of single curve gears, the diameter of this circle is .968 of the diameter of pitch circle. 5. With dividers set to one-quarter of the radius of pitch circle, draw arcs forming sides of teeth, placing one point of the dividers in the base circle and with the other point describing an arc through a point in the pitch circle that w^as made in laying off the parts equal to one-half the circular pitch. Thus with A as centre, an arc is drawn through B. 6. With dividers set to one-sixth of the widest part of tooth space, draw the fillets for strengthening teeth at the roots. These fillet arcs should just join the whole depth circle to the sides of teeth already described. Single curve or involute gears are the only gears that will run at varying distances of axes, and transmit un- varying velocity. This peculiarity makes involute gears especially valuable for driving rolls or any rotating pieces, the distance between whose axes is likely to be changed. The assertion that gears crowd harder on bearings when of involute than when of other forms of teeth, has not been proved in actual practice. It is an excellent practice to cut out the drawings of a pair of gears, that have been made on Bristol-board, and test their accuracy in running together. MECHANICAL DRAWING. 89 RACK TO MESH WITH SINGLE CURVE GEARS HAVING 30 TEETH AND OVER. This gear is made precisely the same as the one last described. It makes no difference in which direction the construction radius is drawn, so far as obtaining form of teeth and making gear is concerned. Here the radius is drawn perpendicularly to pitch line of rack and through one of the tooth sides, B. A semicircle is drawn on each side of the radius of the pitch circle. The points A and A ' are each one-fourth the radius of the pitch circle distant from point B^ and correspond to the point A in the last figure. In this construction draw the lines BA and BA '. These lines form angles of 75^° (degrees) with radius BO. Lines BA and BA' are called lines of pressure. The sides of the rack teeth are made perpendicular to these lines. A Back is a straight piece having teeth to mesh with a gear. A rack may be considered as a gear of infinitely long radius. The circumference of a circle approaches a straight line as the radius increases, and when the radius is infinitely long any finite part of the circum- ference is a straight line. The pitch line of a rack, then, is a straight line just touching the pitch circle of a gear meshing with the rack. The thickness of teeth, addendum, and depth of teeth below pitch line are cal- culated in the same manner as for a wheel. A rack to mesh with a single curve gear of 30 teeth or more is drawn as follows : 1. Draw straight pitch line of rack; also draw ad- dendum line, working depth line and whole depth line, each parallel to the pitch line (see figure) . 2. Point oft' the pitcli line into parts equal to one- half the circular pitch, or = t. 90 MECHANICAL DRAWING. Fig.r RACK TO MESH WITH SINGLE CURVE GEAR HAVING 30 TEETH AND OVER. MECHANICAL DRAWING. 9I 3. Through these points draw lines at an angle of 75i^ with pitch lines, alternate lines inclined in opposite directions. The left-hand side of each rack tooth is perpendicular to the line B A, The right-hand side of each rack tooth is perpendicular to the line B A' , 4. Add fillets at bottoms of teeth equal to one-sixth of the width' of spaces between the rack teeth at the ad- dendum line. ANGLES FOR RACK TEETH. To get the proper angle for rack teeth, draw a semi- circle on a line A B. With centre A^ and radius equal to one-quarter of A B draw radius, cutting semi-circle at C. A straight line through A C will form an angle of 75|^ with the line A B. To get the angle for sides of a tool for planing out rack teeth proceed as follows : On line O B describe a circle. From B lay off on the circumference chords B A and B C, each equal to one-fourth of O B. Angle A O C is 29^ — the proper angle for the point of the tool. Make end of rack tool .31 of circular pitch, and then round the corners of the tool to leave fillets at the bottom of rack teeth. Thus, if the circular pitch of a rack is i^", and we multiply it by .31, the product, .465", will be the width of tool at end, for rack of this pitch, before the corners are taken of^\ GEARS AND RACKS TO MESH WITH GEARS HAVING LESS THAN 30 TEETH. The construction of the rack is similar to that just described. (See Figure Gear 2 P., 12 teeth in mesh with rack) . The curve on face of tooth, or that part outside of pitch circle, may be constructed as for a gear having 30 92 MECHANICAL DRAWING. teeth or more, but \\\^ flanks^ or curve of tooth inside of pitch circle, are drawn as follows : In gears of 12 and 13 teeth the flanks are parallel for a distance, inside the pitch circle, of one-third of a diameter pitch (^/^ ^) gears of 14, 15 and 16 teeth, one-fifth s; 17 to 20 teeth, one- sixth s. In gears of more than 20 teeth the parallel con- struction is omitted. The involute tooth of this gear of 12 teeth is drawn as follows : Draw three or four tangents to the base circle, iV jj\ kk' n\ letting the points of tangency on the base circle i\ j\ k\ /', be one-fourth of the circular pitch apart ; the first point, /', being distant from / one-fourth of radius of pitch circle. With dividers set to one-fourth the radius of pitch circle, placing one point on /', draw the arc a' ij; with one point on/', radius^*', drawy'y^y with one point on k' draw kl. Should the addendum circle be outside of / the tooth side may be completed with the last radius //'. The arcs a ' i, ij\ Jk, kl^ together form a very close ap- proximation to a true involute from the base circle ij'k'T . The exact involute for gear teeth is the curve made by the end of a band when unwound from a cylinder of the same diameter as the base circle. With diameter equal to the distance between the ends of two adjacent involutes, where they meet the base cir- cle, draw a circle about centre of gear. Lines from these points tangent to the ciicle form part of the flanks of teeth. From the whole depth circle, draw fillets with radius equal to J^ widest tooth space. These will butt into the parallel lines about ^ s from the base circle. This method is conventional, depending upon the judgment of the designer, to eflect the object to have spaces as wide as practicable just inside base circle and then strengthen flank with as large a fillet as will clear 94 MECHANICAL DRAWING. addenda of any gear. If flanks in any gear will clear addenda of a rack, they will clear addenda of any other gear, except internal gears. An internal gear is one having teeth on the inner side of a rim or ring. The foregoing operation of drawing tooth sides, al- though tedious in description, is very easy of practical application. The faces of teeth of rack are rounded oflf by an arc or radius of i ^ pitch, with centre in working depth line. BEVEL GEAR BLANKS. The pitch of Bevel Gears is always figured at the largest pitch diameter. Most Bevel Gears connect shafts that are at right angles to each other. The following directions apply to any angle, but the sketch is made with axes at right angles. Having decided upon the pitch, numbers of teeth and angle of shafts : (The sketch is made for gears i.i and 2.2 inches diameter.) Draw axes AOB, COD, Fig. i8. At a distance from AOB, equal to one-half the diameter of the gear, dra'w a line parallel to AOB. At a distance from COD, equal to one-half the diameter of the pinion, draw a line par- allel to COD. From the point ;/, where these two parallels meet, draw perpendiculars to AOB and COD, On these perpendiculars lay oft' the pitch diameters, // of the gear, and ;;/// of the pinion, the point / being com- mon. From y, ;/ and ;;/ draw lines to O These lines give size and shape of pitch cones, and are called Cone Fitch Lines. Through points ;;/, / and y, draw lines mx, iy and jz perpendicular to Cone Pitch Lines. On these lines, from cone pitch line, lay off' distances MECHANICAL DRAWING. 95 for addenda, working depth and whole depth of teeth. From the points so obtained, draw lines to the centre O. These lines give the height of teeth above Cone Pitch Lines, and the whole and working depths of teeth. The teeth become smaller as they approach O and be- come nothing at that point. It is quite as v^ell never to have the length or face of teeth, imn^ longer than one- third the distance Om^ nor more than two and a half times the circular pitch. Having decided upon the length of face, draw limit- ing lines 7n'x\ i'j' and j'^z'. We have now the outline of section of gears through their axes. A straight line drawn through the largest diameter of the teeth, perpendicular to axis of the gear, is called the largest diamete?^ In practice, these diam- eters are obtained hy measuring the drawing. To obtain data for teeth, w^e need only make drawing of section of one-half of each gear. We first draw centre lines AO, BO and lines ^i;// and €d, then gear blank lines as in the case just described. (See Fig. 17.) To obtain shape of teeth in bevel gears, we do not lay them off on pitch circles in same way as in spur gears. S = .200" D'= .400' 8+/= .231" DM-/ - .431" D"'= .266' S'+/ =.165' D"'+/ =.298' BEVEL GEARS. FORM AND SIZE OF TEETH. MECHANICAL DRAWING. 9^ A line running from a point on cone pitch line to 'Centre line of a bevel gear, perpendicular to this cone pitch line, is the radius for circle upon which to draw outlines of teeth at this point. Hence Ac is the geometrical pitch circle radius, for large end of teeth, and AW the geometrical pitch radius for small end of teeth of wheel. To avoid confusion, the distance A'c' is transferred to Ac' , For the pinion we have the geometrical pitch circle radius Be for large end of teeth, and the radius B'c for small end of teeth. Transfer distance B'c' to line Be"''.'' About A^ draw arc cn7n, and upon it lay off spaces equal to the thickness of tooth at pitch line, and draw outlines of teeth as previously described. We have now the shape of teeth at large end, repeat this operation with radius Be about B, and we have form of teeth, at large end of pinion. Upon arc of radius A'e' we get shape of teeth of small end of gear, and upon arc of radius B'e' we get shape of teeth at small end of pinion. The sizes of tooth parts at small end may be taken directly from the diagram, or they may be calculated as follows : Dividing the distance Oe'^ which, for example, may be 2 inches, by Oe, which may be three inches, we get Yz or .666 for a ratio. Multiplying outside sizes by .666, we get the corresponding inside sizes. Thickness of teeth at outside being .314 inchj fi of it gives us .209 inch as thickness of teeth inside. When cutting bevel gears with rotary cutters, tlie angle of cutter head is set the same as angle of working depth ; thus : To cut the gear we have the cutter travel * Tredgold's method from Rankine, App. Mech. p. 448. 