Glass, Book. COPYRIGHT DEPOSIT "The dexterous hand and the thoughtful mind find their strength in union alone" ROGERS' '^ Drawing and Design An Educational Treatise RELATING TO LINEAR DRAWING; MACHINE DESIGN; WORKING DRAWINGS; TRANSMISSION METHODS; STEAM, ELECTRICAL AND METAL WORKING MACHINES AND PARTS; LATHES; BOILER AND PARTS; INSTRUMENTS AND THEIR USE; TABLES, Etc. THEO. AUDEL & CO., 72 FIFTH AVENUE, NEW YORK PUBLISHERS -COPYRIGHTED BY THEi. AUllL NEW YORK, 1913 ALL RIGHTS RESERVED ^0. (M3 _©CI,A85128 5 ■35 3 ^"2^ \ iiniiiiiiiiiiiiiitt 3 I " One peculiar feature of the draftsman's opportunity is that it takes hold of all the mechanical occupations, and of one almost as much as of the other. It is not in the least monopolized by the machinist, and it is not the necessity of his shop more than of the others. The pattern maker certainly has quite as much to do with working drawings, and why not also the molder, the black- smith, the boiler maker, the carpenter, the coppersmith and all the rest? It will be to the immense advantage of the workers in any of these lines, and to the young man a most presumptive means of advancement, to be not only able to read drawings, but to make them." — American Machinist. PREFACE In a report to the Bridge Commissioners, as to the progress being made in the construction of the steel cables designed to support the immense weight of the (N. Y.) East River Suspension Bridge, Chas. G. Roebling, C. E., used these impressive words, quoted, as printed in The Sun : " Further, Mr. Roebling said the work of placing four cables nineteen inches in diameter across the river, was one that REQUIRED A CERTAIN DELIBERATION. No ERROR OP ANY KIND MUST BE MADE. Although all the men that could be utilized in the work have been employed, yet the progress made appears to be slow. Laymen might, from this, infer that the work is lagging, but the Commissioners should know that this was not the case. The work will proceed, Mr. Roebling says, and be finished to the perfect satisfaction of the Commissioners." These emphasized words have been frequently in the mind of the author of this work, relating to drawing and design. During the long months required by its preparation the greatest of care has been taken to avoid error of any sort, and the utmost deliberation has been given to the careful presentation of each subject. This has been called "the age of illustration ;" the truth of the saying is evident on every side from the daily illustrated newspaper to the blue print in the hand of the iron worker. In illustrations of whatever nature we come back to the L. B. & T. elements — length, breadth and thickness — and to linear drawing as the foundation of all drawing whether industrial or artistic ; for linear drawing has for its object the accurate delineation of surfaces and the construction of figures obtained by the studied combination of lines. We must come back to first principles in all knowledge, as the ball comes back to the hand of the skillful thrower, so that on the next attempt it may be projected still further upward. PREFACE. The ability to draw is like an added sense whose value could be somewhat determined, if those engineers and others who are skilled in the art, would name the sum of money for which they would part with its knowledge — for — " A chance sketch — the jotted memoranda of a contemplative brain often forms the nucleus of. a splendid invention. An idea thus preserved at the moment of its birth may become of incalculable value when rescued from the desultory train of fancy and treated as the sober offspring of reason." This quotation is from the one who wrote also the noble sentence — " Thou hast not lost an hour whereof there is a record ; a written thought at midnight will redeem the livelong day." From its inception to its closing page the main idea of the author has been to instruct, to impart knowledge of drawing and design with special reference to a considerable degree of method and completeness ; his aim has been to educate, or to draw out, and develop harmoniously the mental powers — to train to a certain result the various processes described and to nurture an abiding interest in the student's mind of a noble and ancient art. "First, the blade, then the ears, then the ripened corn appears " expresses what has been the attempted order of instruction. The power to draw is akin to that, and, to the engineer and mechanic, second only to the power to read ; one needs not only to read the printed page but also to read a blueprint or a rightly drawn and porportioned sketch ; there should be many thousand good draughtsmen scattered about and around before there is one professor or regular instructor m the art ; for to the average man drawing should be looked upon as a help in his daily avocation rather than as a staff to lean upon for life-long employment. PREFACE. There is a current saying, "one never sees an old draughtsman." This is more true in the United States — the home of Opportunity — than in older countries ; its meaning is that the position in the draughting room is but a stage in the development of the Engineer, the Superintendent, the Manager in engineering works. A good knowledge of draughting is a round on the ladder of preferment ; a second round is a fair working knowledge of tJie mathematics and theory of mechanism, for the foundation of all accurate attainment in drawing and design are laid in these two fundamental sciences. It may be well always to remember that " Education does not consist merely in storing the head with materials ; that makes a lumber room of it ; but in learning how to turn those materials into useful products ; that makes a factory of it ; and no man is educated unless his brain is a factory, ^vith storeroom, machinery and material complete." To this may be added that the helpful value of a teacher or instructor cannot be overestimated ; man was not created to do his appointed work alone, he needs all assistance and aid possible — to help and be helped — is the universal law of progress, and especially is this true in the first beginnings in the art of drawing ; afterwards the student may be supposed to have acquired a real interest in his stimulating and useful endeavors. It is an odd thing that the preface, which is always understood as something going before, is often written last, hence these few long paragraphs are prepared to close the long and rather pleasant task of the author of the book ere it is delivered iyt toto to the Printer, Binder and to the management of its most excellent and reliable Publishers for its introduction to those for whom it is designed. With these views and to further such ends this book has been prepared, and with such aims more or less successfully attained, the volume is now committed to the kind favor of its patrons. TABLE OF CONTENTS PAGES Plan of the Work Abbreviations and Conventional Signs - 25-26 Useful Terms and Definitions - - - 27-40 Drawing Board, T-Square and Triangles - 41-51 Lettering 52-64 Shade Lines 65-77 Section Lining - 77-8 1 Geometrical Drawing - - - - 83-110 Isometric Projection 111-120 Cabinet Projection 121-127 Orthographic Projection - - . . 128-161 Development of Surfaces - - - 162-179 Working Drawings " 181-187 Tints and Colors 188 Tracing and Blue Printing - , - - 189-195 Reading of Working Drawings - - 196-198 Machine Design - - - - . - 199-228 PAGES Physics and Mechanics - - - - 212-228 Material Used in Machine Construction 214-215 Screws, Bolts and Nuts - - - - 228-242 Rivets and Riveted Joints - - - 243-251 Power Transmission 253-255 Shafts and Bearings 256-266 Belts and Pulleys, ----- 266-277 Gear Wheels 278-304 Metal Working Machines - - - 305-332 Dies and Presses - - . - - . 308-314 Drilling AND Milling Machines - - 31S-319 The Lathe 320-332 Engines and Boilers - - - - 333-389 Electrical Machines - - - - - 391-407 Drawing Instruments .... 408-426 Logarithms 435-460 Tables and Index - - - - . - 461-486 THE SCOPE AND PLAN OF THE WORK, The special mission of this book can almost be gathered from its title page and the preface. It is intended to furnish gradually developed lessons in linear drawing applied to the various branches of the mechanic arts. The work is comprised within some twelve divisions or general subjects ; the first of which consists of Abbreviations and Conventional Signs, Useful Terms and Definitions with illustrations. The second section relates to the Drawing Board, T-square and Triangles and their use, lettering, shade and section lining, etc. The third division is devoted to Geometrical Drawing ; the subject is preceded by many valuable definitions, axioms and examples of postulates and followed by many illustrations, largely based upon the problems solved by Euclid more than twenty-two centuries ago. The fourth division relates to the Development of Surfaces and Isometric, Cabinet and Orthographic projections. The fifth section relates to Working Drawings embracing Tracing, Blue Printing, Dimensioning, Reading of Drawings, etc. The foregoing portions comprise "Part One" of the work and relate almost exclusively to Drawing and Definitions. "Part Two" is devoted to Machine Design, Transmission Methods, Metal Working Machinery, Engines and Boilers, Electrical Machines, etc., which embrace the sub- divisions six to ten. Each one of these sections is preceded by explanatory matter, and accompanied by illustrations of the different machines, with working directions for proportioning and designing. "Part Three," in addition to Drawing Instruments and their U^^e and the Index, contains tables, of the utmost value, for use in connection with the preceding sections, especially so, as the basis of the work is planned to be largely mathematical. ROGERS' DRAWING AND DESIGN. The making of a book of any considerable scope and value is either — as in olden times — the life work of a single author, or as, at the present, the combined efforts of several individuals, whose united efforts produce it in a much shorter time, and it must be hoped, in greater perfection. Although no more than a year has elapsed from the opening subject of "Abbreviations and Conventional Signs" to the closing reference — Index — pages, in no sense should the work be considered as being hasty or superficial, for the outcome of the combined efforts of those named below, is worthy of praise for having produced a thoroughly scientific and helpful book. First of all, to Mr. John Weichsel, M. E., instructor in drawing and design in one of the foremost technical institutes of New York City, is due the credit of furnishing most of the drawings and diagrams used throughout the work, with the text accompanying each ; the book itself is the highest testimonial to the admirable and thoroughly technical character of Prof. Weichsel's work. Mr. Henry E. Raabe, M. E., has been the technical editor throughout the period of the prepara- tion of the text and the arrangement of the illustrations in their appropriate places. Many of the drawings, explanatory notes, and "cuts" are also his own production. Messrs. Sutherland & Graham, Engravers, and George Byron Kirkham, Artist, are entitled to thanks for many designs and illustrations, as well as for professional advice and suggestions in several details of the "lay-out" of the volume. Mr. P. Hetto, of the U. S. Navy, an accomplished draughts- man and scholar, has read the "proof" of each separate page with critical care, and to him should be accorded praise for the almost perfect freedom from errors of any kind which marks the completed volume. Mr. H. Harrison, of the L. Middleditch Press, has used his wide experience in the typographical arrangement of the work; in this he has been aided by Mr. Henry J. Harms in overseeing the final issue and printing of the book ; the excellence of their work is evident on every printed page. It may be added that the kind and experienced editor-in-chief has combined and added to, and in some cases, taken from, the "matter" submitted by the foregoing named persons and others and the result of the whole, is now ofTered with confidence to the patronage of the Mechanical World, by The Publishers. ABBREVIATIONS AND CONVENTIONAL SIGNS* In order to simplify the language or expression of arithmetical and geometrical opera- tions the following conventional signs are used : The sign -f- signifies plus or more and is placed between two or more terms to indicate addition. Example: 4 -|- 3, is \ plus 3, that is, 4 added to 3, or 7. The sign — signifies minus or less and indicates subtraction. Example : 4 — 3, is 4 minus, that is, 3 taken from 4, or i. The sign x signifies multiplied by and placed between two terms, indicates multiplication. Example : 5 X 3, is 5 multiplied by 3, or 15. When quantities are expressed by letters, the sign may be suppressed, thus we write indifferently, a x b or ab. The sign : or (as it is more commonly used) -4- signifies divided by, and, placed between two quantities, indicates division. Example : 12 : 4, or 12 -7- 4 or — is 12 divided by 4. The sign = signifies equals or equal to, and is placed between two expressions to indi- cate their equality. Example : 6 + 2 = 8, meaning that 6 plus 2 is equal to 8. ABBREVIATIONS AND CONVENTIONAL SIGNS. The union of these signs : :: : indicates geometrical proportion. Example : 2 : 3 : : 4 : 6, meaning that 2 is to 3 as 4 is to 6. The sign /^Z indicates the extraction of a root ; as, -y/ 9 = 3, meaning that the square root of 9 is equal to 3. The interposition of a numeral between the opening of this sign, V, indicates the degree of the root. Thus: ^^27 = 3. expresses that the cube root of 27 is equal to 3. The signs < and > indicate respectively, smaller than and greater than. Example : 3^4 = 3 smaller than 4 and, reciprocally, 4 > 3 = 4 greater than 3. Fig. signifies figure ; and pi., plate. USEFUL TERMS AND DEFINITIONS. Lines, Angles, Surfaces and Solids constitute the different kinds of quantity called geometrical magnitudes. LINES AND ANGLES. A surface is that which has extension in length and breadth only. A solid is that which has extension in length, breadth and thickness. An an^le is the difference in the direction of two lines proceeding from the same point. A point is said to have position without magni- tude ; it is generally represented to the eye by a small dot A line is considered as length without breadth or thickness ; it denotes the direction between two points. Lines are principally of three kinds : (i) right lines, (2) curved lines, (3) mixed lines. FiG.l. A right line, or as it is more commonly called, a straight line is the shortest line that can be drawn between two given points, as above in Fig. i. A curved line is one of which no portion, how- ever small, is straight ; it is therefore longer than a straight line connecting the same points ; a straight line is often called simply a line, and a curved line a curve, a regular curved line, as Fig. 2, is a portion Fig. 2. of the circumference of a circle, the degree of curva- ture being- the same throuo-hout its entire leng-th ; an irreo^tilar curved line has not the same degfree of curvature throughout, but varies at different points. A -waved line, shown in Fig. 3, may be either regular or FiQ. 3. irregular; the illustrat.on shows the former, the inflections on either side of the dotted line being equal. Note. — There are other lines used in common drawing-room defini- tions, viz.: Broken, etc. Bi oketi — One composed of different successive straight lines. Center — A line used to indicate the center of an object. Conshuction — A working line used to obtain required lines. DoUed — A line composed of short dashes. Dash — A line composed of long dashes. Dot and Dash — A line composed of dots and dashes alternating. Dimension — A line upon which a dimension is placed. Full — An unbroken line, usually representing a visible edge. Shade — A line about twice as wide as the ordinary full line. 27 28 ROGERS' DRAWING AND DESIGN. Mixed lines are composed of straight and curved lines, as Fig. 4. FiQ. 4. Parallel lines are those which have no inclina- tion to each other — Figs. 5 and 6 being everywhere equidistant ; consequently they never meet though produced to infinity. If the parallel lines shown in Fig. 6 were produced they would form two circles having a common center. Fig. 5. Fig. 6. Horizontal lines are lines parallel with the hori- zon, as in Fig. 7. Vertical lines are often called plumb lines as they are parallel to a plumb line suspended freely in a still atmosphere. A horizontal line in a draw- ing is shown by a line drawn from left to right across the paper ; a vertical line in a drawing is represented by a line drawn up and down the paper or at right angles to a horizontal line, as in Fig. 7. HORIZONTAL Fig. 7. Inclined or oblique lines occupy an inter- mediate between -horizontal and vertical lines as shown in Fig. 7 ; two lines which converge towards each other and which, if produced, would meet or intersect are said to incline towards each other. ROGERS' DRAWING AND DESIGN. 29 Perpendicular lines. Lines are perpendicular to each other when the angles on either side of the point of meeting are equal. Vertical and horizontal lines are always perpen- dicular to each other, but perpendicular lines are not always vertical and horizontal ; they may be at any inclination to the horizon, provided that the angles on either side of the point of intersection are equal, as X Y and Z in Fig, 8. / / s. «•■ -9 1 1 i i f \ / ■p r -7 <>•• ■^*-> e-^ Fig. a. Fig. 10. Angles. Two straight lines drawn from the same point, diverging from each other form an angle, as shown in Fig. 9 : the angle is the differ- ence in the direction of two lines proceeding from the same point. Note. — Mechanics' squares, if true, are always right-angled. A right angle is formed when two straight lines intersect so that all angles formed are equal, as shown in Fig. 10. An obtuse angle is greater than a right angle, \,^o^'"^'^f^ Fig. II. Fig. U. An actite angle is smaller than a right angle, Fig. 12. Obtuse and acute angles are also called oblique angles ; and lines which are neither parallel nor perpendicular to each other are called oblique lines. The vertex or apex of an angle is the point in which the including lines meet. An angle is commonly desig- nated by three letters and the letter designating the point of divergence is always placed in the middle. The magnitude of an angle does not depend upon the length of the sides but upon their divergence from each other. Fig. 12. Fig. 13. 30 ROGERS' DRAWING AND DESIGN. STRAIGHT SIDED FIGURES. A surface is a magnitude that has length and breadth without thickness; as a plane surface, or, the imaginary envelope of a body. A plane is a surface such that if any two of its points be joined by a straight line, such line will be wholly in the surface. Every surface which is not a plane, or composed of plane surfaces, is a curved surface. Fig. 14. Fig. 15. Fig. 18. A plane figure is a portion of a plane terminated on all sides by lines either straight f)r curved. A rectilinear figure is a surface bounded by straight lines. Polygon is the general name applied to all rectilinear figures but is commonly applied to those having more than four sides. A regular polygon is one in which the sides are equal. A triangle is shown in Fig. 13. When sur- faces are bounded by three straight lines they are called triangles. Fig. ir. An equilateral triangle has all its sides of equal length, and all its angles equal, as appears in the illustration. Fig. 13. Fig. 18. An isosceles triangle has two of its sides and two of its angles equal, as illustrated in Fig. 14. ROGERS- DRAWING AND DESIGN. 31 A right-angled triangle has one right angle, Fig. 15 : the side opposite the right angle is called the hypothenuse ; the other sides are called respec- tively the base and perpendictilar. The altitude of a triangle is the length of a perpendicular let fall from its vertex to its base. iTia. n. Fig. 20. Fio. 21. Fio. 22. A quadrilateral is a figure bounded by four straight lines. If the opposite sides of a quad- rilateral are paralleled it is* called a parallelogram. Note. — The superficial conlents of a triangle may be obtained b}' multiplying the altitude by one half the base. A parallelogram, in which the four sides are equal, and form right angles with each other, is called a square. Fig. 16. There are three kinds of quadrilaterals : The trapezium, the trapezoid, and the parallelogram — as above. The trapezium has no two of its sides parallel, Fig. 17. The trapezoid has only two of its sides parallel, Fig. 18. There are four varieties of parallelograms : The rhomboid, the rhombus, the rectangle and the square. The square is an equilateral rectangle, Fig. 16. A rhombus is a parallelogram as shown in Fig. 19, one in which the four sides are equal, but none of the angles are right angles. A rectangle is a parallelogram which has its op- posite sides parallel, and all its angles right angles, Fig. 20. A rhomboid is a parallelogram in which the adjacent sides are unequal, and none of the angles are right angles, Fig. 21. 32 ROGERS' DRAWING AND DESIGN. A diagonal is a straight line, which joins two opposite angles of a polygon, Figs. 17, 22. A pentagon is a polygon bounded by five straight lines, Fig. 23. If the sides and angles formed by them are equal the figure is called a reo-- tilar pentagon. Fig. 24. Fig. 23. Fig. 24. Fig. 2.-). Fig. 26. Fig. 27. Fig. 2/i. A hexagon is a polygon bounded by six straight lines. Fig. 25 illustrates a regular hexagon. A heptagon is a polygon bounded by seven straight lines. Fig. 26 illustrates a regular hep- 4agon. An octagon is a a polygon bounded by eight straight lines. In Fig. 27 is shown a regular octagon. A decagon is a polygon bounded by ten straight lines. Fig. 28 illustrates a regular decagon. Fig. 29. A dodecagon is a polygon bounded by twelve straight lines. In Fig. 29 is shown a regular dodec- agon. Note. — Polygons of more than eight sides are rarely used in me- chanical drawing. Their most frequent application occurs in laying out of the hubs of large sectional wheels. ROGERS" DRAWING AND DESIGN. 33 A convex surface is one that when viewed from without is curved outward by rising or swelHng into a rounded form, Fig. 30. CONVEX. Fig. 30- A double convex stirface is regularly protuberant or bulging on both sides. Fig. 31. Concave means hollow or curved inward ; said of an interior of an arched surface or curved line in opposition to convex, Fig. 31. CIRCLES AND THEIR PROPERTIES. A circle is a plane figure bounded by one uni- formly curved line, all of the points in which are at the same distance from a certain point within, called the center ; the circumference of a circle is the curved line that bounds it ; the diameter of a circle is a line passing through its center, and terminating at both ends in the circumference ; the raditis of a circle is a line extending from its center to any point in the circumference ; it is one-half of the diameter ; all the diameters of a circle are equal, as are also all the radii ; an arc of a circle is any portion of the circumference ; the fixed point about which the circle is drawn is called the center of the circle ; any straight line, drawn within the circle, connect- ing any two points in the circumference without passing through the center, is called a chord. A semicircle is the half of a circle and is bounded by half the circumference and a diameter ; a segment of a circle is any part of its surface cut off by a straight line ; a sector of a circle is a space included between two radii and the arc they inter- sect. See Fig. 32. Note. — Radius is derived from the Latin word ray, meaning a di- vergent line, the plural in Latin is radii ; the English word for the plural term is radiuses. 34 ROGERS' DRAWING AND DESIGN. A quadrant is a sector equal to one-fourth of the circle ; the two radii bounding a quadrant are at right angles. A tangent to a circle or other curve is a straight line which touches it at only one point. Every tangent to a circle is perpendicular to the radius drawn to the point of tangency. A degree. The circumference of a circle is sup- posed to be divided into 360 equal parts called degrees and marked (°). Each degree is divided into 60 minutes, or 60'; and for the sake of still further mi- nuteness of measurement, each minute is divided into 60 seconds, or 60". In a whole circle there are, therefore, 360X60X60^1,296,000 seconds. The annexed diagram, Fig. 32, exemplifies the relative positions of the Sine, Tangent, Co-Sine, and Co-Tangent of an angle ; the co- in co-sine and co-tangent is simply an abbreviation of the word, complement. The circumferences of all circles contain the same number of degrees, but the greater the radius the greater is the absolute measures of a degree, and every circumference is the measure of precisely the same angle. Thus if the circle be large or small, the number of the division is always the same, a degree being equal to -jiirth part of a circle ; the semicircle is equal to 180° and the quadrant to 90°. The sine of an arc is a straight line drawn from one extremity perpendicular to a radius drawn to the other extremity of the arc, Fig. .32 ; the co-sine of an arc is the sine of the complement of that arc, as shown in the same figure. The tangent of an arc is a line which touches the arc at one extremity and is terminated by a line passing from the center of the circle through the other extremity of the arc, Fig. 32 ; the co-tangent of an arc is the tangent of the complement. For the sake of brevity, these technical terms are contracted thus : for sine, we write sin.; for co-sine, we write cos.; for tangent, we write tan., etc. Fig. 33. Concentric circles are those which are de- scribed about the same center, Fig. 33. ROGERS' DRAWING AND DESIGN. 35 Mccentric circles are those which are described about different centers, Fig. 34. Fig. 34. Fig. Sj. Eccentric circles are two or more circles whose centers lay within the circumference of one or more of these circles, but do not form a common center about which they could all be described. Figs. 34, 35- 36. Fig. 36. Fio. 37. The centers of eccentric circles may also lay out- side of each other's circumference, as in Fig. 37, or the center of one circle may lay outside of the other's circumference, while the former circle may be either wholly or partly within the circumference of the latter, as in Figs. 38 and 39. Fig. 38. Fig. 39. In another instance the center of one circle may lay on the circumference of the other, as in Fig. 40. Fig. •40. The distance between the centers of eccentric circles is called the radius of eccentricity. 36 ROGERS' DRAWING AND DESIGN. If two circles lay in a position as indicated in Fig. 41, they are not regarded as eccentric circles, but are treated as two independent figures. Fig. 41. A Parabola is a curve, described by a point, moving so, that its distances from a straight line, and a fixed point are always equal, Fig. 42 ; the Fig. 42. Straight line is called the Directrix, and the fixed point is called the Focus of the parabola ; a straight line drawn at right angles to the directrix, and pass- ing through the focus, is called the Axis. A Hyperbola is a curve from any point of which, if two straight lines be drawn to two fixed points, their difference shall always be the same. See Fig- 43- ^ Fig. 44. An Mllipse is a curve, described by a point, mov- ing so, that the sum of its distances from two fixed points is always constant ; the two fixed points are called.- the Foci, Fig. 44. ROGERS' DRAWING AND DESIGN. 37 SOLIDS. A solid has the parts constituting its substance so compact or firmly adhering as to resist the im- pression or penetration of other bodies ; it has a fixed form, is hard, firm and unHke a fluid or liquid ; it is not hollow, hence sometimes heavy. A conic section is a curved line formed by the intersection of a cone and a plane. Intersection of solids is a term used to de- scribe the condition of solids virhich are so joined and fitted to each other as to appear as though one passes through the other By the envelope of a solid is meant the surface which encases or surrounds it. A prism is a solid body whose ends or bases are equal and parallel plane figures, and whose sidt s are parallelograms. The shape of a prism is always expressed by the form of its bases. A triangular prism is a prism with the trian- gular bases, as shown in Fig. 45. A quadrangular prism is a prism with quad- rangular bases. Fig. 46. A pentagonal prism is a prism with pen- tagonal bases, Fig. 47. A hexagonal prism is a prism with hexagonal bases. Fig. 48. Fig. 45. Fiu. 4U. Fio. 4T. A cube'is a quadrangular prism whose b;ises and sides ar ' all equal and form perfect squares, Fig. 49. Fic;. 4S. Fli:. 41), Flti. .ill. A cylinder is a solid, bounded by two equal circular surfaces or bases, and one continuous curved surface, Fig. 50. All cross sections of a cylinder are equal to the bases. 38 ROGERS' DRAWING AND DESIGN. A cone is a solid bounded by a circular base, and one curved surface, extending from the circular base to a point opposite it, Fig. 51. Fig. 51. PiQ. 52. Fig. 53. Aright cone. If a perpendicular, droppedfrom the apex of the cone to its base, meets the center of the base circle, the cone is called a right cone, Fig. 52. The perpendicular in this case is called the Axis of the cone. An oblique cone. If the perpendicular falls alongside the center of the base circle, or entirely outside of its circumference, the cone is called an oblique cone. Fig. 53. A truncated cone. A cone, cut off in the man- ner shown in Fig. 54, is called a truncated cone, If an oblique cone is cut in the above manner, it is called an oblique truncated cone. Fig. 55. If a cone is cut by a plane, parallel to the outline of its surface, vertically opposite the center line of the cutting plane, as shown in Fig. 56, the outline of the section is 2. parabola. Fig. 57. Fig. 54. Fig. M. Fio. 57. ROGERS" DRAWING AND DESIGN. 39 If the cutting plane forms a smaller angle than the parabola, with the outline of the side on which A Fig. 59. it is cut, as shown in Fig. 58, the section is a hyper- FiG. 60. Fig. 61. bola. Fig. 59. If the cutting plane forms a greater angle than the parabola, with the surface, so that the cone is cut in the manner shown in Fig. 60, the s?ction is an ellipse, Fig. 61. t^ pyramid is a solid, whose base is a polygon, and whose sides are formed by triangles. The point in which all the lines of the triangular sides meet, is called the apex of the pyramid. APEX Fig. 62. Fig. 63. Fig. 64. Fig. 65. Pyramids are classified as triangular, quadrangu- lar, pentagonal, hexagonal, etc., pyramids, depend- ing upon the shape of their base, Figs. 62, 63, 64. If the base of a pyramid forms a regular polygon, and a perpendicular dropped from the apex, to the base, passes through the center of the base, it is called a right pyramid. Fig. 65. The altitude of a pyramid or a cone, is the per- pendicular distance from the apex to the base. Figs. 66, 67. The altitude of a prism or a cylinder is the per- pendicular distance between the bases, Figs. 68, 69. 40 ROGERS' DRAWING AND DESIGN. A truncated pyramid is the part remaining, after the apex is cut away, Figs. 54 and 70. A truncated cone or pyramid is also called the frustrtim of the cone or pyramid. Fig. tj6. Fig. 70. Fig. 67 Fig. 68. Fig. Fig. 71. A sphere is a solid, bounded by a uniformly curved surface, any point of which is equidistant from the center, Fig. 71. A polyhedron is a solid, bounded by polygons. There are five regular polyhedrons — as follows : A tetrahedron is a solid, bounded by four equi- lateral triangles, Fig. 72. A hexahedron is a solid, bounded by six squares ; the common name for this solid is cube, Fig. 49. An octahedron is a solid, bounded by eight equilateral triangles. Fig. y^- \ Fig. 72. Fig. 73. A dodecahedron is a solid, bounded by twelve regular pentagons. An icosahedron is a solid, bounded by twenty equilateral triangles. DRAWING BOARD, T-SQUARE AND TRIANGLES. The problems explained in the following para- graphs are but a small part of the great number of problems that may be executed with the aid of the tee-square and triangles. Fio. 74. In fact, all drawings, embracing straight lines only, may be drawn with the aid of the above instru- ments, provided the nature of the drawing does not call for greater accuracy or for lines other than straight ones. In the latter case, the mathematical instruments described hereafter need to be em- ployed. The paper on which it is intended to make a drawing, is generally fastened, by means of thumb- tacks, to a specially made board called a drawing board, Fig. 74. The drawing board should be made about 2 inches longer and 2 inches wider than the paper. It should be made of well-seasoned, straight-grained pine, free from all knots ; the grain should run lengthwise of the board. The edges of the board should be square to each other and perfectly smooth in order to provide a good working edge for the head of the tee-square to slide against. A pair of hard-wood cleats is screwed to the back of the board. The board should be about three- quarter inch in thickness. The cleats, fitted at the back of the board, at right angles to its long- est side, may be about two inches wide and one inch thick. Such cleats will keep the board from warping through changes of temperature and moist- ure. 4i 42 ROGERS' DRAWING AND DESIGN. Fig. ' Fig. 76. All lines parallel to the longer edges of the board are called horizontal lines. For drawing such lines an instrument is used, called a tee-square, Fig. 75. A tee- square consists of two parts, the head and the blade, which should be square to each other. The blade should be as long as the drawing board. It should be made of well-seasoned, fine-grained hard wood, and as light as its proper use will permit. The head may be made of any kind of well-seasoned wood. The blade should be laid on the face of the head and there fastened to it with four or five screws. The tee-square should be used with its head held firmly against the left hand edge of the board. Any number of hori- zontal lines may be drawn by sliding the tee-square up or down, Fig. 76. Another kind of a tee-square is shown in Fig. 'J 'J. The blade of this tee-square is fastened to the head by means of a square-necked bolt and a fly-nut. The blade may be so adjusted as to form any desired angle with the head. This tee- square is called the adjustable tee-square. ROGERS' DRAWING AND DESIGN. 43 Fig. 77. Fig. 78. After setting the blade at the desired angle to the head, we can draw any number of parallel lines at that angle, by sliding the tee-square up or down. Fig. 78. For drawing lines other than horizontal ones, set squares or triangles are used. They are made in various styles, some being cut out of a single piece of wood, while others are framed together of three or more pieces. Two triangles will be required for ordi- nary purposes. One should have one angle of 90 degrees, that is a square angle or a right angle, and two angles of 45 degrees each, that is equal to one-half of a right Fig. 79. Fig. 80. 44 ROGERS' DRAWING AND DESIGN. angle ; the two short sides of the triangle are of equal length, Fig. 79. The other triangle should contain one angle of 90 degrees, one angle of 30 degrees (that is equal Fig. 81. to one-third of a right angle) and one angle of 60 degrees (that is equal to two-thirds of a right angle.) In this triangle the shortest side is equal to just one half the longest side, Fig. 80. The first triangle is called the 45-degree triangle, the second, the 30-degree triangle or the 60 degree triangle. These triangles may be made of wood, hard rubber, or celluloid, of which materials it is also preferable to make the tee-square for many reasons. Triangles made of straight-grained well-seasoned hard wood will be found most satisfactory. By placing the tee-square in position on the drawing board, with its head against the left-hand edge of the board, and plac- ing either triangle with its short side to the edge of the tee-square, we may draw lines parallel to the short side of the drawing board, which we will call vertical lines. Fig. 81. Any number of vertical lines may be drawn by sliding the triangle in this posi- tion along the edge of the tee-square. The manner in which the head of a tee- square is united to the blade determines its adaptability or otherwise to the use made of it ; in some the head of a tee- square is rectangular in section, and the blade mortised into it ; in others the blade is dove- tailed and let into the head of a tee-square for the whole of its thickness ; the method spoken of on page 42 is, however, the most approved. ROGERS' DRAWING AND DESIGN. 45 Keeping the tee-square in position and placing against its blade one edge of the 45-degree angle, we may draw a line making an angle of 45 degrees with a horizontal line, Fig. 82. Such a line is called o o o Fig. 82. a 45-degree line, or we may write it 45° line, the small circle at the top, placed after the number meaning degree. By placing one edge of the 30-degree angle against the blade of the tee-square, held in position on the board, a line making an angle of 30 degrees with a horizontal line, or simply a 30-degree line, Fig. 83, may be drawn ; in a similar manner a 60-degree line may be drawn with the 60-degree angle of the triangle. By combining the two triangles as in Figs. 84, 85, a 15-degree line and a 75- degree line may be drawn. We may draw a line or lines parallel to any given line in our drawing, by the use of the two triangles in the following manner : Place one of the triangles with one of its edges exactly on the given line ; place the longest edge of the second triangle against the longer one of the two remain- ing edges of the first triangle; then hold the second triangle in place and slide upon it the first triangle. Fig. 86. Using the triangles in a similar manner we may draw a line or lines at a right angle to anv triven line, thus — Place one edge of either triangle exactly on the given line ; place the longest edge of the second triangle exactly to the longest edge of the first triangle. Hold the second triangle in place and turn the first triangle so that one edge will form a O 46 ROGERS' DRAWING AND DESIGN. right angle with the given line, as in Fig. 87. By placing one edge of the right angle of either tri- angle on the given line, as the first operation, the first trianele will not have to be turned. 6 By sliding the first triangle upon the second one, any number of lines may be drawn which will be at right angles to the given line. With a knowledge of the preceding rules a great ROGERS' DRAWING AND DESIGN. 47 variety of figures may be drawn. In the following we will show how a square, an equilateral triangle, a hexagon and an octagon may be drawn by these simple means. GIVEN LINZ Flu. 86. Fio. 87. ROGERS' DRAWING AND DESIGN. Let it be required to draw a four-sided plane figure, all sides of which are equal and all its angles right angles. Such a figure is called a square. If the sides of the required square should be parallel to the edges of the drawing board, we then draw a horizontal line by means of the tee-square. Fig. 88. On this line we mark two points, the dis- tance between them being equal to the side of the required square. By means of either triangle draw vertical lines through the two points on the first ROGERS' DRAWING AND DESIGN. 49 line ; make one of the vertical lines equal to the first line, that is equal to a side of the square, by drawing a 45-degree line from the foot of one of the vertical lines, to meet theother vertical line, and move the tee-square to the point of intersection of these two lines, where a horizontal line is drawn, meeting the other vertical line and forming the con- cluding side of the required square. If a square is to be drawn on any given line, which is neither horizontal nor vertical (such lines are called oblique lines) we will proceed as follows : Fig. 89, on the given line mark two points, the dis- tance between them being equal to a side of the re- quired square. Through these two points draw lines at rig-ht angfles to the first line. Make one of these sides equal to a side of the required square and draw through the end of it a line parallel to the first line. This line will form the concluding line of the square. Let it be required to draw a square, when the length of its diagonal only is given, Fig. 90. Place the longest edge of the 45-degree triangle exactly on the given diagonal. Place the tee-square with an edge of its blade against one of the short sides of the triangle. By sliding the triangle upon the tee-square held in place, draw two lines through the ends of the given diagonal with the other short edcre of the triano-le. These will form two sides of the square. Bring the triangle back to its first posi- FlG. 90. tion upon the diagonal, hold it in place and remove the tee-square, placing it now against the other short edge of the triangle. Sliding the triangle upon the tee-square, draw the two remaining sides of the square, as before. 50 ROGERS' DRAWING AND DESIGN. To draw an equilateral triangle upon a given line. The angles in an equilateral triangle are 60- degree angles, Fig. 91. Fig. 91. Place the edge of the blade of the tee-square ex- actly on the given line. Place one edge of 60-de- gree angle of the 60-degree triangle to the edge of the tee-square, and draw lines making an angle of 60 degrees with the tee-square, through both ends of the given line. These lines, with the given line, will form the required triangle. To draw a regular hexagfon : Draw a line, AB, Fig. 92 ; set off from any point, O, on the line AB, two distances, AO and OB, each equal to a side of the required hex- agon. Through the points A, O and B draw six parallel lines making angles of 60 degrees with the line AB ; three lines, AE, DOC and FB in one direction, and the other three, CB, EF and AD in the other direction. Join the points E and C and D and F. A E C B F D is the hexagon required. To draw a hexagon on a given line. Let AB be the given line. Fig. 93. Draw the lines AC and BE at an angle of 60 degrees to the given line, in one direction, and the lines AF and BD, at the same angle, in the other direction. At the points A and B on the given line, draw two lines at right angles to this line, these lines cutting the lines, EB and AF, at the points E and F. Join E and F ROGERS' DRAWING AND DESIGN. 51 and through the points E and F draw the Hnes EC and FD at 6o-degree angles to the given Hne, EC cutting the line AC at C and FD cutting the line BD at D. Then A C E F D B is the required hexagon. To draw an octagon on a given line : Let AB be the given line, Fig. 94. At the points A and B draw lines at angles of 45 degrees to the criven line, AC, in one direction, and BH in the other direction. Make AC and BH each equal to the line AB. Through C draw the line CF parallel to BH and through H the line EH parallel to AC. Draw lines through C and through H at right angles to the given line ; the line CD cutting the line BD Fig. 94. at D and the line HG cutting the line AG at G. Through D draw the line DE parallel to CF and cutting EH at E ; then draw through G the line GF parallel to EH and cutting CF at F. Join EF and ACDEFGHBis the required octagon. 52 ROGERS' DRAWING AND DESIGN. ALPHA BETA ANTiqVA 2rasE5?;Siso f Fig. 95. ROGERS' DRAWING AND DESIGN. 53 LETTERING. Let it be said that lettering is intended to convey to the mind of the observer a simple but attractive impression of what the drawing is to express. When the information necessary to the reading of a drawing cannot be expressed by lines and scale dimensions, it must be indicated in the form of printed explanations, remarks, etc., as explained and illustrated in the following pages. Whole volumes have been published upon this most fascinating subject. When writing was the universal mode of expression, that is, before the in- vention of printing — the art of lettering was one of the fine arts. Many manuscripts are now extant whose titles are made upon vellum in inks of gold, scarlet, blue and other gaudy colors ; these have added vastly to the value of the books and aided in their preservation through the long centuries. The illustration upon the opposite page is given as a specimen of one of these ancient {^'antique") alphabets. To do good lettering is not an easy task, and unless the student is already experienced he should devote much time to practicing the art, working slowly and bearing in mind that much time is required to make well-finished letters. Lettering of various styles are in use, some quite simple and others difficult, especially in cases where an ornamental heading is required, but it must be remembered that a drawing is primarily made to convey an idea, and not for an ornament. The character and size of the letters on all work- ing drawings should be in harmony with the draw- ing on which they appear. It is desirable to have all lettering on a drawing made in the same style, only differing in size or finish of details. Capital letters should always be sketched in pencil, especially by the beginner, and inked in afterwards ; the lettering used on mechanical draw- ings is usually of the simplest character, the letters being composed of heavy and light strokes only ; for headings, titles of large drawings, where com- paratively large lettering is required, it will be most appropriate to use large letters. The title should be conspicuous, but not too much so ; sub-titles should be made smaller than the main-title. The " Scale " and general remarks placed in the margin of the drawing or near the title should come next in size. All explanations and remarks on the views should not be larger than one-eighth inch. The examples of lettering given as illustrations are briefly explained on page 63. 54 ROGERS' DRAWING AND DESIGN. n " "IS ''■ .S " :::5 ::::: ::ii:::2::i5 : :::: : — hv- "]a:_:_!.__ - >- ^-1^ — :[?3i:- ::3i::::i: " II ^ 11^ :: III - ~ -I 3t-iC "^-t X :: zz-l-Az ±S.i:i:: i:::^::±:: :::ii :s^:f3::::::::::: ^ TT Un^ N U1 I I !SI IS m m a ^~^ s s » 5 tzt Fig. 90. E ;^ iii IS I I s FW ? B z E SB S a ^ 5 ^ ZuS Fig. 97. ROGERS' DRAWING AND DESIGN. 55 a Fig. 98. (See page 63.) Fig. 99 shows one of the devices in use for facilitating the laying out of letters ; this instrument is known as the lettering triangle and may be made of metal, hard rubber, celluloid, etc. The broken line a a contains lines of different inclinations by which the slanting parts of those letters, shown on the triangle, may be laid out. The highest inclined line may be used for the slanting strokes of the letter K ; the inclined line, situated next to the highest is intended to be used for the letters N, X and Y; the next inclined line is to be used for the drawing of the letters A, M and V and the lowest inclined line is used for the letter W. Other triangles and templates have been made for laying out lettering of different character, of which the example given is one of several in com- ^ mon use. Fig. 99. 56 ROGERS' DRAWING AND DESIGN. Fig. 100. The above figure, loo, shows a pen made spec- ially for round writing- upon drawings ; while nearly all lettering is executed by a common writing pen this device deserves a description. The " nibs " of the pen are cut off, leaving a short, straight line at the point ; the width of this point is equal to the greatest thickness of a line which may be desired for the letters ; these pens are manufactured in various widths and numbers ; No. I is made for a stroke of about ys" in thick- ness ; the highest numbers are made with a point nearly like the ordinary writing pen point. The No. I pen may be used for capital letters, about one inch high and for small letters about }4" high ; the pen is always held parallel to the line ad at 45°, Fig. loo. A motion of the pen in the direction of the line a b produces a fine line ; all strokes, light or heavy, are made by means of the whole width and not by only one edge of the pen ; heavy strokes are made with the pen moving in the direction c d, Fig. lOo, with the whole width of the pen. The pen should move smoothly over the paper without any special pressure being brought to bear on it. Vertical strokes produced by a downward motion of the pen will not be quite so wide as the line c d. All strokes should be executed with an unaltered position of the pen "nibs," which must remain par- allel to the direction of the line ad and inclined about 45°. Letters containing circular curves are made with the pen in the same position ; a circle should not be made by one continuous motion of the pen ; it should be formed of two semicircles, taking care to smoothly join the two semicircles. ' It is well for the beginner to lay out a number of squares in pencil and to practice the circular strokes within the squares ; the completed circle should be contained within the square. The light strokes will be parallel to the diagonal of the square, the vertical stroke should be parallel to the vertical sides of the square. The attractive appearance of the lettering will entirely depend upon the correct- ROGERS' DRAWING AND DESIGN. 57 ness of connecting the semicircles and straight lines of which the letters are composed. The size and thickness of the writing depend on the width of the pens and cannot be arbitrarily ex- ecuted by means of the same pen, without distort- ing the regular form of the characters. The pen must at all times be kept clean, as other- wise no clean-cut line can be obtained. The ink should be kept only on the outside or upper side of the pen, and its bottom side should be kept perfectly dry. As soon as the draughtsman notices that the bottom of the pen becomes wet he should cease writing with it, as it will produce an uneven line. The letters should be made with plain, even, clear-cut lines, well proportioned in all parts and especially well spaced. A special device, called an " inkholder," is used in order to keep a sufficient quantity of ink on the upper side of the pen. Free hand lettering should only be taken up after the student is proficient in mechanical lettering ; pencil guide-lines for letters and words should be drawn ; larger letters may first be penciled in very lightly, and an ordinary writing pen may be used for inking them in. A 3 CDEF'GMlJ^LJyrjVOP QRSTU VWXYZ /S3436?'890 abcdey^hiJklrrirLopqrst-uvzoacyJZ COJ^J^ECTJJVGROD. Fig. 101. (See page e,3.) Letters should be so placed as not to interfere with the lines of the drawing and should clearly point out the part intended to be described. When single letters are used, they should be inked in be- fore the shade or section lines are drawn ; it is a good plan to start with the middle letter of the in- scription and work in both directions. The use of both the writing and drawing pen enables the lettering to be done in a much shorter time ; when the ruling pen is employed it is in con- nection with the tee-square and the set square. The appearance of a drawing will often be helped by a border put on in connection with the lettering. 58 ROGERS' DRAWING AND DESIGN. The four principal styles of letters used in mechan- ical drawing are Block, Roman, Old English and Script, each of which will be found illustrated under tendency is in the direction of simply designed letters, legibility being considered of vital impor- tance. ABCDEFGHIJKLMN O P QRST UVWXYZa. s^B Cn£:rGHI.JKLM NOPQRS TUVWXYZ abcdefghi/klmnopgrs t uvwxyz J234567a90i 1234567890^ Fig. 102. (See page es.) I this section of the work ; it will be found upon in- vestigation that most letters in use to-day are founded upon one of these four styles ; the modern It will familiarize the student with the standard alphabets in Roman, Block, Old English and Script to copy the several specimens given. ROGERS' DRAWING AND DESIGN. 59 The space between the nearest parts of all letters should be exactly alike ; this rule also applies to the space between each word ; between the words, of course, should be wider than the letter spaces. A reasonable space (never less than one-third the height of the letters used), should be left between lines of words. Fig. 103. (See paae os.) 60 ROGERS' DRAWING AND DESIGN. Mathematical accuracy should be aimed at as a rule in all lettering executed for mechanical drawings. A knowledge of punctuation, spelling, capitalization and paragraphing is essential in for the bottom and then the letters should be sketched with the utniost care ; the outlines may be ruled with a ruling pen, if desired, and the curved lines drawn with a compass ruling pen. Fig. 104. (See page r,s.) i.n.ffl.iv:y.vi.vn.Yni.K.x.ix.isx.xL.L.xc.c.D.M. ^ ■r Fia. 105. -5v. J'^ so , ■^oc JHo. -fcee- this work ; if unfamiliar with these subjects the student should acquire a. thorough knowledge of them. Perfectly horizontal ruled lines should first be drawn, one for the top of a line of letters, another The heavy or shaded stems of letters should all be of the same width ; after the outlines have been carefully penned in, the unfilled spaces may be ''brushed" in with either liquid India ink or very black water color. ROGERS' DRAWING AND DESIGN. 61 3 4 3 JBCDEFGHIJFCULrOPQRSTUF ffXYZ &. abcdef'gh ijklmnojjq rfstuvwxyz. 6 7 S 9 Fig. lOB. at 34 Fig. 107. 62 ROGERS' DRAWING AND DESIGN. ''Mlf^ Fig. 108 ROGERS' DRAWING AND DESIGN. 