9 98 MECHANICAL DRAWING. in the direction Op. The angle AOp is called the " cut- ting angle," being measured from the axis of the gear. In this method the angle of face of pinion is the same as cutting angle of gear, and face angle of gear is the cutting angle of pinion, and clearance is the same inside as outside. EPICYCLOIDAL TEETH. An epicycloid is " a curved line gen- erated by a point in the circumference of a circle, which rolls ^ on the circumference ^ of another circle, S either internally or ^s externally." — (Wor- cester.) Hence an epicy- cloidal tooth has parts of epicycloids for the curves of its faces. In the sketch, hav- ing determined on one-inch pitch, and found radius of roll- ing circle from dia- gram, the point ^was selected for a starting point, and the centre of the rolling circle having its centre at r' rolled on the pitch circle through four stations as shown, MECHANICAL DRAWING. 99 and the epicycloid traced by o found to be ox^ for the face of the tooth, and in the same manner, the epicycloid ox developed for the flank of the tooth. Four positions of the centre of the rolling circle, and four of the point are given in each case. By the diagram, thickness of tooth, / = .48 pitch. Length of tooth, I = ,^ F. Three-tenths of this dis- tance out from the pitch circle determines a point in the addendum circle, and four-tenths pitch in from the pitch circle gives a point in the w^hole depth circle. This allows -^-^ P for clearance. The curves obtained for one side of the tooth may now be reversed at a distance of .48 P^ and we have the out- line of a tooth that may be duplicated around the wheel. "It is considered desirable by millwrights, with a view to the preservation of the uniformity of the shape of the teeth of a pair of wheels, that each tooth in one wheel should work with as many different teeth in the. other wheel as possible. " They, therefore, study to make the numbers of teeth' in each pair of wheels which work together, such as to be prime to each other, or to have their greatest com- mon divisor as small as is possible consistently with the purposes of the machine. " The smallest number of teeth which it is practicable to give a pinion is regulated by the principle, that in order that the communication of motion from one wheel to another may be continuous, at least one pair of teeth should always be in action ; and that in order to provide for the contingency of a tooth breaking, a second pair, at least, should be in action also."* The least number of teeth that can usually be em- ployed is as follows : * Rankine. lOO MECHANICAI. DRAWING. Involute teeth, 25; epicycloidal teeth, 12; cylindrical teeth, or staves, 6. The Arc of Contact on the pitch lines is the length of that portion of the pitcJi lines w^hich passes the pitch point during the action of one pair of teeth ; and in order that two pairs of teeth, at least, may be in action at each instant, its length should be double the pitch. It is divided into two parts, the arc of approach and the arc of recess. In order that the teeth may be of length sufficient to give the required duration of contact, the distance moved over by the point on the pitch line, during the rolling of a rolling curve to describe the face and flank of a tooth, must be, in all, equal to the length of the required arc of contact. Link of Pressure. When one body presses against another, not attached to it, the tendency to move the second body is in the direction of the perpendicular at point of contact. This perpendicular is called the line of pressure. The angle that this line makes w ith the path of the impelling piece is called the a^igle of pressure. In the case of gearing, the line of pressure makes an angle with the line of centres of 75^ to 78^. PROPORTIONS OF GEAR WHEELS. Much of the study and w^ork on gears by engineers and manufacturers has been devoted to the improvement of the shape of the teeth of gear w^heels, and a large part of the illustrative chart is likewise occupied with representations of some of the most important of the results of this study and work. A matter often left hap- hazard and "rule of thumb" design is the proportions of gear wheels. In the sketch representing epicycloidal teeth, the de- MECHANICAL DRAWING. lOI sign of a gear wheel is completed and the proportions so graphically represented as to enable a student with very small labor to design any gear. The proportions of a gear of less than 12 inch diameter are of compara- tively small interest to the draughtsman or designer since they have been so often designed and manufactured that fairly perfect ones may be obtained of a dozen different manufacturers. When, however, we need a gear wheel of from 2 to 10 feet diameter, we are, for various reasons, inclined to bestow considerable care on its design. The proportions given in the sketch were compiled from the statements of three authors, and modified some- what by personal experience. The pitch of the gear is here made the basis of all dimensions. As to the relative strength of the different parts of a gear wheel, there is a wide difference of opinion ; some holding that the teeth should be the weakest part and others contending that all parts should be equally strong — the latter having in mind the principle which the Deacon had when he built his celebrated '' one boss shay." TEETH OF GEAR WEELS. There are at least two good reasons why the teeth should be made the weakest part of a gear. The teeth are the smallest part of the gear, and in falling — after having been broken —are least likely to damage the ma- chine of which the gear is a part. Also, if but a few teeth or cogs are broken out, they may be easily replaced by pins with small loss of time — after which a new and better gear may be substituted. The value of a tooth in transmitting power is a sub- ject of great importance in this study of gears. ♦9 I02 MECHANICAL DRAWING. The following formulas for the strength of teeth are from Thomas Box's Practical Treatise on Mill Gearing. The conclusions drawn from them and also the com- parison of the different forms of gear teeth, which fol- lows, are by Mr. Geo. B. Grant, a well-known authority on the subject of Gearing. STRENGTH OF A TOOTH. For worm gears, crane gears, and slow moving gears in general, we have to consider only the dead weight that the tooth can lift with safety. If we use a factor of safety of lo, we can use the formula JV = 350 c. /. — in which IV is the weight to be lifted, c is the circular pitch, and / the face, both in inches. For wooden cogs, substitute 120 for 350 in this formula. When the pinion is large enough to insure that two teeth shall always be in fair contact, the load, as found by this rule, may be doubled. Example. A cast-iron gear of 3 in. circular pitch, 6 in. face, will lift W = 350 X 3 X 6 = 6,300 lbs. HORSE POWER OF A GEAR. For very low speeds we may use the formula IfJ^ for low speed = -0037 ^^> ^^ ^ /> in which d is the pitch diameter, c the circular pitch, and / the face, all in inches, and // is the number of revolutions per minute. The horse power of a gear 3 ft. in diameter, 3 in., pitch and 10 in. face, at 8 revolutions per minute, is If I" = .0037 X 36 X 8 X 3 X 10 = 33 MECHANICAL DRAWING. I03 For ordincDj or high speeds we must use the formula HP^ ,012 c'f VJTt, When in doubt as to whether a given speed is to be considered high or low, compute the horse power by both formulae and use the smaller result. For bevel gears the same rule will apply, if we use the pitch diam- eter and the pitch at centre of face. The rules given above for the horse power of gears apply only to cast gears. One of the chief sources of weakness in a cast gear is that the continual pounding of the teeth on each other crystalizes the metal so that its strength is greatly decreased long before it is worn out. There are no recorded tests on the horse power of cut gears, and, consequently, we can only proceed by judgment and inference. Thus w^e may say that the weakening source — pounding — of the cast gear is absent with the cut gear, and from that point of view the latter would be stronger. In the absence of proof to the con- trary, we may assume that the rule that applies to cast gears for slow speeds, where impact need not be con- sidered, can safely be applied at higher speeds to cut gears where there is no impact to be allowed for ; and we have the formula : Horse power of cut gears at ordinary speeds = .0037 dncf ; or, to be within bounds of safety, -^"^^^^^^-^ A COMPARISON OF EPICYCLOIDAL WITH INVOLUTE TEETH. This matter might easily cover many pages, but is condensed to the following points : Adjustability. Involute — single curve— teeth alone can possess the remarkable and practically invaluable property, that they are not confined to any fixed radial I04 MECHANICAL DRAWING. position with respect to each other, for, as long as one pair of teeth remains in action, until the next pair is in position, the perfect uniformity of the action of the curve is not impaired. Epicycloidal teeth must be put exactly in place and kept there, and the least variation in position, from bad workmanship in mounting, or by wear or alteration of the bearings in use, will destroy the uniformity of the motion they transmit. Uniformity. The direct force exerted by involute teeth on each other, is exactly uniform, both in direction and amount, and this property insures a uniform wear- ifig action of the teeth, a nearly uniform thrust on the shaft bearings, and a steadiness and smoothness of action that cannot be claimed for epicycloidal teeth under any circumstances. The direct pressure acting between epicycloidal teeth is variable in amount afid very vari- able in direction, and consequently the friction and wearing action between the teeth, as well as the thrust on the bearings, is variable between wide limits. Friction. This measure is always in