63 In Figures 96 and 97 letters and numerals. are examples of block The squares are laid off in fine pencil lines and the desired letters may be sketched on the drawing in pencil and then inked in with pen and triangle ; when the inking is completed all the pencil lines are erased. This is a rudimentary form of letter that can be made with the aid of cross section paper. In Figs. 98 and 103 are shown two styles of free- hand lettering. The vertical letters are more diffi- cult to draw than the slanting ones. When making these letters two fine pencil lines should always be drawn, one at the top and one at the bottom of the letters and sometimes it is very convenient to rule a third guide line midway between the two others. These examples exhibit a form of lettering known as round writing ; the easy way to master it and its artistic appearance, combined with the rapidity with which it can be written, are its principal merits. In the upper part of Fig. 102, page 58, is shown another example of the block letter ; this is very distinct and readily executed by the aid of the drawing pen. In the lower part of the figure (102) are shown Italic letters and numerals; the proper angle for their slant is 23°. In Fig. 104 are shown ornamental letters based upon the Roman ; in the Roman letters the square is taken as the basis of construction; W takes the whole square, its height and width being equal ; I is one-quarter as wide ; A five-sixths, etc. In Fig. 105 are shown the form and proportions of the Roman numerals and their value in the Arabic method of expressing numbers. In Fig. 106 is given another example of Italic letters and numerals. In Figs. 107 and 108 are given illustrations of script letters and Figs. 109 and 1 10 will suggest to the student still other forms. m The letters shown in Fig. loi are constructed a simple form convenient for remarks, etc., needed to be placed in the margin of the drawing. 64 ROGERS' DRAWING AND DESIGN. ffl 1 C^ I 2 OD \P i <» s t 5 Fro. 109. [F (S) [^ § q] D° © i 1^ ^^ ^ 1 d Da D J) K [L T 1 W M H V gj Od a D n Ds - It DD W \w n ^ IG- 110. 1 © S) (D DQQ ROGERS' DRAWING AND DESIGN. 65 SHADE LINES. In instrumental drawings shade lines are used for the purpose of making the reading of a drawing easier than if all lines were of the same thickness. By means of these, the draughtsman knows with- out referring to any other view of the object whether the part looked at is above or below the plane of the surface ; for instance, the rectangles in Fig. 1 1 1 represent square projecting pieces, whereas the the plane of the paper, and also with all vertical and horizontal lines of the drawing, and to come from the upper left-hand corner of the drawing ; the direction of the light being indicated by the slanting edge of the 45° triangle as shown in Fig. 1 13. All the rays of light are not supposed to be emanating from one and the same point, but from a large and distant source of light and are thrown in parallel lines. The shade lines are the edges of such surfaces as rectangle differenc In ore uniform, assumed such a w Fio. 111. ales ; the de lines. may be light are ction, in rees with s m r e bein ler th; and t( to COl ay as t^ig. II y made it the 3 avoid ne in a to mak 2 reprc appan shading confus single 2 an an isent sc ;nt by t y on di ion the invaria gle of I uare h( he sha ■awlngs rays of Die dire 15 deg Fio. 112. " relief " and aid the reader or the student in under- standing the true character of the object with greater facility than could be done on drawings with all lines of one and the same thickness. 66 ROGERS' DRAWING AND DESIGN. The following rules should be strictly adhered to by the student in shading drawings : (fl). The rays of light are assumed to make an angle of 45 degrees with the plane of the paper and to come from the upper left-hand corner, at an angle of 45° with all horizontal and vertical lines as previously mentioned. (b). Each view of the subject should be consid- ered as a top view for the purpose of shading ; its top part will thus be exposed to the light. (c). Lines representing edges which cast shadows are to be drawn in heavy lines. (d). All the edges formed by the intersection of a light and dark surface or two dark surfaces, are to be shaded. Figs. 114 and 115 show two views of a square block or rectangular prism. In the top view, abed, the rays of light fall upon the rear side of the object, b c, upon the top, abed, and upon the left side, a b ; the light does not reach the front and right sides, a d, and, d c ; hence they are dark sur- faces ; the edges a d and d c, separating the light surface, abed, from the dark surfaces are there- fore shade lines. The explanations given in regard to the top view can also be applied to the other view of the same object and the lines e h and h g will thuS be ob- tained as shade lines. Fig. 113. ROGERS' DRAWING AND DESIGN. 67 Figs. ii6 and 117 represent a hollow square- b prism ; its top view shows the shade lines a d and d c, that is, the right and bottom sides of the view c upon which the rays of light do not fall ; the lines b a d f ? u Fig. U4. a A f Fui. 116. 9 i : m ^ m r 8 M P n Fiu. iir. 1 e f and f g are also shade lines for the same reason ; e h the vertical section r i p shows the right side p Fig. 115. and bottom r p, also the vertical line k 1, as shade lines. 68 ROGERS' DRAWING AND DESIGN. Fig. 118. Fig. 119. Another example of shade lines is given in Figs. ii8 and 119; in this case the top view shows a hexagonal hole, offering a good opportunity to con- sider the shading of objects with inclined planes. It will be plainly seen that the lines c d and b c must be shade lines and the lines a f and f e, situ- ated directly in the way of the rays of light, are light lines. The two remaining sides b a and d e of the hexagonal hole cannot readily be put down as shade or light lines. In order to find out^ their nature draw a number of 45" parallel lines oVer the figure in question and those faces of the hexagonal hole reached by the rays of light, represented by the 45° lines. It will be seen that the edge b a intersects the ar- rows of light ; the face a b will therefore be a dark surface, and consequently must be shaded. It will also be noticed that the rays of light fall directly upon the face, an edge of which is the line e d, thus making this latter a light line. In order to illustrate more clearly the shading of lines belonging to planes, which are inclined in va- rious degrees to the direction of the light, the top view of a cube placed in three different positions is shown in Figs. 120, 121 and 122. In Fig. 120 the edge a b makes an angle of more than 45° with a ROGERS' DRAWING AND DESIGN. 69 horizontal line ; parallel lines drawn at an angle of 45° representing the rays of the light will strike the side of which a b is an edge ; this side must there- fore be a light surface and the line will be a light one. In Fig. 12 1 thelineab forms an angle of 45 degrees with a horizontal line showing that the side of the cube of which a b is an edge is placed in a position than 45 degrees with a horizontal line. It will be observed that the side of which a b is an edge can- not receive any direct light, as the rays are broken by the edge of which b is the highest point; hence the edge b a is a shade line. In Fig. 123 are shown the front and bottom views of a cylinder ; the shade lines on the front are easily Fio. 120. Fig. 121. Fig. 122. parallel to the direction of the light ; for this reason said side is considered a ligrht surface and the line a b is a light line. This is done in every case in which the line in question forms an angle of 45 degrees with a horizontal line. The same is true for the line c d, which is parallel to a b. In Fig. 122 the edge a b forms an angle of less determined, in a similar manner to foregoing cases where a drawing has been made of a rectangle. Many draughtsmen, however, will only shade the plane sides of the cylinder, claiming that shade lines are intended to represent edges only; ac- cording to this view the bottom line of the eleva- tion of the cylinder should only be shaded. 70 ROGERS' DRAWING AND DESIGN, At times a tendency has been noticed to give the shade lines a more general application. As shade lines are always found on all right and bottom sides of those parts situated in front of the surrounding surface, it is very easy to recognize this condition in every similar case, whether the part in question be bounded by plane or by cylindrical surfaces. It is therefore recommended that shade lines be drawn in each case, independently of the character of surface, with very few exceptions, when a rigid adherence to this rule will tend to produce a bad effect. The shaded portion in the bottom view of the cylinder, Fig. 123, shows the manner in which circles are to be shaded when they represent projections of cylinders or circular holes. Fig. 124 shows that the circle is shaded between the points of tangency of the two 45-degree lines a b and f g ; the heaviest part of the shaded circle is near the 45-degree line c d e, passing through the center of the circle. The thickness of the lines should gradually decrease from e toward b and to- ward g ; to obtain the best result with neatness is to shift the center point of the compasses along the line c e, a distance equal to the thickness of the de- sired line. With the same radius used to describe the original circle describe now part of another circle, being careful not to run over the first one and to stop when the two lines coincide. I ? Fig. 123. The shade line made in this way must not be drawn too heavy ; to assure the success of this op- ROGERS' DRAWING AND DESIGN. 71 eration it is necessary to have a very sharp needle point in the compass in order not to cause too large a hole in the center of the original circle. The shifting of the center may be avoided when drawing the shade line in the following manner: When the original circle is drawn keep the center- FlG. 121. point in the center and without changing the radius put the pen point in motion in the direction of the part of the circumference to be shaded. The pres- sure upon the pen is gradually increased as it ap- proa'ches the heaviest part of the shade line and then gradually diminished. Figs. 125 and 126 show the manner of shading a circular hole. The directions for this operation are the same as for shading the projections of a cylinder base, except that the opposite half of the circle is FiQS. 125 AND 126. shaded, which is done by shifting the center in the opposite direction. 72 ROGERS' DRAWING AND DESIGN. In order to make the conventional way of putting in shade lines more easily understood, a few illus- trations are added for the benefit of the student. In Fig. 127, the position of the shade lines in the top view is quite plain ; in the front view, Fig. 128, the bottom line a b of the smaller block placed in throws a shadow upori the corresponding part of the front face of the larger block and the line a b is therefore a shade line. Similar cases are given in Figs. 133 to 138. In the front view. Fig. 136, a portion a c of the bottom line a b is shaded and in Fig. 138 the whole a h 1 a h Ui^J i J i Figs. 127 and 128. Figs. 139 and 130. Figs. 131 and 132. the middle of the top face of the larger block is a light line. In Figs. 129 and 130, the small block placed on top of the larger one so that the front faces of both are in one plane shows a light line a b. In Figs. 131 and 132, the smaller block is moved forward so that its lower face a b is the front edge. of the bottom line a b is a shade line. Fig. 139 shows the front and top views of a prism. The shading of the top view does not present any new points. In the front view the face e b c f is dark ; the edge e b separating the light face c^a b c from the dark one and consequently e b must be a shade line ; e f is also a shade line for reasons ex- ROGERS' DRAWING AND DESIGN. 73 FiciS. 133 AND VU. Figs. 1.3.5 and 136. plained previously. The line b c of the upper base a c should also be shaded ; this would make part of the straight line a c heavy and the greater part light, which would produce a very odd effect and therefore the whole line, in similar cases, is drawn light. The line c f is made h e a V }' b )' many draughtsmen, as the surrounding of dark surfaces by heavy lines Figs. 1.37 and 138. L_ a b brings out such surfaces much stronger. From the various examples given above it will be seen that no special rules can be given for shading, that is, rules that would cover all cases likely to arise.' The conventional practice introduces a great variety of exceptions to any rules designed for this 7^ b i e Fid. 139. purpose. The draughtsman has to keep in mind the true purpose of putting in shade lines and place such lines where and whenever, in his opinion, they will serve as an aid to the understanding of the drawing. 74 ROGERS' DRAWING AND DESIGN, PARALLEL LINE SHADING. Plane surfaces are shaded by a number of par- allel lines running parallel to the length of the plane which is to be shaded. If the plane is to be repre- sented very light, it may be left blank or coveretl with very fine parallel lines, as shown in Fig. 140. A dark plane is shaded iiy a number of heavy parallel lines, Fig. 141. which does not receive any direct light. The heavy lines become lighter gradually and are drawn very fine near the midd'e of the cylinder; after this the lines are again dra.vn slightly heavier up to the side of the cylinder, which is nearest to the source of the lis>ht. The shadintr lines near the liofhter side of the cylinder should never be as heavy as the heaviest lines on the dark side of the cylinder ; this is illustrated in Figs. 143, 144 and 145. The surface Fig. 140. Fig. 141. Fio. 142. If the plane is parallel to the plane of the paper, the shading lines should be drawn with equal spaces between them throughout the full width of the plane. If the shaded plane is inclined to the plane of the paper it is shaded by a number of lines, with the spaces between these lines graciually increasing, while the thickness of the lines gradually decreases as may be seen in Fig. 142. A cylinder is shaded by a number of parallel lines, whjch are heaviest near to the side of the cylinder near the middle of the cylinder is often left blank, as it is difhcult to produce the effect of a light tint which is desirable at that place. A hollow cylinder or a concave surface is shaded similar to a cylinder, as shown in Fig. 146. The view of the sleeve nut shown in Fig. 147 illustrates the manner in which conical surfaces are shaded. Some draughtsmen do this by drawing the shading lines parallel to the outside elements of the cone. A somewhat better result is produced, how- ROGERS' DRAWING AND DESIGN. 75 ever, by drawing the lines slanting and tapering to the vertex of the cone, virhich is to l^e shaded. Wherever possible an ordinary pin may be put into the board exactly in the vertex of the cone. Tlie Fir.. 146. ruling edge of the triangle is thus easily kept against the pin, securing the proper direction for the tapering shading lines. 76 ROGERS' DRAWING AND DESIGN. In Fig. 148 at a b is shown a cylinder placed in a horizontal position, which is slightly rounded at the end, so as not to have any sharp edge. This Is also indicated by shading lines drawn at right angles to the shading lines of the cylinder. FiQ. U" Fig. 148. Fig. 150. Fig. 151. ROGERS' DRAWING AND DESIGN. 77 Fig. 149 shows how this may be done by the aid of curved lines ; however the time .required for this method does not recommend it for ordinary work- ing drawings ; the same figure includes a spherical surface and shows how such surface may be shaded. Fig. 150 shows the shading of a curved cylinder. Fig. 151 shows a method of representation of knurled surfaces. The spacing of the inclined lines varies, being closer near the sides of the figure. In conclusion let it be noted that the best effects are, as a rule, produced by the fewest lines ; draw- ings executed to small scale will look best with a shading that does not include any very heavy lines ; larger scale drawings require the use of very heavy shading lines. In ordinary working drawings shading is, as a rule, but very little employed ; it is, however, some- times done to shade the surface of shafts and even bolts as well as other cylindrical parts of small diameter by a few conveniently placed lines. SECTION LINING. It is sometimes necessary to make use of a sec- tion in order that certain details, which would otherwise be hidden, may be shown in a plain. Fju. ir,:;. 78 ROGERS' DRAWING AND DESIGN. concise manner. The method used in shops, and the best for most purposes, consists of drawing parallel lines within the section, which lines are usually inclined 45 degrees. By changing the direction of these lines a clear distinction may be made between different pieces in the same view, which may be in contact. A difference in material is shown by a variation of the character of the sectioning, see Figs. 152 and 153. The section lines are best drawn from left to right or from right to left, usually inclined 45 degrees and about one-sixteenth inch apart. For large drawings the spaces between them may be as much as one-eighth inch. Placing the lines too near together makes the work of sectioning much harder ; the lines should not be drawn first in pencil, but only in ink, as the neat appearance of the drawing depends largely upon the uniformity of the lines in the section and these lines are to be spaced by the eye only. The process consists simply in ruling one line after another, sliding the triangle along the edge of the tee-square for an equal distance after drawing each section line. Figs. 154-162 inclusive, are examples of section lining quite generally used. Fig. 154. Fig. 155. ROGERS' DRAWING AND DESIGN. 79 Fig. lot). c Fig. 15s. Fig. 157. Fig. 159. 80 ROGERS' DRAWING AND DESIGN. Cast iron is indicated by a series of parallel lines of medium thickness, equally distant apart as shown in Fig. 154. Wrought iron is sectioned in the same manner as cast iron except that every alternate line is a heavy line, Fig. 155. Cast steel is sectioned by drawing two lines, of medium thickness close together, and the third line about one and one-half times as far from the second as the second is from the first and so on as shown in Fig. 156. Brass is sectioned by parallel lines similar to cast iron, except that every other line is broken; see Fig. 157. Babbit is sectioned like cast iron in both directions, forming little squares, Fig. 158. Wrought steel is sectioned by two light lines and one single heavy line. The light lines should be drawn similarly to those in Fig. 156 for cast steel, and the heavy line should be about one and one-half times as far from the light lines as the distance between them, as shown in Fig. 159. Wooden beams are sectioned by a series of rines and radiating lines in imitation of the natural appearance of a cross section of an oak tree. Fig. 160. A beam or board is represented by lines run- ning similarly to those of the grain in an oak board. Fig. 160. Brick and stone are represented as shown in Figs. 161 and 162. Xbin strips of metal Vikc the stct'ion of boiler plates may be sectioned in the ordinary way by the Fig. 160. usual section lines ; but as this requires consider- able work and produces an ill effect in the drawing, It is often better to fill in the whole sectional area with solid black. In this case a white line must be left between the adjoining pieces ; this method is recommended only for small sections, see Figs. 163 and 164. ROGERS' DRAWING AND DESIGN. 81 w///////////////////mw//. w//mm//y/M//////////A %^.^%^^^^^%%^^ ^m^^m^^M^MM^ Fig. 161. Fia. 162. Fig. 163. Fio. 164. s&. ! lllllllMllltliiiii IIMIIIIM^^ ■— ^^^^^ e iiiiiiiiiiiiiiiiiiii GEOMETRICAL DRAWING. Geometry is the science of measurement ; it is the root from which all mechanical drawings issue ; the principles involved in the following problems, make up the fundamental bases of all instru- mental drawing, as well as all "laying out" of work in the shop, where great accuracy is required. The elementary conceptions of geometry relate to the simple properties of straight lines, circles, plain surfaces, solids bounded by plain surfaces, the sphere, the cylinder and the right cone. Higher geometry is that part of the science which treats of the relations of these to lines, circles, surfaces, etc. Some geometrical terms have already been described, to these are now added a few relating to the more advanced parts of this oldest and simplest of sciences. An axiom is a self-evident truth, not only too simple to require, but too simple to admit of dem.onstration A proposition is something which is either proposed to be done, or to be demonstrated, and is either a problem or a theorem. A problem is something proposed to be done. A theorem is something proposed to be demonstrated. A hypothesis is a supposition made with a view to draw from it some consequence which establishes the truth or falsehood of a proposition, or solves a problem. A lemma is something which is premised, or demonstrated, in order to render what follows more easy. A corollary is a consequent truth derived immediately from some preceding truth or demon- stration. A scholium is a remark or observation made upon something going before it. A postulate is a problem, the solution of which is self-evident. 8.1 86 ROGERS' DRAWING AND DESIGN. EXAMPLES OF POSTULATES. Let it be granted — I. That a straight line can be drawn from any one point to any other point ; n. That a straight line can be produced to any distance, or terminated at any point ; in. That the circumference of a circle can be described about any center, at any distance from that center. AXIOMS. L Things which are equal to the same thing are equal to each other. /, n. When equals are added to equals the two or more wholes are equal. [ in. When equals are taken from equals the remainders are equal. IV. When equals are added to unequals the wholes are unequal. V. When equals are taken from unequals the remainders are unequal. VI. Things which are double of the same thing, or equal things are equal to each other. VII. Things which are halves of the same thing, or of equal things, are equal to each other. VIII. The whole is greater than any of its parts. IX. Every whole is equal to all its parts taken together. X. Things which coincide, or fill the same space, are identical, or mutually equal in all their parts. XI. All right angles are equal to one another. XII. A straight line is the shortest distance between two points. XIII. Two straight lines cannot enclose a space. The tools used in geometrical drawing are the compass, with pencil and pen points, the ruling pen, straight edge and scales ; in the following pages will be found a series of exercises which have been selected with a view to their importance in their application in problems of accurate drawing. ROGERS' DRAWING AND DESIGN. 87 EXERQSES IN GEOMETRICAL DRAWING. To bisect a given straight line; that is, to divide it into two equal parts. Let AB be the given line, Fig. 165. / / / / B ;c Fig. ]&5. From A as a center with a radius g-reater than one-half of the given line AB, describe the arc i 2, From B as a center, and with the same radius, describe an arc, cutting the former at i and 2 ; then through the points of intersection draw the line 1C2 and it will divide the line AB into two equal parts at the point C. To bisect a given angle ; that is, to divide a given angle into two equal angles. Let ACB be the given angle. Fig. 166. With the vertex C as a center, and any radius, describe an arc cutting both sides of the given angle at I and 2. From i and 2 as centers, with any radius, describe arcs cutting each other at 3. 88 ROGERS' DRAWING AND DESIGN. Through this point of intersection draw the line 3C and it will bisect the angle as required. To divide a given angle into four equal parts. Let ACB be the given angle, Fig. 167. Bisect the given angle as described in Problem 2 by the line 3C. Bisect the angles 3CB and 3CA by the lines C4 and C5 and these lines divide the angle into four equal angles as required. To trisect a right angle ; that is, to divide it into three equal parts. Let ABC be a right angle, Fig. 168, that is, an angle with the sides perpendicular to each other. From B as a center with any radius, describe an arc cutting the sides of the angle at i and 4. B Fig. 16S. '4 *^ With the same radius and with 4 as a center, describe an arc cutting the former at 2. From i as a center with the same radius, cut the arc at 3. Through the points 2 and 3 draw the lines 2B and 3B and they will divide the angle into three equal parts as required. ROGERS' DRAWING AND DESIGN. 89 To draw a line perpendicular to a given straight line from a given point in that line ; that is, to erect a perpendicular to the given line at a given point in that line. Let AB be the given line and C the given point in that line, Fig. 169. 3. B C FUi. IfiH. With any radius set off on each side of the point C, equal distances, as Ci and C2. From the points I and 2 as centers, with any radius greater than Ci or C2, describe arcs cutting each other at 3. Through the point of intersection draw the line 3C, which will be perpendicular to the line AB. To draw a perpendicular line to a straight line, from a given point without that line ; that is, to drop a perpendicular to a given line from a point with- out it. Let AB be the given line and C the given point, Fig. 170. 1 ^^. \ \ / D B f / / Fig. 170. From C as a center with any radius extending below the line AB describe an arc i 2, cutting AB at I and 2. From i and 2 as centers, with the same or any other equal radii, describe arcs cutting each other at 3. Through the point C and the point of intersection 3 draw the line 3DC ; then the line CD will be perpendicular to AB. 90 ROGERS' DRAWING AND DESIGN. To drop a perpendicular to a given line front a point which is nearly over the end of the line, Fig. i-ji. Let AB be the given line and C the given point. From any point i on the line AB as a center, with the radius iC describe the arc CE. \ / \ \ \ V\ > / / / Fig. 171. From any other point 2 on the line AB as a center, describe arcs cutting the former arc at C and E. Draw a line through the points C and E and the line CE will be the perpendicular required. Through a given point to draw a straight line parallel to a given straight line. Let AB be the given line and C the given point, Fig. 17:2. From C as a center with any radius describe the arc I, 2, cutting the line AB at 2. z' y' II \ \ \ 7^ \ X \ \:i \ Fig. 172. With the same radius and 2 as a center, describe the arc C3. On the arc 2, i, set off from 2 the chord of the arc 3C, cutting it at i. Through the points C and i draw a straight line DiCE and it will be parallel to AB. ROGERS' DRAWING AND DESIGN. 91 To divide a straight line into any required number of equal parts (^say y equal parts\ Let AB be the given line, Fig. 173. From A draw a straight line AC forming any angle with AB and being of any length. Set the dividers to any convenient distance and set off seven equal divisions on the line AC beginning at A up to the point 7. 6 Jf >• / / / / / •^ / / / ' .^ I I I I I I I X / U Fig. 1T3. Join the points 7 and B by a straight line and draw parallels to it through the points i, 2, 3, 4, 5, 6, and these lines will divide the given line AB into the required number of parts. To divide a given line AB into three and a half equal parts. Let AB be the given line, Fig. 174. Draw a line AC forming any angle with the given line AB. Upon AC set off 7 equal parts, be- ginning at A up to the point 7. — C Join the points 7 and B and through the alter- nate points, 5, 3, I, draw lines parallel to 7B. These lines will divide the given line AB into 3^ equal parts, as required. 92 ROGERS' DRAWING AND DESIGN. To draw upon a straight line an angle which shall be equal to a given angle. Let 1E2 be the given angle and AB the line upon which we intend to draw an angle equal to the given one, Fig. 175. B Fig. 175. From E as a center describe an arc i, 2, with any convenient radius. From any point on the line AB, say from C, as a center, and with the same radius describe the arc 3, 4. From 4 as a center, with a radius equal to i, 2, intersect the arc 4, 3, at 3. A line drawn through the points 3 and C will form with the line AB the required angle. To construct an equilateral triangle, the length of a side being given. Let the straight line AB be the given side, Fig. 176. B Fio. 176. From the points A and B with a radius equal to AB describe arcs cutting each other at C. Draw the lines AC and BC ; then will the triangle ABC be the required equilateral triangle. ROGERS' DRAWING AND DESIGN. 93 To cojisiruci an equilateral triangle, the vertical height or altittide being given. Let AB be the given vertical height, Fig. 177. Through the point B draw a straight Hne CD perpendicular to AB. Through the point A draw another straight line, EF, parallel to CD. From B as center with any convenient radius describe a semicircle cutting- CD at I and 4. From i and 4 as centers, with the same radius, cut the semicircle at 2 and 3. From B and through the points 2 and 3 draw the lines BG and BH ; then GBH will be the required triangle. To construct an isosceles triangle, with a base equal to a given straight line, and each of the two angles at the base equal to a given angle. Let D be the given line and E the given angle, Fig. 178. Fig. 178. Draw a line, AB, equal to the given line D. At the points A and B construct angles equal to the given angle E. Continue the sides of the angles until they meet at C ; then ABC will be the re- quired triangle. 94 ROGERS' DRAWING AND DESIGN. Two sides and the angle between them being given to construct the triangle. Let D and E be the two given lines equal re- spectively to two sides of the required triangle, and F the given angle, Fig. i 79. U Fig. 179. Draw a line, AB, equal to D, and at the point A construct an angle equal to F and make AC equal to E. Join the points C and B by a straight line, and ABC will then be the required triangle. Two sides and the (Lngle opposite one of them being given to construct a required triangle. Let D and E be the two given sides and let E be the side opposite which the angle is to be formed equal to F, Fig. 180. Fig. 181. Draw a line, AB, equal to D. At the point A form an angle equal to F. With the point B as a center and a radius equal to the given line E de- scribe an arc cutting AC at C. Join the points C and B. ABC is the required triangle. ROGERS' DRAWING AND DESIGN. 95 To conslruct a square, the sides of which shall be To construct a square its diagonal being given. equal to a given line. (See definition, page 31.) Let AB be the given line, Fig. 181. Let BD be the given length of a diagonal, Fig. 182. At the point A erect a perpendicular AD (see Bisect the diagonal BD at the point P by the page 89) equal in length to AB. straight line AC. 1 D \ c_ \ N ^' \ \ \ \ / k" . C /P\ \ \ y- \ \ \ A \ \ B \ \ / \ / / \ \ X ^AV '\ B 1 1 Fu;. 1S2. Fig. 1»1. From the points B and D as centers, with a radius equal to AB, describe two arcs cutting each From P as a center with a radius equal to PB, or other at C. Connect D and C by a straight line PD, cut the line AC at the points A and C. Join and B and C by a straight line, and ABCD is the points AB, BC, CD and DA, and ABCD will the required square. be the required square. 96 ROGERS' DRAWING AND DESIGN. To construct a rectangle whose sides shall be equal to two given lines. (See definition, page 31.) Let AB and CD be the given lines, Fig. 183. Draw a straight line EF equal to AB, from E draw EH penpendicular to EF and equal to CD. D B Fig. 183. From H and F as centers with radii equal to AB and CD describe arcs intersecting at G. Join the points FG and HG; then EFGH is the required rectangle. To construct a parallelogram when the sides and one of the angles are given. (See definition, page 31.) Let AB and CD be the given sides and O the given angle, Fig. 184. Draw a straight line, EF, equal to AB. At E draw an angle equal to the given angle O. Make the side, HE, of this angle equal in length to CD. B Fig. 184. From the point F with a radius equal to CD and from H with AB as a radius describe arcs in- tersecting at G. Join HG and FG. EFGH is the required parallelogram. ROGERS' DRAWING AND DESIGN. 97 To construct a parallelogram when the sides and one of the diagonals are given. Fig. 185. Let CD be the given diagonal and AB and EF the lengths of the two sides. E- A- Fig. 185. Draw a line, GK, equal to the given diagonal CD. From G and K as centers, with radii equal in length to AB and EF describe arcs intersecting at L and H. Join GL, LK, KH and HG. GHKL is the required parallelogram. To find the center of a given arc, its radius being given. (See definition, page 33.) Let AB be the given arc and E the radius, Fig. 186. Fig. 186. From any two points A and B on the given arc, as centers, with a distance equal to the radius E describe arcs intersecting at C ; then C will be the required center. 98 ROGERS' DRAWING AND DESIGN. To find the center and to describe the circle, three of whose points are given ; that 7S, to describe the circumference passing through three given points. Fig. 187. Let A, B and C be the given three points, Fig. 187. With A, B and C as centers and any convenient radius, draw arcs cutting each other at D and E and at K and L, and through the points of their in- tersection draw lines KO and DO ; the intersection of these lines at O is the required center. With O as a center and OA as a radius, describe the re- quired circle. Fig. 188. To draw a tangent to a circle, passing through a given point on the circumference. (See definition, page 34.) Let A be the given point on the given circum- ference, Fig. 188. From A to the center O of the circle, draw the radius AO. Through A draw the line BC perpen- dicular to AO. The line BC is the required tangent. ROGERS' DRAWING AND DESIGN. 99 To draw a tangent to a circle from a given point without the circumference. Let A be the given point and C the center of the given circle. Fig. 189. Fin. 1X9. Join AC and bisect it at O. From O as center, with a radius equal to OC or OA describe a semi- circle, cutting the given circle at D. The required tangent is a line passing through A and D. To draw lines tangent to two given circles. Case I. — From O, the center of the larger circle, Fig. 190, draw any radius OE on which set off from E, a distance EG equal to the radius of the smaller circle. With O as a center and OG as radius de- .scribe the circle GHI and draw tangents PH and Fio. Ifll). PI to this circle from the center P of the other circle. (See the preceding problem.) From O and P draw perpendiculars to these tan- gents and continue them until they cut the given circles at AB and CD. Join the points. The lines AB and CD are the required tangents. 100 ROGERS' DRAWING AND DESIGN. To draw lines tangent to two given circles : Case II. — From O, the center of one of the given circles, Fig. 191, draw any radius OE and lengthen it outside of the circle up to G, making the distance EG equal to the radius of the other circle. From O as center and OG as radius, describe the circle GHI ; draw tangents PH and PI to this circle from the center P of the other circle. Draw perpendiculars to these tangents from O and P and they cut the given circles at the points A BCD. The lines joining the points A and B and C and D are the required tangents. To inscribe a square in a given circle; that is, to draiv a square within the circle, with all the vertices of its angles resting on the circumference. Let ABCD be the given circle. Fig. 192. B Draw two diameters, AC and BD, at right angles to each other. Draw the lines AB, BC, CD, and DA, joining the points of intersection of these di- ameters with the circumference of the circle ACBD. ACBD is the required square. ROGERS' DRAWING AND DESIGN. 101 To describe a square about a given circle. Let EGHF be the given circle, Fig. 193. Draw two diameters, FG and EH, at right angles to each other. At the points EGHF, where these F Fig. 193. diameters intersect the circumference of the given circle draw lines perpendicular to these diameters. These lines will intersect each other at ABCD. which is the required square. To inscribe a hexagon in a given circle. (See definition, page 32.) Draw a diameter AB in the given circle, Fig. 194. From A and B as centers, with a radius equal to the radius of the given circle, describe four arcs cutting the circumference of the circle at DEF and G. Join these points by straight lines. ADEBFG is the required hexagon. B G F Fig. 194. To divide the circumference of the circle into six equal parts. We set the dividers to equal the radius of the circle and get the required result by stepping the radius six times around the circle. 102 ROGERS' DRAWING AND DESIGN. To construct a hexagon upon a given line. Let AB be the given line and let it equal in length a side of the required hexagon, Fig. 195. From A and B as centers describe arcs cutting each other at G, the radii of the arcs being equal to AB. Fig. 195. From G as center with the same radius de- scribe a circle. With the same radius set off arcs cutting the circumference at CEF and D. Join these points by straight lines and they will form the sides of the required hexagon. To describe an octagon in a given square. (See definition, page 32.) Let ABCD be the given square, Fig. 196. Draw the diagonals of the square cutting at E. Fig. 19K. From ABC and D as centers, with a radius AE, describe arcs cutting the sides at GH, etc. Join the points so found to complete the required octagon. ROGERS' DRAWING AND DESIGN. 103 To describe an octagon on a given litte, one side of the octagon being given. Let AB be the given side, Fig. 197. Lengthen the line AB both ways. Erect perpen- diculars to this line at A and B. Fig. 197 Bisect the external angle at A by the line AH, and the external angle at B by the line BC. Make AH and BC equal to AB. Draw HG and CD par- allel to AE and equal to AB. From G as center, with a radius equal to AB, cut the perpendicular AE at E, and from D as center with the same radius cut the perpendicular BE at F. Complete the octagon by joining GEE and D. To draw a regular polygon of any number of sides on a given line. (See definition, page 30.) Let C5 be the given side of the required poly- gon, Fig. 198. "n /p o c Fig. 19S. Lengthen the line C5 to O. With C as center and a radius equal to C5 describe the semicircle O I 2345, and divide this into as many equal parts as there are sides in the required polygon. Join C with 2, 3, 4, etc., by straight lines. With 2 as a center and a radius equal to C5 describe an arc cutting the line C3 at D. With D as center, and with the same radius draw an arc cutting the line C4 at E, and so on. Join the points C2D, etc., to form the required polygon. 104 ROGERS' DRAWING AND DESIGN. To inscribe a regular pentagon in a given circle. (See definition, page 32.) Draw two diameters AC and DB at right angles to each other, Fig. 199. Bisect the radius OB at I. With I as center and a radius equal to I A describe an arc cutting the di- ameter DB at J. Fio. 199. A straight line joining A and J is equal to one side of the required pentagon. With arcs of a radius equal to AJ set off on the circumference the points where the sides of the pentagon will ter- minate. To inscribe a regular polygon of any number of sides, within a given circle. Fig. 200. Draw two diameters AC and D7 within the given circle. Fig. 200, at right angles to each other. ROGERS' DRAWING AND DESIGN. 105 Divide the diameter D7 into as many equal parts as there are sides in the required polygon. Let it be seven in this case, at the points 123456. Lengthen the diameter AC making AK equal to three-fourths of the radius of the given circle. Through K and 2 draw a straight line cutting the circumference at L Join the points D and I by a straight line, and it is equal in length to one side of the required polygon. Set the dividers to equal this side, and set off the other sides around the cir- cumference. To describe an octagon in a circle. Draw two diameters at right angles ; these diam- eters divide the circumference into four equal arcs. Bisect these arcs to complete the octagon. To drazv an oval by circular arcs. Let CD be the major axis and AB the minor axis of the oval, Fig. 201. Find the difference of the semi-axes and set it off from O to e and f on CD and AB. Bisect ef and set off one-half of it from e to g and draw gh parallel to ef. From the center O on CD lay off the distance Oi equal to Og and draw hi ; through the points i and g draw the lines Ri and gR parallel to gh and hi. With Cg as a radius and the points g and i as centers, draw the arcs jCm and nDp ; with RA as a radius and R and h as centers draw the arcs jAn and mBp meeting the small arcs in the points j and n and m and p. The figure AnDpBmCj is the required oval. 106 ROGERS' DRAWING AND DESIGN. MECHANICAL METHOD. Draw a line AB equal to the major axis of the y^ "^ ^^\ / ^ ^ \ required ellipse, Fig. 202. % ^^^--'''^ Bisect the line AB at E. At E draw a line CD \ perpendicular to AB. Make ED equal to EC and A / \ equal to one-half the minor axis. Set the compass / \ ^^ \ to a distance equal to AE or EB, and with C or D / \ ^^ \ as center, describe an arc cutting the major axis at / A \ F and G. F and G are the foci of the ellipse. // \ \ Fasten the ends of a string, whose length is equal / \ ^ to the length of the major axis, AB, at thfe foci F to / \ ^ and G. This may be done by fixing pins at the foci and by providing the ends of the strings with ^^ \ ""x \ loops. \ ""^N \ , Trace a curve with the point of a pencil H \ ^^ \ / pressed against the string so as to keep it stretched. \ ^^^ \ / The curve thus traced will be the required ellipse. \ ^'1 / GEOMETRICAL METHOD. ^ / Draw a rectangle ABCD enclosing the axes of \ ^ r» / the ellipse, Fig. 203. \. y^ Let EF be the major axis and HJ the minor ^^"■-^^■^ ^^^.^ axis. Divide AB, DC and EF into a like number of equal parts making the number an even one. Fir.. 202. The greater the number the more accurate will be To draw an ellipse, the major and the minor axes the resultant ellipse. Let the number in this case being given. (See definition, page 36.) be 8. ROGERS' DRAWING AND DESIGN. 107 From 2, 4, 6, and from corresponding points in DF draw lines to H. From tlie points placed on KB and FC draw lines to J. From J and H draw lines through 5, 3 and i and through correspond- ing points on LF to meet those already drawn. Through the intersection of 2H with Ji, 4H with J3, etc., draw the outline of the ellipse. Finish carefully in pencil, freehand, and then ink in with aid of an irregular curve. To describe a parabola, the base BC and the alti- tude EF being given. (See definition, page 36.) On the given line BC construct the rectangle ABCD with an altitude or height EF, Fig. 204. From F the middle point in BC erect the perpen- dicular EF ; divide AB and BF into the same num- ber of equal parts, say four. In like manner divide DC and FC. Draw lines from 2, 4 and 6 on AB and from corresponding points on DC to E ; from 5, 3 and I and from corresponding points in FC draw lines parallel to EF, meeting the lines drawn to E from 2, 4, 6, etc. Through the intersection of 5 with 6, 3 with 4 and I with 2 and corresponding ooints, draw the curve of the parabola. 108 ROGERS' DRAWING AND DESIGN. To describe a hyperbola, the transverse axis, the altitude and the base being given. (See definition, page 36.) Let FI be the axis of the hyperbola, EI its alti- tude and BC its base, Fig. 205. F ///I 1\\ //// >\\ /'/111 ' \ \ 1 1 , \ / ' ' i I I \ \ \ / ' ' ' ^ \ 5 3 1 E Fig. 20.5. On BC construct a rectangle ABCD with EI as its altitude. Divide AB and BE into the same number cf equal parts, say 5. Divide DC and EC in like manner. From F draw lines to the points of divi- sion on BC. From the points of division on AB and DC draw lines to I. Through the intersection of 8 with 7, 6 with 5 and corresponding points, draw the curve of the hyperbola. Fig. 20fi. To construct a spiral composed of arcs of various radii. Let ABC be a small equilateral triangle, Fig. 206. Note — A spiral is a curve described about a fixed point, and which makes any number of revolutions around that point without returning into itself. ROGERS' DRAWING AND DESIGN. 109 Lengthen the sides AB, BC and CA. With B as a center, BA as a radius, describe the arc AG meeting the line BC prolonged at G. With C as center and CG as radius, describe the arc GE meet- ing the line AC prolonged at E. With A as center and AE as radius, describe the arc EF meeting BA prolonged at F, and so on, using successively the points BCA for centers. By using any regular polygon in the same man- ner, that is, lengthening its sides and taking the angular points of such figure for centers success- ively in order, as in the above problem, a different spiral may be formed. To draw the outline of a snail by circular arcs. Let C be the axis or center of rotation upon which the snail is fixed, Fig. 207. The point B nearest to the center and the point A most distant from the center being also given. From the center C describe a circle whose diam- eter shall be equal to one-third of AB and divide the circumference into any number of equal parts, as I, 3. 4. etc. Draw through each of these points tangents to this circle. Then from the point i as center, lA as radius, draw the arc 1-2' and from 2 as center, 2-2' as radius, describe the arc 2-3'; and from 3 draw the arc 3-4' and so on, taking in order the points I, 2, 3, 4, etc., as centers. Note. — The .<;nail is a mechanical movement u.secl for a great variety of purposes, as in time-pieces, rlrop ii\otions, etc. 110 ROGERS' DRAWING AND DESIGN. To draw the outline of a heart-wheel. 6 Fig. SOS. No'rE. — The heart-wlieel is a popular mechanical device producing uniform reciprocating motion. Let C be the axis or center of rotation, upon which the heart-wheel is fixed, Fig. 208, and let AB be the required extent of the rectilinear motion, A being the nearest point to the center and B the most distant. From the center C with a radius equal to CB describe a circle. Divide this circle into any number of equal parts, say 12, and through the points of division draw radii Ci, C2, C3, C4, etc. Divide the line AB into half the number of equal parts, the circle is divided into (in this case six), as i, 2, 3, etc. Then from the cen- ter C with the distance Ci on the line AB, describe an arc cutting the first radius at the point D ; then take the other divisions on the line AB and in succession with them from the center C draw arcs, cutting their respec- tive radii Ci, C2, C3, etc., at the points DEFG and H, which are the points in the required heart-wheel curve, its highest point being C and its lowest A. The construction of various machine parts in- volves many problems similar to the preceding ; these will be introduced when treating of the design of mechanical motion and the construction of parts of various machines. ISOMETRIC, CABINET AND ORTHOGRAPHIC PROJECTIONS AND DEVELOPMENT OF SURFACES. The word projection means to throw forward, and in mechanical drawing it is the projecting or throwing forward of one view from another ; in drawings the lines in one view or plan may by this system be used to find those of others of the same object, and also to find their shape or curvature as they would appear in other representations. Isometric projection is that in which but a single plane of projection is used. Cabinet projection is somewhat like isometric projection; the cabinet projections are : i, a horizontal line ; 2, a vertical line and 3, a 45-degree line ; all measurements on the drawing must be laid off parallel to these axes ; cabinet projection is one of several systems of oblique projection. Orthographic projection. The primary geometrical meaning of the word orthographic is ''of or pertaining to right lines or angles," hence all the projecting lines are either horizontal or vertical. Drawings made up in this manner will be easily understood by many people unacquainted with the special methods of drawing generally used in mechanical branches. Development of surfaces will be defined and illustrated under its own chapter, page 162. Objects represented as thus described give a clear understanding of all their dimensions, and approximately show them as they appear to the eye of the observer ; the method of representing objects as they really appear to the e\ e is called perspective drawing. This latter method, however, presents so many difficulties of construction, that various other means have been devised, all aiming to give the advantages of perspective, and avoiding at the same time the difficulties of construction. These methods, also called false perspective, are described under the heading of isometric projection, and will be further explained in the following chapter under the title " Cabinet Projection." 113 114 ROGERS' DRAWING AND DESIGN. ISOMETRIC PROJECTION. Figure 209 shows a solid figure, a cube, with equal sides and resting on one of its corners ; the lines ac, ab and ag are called isometric axes ; these axes form an angle of 1 20 degrees with each other. Fig. a». They may be drawn by the 30° and 60° triangles ; the lines ac and ab forming angles of 30° with a horizontal line ; ag is a vertical line. All the lines in this figure are parallel to these axes, viz.: all the lengths are parallel to ab and all the widths are parallel to ac. Fig. 210. ROGERS' DRAWING AND DESIGN. 115 The method of thus representing objects is called isometric projectioti ; drawings made in this manner show very clearly, with one view, the object as it appears when looked upon ; all the sizes of the object are drawn full size, or made to one scale, parall(;l to the isometric axes. With these rules in mind several objects will be represented in isometric projection in order to ex- plain its principles. To draw a square block, 4 by 2" by 2" , Fig. 210. First draw the isometric axes, ab, ac and ad ; ab is a vertical line whereas ac and ad are lines form- ing angles of 30° with a horizontal line ; make ab equal to 2 inches, ad equal to 4 inches and ac equa to 2 inches; from c draw cf, parallel and equal to ab, and from d, draw dh, also parallel and equal to ab. Join the points b and f ; and the line bf will be equal and parallel to ac. Then join the points b and h and the line bh will be equal and parallel to ad ; from the point f, draw the line fg equal and parallel to bh, then draw the line gh, which will be equal and parallel to bf ; from the point of intersection g draw the vertical line gk, from c and d draw the lines cb and dk, respect- ively, and parallel to ad and ac. To draw a rectangular frame made 0/ wood y^' thick, the outside dimensions being 16" long, 8" wide and 2" deep, as shown in Fig. 211. First draw the isometric axes ab, ad and ac : make the line ab equal to the depth required, or 2", the line ad equal to 16" or the length desired for the frame, and finally the line ac equal to 8' or the width. FlG.I-'ll. 116 ROGERS' DRAWING AND DESIGN. Now draw the lines cf and de equal and parallel to ab and then draw the lines fb and eb, equal and parallel to ac and ad, respectively. From the point f draw ft equal and parallel to ad. Next join the points e and t and the line et will be parallel and equal to ac. Now make mb and bn, tq and tr, each equal to ^" for the thickness required in this example ; draw the lines mp and gr parallel to eb and also draw the lines qk and hn parallel to bf. The two lines gr and qk intersect at s ; from this point s, draw a vertical linesu, parallel and equal to ab. From u draw a line parallel to ad in the direction of ac and cutting this latter line ; also draw from u a line parallel to ac in the direction of mp and cutting said line. Note. — For objects as represented in figures 210 and 211 an iso- metric projection is desirable, but when the objects to be drawn contain curved surface lines the application of the above described method is limited. Fig. 212. To draw a right cylinder in a horizontal position, as shown in Fig. 212. Draw a square abed, Fig. 213, with sides of exactly the same length as the diameter of a circle whose sur- face is to be the base of the required cylin- der. Within this square draw ' a circle efgh, tangent to the square and its diame- ter equal to that of the base of the cylinder. Next draw the diagonals ad and be, cutting the circle at the points eghf ; join the points e and g, g and h, h and f, e and f by straight lines and extend these lines until they meet the sides of the square. These lines cut off equal lengths of the sides of the square in its four corners, so that ai^a2=d3^ d4, etc. Now suppose that the required cylinder is placed in a square prism, so as to exactly enclose the cyl- inder as shown in Fig. 214. ROGERS' DRAWING AND DESIGN. 117 It is evident that the prism will have two ends equal to the square shown in Fig-. 213, and that the length of the prism will be equal to that of the cylinder. Then draw the prism in isometric projection as explained on page 115; draw the diagonals AD and BC in the end ABCD of the prism and lay out the isometric pro- jection of the circle which is to form the base of the required cylinder. Set off on the line AC a distance Ai equal to ai in o I 3 A C, m \ / / /h N V \f ) / \ n a k Fio.213. Fig. 213; from the point D on the line DB, set off the distance D4, equal to ai, in Fig. 213, or equal to Ai in Fig. 214. 