favor of the involute, although the advantage is usually claimed for the epicycloid, both as to maximum and average values, and as this is an important point, it should have great weight in deciding between the two forms of teeth, for the element of friction is of chief importance in deter- mining the life of a gear in continual and heavy service. Thrust on Bearings. Here the advantage is with the epicycloidal tooth, but not by a large amount, and is not a matter of first consequence. Strength. The greatest strain comes at the root of the tooth, and as the involute tooth spreads more than the epicycloidal tooth, it is stronger at that point. MECHANICAL DRAWING. I05 GEAR DESIGNING. After the cross-sectional area of the tooth is deter- mined on, the proportions for the rest of the gear are easily arrived at by looking at the sketch and accom- panying diagrams. In the right hand diagram, the vertical column of figures indicate circular pitch. The horizontal lines are simply divisions of the pitch — or pitches. The inclined lines show the variations of wheel di- mensions corresponding to the pitch. Thus, small r, the radius of rolling circle, for a 4 inch pitch gear is ^ of 4 inches, or 31^ inches; for 3 incii pitch, r is 2S/^ inches; for 3 inch pitch, r is i^ inches, and for i inch pitch, r \^ 1/^ of an inch. This matter of the size of the rolling circle is a very important one. Its size may be increased until the flanks of the teeth are straight radial lines, or decreased until the face of the tooth is an arc of as small a circle as the rolling circle itself, /. e.^ an epicycloid which nearly coincides with an arc of a circle equal to the rolling circle. Looking now for the thickness of rim, we find that it is given in the left hand diagram as <^ = i^ inch + .4 P^ and this for 4 inches P is 1.735 inches, or about i^ inches, for 3 inches pitch d is i^, for 2 inches \\ of an inch, and \\ of an inch for i inch pitch. This rim is in- creased from the edge or side of the wheel, toward the centre, until it is 1.2 ^ thick, where it is reinforced by a central rib d wide and d thick. This rim has had its teeth stripped from it, and therefore is strong enough, though it looks light. For computing the strength of arms we must have the pitch diameter and width of face of wheel given. De- CO CO — o [^ ^ PR0PQRTI0N5 OF GEAR WHEELS MECHANICAL DRAWING. IO7 noting the face of the wheel by b, half the pitch diameter by R, and the required depth of the arm at the hub by h^ the following formulae for arms are considered good : For 4 arms h^=^ .61 V]^ For 6 arms /^ = .5 V^ For 8 arms h = .46 V^^ For 10 arms /i = .443 V^^ For 1 2 arms /i = .438 F^ The depth of the arm at the rim should be ^ of the hub depth. The term depth is here used to denote the dimension of the arm in a plane at right angles to the axis of rota- tion. The arm is d thick. To strengthen the arm against side thrust and twisting, a rib a is put on each side of the arm, nearly as wide as the face of the gear. Its thickness is .7 ^. The fillet between the arms at the hub should never be less than 5^ d deep. If the rib a is not used, the thickness of the arm should be increased to 1.2 d. The thickness, IV, of the hub should be 4|f inches for a lo-inch shaft and J^ an inch less for each inch decrease in diameter of the shaft down to a three-inch shaft, where W is ijV inches. For a 2-inch shaft IV is '^ of an inch, and ^ inch for a i-inch shaft. These explanations, with the diagrams and previous designs of tooth forms, should enable the student to design gear wheels correctly and with ease. It may be well to add here that no amount of " book wisdom '' will supply a want of practical knowledge of the subject in hand, nor will any amount of " finger wisdom " enable a mechanic to design the gear he fault- lessly makes. MECHANICAL DRAWING. IO9 Experience, judgment and " common sense," as it is called — though it is rather uncommon — are necessities to the designer. These coupled w^ith " book wisdom" ought to make a good designer. If in addition to these qualities he has plenty of " finger wisdom " and a natu- ral mechanical ability, his designs should be as nearly perfect as our civilization demands. 10 STRENGTH OF MATERIALS. For the student's convenience, a few tables, diagrams and alphabets are here provided. Strength is the resistance a body opposes to a perma- nent separation of its component parts. Beams of the same material vary greatly in strength, and they some- times break under one-fourth the load corresponding to the figures given as their breaking strength. A large factor of safety is hence advisable. A solid cylinder varies in strength as the cube of its diameter. The formula for this case becomes, where fixed at one end and loaded at the other. Load = 1.7 X 6 X /. Where R = stress at the breaking point, d the dia ni- ter and / the length. If the cyhnder be uniformly loaded it will carry twice as much load ; supported at the ends, and loaded in the middle. Load becomes quadrupled ; supported at both ends and uniformly loaded, it is 8 times as great. A beam supported at one end and fixed at the other, and loaded uniformly, has the same strength as the last case, as has also a beam