118 ROGERS' DRAWING AND DESIGN. Through the point i draw a line parallel to AB, cutting the diagonals AD and BC at the points e and f ; through the point 4 draw also a line parallel to CD intersecting the diagonals at the points g and h ; draw the line mn through o, parallel to AB, cutting AC at m and BD at n ; draw a line parallel to AC through the same point O, cutting the line CD at 1 and the line AB at k. The points k, e, m, g, 1, h, n and f are points through which the required circle drawn in isometric projection will pass. The curve thus obtained is evidently not a circle, but has the form of an ellipse, its minor axis being eh and its major axis fg. This ellipse may be drawn by any method explained in the section pertaining to Geometrical Drawing. The other end of the cylinder, which is to be in- scribed in the figure KLMN, may be drawn in the same manner as already explained, and the ellipse GF will be obtained. In order to complete the isometric projection of the cylinder draw the lines Gg and Ff, joining both faces of the cylinder ; these lines are to be drawn through the ends of the major axis of both ellipses and they are tangent to these two curves. To draw a pattern of a crank, shown in Fig. 21^, isometric projection. The pattern consists of two cylinders joined by a board. The larger cylinder into which the shaft will fit is 3" in diameter and 25^" long; the smaller cylinder to which the crank pin is to be fitted, is 2" in diameter and 2 " long. The distance between the center lines of the two cylinders is 5". Proceed as follows : Describe a circle 3" in diameter, as in Fig. 216, and draw a square around it, and within the square draw two diagonals and other lines as in Fig. 213 ; draw the isometric projection of a prism having Fig. 216 as a base and a length equal to 21^"; said prism is marked ABCDdab and its hiddqn parts are not shown. In this prism lay out the isometric projection of the larger cylinder, whose front face will be the ellipse klNcjM. Fig. 218 shows only a small part of the ellipse forming the rear end of the cylinder and this small visible part is represented by mi. Through the center of the first ellipse draw the line MN parallel to CD and the line kc parallel to AD ; then draw the line eg through the point c and parallel to Aa and equal to t^/^". The point g indicates the place where the board, connecting both cylinders, is fastened to the first cylinder. The board intersects the cylinder, form- ing an additional ellipse, or more properly, a part ROGERS' DRAWING AND DESIGN. 119 of an ellipse, represented in Fig. 21S by uge ; this part of the ellipse is exactly equal to the part jcN of the ellipse McNk, — -f — ■"»;" /•- Fm. 217 .i.._ M r^ zi'- FlQ. 216. Fio. 215. and may be constructed by drawing from different points of the curve, jcN, a number of lines parallel and equal to eg. Fio. 218. 120 ROGERS' DRAWING AND DESIGN. The line uj is a tangent to both of these curves. From the point g draw the Hne gf parallel to ck and equal to 2^'; through f draw the line hp parallel to CD so that hf is equal to fp, each of these being equal to one inch ; from the line hp draw the isometric projection of the prism hpsnto, which is to enclose the smaller cylinder. The base of the latter is shown in Fig, in Fig. 218 by the ellipse vfvv. The length of the prism is to be equal to 2". When the small cylinder has been drawn in isometric projection within this prism draw the line vw through the cen- ter of the ellipse vfw and parallel to hp ; draw the line vr through the point v, the distance vr being made equal to one inch and through the point r draw the line rm, tangent to the ellipse mi. The lines wu and ve are both tangent to the ellipse uge. The hidden parts of the object are not indicated in Fig. 218. Fig. 219 represents a tool chest drawn in isometric projection It is given here as an example of a large class of objects well adapted for representation by this method. ROGERS' DRAWING AND DESIGN. 121 CABINET PROJECTION. Cabinet Projection is somewhat similar to Iso- metric Projection ; its difference consists in selecting three axes to which all measurements of the object are drawn parallel ; see a, b, c, following : Fin. -iSS. The axes for cabinet projection are : i, ime; anc a 45' a hori- line, as zontal line ; 2, a vertica shown in Fig. above. It is to be remembered that : a. All horizontal measurements, parallel to the length of the object must be laid off parallel to the horizontal axis, in their actual sizes. b. All vertical measurements, parallel to the height of the object, must be draAvn parallel to the vertical axis, in their actual sizes. Fig. 221. c. All measurements parallel to the thickness of the object must be laid off on lines parallel to the 45° axis, in sizes of only one-half of the actual cor- responding measurements. It is not essential which side of the object should be considered its length and which side its thickness. 122 ROGERS' DRAWING AND DESIGN. To draw a cube in cabinet projection, as^hown in Fig. 221. Suppose each side of the cube to be 3" long. Draw the three axes: ab=horizontal axis, bd^ vertical axis, and bc=axis inclined 45°. On the line ab set off, from the point b, the distance b]=3"; on the line bd, from b, lay off b2^3"; and on the line be measure off f'^ij^". A vertical line drawn Fig. 222. through point i parallel to the vertical axis bd and a horizontal line drawn through point 2 parallel to the horizontal axis ab will intersect at point 3 and thus complete one face of the cube b-2-3-1. Now, through the point 4 draw a vertical line parallel to bd and through point 2 draw a line in- clined 45° with the horizontal ; these two lines intersect at the point 5 and complete the side b-2-5-4 of the cube. The remaining lines are drawn in a similar man- ner, parallel to the axis, from the points 3 and 5, intersecting at the point 6 and showing the top of the cube 3-6-2-5. Fig. 223. Next draw through tlie point 4 a horizontal line and parallel to ab and through the point i a line in- clined at 45° and parallel to be ; these two lines cut at the point 7 ; join the points 6 and 7 by a straight line and cube is complete. ROGERS' DRAWING AND DESIGN. 123 In Fig. 222 is shown in cabinet projection the frame represented in Fig. 211. The length of the frame, 16 inches in actual measurement, is repre- sented here on the 45° axis by only one-half of its actual size or 8 inches long ; all the other measurements are equal to the actual sizes of the object, as described on page 122. zontal axis ; 3, parallel to the vertical axis, that is, in a standing position. The first position of the cylinder being the most convenient for drawing it in cabinet projection ; it will be considered here before the others. m (e v \ \ \ ^ p ( n Fig. 224. B Fio. a'). To draw a right cylinder in cabinet projectioti, its base to be the circle shown in Fig. 22^. The cylinder may be placed in the following po- sitions : I, parallel to the 45° axis; 2, parallel to the hori- ExAMPLE I. — Draw in cabijiet projection, the prism abcgfedh. Fig. 22 j, enclosing the cylinder ; the face of the prism, abcg, will contain the visible base of the cylinder; which is shown in Fig. 223 by the circle kl, which is equal to it. \^ 124 ROGERS' DRAWING AND DESIGN. In the rear end of the prism draw the circle nm for the other end of the cylinder, and draw the lines kn and Im tangent to both circles ; this completes the cabinet projection of the cylinder. It is advisable to select this position for all cylin- ders, as much as possible, when they are to be drawn in cabinet projection, as the circles repre- senting the faces of the cylinder may be drawn by circles and the dra\/ing of ellipses is avoided. Example II. — Describe the circle forming the base of the required cylinder. Fig. 22^, within the square ABDG, Fig. 22^, and draw ihe diagonals BG and AD, cutting the circle at the points hefg. Through the points e and h draw the line ehi and through the points fg the \\n^ fg2 ; the distance Ai will be equal to the distance B2. Now, . assuming that the cylinder is contained within a rectangular prism, each end of which is equal to the square shown in Fig. 224 and the length of which is equal to that of the cylinder, draw this prism in cabinet projection as shown in Fig. 225. Lay out the axes ab, bd and be ; make ab equal to the length of the prism, that is, equal to the length of the required cylinder; make bd equal to AG, Fig. 224, and be equal to }4 of the distance AB In Fig. 224. Through the point c draw a vertical line ce equal and parallel to bd, then join the points d and e by a straicfht line thus forming- the figure bdec, which will be one end of the prism ; from the points a and f draw the lines ah and fg, each equal and parallel to be ; then -draw the line gh equal and parallel to af. The figure afgh thus obtained is the other end of the prism. Now, lay out one face of the cylinder within bdec. In order to do this draw the diagonals dc and eb, set off from b on the line be the distance bi equal to one-half of the distance Ai in Fig. 224 and on the same line be, Fig. 225, point off the distance 2c from the point c and equal to bi. Through the points i and 2 draw vertical lines which will intersect the diagonals eb and cd ; the points of intersection thus obtained together with the points 4, 5, 3 and 6 — it is evident how these points are found — define the position of the curve which will represent the circle forming one face of the cylinder as it appears in cabinet projection. The curve within afgh is to be drawn in a similar manner for the other end of the cylinder. Two horizontal lines, each tangent to both these curves, will complete the cabinet projection of the cylinder. ROGERS' DRAWING AND DESIGN, 125 Example III. — From the drawing in this figure 226, it is evident that the construction in this case is exactly the same as in case 2. From the above problems it will be seen that v \ :---^,<" ^ ^=::^^^ ==^ \ •^^ \ y ^ Fig. 226. objects with circular forms which are to be drawn in cabinet projection should be placed preferably with all or most of its circles as in the cylinder rep- PiG. 23T. resented in Example I ; in this position, as already previously stated, all circles in the object will be represented by their actual sizes in the cabinet pro- 126 ROGERS' DRAWING AND DESIGN jection and in this manner the construction of diffi- cult curves may be avoided. Isometric projection does not offer this advan- tage as in that method, all circles, without exception, will appear as ellipses ; consequently, cabinet pro- jection has a distinct advantage, and is therefore oftener employed when a drawing of an object in false perspective is required. As an illustration of the principles explained in the preceding pages the cabinet projection of the pattern for a crank, shown in isometric projection in Fig. 215, will be given in Fig. 227. At a glance it will be seen that the cabinet pro- jection of this object can be drawn in much less time than its isometric projection. It is, however, necessary to bear in mind, that, whereas all meas- urements in isometric projection are equal to the actual sizes of the object, those in cabinet projec- tion which are parallel to the 45° axis are drawn equal to y^ of their actual size. Figs. 228, 229, 230, 231, and 232 represent addi- tional illustrations of objects drawn in cabinet pro- jection. Note. — The thorough knowledge of cabinet and isometric projec- tions will be of great advantage, both to the student and the mechanic, as they will thereby be enabled to represent different objects in drawing in such a manner as to be easily understood by persons who would not understand a mechanical drawing executed in another, though perhaps a more generally approved manner. ROGERS' DRAWING AND DESIGN. 187 Fia. 229. Fig. 230 Fui. 231. Fig. 232. 128 ROGERS' DRAWING AND DESIGN. ORTHOGRAPHIC PROJECTION. Isometric drawing and cabinet projection, while showing the object as it really appears to the eye of the observer, are neither of them very convenient methods to employ where it is necessary to measure every part of the drawing for the purpose of repro- ducing it in the shop. All shop drawings, or working drawings as they are usually termed, are made by a method known as orthographic projection ; in isometric or cabinet projections, three sides of the object are shown in one view, while in a drawing made in orthographic projection, but one side of the object is shown in a single view. To illustrate this, a clear pane of glass may be placed in front of the object intended to be repre- sented. In Fig. 233 a cube is shown on a table ; in front of it, parallel to one face (the front face) of the cube, the pane of glass is placed. Now, when the observer looks directly at the front of an object from a considerable distance, he will see only one side, in this case only the front side of the cube. The rays of light falling upon the cube are re- flected into the eyes of the observer, and in this manner he sees the cube. The pane of glass, evi- dently, is placed so that the rays of light from the object will pass through the glass in straight lines, to the eye of the observer. The front side of the object, by its outline, may be traced upon the glass, and in this manner a figure drawn on it (in this case Fig. 2:«. a square) which is the view of the object as seen from the front. This view is called the front eleva- tion. - One view, however, is not sufficient to show the real form of a solid figure. In a single view two ROGERS' DRAWING AND DESIGN. 129 dimensions only can be shown, length and height ; hence the thickness of an object will have to be shown by still another view of it, as the top view. Now, place the pane in a horizontal position above the cube which is resting on the table, Fig. Fig. ast. 234, and looking at it from above, directly over the top face of the cube, trace its outline upon the pane ; as a result, a square figure is drawn upon the glass, which corresponds to the appearance of the cube, as seen from above. This square on the glass is the top view of the cube, or its "//aw." In Fig. 235 is shown the manner in which a side view of the cube may be traced ; the glass is placed on the side of the cube, which rests on the table as before, and the outline of the cube on the glass in this position, is called its ''side elevation.'' Usually either two of the above mentioned views will suffice to show all dimensions and forms of the object. In complicated pieces of machinery, how- ever, more views, three and even more may be re- quired to adequately represent the proportions and form of the different parts. A drawing which represents the object as seen by an observer looking at it from the right side is called the right side elevation and a drawing show- ing the object as it appears to the observer looking at it from the left side is called the left side eleva- tion. A view of the object as seen from the rear is called the rear view or rear elevation, and a view from the bottom, the bottom view. The different views of an object are always ar- ranged on the drawing in a certain fixed and gener- ally adopted manner, thus — The front view is placed in the center ; the right side view is placed to the right of the front view, 130 ROGERS' DRAWING AND DESIGN. and the left side view to the left ; the top view is placed above the front view and the bottom view below it. The different views are placed directly opposite each other and are joined by dotted lines called projection lities. Fig. 23.1. By the aid of projection lines, leading from one view to the other, measurements of one kind may be transmitted from one view to the other ; for in- stance, the height of different parts of an object may be transmitted from the front view to either one of the side views ; in like manner the length of different parts of the object may be transmitted by the aid of projection lines, to the bottom view and top view. It is often desirable to show lines belonging to an object, although they may not be directly visible. In Fig. 236 the top view and the bottom view show plainly that the object is hollow ; looking at the object from the front or from the sides, however, the observer could not see the inside edges of the object, except it were made of some transparent material. For mechanical drawing, we may assume that all objects are made of such material, transparent enough to show all hidden lines, no matter from which side the object is observed. It is the gen- eral practice to draw the hidden edges by lines made of dashes — dash lines — as in Fig. 236. In the following articles the student will find a number of exercises on the application of ortho- graphic projection. Note. — Mechanical drawing is used mainly to represent solids, but solids are bounded by surfaces -whicb in turn are bounded by lines which by themselves are limited by points ; views of a solid can there- fore be found hy drawing the views of its limiting points, lines and surfaces, according to the principles of orthographic projection. ROGERS' DRAWING AND DESIGN. 131 Tof3 Vi ew Lefl 'SideVitu/ t I Front View I • HigJii Side Vteu JBoitom View Fig. 236. 133 ROGERS' DRAWING AND DESIGN. Draw the front view, left side view and top view of the rectangtilar prism showji in .Fig. sjj. Fig. 238 shows the drawing of the prism in the three required views ; the lines showing the dimen- sions are made by long dashes drawn very thin. FlO. 237. It is important to remember that dimension lines must be drawn parallel to the distances, the size of which they are intended to show. The dimension lines terminate in arrow heads drawn with an ordi- nary writing pen. If a dimension line is carried outside of a view, short auxiliary dotted lines are employed to join the part of the object to which the dimension line refers. The dimension line is left open near the middle, where the figure denoting the measurement is placed. These figures should be written very plainly and placed so as to read along the dimension line ; for horizontal lines from /v < 1 • : • t 1, ^ y" — — •? i^G. S38. the bottom of the drawing, and for vertical lines from the right hand side of the drawing. The inch is marked " the foot ' — for example : i foot, 3 inches is represented by i' 3". More infor- mation concerning dimensions will be found in the chapter treating on working drawings. ROGERS' DRAWING AND DESIGN. 133 Draw a front view, top view and right side view of the wedge showji in Fig. 2^0. Draw the front view first. Lay off a straight line, on which mark two points 3" apart ; through the point on the right erect a perpendicular, which make one inch long ; two sides of the right-angled triangle forming the front view of the wedge are thus found. Join the two ends of these sides by a straight line and the front view is complete. The student will draw the side view and the top view in correspond- ing positions to the right side and above the front view, as in Fig. 239. Fig. 240. Fig. 239. 134 ROGERS' DRAWING AND DESIGN. Draw a front view, both side views and top view of the object shown in Fig. 2^1. Fig. 242 shows the required views of the object. The edge ab which is visible in the right side view will not make the understanding of the view more difficult. Whenever the view is so complicated that any additional lines would only tend to obstruct a clear conception of the object, it is advisable to carry the dimension lines outside of the view. Dimension lines must necessarily be of three Fia. 241. is hidden in the left side view and therefore is represented by the dash line cd. It may often be possible to put in the dimension lines within the views, when the object is not of a complicated nature, and when the dimension lines kinds: i, parallel to the lengths of the different parts of the object ; 2, parallel to the width of these parts, and 3, parallel to the height. The dimension line must always be parallel to the line or edge whose length it represents. ROGERS' DRAWING ANQ DESIGN. 135 ; V d --...^ , a P-- ^ Jf- — --^ > V \ t Fig. 243. 136 ROGERS' DRAWING AND DESIGN. Draw a front view, side view and top view of the model shown in Fig. 24J. As the object to be drawn has the same appearance from either right or left side, it does not matter which side view is to be drawn. Fig. 243. The construction of the views is so obvious that no explanation need be offered with the drawing shown in Fig. 244. It will be noticed that this figure, as well as all others in this chapter, are shown with lines representing the sides of the dif- ferent parts of the object. Draw two elevations {a front view and a side view^ and a top view of a hexagonal prism f long and 2y2" between any tivo parallel sides. A X"" ■V. Fig. ;244. It is evident that a hexagonal prism has six faces and of these three are parallel to the remaining three faces. The distance between any two parallel faces or sides is the same ; in this case it is equal to 23^ "; let us draw the top view of the prism first of all. ROGERS' DRAWING AND DESIGN. 137 N Draw two lines, AB horizontal and CD vertical, Fig. 245, intersecting each other at the point O. If it is intended, as in this case, that the intersection, O, of these two lines should coincide with the center of the view which is to be drawn, then these lines are called center lines ; the use of center lines in projection drawing is very extensive. Make the line CO equal to OD and each equal to 1%", so that the line CD is equal to 2j4", the distance between the parallel sides of the prism. Through C and D draw the lines eCd and aDb parallel to AB ; then through the point O draw two 60-degree lines eb and ad, cutting the lines eCd and aDb at the points e, b, a and d. Through these points draw the re- maining sides of the hexagon, parallel to the lines eb and ad. The hexagon, aAedBb shows the top view of the prism. To draw the front view proceed as fol- lows : Through the points Aab and B draw the vertical lines AE, aH, bj and BF. Draw the horizontal line NGP, make PF equal to 5", the height of the prism and through the point F draw the hori- zontal line KEF ; then the figure, EHJ FPG is the front view of the prism. It will be noticed that the front view shows three faces of the prism : HEGS, HSRJ and JRPF, the faces HEGS and f ^ K M Fig. 245. E H 138 ROGERS' DRAWING AND DESIGN. JRPF appear narrower than the face HSRJ, the latter being situated right in front of the observer and parallel to the plane of the paper is seen in its true size, while the other two faces seen in the front view being in an inclined position relative to the front face appear narrower than their true width. The side view KNTLM shows only two faces of the prism. The dis- tance KM is equal to CD, the edge LT corresponding to the edge marked by the letter A in the top view, cuts the line KM into two equal parts, KL and LM. To draw the top view, front view and side view of a hexagonal pyramid 5" high, each side of the hex- agonal base being equal to i%". The top view of the pyramid must be drawn first. Fig. 246 shows the required views. The top view appears as a regular hexagon, in which all diagonals are drawn by lines as heavy as the sides, as these diagonals show the edges of the faces of the pyramid. The center of the hexagon where all the diagonals meet represents the vertex of the pyramid. The front view and the side view are drawn in the manner explained in the construction of these views of the hexagonal prism, the edges of the faces in this case all meeting in the vertex which is placed 5" above the middle of the line representing the base. ROGERS' DRAWING AND DESIGN. 139 To draw a top view ajid a front view of an octag- onal prism. Let each side of the octagonal bases be equal to one inch and let the height of the prism be 8". To complete the front view, intersect these lines by two horizontal lines 8" apart. The side view of this figure is identical with the front view. Draw the top view first. Fig. 247 shows the re- quired views. The top view is an octagon, each side of which is equal to one inch. The front view is drawn by projecting vertical lines from the points a, b, c, and d of the octagon. These vertical lines form the vertical edges of the faces of the prism. Fig. 248 shows three views of a sphere, each of which appears as a circle. The lines, AB, CD, EF and GH are center lines. They are composed of long and short dashes, alter- nating, and are usually extended indefinitely beyond the outlines of the views. 140 ROGERS' DRAWING AND DESIGN. Center lines are drawn through the middle of the view in all cases where such a line will divide the view into two perfectly equal parts so that one part will have all its details situated opposite the corre- sponding details of the other part, so that if the paper on which the view is drawn is folded along the center line, all parts in one half of the view will cover exactly all corresponding parts in the other half of the view. We say then that the view (or object) is sym- metrical with regard to the center line. In Fig. 245 the top view and the front view are symmetrical with respect to the center line CD. The top view, however, may be folded along the line AB, and in this case the lines of the hexagon on one side of the line AB will exactly cover the lines in the other half of the hexagon ; we see then, that the hexagon is symmetrical in regard to the center line AB also. In all cases where a view is symmetrical in re- spect to two lines, both of these lines must be drawn. Wherever the view is symmetrical to one line only, not more than one center line must be drawn ; in Fig. 248 all views are symmetrical to both horizontal and vertical center lines ; center lines continued from one view to the other show that the views belong together, just as projection lines would indicate the same. B Pig. 248. A center line should never be used as a dimension line, but such lines may be laid off from the center line on both sides of it. ROGERS' DRAWING AND DESIGN. 141 Fro. 249. In Fig. 2^g is shown the top view (or plan) and front view {or elevation) of a cylinder, j" high and i%" in diameter. The top view is a circle ij^" in diameter, the front view a rectangle 3" by i^". All side views of the cylinder are the same. As in all figures standing on a base of an irregu- lar shape, the top should be drawn in this case be- fore the front. The width of the front view is determined by projection lines from the top view ; observe that the top view has two center lines, a horizontal and vertical one ; the front view has only one line of symmetry, the vertical. Draw the front view and side view of a cylindrical pipe 8" long^ outside diameter f, inside diameter j" ; in Fig. 250 the required views are shown. The two dash lines in the front view show the inside walls of the pipe, which are represented in the top view by the smaller circle. Fig. 250 may also represent two views of a pipe into which a cylinder has been inserted. We have here an interestinof illustration of a case where two views of an object, a front view and a top view, do not define sufficiently the true character of the object represented. A similar difificulty may arise with most hollow objects, and it is evident that some method must be devised to overcome any Fig. 230. 142 ROGERS" DRAWING AND DESIGN. such misunderstanding as to the true nature of the object represented. This may be done by representing the front view of the pipe as if it were cut in half like the cylinder Fio. 2r,i. shown in Fig. 251 ; a front view of such a pipe cut in half is shown in Fig. 252 ; the top view is that of a whole pipe. The line 1-2 shows the manner in which the cylinder is supposed to be cut, and is called the line or plane of section. The front view in Fig. 252 we call the section view or section on 1-2. The line of section should be mads up of dashes alternating with two dots. cSection on /-%. Fio. 252. ROGERS' DRAWING AND DESIGN. 143 The inner part of the material of the pipe ex- posed by cutting, is covered by lines about xVth I inch apart and inclined 45 degrees. Fig. 253 shows the same pipe with only a portion of its upper half cut away; in Fig. 255 is shown this partial section of the pipe. Fig. 2.56. Fig. au. 144 ROGERS' DRAWING AND DESIGN, In Fig. 254 is shown still another way of cutting the pipe, and in Fig. 256 appears the corresponding front view, with a similar partial section. Within the pipe de- scribed in the preceding problem {8" long, ^" outside and j" inside diameter^ is placed another pipe 8" long, j" outside diameter and 2" inside diameter. Draw the top view and section of these two pipes. The top view (Fig. 257) shows three cir- cles, 4', 3" and 2" in diam- eter ; the section on the line AB shows one-half of one pipe within the half of the other pipe. The section lines in the one pipe run in a different direction from those in the other ; this is done in order to show more dis- tinctly that there are two separate pipes. I i m Fig. 257. Draw two views of a cylindrical ring. Fig. 258. Fig. 258 shows the plan and section of such a ring. The drawing does not require any special explanation. ROGERS' DRAWING AND DESIGN. Draw two views of the cylinder with square Jlange shown in Fig. 2^g. Let the side of the cylinder be lo" long (entire leno-th) outside diameter 4", inside diameter 3", and the flange 6" by 6" and ^'2" thick. The flange has four bolt holes, each i/^" in diam- eter. The top view and section of this figure are shown 60. in Fig, Fig. 259. 146 ROGERS' DRAWING AND DESIGN. Fig. 261 shows the top view and two sections of a bed plate. view, is a section parallel to the short side of the bed plate and is called a cross section or a lateral section. Jiongitudina? o degrees to the lower edge of the drawing board. Let it be required to draw the front view and top view of tliis object. Draw the top view first. To do this, draw the rectangle A BCD, AD par- allel and equal to BC, 4" long, and DC equal and parallel to AB, 2" long ; AD is inclined 30° and DC 60° to a horizontal line. Fig. 266. To draw the front view, draw the horizontal line FM, and through the points A, B, C and D draw vertical lines meeting the line FM at the points F, H, K and M. On the side AF set off the distance FE equal to 3"; through the point E draw the horizontal line EG cutting the line BH at G. On the line MC set off the distance MN equal to 2" and through the point N draw the horizontal line NL, cutting the vertical line DK at L, join the points E and L, G and N. This completes the required front view; EF and GH are the two longer vertical edges of the object, GH being hidden. LK and MN are the two shorter vertical edges of the object, both visible. 152 ROGERS' DRAWING AND DESIGN. In Fig. 267 is shown the front view, top view and side view of the object just drawn. In this instance the object is placed with its longer vertical edge nearer to the observer ; otherwise the position of the object is exactly the same as described in the preceding exer- cise. _/ The side view is drawn in a manner similar to the front view, by lines projected from alt points (corners) of the top view. No doubt the student has noticed that in drawing an object placed at an angle to the lower edge of the drawing board, but having two faces parallel to the plane of the paper, we draw the top view first ; that is, the view of it, which being parallel to the board, will appear in its simplest outline, with all lines drawn in their true length and posi- tion. ROGERS' DRAW1NC3 AND DESIGN. 153 Fig. 288. To draiu the front view and top vieiu of a hex- agonal prism, standing upon a horizontal plane and having two of its parallel vertical sides, parallel to the lower edge of the drawing. Let each side of the hexagon forming the bases of the prism be equal to one inch and the height of the prism be 4"; the top view is drawn first ; it is a regular hexagon, length of each side being i", Fig. 268. The front view is drawn by projecting lines from the corners of the hexagon shown in the top view, these lines making the vertical edges of the prism, and then intersecting these lines by two horizontal lines 4" apart, thus forming the top and the bottom of the prism. If an object is to be drawn, placed so that it is inclined to the plane of the paper, but having its front face parallel to the lower edge of the drawing board, the front view is drawn first. As a rule it will be observed that that view is draivn first, which is drawn easiest, and especially the view which shows the object in its true form ; the other views are drawn by projection from the different points of the view completed. ROGERS' DRAWING AND DESIGN. Let it be 7^equired to dratv the top view and front view of the same prism as in the same exer- cise, but placed so that tivo of its parallel vertical sides are paral- lel to the lower edge of the draw- ing board and t lie base inclined to the plane of the paper at an angle of 30". Draw the front view agd top view of the prism, Fig. 269, showing the prism standing in a vertical position ; WVZY is this front view and TORSU is the corresponding top view. To draw the front view of this hexagonal prism with its base inclined at an angle of 30 de- grees, draw a line AF making an angle of 30 degrees with a horizontal line. Upon this line erect the rect- angle which is exactly equal to the front view of the hexagon in its vertical position, as shown in the same figure by WVZY. ROGERS' DRAWING AND DESIGN. 155 To draw the top view, extend the horizontal lines of the top view RS, TU indefinitely ; then draw vertical lines through the points B, C, D and E; tliese lines intersect the horizon- tal lines RSK and TULO and the center line OGNP in the points, LGJINM forming the upper face of the prism in the required top view. To complete the top view draw vertical lines through H and F cut- ting the line JK at the point K, the line GP at the point P and the line LO at the point O. To draw a hexagonal pyramid, having the sizes of the hexagonal prism in the preceding exercise, and placed in the same position. The construction in this case is exactly the same as in the last exer- cise ; Fig. 270 shows the required drawing. Fig. 270. 156 ROGERS' DRAWING AND DESIGN. S E ROGERS' DRAWING AND DESIGN. 157 To draiv the fop z>ic-a' and fro)it vieio of a cylinder 7^'/iose axis makes an angle of 60° zvit/i a horizontal line and u? dined filane. X "■- ^ --' \ V ^'^N ^^"\-'''' V- /\ ""v ^-'■■' \ ^v / ^ X f \ \ w / X \ y \ ^» / \ '^ - ' . «/ ^. ^'^< ^ y \- '^' ^-x /\ \ \ ' \ >■ / ^ ". \ - / \ \ / \ \ \ / \ \ ^ X V / '' ^. > N V- — ^-J<-^ _— ^^— 1 — -^- ^T -X v. \» " \ s2l i.X.^>.-.-. X :::/ / V V 1 * \ ■* ^ /^ f-x --M^---v-Y-v 7< ■«/ \ i \ \ \ ^ ^\^ y^ \ ^ i ^^ / ^^^ y' \ 1 *^^ ^v'^ X \. 1 \ ^/K^ **^\" T^~"\" 'yTy^'^ \^^_j__,>\ 4 4', 5"5', 6"6', 7"/' ; Fid. 2H1. the cylinders is represented in the front view by two 45-degree lines, ad and dg. To develop the pipes divide the circle in the end view, Fig. 282, into any number of equal parts, in this case let it be twelve parts. the line 4"4' just meets the lines of the section in the point d. The line 5"5' cuts the lines of the section in the points e and c, the line 6"6' cuts the section lines in the points f and b and the line ""7' cuts the lines of the section in the points g and a. 166 ROGERS' DRAWING AND DESIGN. Draw vertical lines through the points a, b, c, d, e, f and g. After all these lines are drawn we have all that is necessary to complete the development of the cylindrical surfaces. the opening, into which the vertical cylinder will fit. The rectangle ABCD has one side AB equal to the length of the horizontal cylinder, Fig. 282 ; the I 1 ^^ 1 ^---.^^ 2- 2 /^ ia\ 3" 4 d 3 4 4 ii\ C- -H 5" c/ ;k N. e 1; 5' -\^ \6 trr^J. 9/ /\ 1 6^ .... b/ !J >; 6' /I i ^"^ T a h 1 i m| g n T ! Fig. 282. Fig. 283 shows the development of the horizontal cylinder ; the rectangle ABCD is equal to the cyl- inder surface. The curve ODGL is cut out within the rectangle for the joint which is the outline of other side AD is equal to the circumference of the circle, showing the end view of the horizontal pipe, Fig. 282. The twelve divisions marked on the cir- cle are set off on the straight line AD (Fig. 283) so ROGERS' DRAWING AND DESIGN. 167 cl et ^^ ot n Fig. 283. 168 ROGERS' DRAWING AND DESIGN. that together they are equal to the circumference of the circle. The outline of the opening for the intersection of the horizontal pipe with the vertical branch is laid out in the middle of the rectangle ABCD in the following manner : On the middle line 6'6 are set off the distances 6'0 and 6G each equal to g;' There still remains to be drawn the development of the vertical branch of the tee-pipe ; this is found in the same manner as the horizontal part, i. e., by laying out the surface of the vertical cylinder ; that is, by making it equal in length to the circumfer- ence of the circle showing the end view of the cylinder. The development is shown in Fig. 284. l[ aj 3 4| 5| 6l 7| 8 9 to 1 12 J c ~-^ '^ J "~^ ^1 ""E* D\ E\ . /g y ry /n Fig. -^4. K\ l" m\ N X^ (or a;") in Fig. 282, on the lines 5'5 and ;'; are set off the distances 5'P, 5F, 7'N and 7H each equal to the distance 6'f, Fig. 282 (or b6"). The distances 4'R, 4E, 8K and 8'M are set of? on the lines 8'8 and 44, to equal the distance es' (or C5") of Fig. 282. The lines 3'3 and 99 are touched by the curve of intersection in their center at points D and L. On the line AB are set off the twelve parts of the circumference and in each one of these divisions is erected a perpendicular to the line AB; on these perpendiculars are laid off successively the length of the vertical lines drawn on the surface of the vertical branch ; the lines AC, iD, 2E, 3F, G4, 5H and 6J in Fig. 284, are equal correspondingly to ROGERS' DRAWING AND DESIGN. 169 the lines ah, bi, cj, dk, el, fm and gn in Fig. 282. Thus one-half of the development ACJ6 is con- structed ; the other 6JP12 is exactly equal to the first part. The method employed in these cases may be ap- plied to nearly all developments of cylindrical sur- faces ; it consists in drawing on the surface of the cylinder, which is to be developed, any number of equidistant parallel lines. The cylindrical sur- face is then developed and all parallel lines drawn in it. By setting off the exact lengths of the parallel lines a number of points are obtained, through loliich may be traced the outline of the desired development. It has been noted in Fig. 282 that the intersec- tion of two cylinders of equal diameters — their arcs intersecting each other — will always appear in the side view as straight lines at right angles to each other. If one cylinder is of a smaller diameter than the other then the intersection will be a curve. Now, let it be required to find the intersection of two cxUndrical surfaces when the smaller cylinder passes through the larger one, their axes intersecting each other. The front view, top view and end view of such two intersecting cylinders is shown in Fig. 285. The larger cylinder is marked by the letter B in all views, the smaller one bv the letter A. Divide the circle in the top view into any number of equal parts, say twelve, marking the divisions of tlie numbers i, 2, 3, 4, etc. Through these points draw vertical lines cutting the small cylinder in the side view in the points t, u, v, w, x, y and z. Divisions exactly like those made by these points will be set off now on the small cylinder in the end view by the points a, b, c, d, etc., through which vertical lines are drawn cutting the larger cylinder in the points e, f, g, h, etc. Extend the line u 12 down- Ward until it meets a horizontal line drawn through g. These lines give the point k at their intersection. The point 1 is obtained by drawing the line v 1 1 downward until it intersects with a horizontal line drawn through the point f ; the point m is obtained by cutting the line 10 w, extended downward, by a horizontal line drawn through the point e. Thus one-half of the required line of intersection, as it appears m the side view, indicated by the points s, k, I, m, is obtained, the other half, m, n, o, p is exactly the same as the first and may be drawn in a similar way. Note. — Such a line of intersection is one which is frequently encountered in mechanical drawing and it is advisable to retain a good idea of its form. In drawing joints or intersections of this kind, it will not be required, as a rule, to lay out the section in the above accurate manner. Keeping in mind the true section of two cylindrical surfaces of different diameters, the student should be ready to sketch the required section freehand, approximately true to suffice for practical purposes. This is done first in pencil and then in ink. 170 ROGERS' DRAWING AND DESIGN. The intersection in this case, as well as in similar cases, where the precise form is required, should be traced through the obtained points carefully and then inked in with the aid of an irregular curve ; for ordinary purposes the section may be represented by an arc of a circle, some- what approaching the curve laid out in pencil. The development of the upper branch of the smaller cylinder is shown in Fig. 287. The line AC contains twelve equal divisions, each equal to one division in the circle in the top view, Fig. 285 ; the length of the line AC is there- fore equal to the circumference of the circle, which represents the top view of the smaller cylinder. Through these divisions on the line AC are drawn perpen- diculars which are made suc- Fiu. 285. ROGERS' DRAWING AND DESIGN. 171 cessively equal to the lines st, ku, Iv, mw, nx, oy, and pz ; in this manner one-half, ADEB of the desired development of the smaller cylinder is the rectangle VWUT, VU being equal to the length of the cylinder and VW to its circumference. It may be divided into two equal parts, one for the upper Fig. 286. Ub O b b] obtained ; the second half BEFC of the develop- ment is an exact duplicate of the first half. The development of the larger cylinder is shown in Fig. 286. The surface of this is represented by half of the cylinder, containing one opening for the upper branch of the smaller pipe, and the otherhalf an exact duplicate of the first one, containing an opening for the lower branch of the smaller cylinder. 172 ROGERS' DRAWING AND DESIGN. To find ihe outline of the opening, draw the cen- ter line C M' M for both halves, and the line HD at right angles to CM in the middle of the first half of the cylinder. On the line VW below the point H set off the distances HG, GF" and FE equal to the distances hg, gf, and fe in Fig. 285; the same distances are The other half of the opening is exactly the same as the first, and is laid off in the same way on the other side of the line M'M. The curve MLKSK' L'M PRDQO is the complete opening for one branch of the smaller cylinder. In the other half of the same figure, a similar opening NN' is laid out for the other branch of the smaller pipe. laid off above the point H, on the same line, so that HG'=HG, G'F=GF and F'E'=FE. Draw lines parallel to HD through the point E, F, G, G', F', and E' and on these lines set forth the distance H S equal to h's (in Fig. 285) GK and G'K' each equal to g'k in Fig. 285 ; FL and F'L' each equal to f'l in Fig. 285 and the distances EM and E'M' each equal to e'm in Fig. 285. Thus one- half, MLKSK'L'M' of the opening is obtained. The rectangular piece, VWTU, with the two openings. MM' and NN', is the required pattern of the larger cylinder. To draiu the development of a four-part elbow. A four-part elbow is a pipe joint made up of four parts, such as is used for stove-pipes ; in Fig. 288, the four parts forming the elbow are AKSI, KXTS, XYZT and YZfd ; of these four parts the ROGERS' DRAWING AND DESIGN. 173 M ^ A A -^y ^\ y\_ sy/^ /\U^2 inches in diameter and an inch or two longer than the paper it is desired to keep and with a tight cover to fit over the outside at one end. MOUNTING BLUE PRINTS FOR THE SHOP. The shop foreman is often put to a great deal of inconvenience because of the rapid destruction, either through becoming soiled or torn, of blueprints which are used at the machines. Some damage is undoubtedly due to careless handling of the prints, but the greater part of the wear and tear cannot be avoided, even with the greatest care, and the spot- ting and creasing soon make the print unusable. To obviate this the blueprints can be fastened on common sheets of pasteboard, but in time the paste- board itself becomes broken and oil-spotted, hence the frequent adoption of the idea of using thin sheet iron as a backing. The prints in common use in the shop are first pasted on pieces of sheet iron, then both sides are varnished over, so as to make the paper oil and waterproof. After being subjected to this treatment, the prints can be hung up near the machines. By thus mounting the prints they are clean and clear, and can be filed away in a small space when not in use ; moreover, they are practically indestructible, because when soiled they can be put under the hose and washed off. 196 ROGERS' DRAWING AND DESIGN. Sheets that are likely to be removed and replaced, for any purpose, as working drawings generally are, can be fastened very well by small copper tacks, or the ordinary thumb-tacks, driven in along the edges at intervals of 2 inches or less. The paper can be very slightly dampened before fastening in this manner, and if the operation is carefully performed the paper will be quite as smooth and convenient to work upon as though it were pasted down ; the tacks can be driven down so as to be flush with, or below the surface of, the paper, and will offer no obstruction to squares. If a drawing is to be elaborate, or to remain long upon a board, the paper should be pasted down. To do this, first prepare thick mucilage, and have it ready at hand, with some slips of absorbent paper I in. or so, wide. Dampen the sheet on both sides with a sponge, and then apply the mucilage along the edge, for a width of ^ -s/g in. It is a matter of some difficulty to place a sheet upon a board ; but if the board is set on its edge, the paper can be applied without assistance. Then, by putting the strips of paper along the edge, and rubbing over them with some smooth hard instrument, the edc/es o of the sheet can be pasted firmly to the board, the paper slips taking up a part of the moisture from the edges, which are longest in drying. TO MAKE DRAWINGS FROM THE PRINTS. To accomplish this, the blueprints may be inked over with " waterproof ink " and when thoroughly dry washed with a solution of oxalate of potash, treated thus the ink lines will remain, and the blue ground will fade and become white and appear similar to an original drawing ; the prints can be bleached by washing them in a saturated solution of oxalate of potash, as above. MECHANICAL DRAWING AND ITS RELA- TION TO PRACTICAL SHOP WORK.* The relation of the drafting room to practical shop work is a vital subject that is constantly forced upon the attention of all by the occurrences of daily work, but each department, the drafting room and the shop, has its well-defined place. In mechanical work we must have first the zdea, or conception of what is wanted, whether the idea comes from the inventor, the draughtsman or the machinist;the draughtsman, by means of the drawing, becomes the interpreter of the idea to the shop. *NoTE. — From an address delivered by L. D. Burlingame, Chief Draugtitsman at ttie Brown & Sliarpe Manufacturing Company, before the Eastern Manual Training Association, at the coiivention held at Boston. ROGERS' DRAWING AND DESIGN. 197 The three important relations hereafter dwelt upon can be stated briefly thus : First — The drafting department as the interpreter /^ the shop —the drawing making plain the meaning and requirements of the designer to the workman. Second— The drafting department as the inter- preter of the shop — the draughtsman, through con- sultation and discussion, making available the practical experience and suggestions of the shop man. Third — The drafting department as the recorder for the shop- — the records of all data and information being so compiled and kept as to be reliable, and quickly available when needed. First. Let us consider the drafting room as an interpreter to the shop : In preparing drawings each piece must be fully and separately detailed, and in many shops, each on a sheet by itself; all particulars of oiling and venting oil holes must be shown, grinding limits given, the depths of tapped holes figured. There must be an indication of when stock is to be allowed for fitting, and of the special kinds of finish on machined sur- faces. All special tools used in manufacturing the piece must be listed below its name, and perhaps a list of operations given either on the drawing or in a separate list. Second. I would earnestly recommend that there be instilled into the minds of technical students the importance of taking advantage of the great mass of mechanical knowledge and the ideas stored up in the minds of the mechanics of the country, in the minds of the men that are actually doing the work, and that the students have it impressed upon them that if they become draughtsmen, one of their im- portant duties will be so to get in touch with the shop as to make this knowledge available, even though it may come to them in crude form from a mind not trained to analyze, to classify and to put ideas upon paper — in other words, that they learn to be the interpreters of the shop. Third. Briefly the ofhce of the drafting room is the recorder for the shop ; here we touch upon the important work of tabulating, listing and classify- ing ; for example : thousands of special tools accumu- late in a large shop ; pro ninent among these are taps, reamers, drills and counterbores, cutters, gauges, etc. ; there are many things to be preserved for ref erence that naturally find their way to the drafting room, such as trade catalogues, photographs, copies of patents and technical journals. The treatment of these in indexing makes all the difference whether they are valuable — of growing value as time goes on — or nearly as worthless. 198 ROGERS' DRAWING AND DESIGN. The importance of the shop side has perhaps been emphasized in what has been said, but I certainly would not belittle the draughtsman, even aside from the high position he often holds as designer and constructor. I agree with the statements made by Prof. Charles L. Griffin; he says: ''The workman of to-day is not permitted to assume dimensions or shapes ; it is his business to execute the draughts- man's orders ; it is, however, often his privilege to choose his own way of doing it, but further than this modern practice does not allow him to go. The drawing is supreme, it is ofificial ; it must be plain, direct and all sufficient." It might be added that to make it so the draughtsman must mentally put himself in the place of the shopman, and antici- pate his needs. The workman will then respect the draughtsman and his work, and will be willing to fol- low implicitly the instructions given on his drawings. TO READ WORKING DRAWINGS. A working drawing should be made, primarily, as plain as possible by the draughtsman ; second, the workman should patiently and carefully study it, so that it is thoroughly understood. In studying a drawing, the object it is intended to represent should be made as familiar as possible to the mind of the student, so that he may fill out in imagination the parts designedly left incomplete — as in a gear wheel where only two or three teeth are drawn in, that he may see, mentally, the whole. Drawings are almost always made "finished size," that is, the dimensions are for the work when it is completed. Consequently all the figures written on the different parts indicate the exact size of the work when finished, without any regard to the size of the drawing itself, which may be made to any reduced and convenient scale. Even in full size drawings this system of figuring is not objectionable. It is a system which should be followed whenever a drawing is made " to work to," for it allows the workman to comprehend at a glance the size of his work and the pieces he has to get made. Figuring makes a drawing comprehen- sible even to those who cannot make drawings. In some figures it is necessary to show end views, also section views, to enable all measurements to be read from the drawing. Fig. 302 represents a blueprint of a bracket-bear- ing, constructed from the drawing, for the Raabe compound oil engine ; the front, end view and plan, with dimensions, are shown ; the scale is full size 1 2"== I ft. MACHINE DESIGN, The study of mechanical drawing not only consists in copying drawings of machinery and dia- grams by accurate measurements and fine finished lines, but it includes the purpose and practical operation of the mechanism designed ; i. e., drawing as a means to an end. The designing of machines requires an extended acquaintance of parts and of similar mechan- isms which have been found suitable for the work required and thus have become standard elements of construction ; to utilize this knowledge is ofte' the lite task of the draughtsman and designer of machinery. It is a matter of common acceptance that machine design depends more upon an acquaint- ance with mechanism and siiop practice than upon a knowledge of the strength of materials and other kindred subjects making up the science of mechanics; this is the reliance, however unscientific it may be, that is depended upon in perfecting the designs for the machinery that is being produced to-day, and there will probably never be another system that, on the whole, will be more satisfactory. It is, however, not sufficient to limit our education to observation of completed working machines ; it is just as necessary to know the theoretical principles and laws of mechanical con- struction ; these have been classified as Theoretical Mechanics or Theory of Mechanism ; a few necessary definitions and general considerations will be found on the succeeding pages Note. — "The correct forms to be given to the raatenais employed in the construction of tools or machinery depend entirely upon Liatural principles. Natural form consists in giving each part the exact proportion that will enable it to fuifiU -ts assigned duty with the smallest expenditure of material, and in placing each portion of the materials under the most favorable conditions of position that circumstances will admit of. " Such natural form is not only the most economical but, strange to say, it is always correct in every :'espect, and is invariably beautiful and lovely in its outlines." — Andrews. ?05 206 ROGERS' DRAWING AND DESIGN. The most successful designer is no doubt born with a love for mechanics and a measure of inventive ability ; if to these inherent qualities be added a retentive memory, a mind trained to observe closely, deliberate carefully and decide wisely, he should be a success. Technical education in itself is of little avail ; but if allied to these other qualities, perfect and round them out, smoothing the way over places that would be otherwise well nigh insurmountable. The cost and results of special machinery depend so much on the ability of the designer that it may be well to consider what his attainments should be ; he should be able to clearly illustrate his ideas- — not nec- essarily a finished draughtsman — and he should have a practical experience in machine shop practice so that to know that the elements of his design can readily be machined, and that no unnecessary trouble be had with the patterns or in making castings from them. He should also know enough of machine design that no illy-proportioned parts disfigure or weaken the structure, and sufficient taste to realize that true art in machinery does not consist of imitating archi- tectural embellishments ; for beauty, as well as for strength and cheapness, castings should be of the simplest shape possible ; rounded corners, especially interiors, straight lines where permissible, with all projections provided for originally, rather than lo appear as afterthoughts, are the principal elements of mechanical beauty. In reference to the particular case in hand, the designer ought to familiarize himself with the methods before employed, if the product has been previously made ; quantities of product expected from the machine, space to be occupied, size, weight, speed, power required, and number likely to be made, should be carefully considered. All notes, deductions, sketches and the like should be carefully preserved, at least until the machine is completed ; that is, actually built, for these sketches may prove to be proof of the most convincing char- acter should questions arise as to mechanical elements considered, even at the time unapproved. It is inconceivable that without shop experience a designer can be highly successful ; the more ex- perience the better, not in one shop alone but several. To succeed requires determination and painstaking hard work ; a mistaken figure, a wrong calculation or blunder of any kind is sure to bring vexation for some one, and possibly a serious loss. Finally, always study simplicity of construction, avoiding as far as possible all special shaped wrenches, etc., and using "more than enough" of iron and steel in all designs, to assure strength and durability. ROGERS' DRAWING AND DESIGN. 