fixed at both ends and loaded in the middle. When fixed at both ends and uniformly loaded, the value of Load is twelve times as great as in the first of the preceding cases. The last statement of the rehitive strength of beams difterently placed is correct for all solid beams. A MECHANICAL DRAWING. Ill wooden beam of triangular section, supported at both ends, is about one-sixth stronger with its base upward than with its base downward. The strongest beam of rectangular section that may be cut from a round log has a depth proportioned to its breadth as 7 to 5. Such a beam is ten per cent, stronger than the largest square beam that may be cut from the same log. The strength of any beam, of whatever material, varies dif-ectly as its breadth and as the square of its depth. This fact is easily remembered in the formula bd^. Hence the transverse strength of a 2 by 6 beam placed edge up is greater than that of a 3 by 6 beam placed side up, in the ratio of 73 to 54. The transverse strength of round iron and steel may be taken as six-tenths the strength of bars of square section having their sides equal to the diameter of the former. A hollow cylinder has a strength exceeding that of a solid cylinder of the same length, weight and volume. Triangular beams of cast iron when the edge resists compression, and their resistance becomes a max- imum when the shape of section and the ratio of tensile strength to resistance in compression, are so related that the beam, when on the point of rupture, is equally liable to break by yielding to either force. The best form for a cast iron '' J-beam" is one in which the area of the lower "flange" is six times that of the upper — since that is about the ratio of the tensile and compressive strengths of cast iron. In general, extending the extreme portions of the sec- tion where stresses become greatest, and restricting the in- termediate part, or the "web," to the size needed to hold the other portions in proper relative positions, will produce forms of beams of greater strength, with a given weight of material, than can be obtained in the cases of rectan- gular, circular or other simple forms of section. Where 112 MECHANICAL DRAWING. the metal has equal strength to resist tension and com~ pression, the top and bottom flanges should be of equal size. This constitutes the Tredgold ^' J-beam," usually made of wrought iron. In many cases the form of sec- tion is determined by convenience, in making up. Thus columns are made up of " L/' '^ U/' "I'* or '' X beams." STRENGTH OF MATERIALS. Materials. Water \ 62 . 5 Wro't Brass 5^3-^ " Copper 543-6 " Iron 481 " ' * Rope ....... Cast Iron I 444 Cast Steel ; 494 Chrome Steel 'j 492 Plate Steel 487 Tin, Cast 460 Zinc ' 435 Phosphor-Bronze, | Wire |. . . . Phosphor-Bronze, | Cast Manganese Bronze .... Aluminum Bronze. .... Ash Wood 47 Cedar ' 35 Oak 53.75 Pine, White 34-62 Pine, Yellow 33. Spruce 31-25 Brick 102 Glass 169 Granite 165 Leather 63 Limestone 197 Marble 167. Mortar j 98 Slate 1178 Rope, Manilla .... I ... . Rope, Hemp I . . . . 035 297 314 270 257 Strength — per sq. in. — lbs. I 8.21 8.69 7.8 7.7 7-9 bo G 7.4 7 30,000 1 00000 34,000 7,500 50,000 50,000; 75,000 90,000 25,000 16,500 125000 88,600 295000 171000^ iioooo 80,000 700 30,000' 30, 000 54,000 675 650 5,000 6,000 .1110000 , 40,000 15,500: . i6,ooO| 800 200 751!- 56H 4611. 70,000 : 95,000 : 14,000; 11.400! 40,000 IIOOOO 130000 8,000 5,90o| 87M I i3,6oo| 6, lOO: 5,775 8,200 5,950 2,000 30,000 15,300 168 75' 75' 11,800 11,400 10,300 300 8,000 10,000 350 2,000 9,000 50 12,000 9,000 15,000 230 130 26 3,065 15,000 120 24,000 90,000 MECHANICAL DRAWING. 113 FACTOR OF SAFETY. The table of '^Strength of Materials " given in this book is composed of figures indicating the ultimate strengths of the materials. Obviously, materials in structures are never intended to be subjected to these strains. By great numbers of tests under a great variety of conditions ''factors of safety" have been arrived at, by w^hich the ultimate strengths are divided, the results being the '' safe w^orking loads." These factors are much greater under moving or "live" than under steady or "dead" loads, and vary w^ith the character of the ma- terial used. The factor of safety for building stone should never be less than ten, and sometimes a far higher value is adopted. It should be remembered that the crushing strength of the mortar or cement used as building ma- terial is often used rather than that of the stone. For machinery the factor is usually 6 or 8 ; for structures erected by the civil engineer from 5 to 6. The follow- ing may be taken as minimum values for the factor of safety. The figures given for materials subjected to shocks are approximate : Material. Copper and other soft metals and alloys Brass, brittle metals and alloys Wro't iron and soft steel Tool and machine steel Cast iron Building stone l^uilding timber 1 .eather Load. 'Dead.' 5 4 3 3 4 10 5 " Live. 10 7 to 10 Shock. 10 10 to 15 8 9 10 to 15 *10 114 MECHi\NICAL DRAWING. HORSE POWER OF SHAFTING. The diagram is one established by J. T. Henthorn^ M. E., to determine the size of good hammered iron shafting necessary to transmit certain powers. The dia- gram is constructed from the formulas '^56 X HP D-. 