207 DEFINITIONS AND GENERAL CONSIDERATIONS. Attraction. This is an invisible power in a body by which it draws anything to itself ; the power in nature acting naturally between bodies, or par- ticles, tending to draw them together ; the attraction of gravitation acts at all distances throughout the universe ; adhesive attraction unites bodies by their adjacent surfaces ; chemical attraction, or chemical affinity, is that peculiar force which causes elemen- tary atoms or molecules to unite. Co-ef&cient is a number expressing the amount of some change or effect under certain fixed con- dition as the co-efficient of expansion ; X\\^ co-efficient of friction ; the word generally means, " that which unites in action with something else to produce the same effect!' Cohesion is that force which binds two or more bodies together. It is that force which the neigh- boring particles of a body exert to keep each other together. Ductility \s that property by which some metals can be drawn out into wire or tubes. Mffort is a force which acts on a body in the direction of its motion. Elasticity is the property possessed by most solid bodies, of regaining their original form or shape, after the removal of a force which caused a change of form. ^Energy is the capacity for performing work ; the kinetic energy of a body is the energy it has in virtue of being in motion ; kinetic energy is some- times called actual energy ; potential energy is energy stored up as that existing in a spring or a bent bow, or a body suspended at a given distance above the earth and acted upon by gravity. The efficiency of a machine is a fraction ex- pressing the ratio of the useful work to the whole work performed, which is equal to that expended. A Factor is one of the elements or quantities which when multiplied together form a product. Force is that which tends to produce or to de- stroy motion ; if a body is at rest anything which tends to put it in motion is a force ; centrifugal force is that by which all bodies moving around another body in a curve, tend to ffy off from the axis of their motion ; centripetal is that which draws, or impels a body toward some point as a center ; force is equiv- alent ^.o push or pull. 208 ROGERS' DRAWING AND DESIGN. Fatigue of Metals. In many cases materials are subject to impulsive loads and a gradual diminu- tion of strength is observed ; in part this deteriora- tion of strength may be due to the ordinary action of a live or repeated load, but it appears to be more often due directly to the gradual loss of the power of elongation in consequence of the slow accumula- tion of \\i^ permanent set ; the latter may be defined as the fatigue of metals. Friction is that force which acts between two bodies at their surface of contact so as to resist their sliding on each other, and which depends on the force with which they are pressed together. Gravity. We can not say what gravity is, but what it does, — namely, that it is something which gives to every particle of matter a tendency toward every other particle. This influence is conveyed from one body to another without any perceptible interval of time. We weigh a body by ascertaining Note. — It appears that in some if not all materials a limited amount of stress variation may be repeated time after time without apparent reduc tion in the strength of the piece ; on the balance wheel of a watch for instance, tension and compression succeed each other for some 150 mil- lions of times in a 3-ear, and the spring works for years without show- ing signs of deterioration. In such cases the stresses lie well within the elastic limits ; on the other hand the toughest bar breaks after a small number of bendings to and fro when these pass the elastic limits. the force required to hold it back, or to keep it from descending ; hence, also, weights are nothing more than measures of the force of gravity in different bodies. Inertia is that property of a body by virtue of which it tends to continue in the state of rest or motion in which it may be placed until acted on by some force. Kinematics. The science that treats of mo- tions, considered in themselves, or apart from their causes ; " Kinematics forms properly an introduc- tion to mechanics as involving the mathematical principles which are to be applied to its more prac- tical problems." LtOad. By the load on any member of a machine is meant the aggregate of all the external forces in action upon it. These may be distinguished as (i) the useful load, or the forces arising out of the use- ful power transmitted, and (2) the. prejudicial resist- ances due to friction, to work uselessly expended, to weight of members of the machine, to inertia due to changes in velocity of motion, and to special stresses caused in the apparatus by changes in its parts through variations of temperature. ROGERS' DRAWING AND DESIGN. 809 There are two kinds of load : first, a dead load which produces a permanent and unvarying amount of straining action, and is invariable during the life of the machine — such, for example, as its weight; and, second, variable or live load, which is alter- nately imposed and removed, and which produces a constantly varying amount of straining action. Everj' load which acts on a structure produces a chang-e of form, which is termed the strain due to the load. The strain may be either a vanishing or elastic deformation, that is, one which disappears when the load is removed ; or a permanent defor- mation or set, which remains after the load is re- moved. In general, machine parts must be so designed that, under the maximum straining action, there is no sensible permanent deformation. The Breaking Load is that load which causes in those fibres which are subjected to the greatest strain, a tension equal to the Modulus of Rupture ; in every case this is equal to the force necessary to tear, crush, shear, twist, break, or otherwise deform a body. Modulus. The primary signification of the Modulus is a measure ; the modulus of a machine means the same as the efficiency of it. " The modulus of a machine is a formula (or measure) expressing the work a given machine can perform under the condition under which it has been constructed"; the words mode, model, mold are kindred terms all formed from the same root-word and meaning some- what the same. Modulus of Resistance is the strain which corre- sponds to the limit of elasticity, compression and expansion each having a corresponding modulus. Modulus of Rtipture is the strain at which the mo- lecular fibres cease to hold together. Modulus of Elasticity is the measure of the elastic extension of a material, and is the force by which a prismatic body would be extended to its own length, sup- posing such extension were possible. The Modulus of a Machine is the amount of work actually ob- tained, divided by the work that should be obtained theoretically. Momentum means impetus or push ; it is the quantity of motion in a moving body ; it is always proportioned to the quantity of matter multiplied into the velocity. Moment is the tendency, or measure of tendency, to produce motion, especially motion about a fixed point or axis. Motion signifies movement ; in mechanics it may be either simple or compound, the latter consists of 210 ROGERS' DRAWING AND DESIGN. combinations of any of the simple motions. The acceleration of motion is the rate of change of the velocity of a moving body, in either an increasing or a decreasing rate. Power is the rate at which mechanical energy is exerted or mechanical work performed, as by a steam engine, an electric motor, etc. Theoretical Resistance is the force which, when applied to any body, either as tension, com- pression, torsion or flexture, will produce in those fibres which are strained to the greatest extent, a tension equal to the modulus of resistance ; or, in other words, it is a load which strains a load to its limit of elasticity. The Practical Resistance often improperly termed merely resistance, is a definite but arbitrary working strain to which a body may be subjected within the limits of elasticity. Ultimate Strength. If the straining action on a bar is gradually increased till the bar breaks, the load which produces fracture is called the ulti- mate or breaking strength of the bar. That ultimate strength is for different materials more or less roughly proportional to the elastic strength. We may insure the safety of a structure by taking care to multiply the actual straining action by a factor sufficiently large to allow, not only for unforeseen contingencies and the neglected causes of straining^ but also for the difference between the elastic and ultimate strength. The actual straining action mul- tiplied by this factor is still termed 2. factor of safety^ and is then equated to the ultimate strength of the structure ; the value of the factor of safety must be determined by practical experience. The Co-efficient of Safety is the ratio between the theoretical resistance and the actual load, or, what amounts to the same thing, the ratio between the elastic limit and the actual tension of the fibres. The Factor of Safety is the ratio between the breaking load and the actual load. As a general rule, for machine construction, the Co-efficient of Safety may be taken as double that which is used for construction subjected to statical forces. The Strength of Materials entering into ma- chine construction is measured by the resistance which they oppose to alteration of form, and ulti- mately to rupture, when subjected to force, pressure, load, stress or strain. Stress is the re-action or resistance of a body due to the load. ROGERS' DRAWING AND DESIGN. 211 Strain is the alteration in shape, as the result of the stress. Tenacity \?> the resistance which a body offers to being pulled asunder, and is measured by the tensile strength in lbs. per square inch of the cross section of the body. Tensile Strength is the resistance per unit of surface, which the molecular fibres oppose to separ- ation. Velocity is the rate of motion ; in kinematics, speed is sometimes used to denote the amount of velocity without regard to direction of motion, while velocity is not regarded as known unless both the direction and the amount are known. — (W. I. D.) Linear velocity is the rate of motion in a straight line, and is measured in feet per second, or per minute, or in miles per hours. Circjilar velocity is the rate at which a body describes an angle about a given point, and is measured in feet per second or per minute, or in number of revolutions per minute, as is a pulley or shaft. Uniform velocity takes place when the body moves over equal distances, in equal times. Variab 'e velocity takes place when a body moves with a constantly increasing or decreas- ing speed. Velocity ratio is the proportion between the movement of the power and the resistance, in the same interval of time. ViS-VlVa, or living force, is a term formerly used to denote the energy stored in a moving body ; the term is now practically obsolete, its place being taken by the word energy. WorU is the overcoming of a resistance through a certain space, and is measured by the amount of the resistance multiplied by the length of space through which it is overcome; the Principle of Work : The foot-pounds of work applied to a machine must equal the number of foot-pounds of work given up by the machine plus the number absorbed by friction. Note. — The simplest possible example of doing work is to raise a weight through a space against the resistance of the earth's attraction, that is to sa)', against the force of gravity. For instance, if a hundred pounds be raised vertically upwards, through a space of three feet, work is done, and, according to the above, the amount of work done is mea- sured by the resistance due to the attraction of the earth or gravity, i.e., one hundred pounds, multiplied by the space of three feet, through which it is lifted. The product formed by multiplying a pound by a foot is called a foot-pound. Thus, in the above instance, the amount of work done is 300 foot-pounds. Had the weight been only three pounds, but the height to which it was raised been 100 feet, the quantity of work done would have been precisely the same, i.e., 300 foot-pounds. 212 ROGERS' DRAWING AND DESIGN. PHYSICS. Physics is that branch of science which treats of the laws and properties of matter and the forces acting upon it ; especially that department of science (known, formerly, as Natural Philosophy) which treats of the causes that modify the general proper- ties of the bodies. The object of physics is the study of phenomena presented to us by bodies ; it should, however, be added that changes in the nature of the body itself, such as the decomposition of one body into others, are. phenomena, whose study forms the more imme- diate object of chemistry. MECHANICS. Mechanics is that section of natural philosophy or physics which treats of the action of forces on bodies. That part of mechanics which considers the action of forces in producing rest or equilibrium is called statics ; that which relates to such action in produc- NoTE. — " The mechanics of liquid bodies is also called hydrostatics or hydrodynatnics , according as the law of rest or of motion are con- sidered. The mechanics of gaseous bodies is called also pneumatics. The mechanics of 7?a?'rf5 in motion with special reference to the methods of obtaining from them useful results constitutes hydraulics." — Webster's International Dictionary. ing motion is called dynamics. The term mechanics includes the action of forces on all bodies, whether solid, liquid, or gaseous. It is usually, however, used of solid bodies only. Applied mechanics is the practical use of the laws of matter and motion in the construction of machines and structures of all kinds. PROPERTIES OF MATTER. The two essential properties of matter, both of which are inseparable from it, are extension and impenetrability. Extension, in the three dimensions of length, breadth, and thickness, belongs to matter under all circumstances ; and impenetrability, ox the property of excluding all other matter from the space which it occupies, appertains alike to the largest body and the smallest particle. The limits of useful knowledge relating to the properties of matter may be found in the three fol- lowing definitions : (a) " An atom is an ultimate indivisible particle of matter." (b) " An atom is an ultimate particle of matter not necessarily indivisible ; a molecule." ROGERS' DRAWING AND DESIGN. 813 (c) "An atom is a constituent particle of matter, or a molecule supposed to be made up of subordi- nate particles." — W. I. D. As no one really knows what matter is in the abstract, not even the most powerful microscope having- shown it, it were wise to rest here. The quantity of matter which a body contains is called its Mass ; the space it occupies, its Volume ; its relative quantity of matter under a given volume, its Density. All bodies have empty spaces denom- inated Pores. In solids, we may often see the pores with the naked eye, and almost always by the microscope ; in Huids, their existence can be proven by experiment ; there are reasons for believing that even in the densest bodies, the amount of solid matter is small compared with the empty spaces, hence it is inferred that the particles of matter touch each other only in a few points. There are also several other properties which are known by experience to belong to all matter, as gravity, inertia, and divisibility ; and others still Note. — The distinction between weight and moment is one impor- tant to have in mind. Weight, in mechanics, is the resistance against which a machine acts as opposed to the power which moves it ; moment, in mechanics, is the tendency or measure of tendency to produce motion, especially motion about a fixed point or axis. which belong not to matter universally, but only to certain classes of bodies, as elasticity, malleability, or the power of being extended into leaves or plates ; and ductility, or the power of being extended in length, as when drawn into wire. The mass of a body, or the quantity it contains is a constant quality, while the weight varies according to the variation in the force of gravity at different places. THE THREE STATES OF MATTER. Matter is any collection of substance existing by itself in a separate form ; matter appears to us in separate forms which however can all be reduced to three classes, namely, solids, liquids, gaseous ; a solid offers resistance to change of shape or shape of bulk, always keeping the same size or volume and the same shape ; a liquid is a body which offers no resist- ance to a change in shape and a gas or vapor is any substance in the elastic or air-like shape. Note. — The difference between a gas and a vapor is one less of kind than of degree. It is important to note that experiment proves that every vapor becomes a_gas at a sufficiently high temperature and low pressure, and, on the other hand, every gas becomes a \apor, at suffi- ciently low temperature and high pressure. 214 ROGERS' DRAWING AND DESIGN. THREE LAWS OF MOTION. As there are three states of matter already de- scribed, i. e., solids, liquids, gaseous, so there are three laws of motion. These are as follows : Law I. " Everybody continues in its state of rest, or of uniform motion in a straight line, except in so far as it is compelled by force to change that state." Law 2. "Change of (quantity of) motion is pro- portional to force, and takes place in the straight line in which the force acts." Law 3. " To every action there is always an equal and contrary reaction ; or the mutual actions of any two bodies are always equal and oppositely directed." The above are " Newton's Laws." Law one tells us what happens to a piece of matter left to itself, i. e., not acted on by forces; it preserves its " state," whether of rest or of uniform motion in a straight line. The first law gives us also a physi- cal definition of " time," and physical modes of measuring it. Law two tells us— among other things, how to find the one force which is equivalent, in its action, to anj* given set of forces. For, however many change of motion may be produced by the separate forces, they must obviously be capable of being compounded into a sin- gle change and we can calculate what force would produce that. Law three furnishes us with the means of studying directly the transference of energj* from one body or system to another. Experi- ment, however, was required to complete the application of the law. MATERIALS USED IN MACHINE CON- STRUCTION. The designer should not only know what provi- sions are to be made for strength, wear and tear, but he should also be familiar with the various mater- ials used in machine construction ; he should know what parts of the design are to be cast, forged, cast in one piece or framed or put together of many pieces and also how the work is done. The principal metals used in machine construc- tion are : Cast iron, wrought iron and steel. Cast iron is a mixture and combination of iron and carbon, with other substances in different pro- portions. The first smelting of the iron ore pro- duces pig iron. Pig iron is very seldom used in construction ; as a rule it is remelted and made into the kind of iron required for construction. The qualities of cast iron depend upon the proportion of carbon contained therein. There are different trades of cast iron : 1st. White cast iron contains only a very small proportion of carbon ; it is very hard and brittle, it is mostly used for manufacturing wrought iron and steel. 2nd. Gray cast iron contains part of the carbon in chemical combination and the rest is mechanically ROGERS' DRAWING AND DESIGN. 215 mixed with the iron in the form of graphite. Gray cast iron is divided into several kinds (mainly three), according to the quantity of carbon in the shape of graphite it contains ; these are Nos. i, 2 and 3. No. i contains the largest and No. 3 the smallest percent- age of graphite. The first kind has a great fluidity when melted and casts well; it has but little strength. The las.t kind has considerable strength and makes the mo.st rigid and massive castings. The great facility of casting this iron into any de- sired mold is the principal reason for its unlimited application and its great utility in machine construc- tion. It cannot be welded or riveted, it is very brittle and it has but little elasticity. These disadvantages cause the designer often- times to select more expensive materials. This iron is mostly used where rigidity and weight are of the utmost importance, as for instance in bed plates, frames, hangers, gears, pipes, etc. Wrought iron is produced by decreasing the quantity of carbon contained in cast iron ; it cannot be cast but can be worked into form by rolling or forging ; it can be welded, punched, riveted, etc., it is flexible and malleable. For shafts it is "cold rolled," thus adding to its strength and elasticity. Steel is refined or nearly pure iron, chemically combined with a certain per cent, of added carbon. Its great elasticity and strength make it the most suitable material for machine construction. Steel is divided into different varieties, according to the amount of carbon contained in it. Steel can be forged like wrought iron and it is fusible. Its hard- ness depends entirely upon the per cent, of carbon contained therein. According to its quality it may be used for cutlery, tools, springs and so forth. In selecting materials for machine construction, the most important properties that must be consid- ered, are : strength, stiffness, elasticity, weight, dura- bility, ease of manufacture and cost. MACHINES. Machines are divided into simple and compound; and machines when they act with great power, take the name, generally of engines, as the pumping en- gine. The simple machines are six in number, viz. : The lever. The wheel and axle. The pulley. The inclined plane. The screw. The wedge. These can in turn be reduced to three classes : I. A solid body turning on an axis. 2. A flexible cord. 3. A hard and smooth inclined surface. 216 ROGERS' DRAWING AND DESIGN. For the mechanism of the wheel and axle and of the pulley, merely combines the principle of the lever with the tension of the cords ; the properties of the screw depend entirely on those of the lever and the inclined plane ; and the case of the wedge is analogous to that of a body sustained between two inclined planes. All machines, however complicated they may be, are combinations of simple mechanical devices; the object in combining them is to give such a direction Note. — Man as a Machine. — " The human body forms an example of a machine. Physiologists calculate the work done by the body in foot tons, a foot ton of work being represented by the energy required to raise one ton weight one foot high. A hard-working man in his day's labor will develop power equal to about 3,000 foot tons, this amount representing both the innate work of his frame involved in the acts of living and his external muscular labor as a hewer of wood and a drawer of water. "A man's heart, in twenty-four hours, shows a return equal to 120 foot tons ; that is, supposing he could concentrate all the work of the organ in that period into one big lift, it would be capable of raising 120 tons weight one foot high. The breathing muscles, in twenty-four hours, develop energy equal to about 21 foot tons, and when are added the actual work of the muscles and that expended in heat production 3,000 foot tons are arrived at as the approximate daily expenditure of energy. ' ' All this power, moreover, is developed on about eight and one-third pounds of food a day, the supply including solid food, water and oxygen. No machine of man's invention approaches near to his own body, there- fore, as an economical energy producer ; and this for the practical rea- son that the human engine gets at its work directly and without loss of power entailed in other appliances that have to transmit energy through ways and means involving friction and other untoward conditions." and velocity to the motion as will enable the ma- chine to do the required work. The study of machines is divided by Reuleaux into the following parts : 1. The study of machinery in general, looked at in connection with the work to be performed ; this teaches what machines exist and how they are con- stituted. 