56 '' ^, which transposed, is HP, or agam, is P = ^ — ^-^ To use this diagram, supposing that it is desired to drive 900 horse power at 304 revolutions per minute : REVOLUTIONS OF SHAFT PER MINUTE. MECHANICAL DRAWING. II5 following the horizontal line representing power and the perpendicular line for revolutions per minute, it is found the nearest diagonal line is one representing 5^' in diameter. Or, supposing there is a shaft "j}^" in diam- eter and its speed is 170 revolutions per minute, and it is required to find the horse power v/hich it will safely transmit. Running along on the horizontal line of 170 revolutions per minute until the diagonal is reached, representing 71^" in diameter, at this intersection is found a perpendicular representing the horse power, which is 1,155. STRENGTH OF LEATHER BELTING. The width of the belts should always be a little less than the face of the pulley ; both are to be determined by the power to be transmitted and the velocity of move- ment. For light work a single thickness only is neces- sary, but for belts from prime movers and in other places- where great power is to be transmitted, double belts are used. For single belts embracing half the pulley, with a velocity of 600 ft. per minute, one horse power can be transmitted for each inch in width of belt, with a max- imum stress on the belt of 50 pounds and pressure on the journals of about 85 pounds per inch of width of belt. J. T. Henthorn, M. E., has given the following for- mula for the strength of double belts, per inch in width, in which D is the diameter of the pulley in feet, 7? the revolutions per minute, and H. P. the horse power : ^X ^^^ ^^ p 450 This formula gives .7 //. P. for a belt one inch wide running on a one foot (diam.) pulley at 100 revolutions per minute, double that for the same belt on a 2 ft. pul- ley at 100 revolutions, triple on a 3 ft. pulley, etc. ii6 MECHANICAL DRAWING. HORSE POWER PER INXH OF WIDTH. 2 3 4 5 6 7 8 9 10 II 12 13 14 15 50 60 70. So- 90. 100 . no 120 140 ISO 160 170 180. 190 200 210 220 230 240 250 260 270 2S0 290 300 310 320 330 340 350 ^\\^ 5^ i ^ h \ ^ C\ i\\ \\\ \\ \\ ^^ A A ^ \v \\\ \\ \ A A ^^ A A \ , w^ A \ A V A A xi A V" k \ \\ \ \\ A \ A X A ^:;^ A] s: a\ i \ v\ A \ \^ N^ A A A A \ /.T \ \\ \ \ \^ s; A \ \ A A A \ \ \ \ \ \ \ \ \ \ \' \^ VS 1 \^ \^ \ \ \ \, \^ \ \A1 \ M \ \ \ \ \ \ \ \ \ \ s \, /? 4 i '? \ 6 7 V \ % l\ /^ \ \ \ I \ \ \ \ \ N \ \ \ \ \ \ \ \ \ \ s \ \ \ \ \ \ \ \ \ ' V 1 1 \ \ \ \ \ \ \ s, \ I \ \ \ \ \ N| V \ V \ \ \ \ \ \ \ \, L \ [ \ \ \ \ ' \ \ \ \ \ \ \ \ \ \ I \ \j \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ V > \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (U;^^Diameter of Pulleys shown by Diagonals. The diagram made from this formula is used as fol- lows : To find the horse power that can be transmitted bv a twelve inch belt on a nine feet pulley running at 190 MECHANICAL DRAWING. II7 revolutions per minute : Following the line represent- ing 190 revolutions and the line representing the 9 feet pulley, we find that they meet on the perpendicular num- bered 12. This means that for this pulley at this speed each inch width of belt will transmit 12 horse power; hence a 12-inch belt will transmit 12 x 12 == 144 horse power. To find the width of belt necessary to transmit 50 H. 7^. on a 4 ft. pulley running at 210 revolutions : Inspection of the diagram shows that a belt i inch wide on this pulley at this speed will transmit 6 H. P.^ whence (50 -i- 6) a belt 8^ inches wide will be neces- sary to transmit 50 H. P. on this pulley at this speed. iiS MECHANICAL DRAWING. 1 1 05 v ^J^^.VjC^tJ^'^cwj •^^ »;5 ^ ^^ ^ ^ ^ i^ ^ cs. -^ ^ bo <-;^ Q^j^ ^ ^ i? 1 ^ ^ < Qi <:\2 ^'^c>^^^^^->^^S;^^ U-A_-^r HANDLE TABLE DIMENSIONS IN INCHES. A B c D E 1 ! F 1 ! G H R 1 i .2 1 8 t\ 1 1 .0333 3 .02 A li tV .25 ^\ H 1 .0416 i .025 .5 li 1 .3 vv A ! .05 ^V .03 .6 2 i A 1 4 2. 8 .0666 tV .04 .8 2i f .5 tV H .0833 ^V .05 1.0 3 i .6 i iV .1 ii .06 1.2 3* i .7 tV U 1 .1166 if .07 1.4 4 1 .8 i 1 ! .1333 u .08 1.6 4* U .9 tV u 1 .15 n .09 1.8 5 u 1.0 « 1 5 .1666 a .10 2.0 5* u 1.1 \i isV i .1833 n .11 2.2 6 14 1.2 i li 1 .2 u .12 2.4 MECHANICAL DRAWING. 119 PROPORTIONS OF NUTS AND B0LT5 DIAM. OF THREADS / \ r^ r : 1 . PER \ / 1 BOLT. INCH. \ /. \^ ' 1 20 tV i i A 18 w if 1 9 f 16 If ii *i tV 14 fi If If i 13 1 I tV j\ 12 i.V 3 1 37 li i 11 lA ItV ii i 10 ItV U 5 i ■ 9 iH ItV If 8 11 11 if u ■ 7 2jV 1+1 II 11 7 2tV 2 1 If 6 2* 2tV \h li 6 2i n lA If 5i 211 2tV lA 11 5 StV 21 IS IJ 5 3M 2}f IJI 2 4i 31 34 ItV 2i 4^ 4,V 3* 15 2i 4 41 3* lU 2i 4 4!;^ 4i 2J 3 3i 53 4-J 2^^^ ^i' MECHANICAL DRAWING. AREAS OF CIRCLES. Area = 3.1416 X Radius^. 131 Diam Area. Diam . : Area. ! Diam Area. 1 Diam Area. 14 .04908 i [^ 20.629 13 132.73 2 989.8 A .0767 H 21.647 V2 143.14 • 36 1017.9 Vi .11045 1 n 22.690 14 153.94 i 2 1046.3 t\ .15033 ' 'A 23.758 2 165.13 37 1075.2 Jf .19635 Vs 24.850 15 176.71 1 2 1104.5 t\ .24850 H 25.967 2 188.69 38 II34.I H .30679 .'s 27.108 16 201.06 2 1164.1 H .37122 6 i 28.274 2 213.82 39 1194.6 H .44179 Ys 29.465 17 226.98 2 1225.4 1 3 .51849 1 ^ i 30.68 2 240.53 40 1256.6 /8 .60I32 1 y. I 31.92 18 25447 2 1288.2 n .69029 Y2 33.183 2 268.80 41 1320.2 I .7854 Yi 34.471 19 283.53 2 1352.6 ni .99402 u 35.785 2 298.65 42 1385.4 i,¥ 1.2272 % 37.122 20 314.16 2 1418.6 i>i 1.4849 7 38.484 2 330.06 ^3 1452.2 ^'A r.7671 ^8 39.871 21 346.36 2 1486.1 iH 2.0739 , '^ 41.282 2 363.05 44 1520.5 ^y* 2.4053 ' Y% 42.718 22 380.13 2 1555.3 1.^8 2.7612 Y2 44.179 2 397-61 45 1590.4 2 3..1416 Ys 45.663 23 415.48 2 1625.9 % 3.5466 ¥ 47.173 2 433.74 46 1661.9 H 3.9761 '» 48.707 24 452.39 2 j 1698.2 Vi 4.4301 8 50.265 2 471.43 47 i 1734-9 Yi 4.9087 Yi 51.849 25 490.87 2 1 1772. H 5-4^19 H 53.456 1 2 510.70 48 , 1809.5 H 5.9396 Yi 55.08S , 26 530.93 2 1847.4 '» 6.4918 Yz 56.745 2 551.55 49 1885.7 3 7.0686 Y^ 58.426 27 572.56 2 1924.4 % 7.6699 Y 1 60.132 2 593. 9^^ 50 1963 5 H 8.2958 '« 1 61.862 28 615.75 2 ; 2002.9 Y% 8.9462 9 63-617 1 2 637.94 51 ■ 2042.8 )o 9.621 1 Y^ 65.397 1 29 660 52 2 ' 2083. % 10.320 y* . 67.201 1 2 683.49 52 ! 2123.7 'i TI.045 H 69.029 30 706.86 2 2164.7 '8 11.793 V-. 70.882 : 2 730.62 53 : 2206.2 4 12.566 % 72.76 31 754.77 2 2248. % 13.364 Y, 74.66 2 779.31 54 i 2290.2 H 14.186 7 8 76.59 32 804.25 , 2 ' 2332.8 /s 15-033 10 78.54 2 829.58 55 ; 2375-8 '•> 15.904 'A 86.590 33 855-30 2 2419.2 % 16.800 II 95.033 2 8S1.41 56 2463. ■f 17.72 % 103.87 34 i)()7 92 2 2507.2 'a 1S.665 12 113. 1 2 <)34.82 , 57 2551.7 5 19.635 Vz 122.72 35 962.11 1 2 1 2596.7 122 MECHANICAL DRAWING. AREAS OF CIRCLES.— Continued. Diam. ! Area. j Diam. A.ea. 1 1 Diam. Area. Diam. Area. 58 2642.1 69 3739.3 . 2 4963.9 90 6361.7 2 2687.8 2 3793.7 80 5026.5 j 2 6432.6 59 2734. ' 70 3848.4 2 5089.6 ' 91 6503.9 2 2780.5 1 2 3903.6 81 5153. 2 6575.5 60 2827.4 71 3959.2 2 5216.6 92 6647.6 2 2874.7 2 4015. I 82 5281. i 2 6720.1 61 2922.5 72 4071.5 2 5345.6 93 6792.9 2 2970.6 2 4128.2 83 54^0.6 2 6866.1 62 3019.1 , 73 4185.4 2 5476. 94 6939.8 2 3067.9 2 4242.9 84 5541.78! ! 2 7013.8 63 3117.2 74 4300.8 2 5607.9 : 95 7088.2 2 3166.9 2 4359.1 85 5674.5 j 2 7163. 64 3217. 75 4417.8 2 5741.4 i 96 7238.2 2 3267.4 2 4477. 86 5808.8 ■ 2 7313.8 65 3318.3 i 76 4536.4 2 5876.5 97 7389.8 2 3369.5 1 2 4596.3 87 5944.7 2 7466.2 66 3421.2 77 4656.6 2 6013.2 ; 98 7543. 2 3473.2 2 4717.3 88 6082 . I 2 7620.1 67 3525.6 78 4778.3 2 6151.4 : 99 7697.7 2 3578.5 2 4839.8 89 6221. I 1 2 7775.6 68 3631.7 1 79 4901.7 ! 2 6291.2 100 7854. 2 3685.3 1 MECHANICAL DRAWING. 123 CIRCUMFERENCES OF CIRCLES. C = 3.1416 X Diameter. Diam. ! ! Circum. Diam. % Circum. Diam. Circum. 71.471 Diam. 1 Circum. M .7854 36.128 i 38 I 119.38 % 1.5708 y i 36.914 23 72.257 K I 120.95 H 2.3562 12 : 37.699 M 73.042 39 122.52 I 3.1416 W 38.484 \ A 73.827 % 124.09 iM 3.927 A 39.270 : y. 74.613 40 125.66 I'A 4.7124 H 40.055 1 24 75.398 Yi 127.23 1^ 5.4978 13 40.841 ' y 76.184 41 128.80 2 6.2832 H 41 .626 % 76.969 . K 130.37 2,14 7.0686 : A 42.412 y 77.754 42 131-95 Vz 7.854 , 1 ^ 43.19- 25 78.54 % 133.52 % 8.6394 1 14 43.982 y 79.325 43 1 135.9 3 9.4248 I H 44.768 ^ li 80. Ill 'A 136.66 M 10.210 A 45.553 y 80.896 44 138.23 'A 10.995 H 46.338 26 81.681 'i 139.8 % II. 781 1 15 47.124 y 82.467 45 141.37 4 12.566 1 W 47.909 Yz 83.252 y2 142.94 M 13.352 { A 48.695 y 84.038 46 144.51 'A 14.137 % 49.480 27 84.823 li 146.08 H 14.922 16 50.265 y 85.608 47 147.65 5 15.708 i H 51.051 Vz 86.394 Yi 149.22 ^4: 16.493 ' A 51.836 ■ y 87.179 48 150.8 '^ 17.279 y 52.622 28 87.965 Vi 152.37 y* 18.064 17 53.407 y 88.75 49 153.94 6 18.849 M 54.192 Yz 89.535 Vz 155.51 ¥ 19.635 i Yi 54.978 y 90.321 50 157.08 ^ 20.420 H 55.763 29 91 .106 % I5S.65 14' 21.206 ! 18 56.549 y 91.892 51 160.22 7 21.991 1 H : 57.334 . A 92.677 'A 161.79 ,^4' 22.776 ! 'A 58.119 ■ y 93.462 52 163.36 >^ 23.562 ' Va 58.905 30 94.248 Vz 164.93 14' 24.347 i 19 I 59.69 Yz 95.819 53 166.5 8 25.133 1 H. i 60.476 31 97.389 V2 168.07 'i 25:918 1 li 1 61.261 Yz 98.96 54 169.64 A 26.703 ! M 62 . 046 32 100 53 'A 171.22 H 27.489 : 20 62.832 ^■i 102.10 1 55 172.79 9 28.274 H \ 63.617 33 103.67 % 174.36 ^4 29.060 A , 64.403 A 105.24 56 175-93 3-2 29.845 % 1 65.188 34 106.81 J/i 177.5 «' 30.630 21 65.973 !•> 108.38 57 179.07 10 31.416 ; \^ i 66.759 35 109.95 A \ 180.64 '4 32.201 ! % I 67 544 Vz III. 53 58 182.21 >^ 32.987 i y* \ 68.330 36 113. 10 ;^ 1 183.78 H 33.772 ■ 22 69 . 1 1 5 1., 114.67 59 185.35 11 34.557 % ' 69.9 37 116.24 Vi 186.92 ''4 35.343 1 A 1 70.686 Yz 117. Si 60 188.49 124 MECIIAXICAL DRAWING . CIRCUMFERENCES OF CIRCLES. Continued. Diam. Circum. Diam Circum. I |Diam. 1 Circum. Diam. Circum. 'A 190.07 % 221 .48 2 252.90 2 284.31 6i 191.64 71 223.05 1 81 254.47 1 91 285.88 % 193.21 V2 224.62 ' 1 256.04 2 287.45 62 194.78 72 226.19 82 257.61 92 289.03 ^., 10.35 2 227.76 2 259.18 2 290 . 6 63 197.92 73 229.34 ' S3 260.75 93 292.17 1., 199.49 2 230.91 2 262.32 2 293 . 74 64 201 06 74 232.48 S4 263.89 94 295.31 % 202 . 63 2 234.05 2 265.46 j 2 296.88 ^^5 204 . 20 75 235.62 85 267.03 95 298.45 2 205.77 2 237.19 2 268.61 2 300 . 02 66 207.34 76 238.76 86 270.18 96 301.59 2 208.91 240.33 2 271.75 2 303.16 67 210.49 77 241.9 87 273.32 97 304.73 2 212.06 2 243.47 2 274.89 2 306.3 68 213.63 7S 245.04 8*8 276.46 98 307.88 •^ 215.2 2 246.61 2 278.03 2 309.45 69 216.77 79 248.18 89 279.60 99 3 1 1 . 02 1/ 72 218.34 2 219.76 2 281.17 2 312.59 70 219.91 So 251.33 90 282.74 _ 100 314.16 [0 019 945 477 8 /-v.