2. The theory of machines, which concerns, itself with the nature of the various arrangements by means of which natural forces can best be applied to machinery. 3. The study of machine design, the province of which is to teach how to give the bodies constituting the machine the capacity to resist alterations of form. 4. T+ie study of pure mechanism, or of kinematics, which relates to the arrangements of the machine by which the mutual motions of its parts, considered as changes of position, are determined. Upon these foundation principles have been con- structed many thousands of machines ; instances are on record where the number of tools and machines have run into the tens of thousands used in a single shop, in another more than three thousand "jigs" were in use ; from this may be perceived the possi- bility of describing but few of the many examples. ROGERS' DRAWING AND DESIGN. 217 STRESSES, STRAINS AND LOADS. The great variety of materials employed in ma- chine construction precludes a complete table of factors of safety for use in practice for various ma- terials under dead and live loads and for machines subjected to sudden and frequent strains of short duration, known as shocks. We give here only a few of the most important materials showing the factors allowed in general practice : FACTORS OF SAFETY. Material MATERIALS. Varying subject to Dead Load. Load. shocks. Cast Iron 6 10-15 15-20 Wrought Iron 4 6 12 Steel 4 7 15 Copper 5 8-10 10-15 Timber 8 10 15 Masonry and Brickwork 15 25 30 The stresses to which constructions and parts of constructions may be subjected are of three kinds, mainly • I. Tensile strain or stress, which has a ten- dency to lengthen the body in the direction of the load. 2. Compressive or crushing strain or stress, which produces a tendency to shorten or crush the body in the direction of the load. 3. Shearing strain or stress produced in a piece of material which is distorted by a load, tending to cut it across. Various metals have a different strength to resist compressive and tensional stresses. Stress is usually measured in pounds per square inch. As mentioned above, when a part is not loaded beyond its limit of elasticity, the stress produced is directly proportional to the strain, so that the stress divided by the strain is a constant quantity for the same material. This constant quantity is called the modulus of elasticity. The modulus of elasticity is found by dividing the stress by the strain. The modulus of elasticity is also called the co- efificient of elasticity. If a cross section of a given bar is equal to A square inches and if this bar is subjected to a load of W pounds which may result in tensile or com- pressive stresses, and if the modulus of elasticity of the material in the given bar is equal to E pounds per square inch, then the strain produced is determined by the formula : W Strain = E X A 218 ROGERS' DRAWING AND DESIGN. MODULUS OF ELASTICITY. Materials. Pounds per sq. in. Cast Iron 1 8,000,000 Wrought Iron (in bars) 29,000,000 " " (in plates) 26,000,000 Steel 30,000,000 Brass (cast) 9,000,000 " (wire) 1 4,000,000 Copper (in sheets) 15,000,000 " (wire) 1 7,500,000 The stress or load per sq. in. of section is called the unit stress. For instance, if a bar is subjected to a load of 2,000 lbs. and the cross section of the bar is equal to 4 sq. in., the unit stress of the bar would be 2000 ^4=500 lbs. As has been said before, strain is the amount of alteration in form of a piece of material produced by a stress to which the piece is subjected. If a wrought iron bar is subjected to pulling stress and is, as a result of this, lengthened titVt of an inch, this change in its form in length or area, as may be the case, is called the strain. The unit strain is the amount of alteration of form per unit of form. It is usually taken per unit of length, and then it is called the elongation per unit of length. We may express this in the following formula : Strain = increase in length of bar original length of bar For instance if a bar 8 ft. long is elongated by jV of an inch when subjected to a load, the strain is equal to tV divided by 96 rsiTT- It is to be remembered that the relation of the proportion between the stress and the strain ip true only within the elastic limit. The smallest load which will cause the rupture of a piece of material is called the ultimate strength of that piece, that is the stress in lbs, per sq. in. which the piece can sustain just before rupture takes place. The following is a table of ultimate strengths : ULTIMATE STRENGTH IN POUNDS PER SQUARE INCH. Material. Tensile. Compressive. Shearing. Cast Iron 19,000 90,000 20,000 Wrought Iron... 52,000 52,000 50,000 Steel 100,000 1 50,000 70,000 Wood 10,000 8,000 600 to 3,000 I ROGERS' DRAWING AND DESIGN. 219 Example i, Fig. 303. A wrought iron bar 2" by 2" in section is subjected to tension by the action of a load ; it is required to find the weight which zvill cause its rupture. The foregoing table of ultimate strengths shows 52,000 lbs. per sq. in. as the tensile stress, and as the given bar measures 4 sq. in., in section, the load required is 52,000x4^208,000 lbs Fig. 308. Fio. aM. Example 2, Fig. 304. A square cast iron block is required to sustain a load 0/80,000 lbs. What must be the Icjigth of a side of this block ? Let us employ a factor of safety, say 5 for cast iron. That is, we will suppose that the load which is to be sustained will be 5 times greater than 80,000, i. e., 400,000 lbs. As the piece of material in question is subject to a compressive stress, we find the ultimate strength of cast iron in compres- sion, go,ooo per sq. in. of section. To find the re- quired area of a section, divide the load, 400,000, by the ultimate strength, 90,000, -Vir°iW-= 4-444 square inches. This is the square section required for the block ; so to find the length of a side, take the square root of 4.444 which is 2.1081 inches, or about 2^^ inches. From this example, as well as from what has been said above, we draw the following conclusions : The resistance to compression, of a piece which is short, compared with its cross section, is calcu- lated by the following formula : Load = Area of Section x Compressive Stress. The compressive strength of materials is gener- ally much more difficult to determine when the ma- terial is of a soft and plastic character which causes them to spread out when under compression. The method here described for the calculation of the compressive strength of materials is true only in the case where the given piece is comparatively short. Longer pieces of material, subjected to com- 220 ROGERS' DRAWING AND DESIGN. pressive stresses are much more difficult to calculate because of other strains arising from the action of the load. The resistance to tension is calculated as in the first example. If a piece having a cross section of A square inches is subjected to a tensile stress by the action of a load of W pounds, and if the ten- sile strength in pounds per sq. in., is uniformly dis- tributed over the cross section, and is equal to f, then the load W = A (area) x f (slre"ngt'h). Resistance to shearing is calculated by the same formula as the resistance to tension, namely : W = A X f . The ultimate shearing strength of metals is usually from 70 to 100 per cent, of their ultimate tensile strength. Stresses induced by bending. When a beam is supported at both ends, the load causes the material, in the upper part, to be com- pressed and that in the lower part to be stretched. We may imagine a horizontal surface separating the compressed part of the beam from the stretched part. We shall call this surface the neutral surface of the beam. The line in which this surface inter- sects a transverse section of the beam is called the neutral axis of that section. It is evident that a beam may have as many neu- tral axes as there are cross sections taken in the beam. The bending stresses occurrino- in a beam supported at both ends will depend not only upon the magnitude of the forces acting thereon, but also on the distances of the line of action of the given forces from any section of the beam under consider- ation. At any point in the length of the beam, the bend- ing action is equal to the sum of all external forces at that point. This is expressed generally as follows : the bending action must be measured by the moments of the forces acting on the beam relative to the given section. The moment of a force is equal to the force multiplied by the length of the perpendicular to the direction of the force, from a point in which the beam is supposed to be fixed. In Fig. 305 the moment of the force induced by the weight of 20 lbs. is equal to 20 times 10 ^= 200 ft. pounds. In Fig. 306, the moment is equal to 20 x 9= 180 ft lbs. The resultant moment of the forces acting on the beam on one side of a given section, referred to that section, is called the bending moment on the beam at that section. For instance, in a beam fixed at one end and loaded at the other with a ROGERS' DRAWING AND DESIGN. 221 Fig. 306- K Jft f Fig. 307. weight of lOO pounds, Fig. 307, the bending moment at a cross section at a distance of 5 ft. from the free end of the beam is 100 x 5=500 ft. lbs. In Fig. 308, a beam supported at both ends is shown, and where a uniformly distributed load of W pounds per unit of length and a concentrated load of W pounds at a distance a from one end is given, n R'^ -X Y iVJC -a- -> 2 w Fig. 308. let it be required to find the bending moment at g section, a distance x from one end. First deter- mine by the principle of the lever, the reactions R and R' of the points of support. The forces to the left of the given section are R, W and w X x. The moments of these forces relative to the section are 222 ROGERS' DRAWING AND DESIGN. R X X, W X (x - a) and -^ and the resultant moment Rx-W X (x-a) - ^^ and this is the required bending mo- ment at the given section. The combined compressive stresses on one side of the neutral axis of any cross section are equal to the com- bined tensile stresses on the other side of that axis. These two equal and parallel forces form a couple, whose moment is the moment of resistance of the beam to bending- of that section. The moment of resist- ance is equal to the bending mo- ment. Suppose that the greatest com- pressive or tensile stress at a given section of a beam is equal to f, then we may express the moment of resist- ance by the product of fz, where z is a quantity called the modulus of the section, depending upon the form of the section of material in consid- eration. The modulus of section or section modulus is sometimes called resisting inches of a section. Form of Section. ^--B--^ i^t- Fig. 311. Section MODULU.S. i }3D' I'll:. ;ji('i. fJB' >, o.//sB^ 9) is. •3./W6 j^., Fig. 312. Section Modulus. _. 3.W6 r>^. 6^ n / ..lJ ^ ^^ D J Fiu. S].5. i -i. ''■"■3- Fj(i. 31B. ROGERS' DRAWING AND DESIGN. 223 Form of Section. Section Modulus. Kii;. :11T. Fig. 318. Fig. :ilH. Section Modulus. :♦ - d- -)i \X J- f6 fy/Mm^ Lt m^rV ends the bottom f///^M(^ \\\\\\\\ will be subject to compression. Example 9, Fig. 330. When a beam, having a uniform cross section from end to end is fixed securely at both ends, the load which the beam is made to carry, being distrib- uted uniformly, as in Fig. 330, the bending moment is greatest at the ends and is equal to tV WL. The bending moment at the center is equal to yi of the moment at the ends, that is, equal to 5V WL, and is contrary to the moments at the ends. If a beam is required to be very stiff, the length Flu. ;J30. should be made as short as possible and the depth as great as circumstances will permit. With the same area of section, the deeper the beam the stronger it will be, provided the breadth of the beam is suf- ficient to prevent 'y///X^///^^ lateral breaking. Various applica- tion of the princi- ples of strength of materials will' be discussed in connec- tion with the design of different parts of machines. Another requisite for successful de- signing is a knowl- edge of the proper- ties of materials commonly used for machine construc- tion. In selecting materials for machine parts the designer must consider their properties in regard to the adaptability for the work to which they are to be subjected ; the strength, stiffness, dura- bility and convenience of working into the necessary form. ROGERS' DRAWING AND DESIGN. 2Ji7 A machine properly constructed, must be able to withstand the stress to which its various parts are to be put, and this depends entirely on their action and endurance, as conditioned by the forms of the parts of the machines. By the word stress we mean a force acting between two bodies or two parts of the same body when subjected to the action of a load. This force is understood to resist the load in preventing it from changing the form of the machine or its parts. The combination of all external forces acting on a part of a structure calls into e.xistence a new force within the structure itself, and this resisting force we call stress. All the external forces are called the load of the machine. The effect of the load is the strain pro- duced in the machine ; the strain is the tendency to change the form of the machine part under the influence of the load. The resistance which is offered by a material to the change of form resulting from the application of a load, combined with its natural power of returning to its original shape after the load is removed, is called its elasticity. When a piece of material deformed somewhat when subjected to a load returns exactly to its original form as soon as the load is removed, the piece of material is said to be perfectly elastic with- in certain limits of a load. When under the influence of a load the piece of material is permanently deformed — that is, does not return to its original form when the load is removed — we say that the limit of elasticity of the material has been reached. Up to the limit of elasticity the stress is directly proportional to the strain ; beyond the limit of elasticity the strain increases taster than the stress until rupture is produced. The loads to which material can be subjected may be divided primarily into two classes : a dead load is one which is applied slowly and remains steady and unchangeable ; a live load is one which con- stantly changes, being either alternately imposed and removed, or varying in intensity and direction. To avoid the danger accompanying an unforeseen intensity of strain, which may produce undesirable deformation or rupture, as may be caused by imper- fect workmanship, poor quality of material or other causes, the parts of a machine are usually made to resist a much greater load than will be brought on them in the regular course. The expected load is supposed to be greater, and for this reason is multi- plied by a number known as the factor of safety. The factor of safety varies for different materials according to their structure and application, as well 228 ROGERS' DRAWING AND DESIGN. as for the same kind of material according to con- ditions to which it may be subjected. For materials, the quality of which is liable to change, the factor of safety must be larger than for materials the quality of which is more uniform and less liable to change through atmospheric exposure or varying temperature. It happens that in some structures the whole load cannot be ascertained with accuracy — in such cases the factor of safety must be increased as a safeguard against unexpected straining action. It may also happen that in some machines the working load may be suddenly increased — for such accidental strains a factor of safety must be allowed. SCREWS, BOLTS AND NUTS. In all working drawings consideration should be given to the manner of uniting the different parts of the machine. Screws play a most important part in machine design, particularly as a means for fastening the different parts together. The representation of bolts and screw threads is consequently of such im- portance that a knowledge of their proportions and the usual method of drawing them, is of great con- sideration to machine draughtsmen ; the exact repre- sentation of a screw thread is somewhat difificult ; it takes both time and care. The proper way to draw a screw thread as it ac- tually appears in a finished screw, is by laying out a curve or curves upon the surface of the cylinder, forming the body of the bolt. This curve is called a helix ; the helix may be defined as a curve gener- ated by progressive rotation of a point around an axis, remaining equidistant from the axis through- out the length of the motion. When a machinist desires to cut a thread upon a cylinder, he will first change the gears of the Lathe to produce the desired number of threads toreach inch of length of the screw ; this being done, the cylinder is put in place on the centers of the lathe and the thread cutting tool is then set to its proper angle. Before proceeding to cut the thread, the tool is moved close to tiic work, so as to trace a fine line upon the surface of the cylinder when the machine is put in motion ; the fine spiral produced upon the surface of the cylinder in this manner, is the helix of the screw. Problem : To dram a helix, the diameter and height of one turn being given. The heiijht of one turn of a helix is called its pitch. Let the diameter of the cylinder be 3" and the pitch 2". ROGERS' DRAWING AND DESIGN. 229 Draw the elevation of the cylinder ABCD above its bottom view i, 2, 3, etc., Fig. 331. The eleva- tion ABCD may be four inches high, that is equal to two complete turns of the helix. Lay off the pitch from the point A upon the line AB equal to A 12 and 12 B. Divide the pitch A 12 into any number of equal parts, for instance in this case 12. Divide the circle into the same number of equal parts. Through the points of division on the circle, draw lines parallel to the line AB and ex- tend them through the full height of the front view ABCD. Through the point i of the divisions of the pitch, draw i-i' parallel to AD, intersecting the vertical line I i' in the point i' which is a point in the re- quired helix. Through the point 2 of the pitch divi- sions, draw the line 2-2' parallel to AD and inter- secting the vertical line 2 2' at the point 2', which is another point of the helix. Through the point 3 of the pitch divisions draw the line 3-3' parallel to AD and cutting the vertical line 3 3' in the point 3', which is a third point of the required helix. Pro- ceed in this manner until the sixth point of the helix is found : it will be situated on the line DC. The points A, i', 2', 3', 4', 5' and 6' determine the position of one-half of a turn of the helix, which may be traced through these points, first in pencil and then inked in. 330 ROGERS' DRAWING AND DESIGN. In the same mariner the second half of the first turn may be completed. The accompanying illus- tration renders a repetition of the above explana- tion unnecessary. The second half of the turn is drawn dotted, as it is on the other side of the cylin- der and cannot'be seen. The second turn may be laid out by the aid of the points of the first turn of the helix in the following manner. Set the com- passes to a distance equal to the pitch and lay ofi the points i", 2", 3", etc., above the corresponding point i!, 2', 3', etc., of the first turn of the helix. A thorough understanding of the above problem is of considerable use, not only for drawing large sized screws, but especially for drawing a worm for worm gears, which will be explained later. A screw with a V-thread, drawn with exact helical curves is shown in Fig. 332. It is made of two helices, one for the top of the thread and the other for the root of it. Fig. 333 shows a screw with a square thread. An examination of the drawing will show that the thread is drawn with four helices ; two helices upon the outside of the cylinder, the top of the thread, and two for the root of the thread. It is evident that the method of drawing screw threads with helices while producing an exact representation of the screw cannot be employed in the shop in drawing machin- ery, where, as a rule, the number of bolts and screws is very considerable. The bolts, nuts, etc., are so numerous on some machines, that it is customary to make separate dol^ sheets, showing all screws necessary for one machine, in all their different sizes and forms. The square thread shown in Fig. 333, would appear, when drawn by straight lines only, as in Fig. 334, and the V-thread shown before would be drawn as in Fig. 335. v^ We have so far considered only right-handed screws. A right-handed screw is one, which passing through a fixed nut and turned in the direction of the motion of the hands of a clock, will advance into the nut. A left-handed screw is one, which to pass through a fixei nut^ must be turned in a direction opposite to the motion of the hands of a clock. Such a thread is shown in Figs. 336 and 337. Screws maybe either single-threaded or double-threaded. If we assume that a screw consists of a cylinder with a coil form- ing the thread wound around it, we may easily define a double screw as a cylinder with two parallel coils Note. — To avoid the difficult and tedious operation of drawing the helices, screw threads are generally indicated by straight lines only. ROGERS' DRAWING AND DESIGN. 231 Fig. 333. Fig. 33:3. Fig. 334. 232 ROGERS" DRAWING AND DESIGN. of thread wound around it. Generally the double- threaded screw is defined as one having two paralled threads. A screw having three parallel threads is called a triple-threaded screw. Double-threaded screws are shown in Figs. 338 and 339. The distance betzveen the centers of two successive threads in a single-threaded screw is called the pitch of the screw. Figs. 340 and 341 ; the pitch is equal to the distance tvhich the screw will advance into a fixed nut during one turn. Fig 341 shows the pitch of a square thread. It is equal to twice the pitch of the triangular thread. Screw threads are generally either triangular or square in section, although some other forms are in use. The triangular thread is called the V-thread. The form of V-thread most commonly used in this country, known as the U. S. Standard thread, is shown in Fig. 342. The U. S. Standard screw, known also as the Franklin Institute Standard, was presented to that Institute by Mr. Wm. Sellers, in a paper read by him in 1864. As a result of this, the Franklin Insti- tute recommended for general adoption by American engineers the following rules and table of standard threads : Table of U. S. Standard Screw Threads. (Outside) Diameter of Screw. Threads per Inch. Diameter at Root OF Thread. Diameter op Tap Drill. \ 20 0.1S5 -h A 18 0.240 \ 3 1(5 0.294 A t\ 14 0.344 S3 7T \ 13 0.4011 il- -h 12 0.454 ¥1 % 11 0.507 rf \ 10 0.620 1" 1 9 0.731 1 1 8 0.837 M H 7 0.940 M li 7 1.065 1 3 If 6 1.160 lA u 6 1.284 I5V If 5i 1.389 m If 5 1.491 n H 5 1.616 i| 2 4i 1.712 i|- 2i # 1.962 2 ^ 4 2.176 'H^ 2| 4 2.426 h\ 3 3i 2.629 2p- H 3i 3.100 H 4 3 3.567 H ROGERS' DRAWING AND DESIGN. 233 The proportion of pitch to diameter is P=o.24 -v/D + 0.625 — o. 1 75 The depth of the thread is 0.65 of the pitch. The table does not give the pitch. To find the pitch, divide one inch by the number of threads. Eight threads to one inch give a pitch of ys". FlO. 3%. FlO. 336. FlO. 337. By the term diameter of the screw is always meant the outside diameter. The diameter measured at the root of the thread is called the inside diameter. In the foregoing table of U. S. Standard Screw Threads, the number of threads to one inch of screw to 4" in diameter. is given from i^ 234 ROGERS' DRAWING AND DESIGN. The third column gives the diameter of the screw at the root, or the inside diameter. The next column gives the diameter of drill to be used for any required diameter of tap or thread. They are ordinarily a little laro-er than the diameter at the root of the thread. Fig. 338. . The screw thread is formed with straight lines at an angle of 60° to each other. The top and bottom of the thread are flattened, each to a width of ^^th of the pitch, Fig. 343. For small diameters of bolts the amount of flat- tening is not made to any particular measure, and in drawing screw threads it may be neglected entirely. For a square-threaded screw, the number of threads per inch is equal to one-half the number on a V- threaded screw. Fig. 339. In a square-threaded screw of U. S. Standard form, the width of the thread is equal to the width of the groove — each equal to one-half the pitch, Fig- 344- The depth of the thread is also equal to one-half of the pitch — that is, equal to the width of the groove. ROGERS' DRAWING AND DESIGN. 235 Figs. 345-350 exhibit the conventional methods of showing; different threads of a bolt. Figs. 340 and 341. Fig. 346 represents a single square-thr aded screw. To draw the screw, first draw the cylinder. Lay off distances each equal to one-half the pitch and through the division points draw lines at right angles to the axis of the cylinder, and cutting the other side of the cylinder, the inclination of the parallel lines in- dicating the thread through the width of the cylinder being equal to one-half the pitch. This method is clearly illustrated in Fig. 346. To Fig. 312. Fig. 345 shows a single V-threaded screw, draw the screw lay out the outlines of the cylinder of the bolt and upon one of its sides set off distances each equal to the pitch. Do the same on the other side of the cylinder beginning at a point one-half the pitch from the end of the cylinder, after which draw the lines for the top of the thread. From the points of division draw lines inclined to each other 60° for 236 ROGERS' DRAWING AND DESIGN. FiQ. 343. Pitch i 1^- i Pitch >k • -k P'fcf* Fig. 3«. the threads, on both sides of the cylinder, then con- nect the roots by straight lines. It will be noticed that these lines are not parallel to the lines connect- ing the tops of the thread. Fig. 347 shows a still simpler method of represent- ing a V-threaded screw. The pitch is laid oft as in the preceding example. The heavy lines represent the bottom of the thread. The method employed in Fig. 348 is still more rapid in delineation and is, therefore, recommended for rapid drawing. Here the heavy lines are used to represent the top of the thread, the fine lines in- dicatinjj the bottom of the thread. In Fig. 349 the fine lines are drawn as long as the heavy lines, which makes the drawing of the thread still easier. A method of indicating screw threads when great haste is necessary and for sketch- ing is shown in Fig. 350. In drawing the thread as illustrated in the last four figures, no particular attention need be given to the number of threads per inch. A note written plainly on the drawing, very near to the representa- tion of the screw, gives the exact number of threads to the inch. Even this may be left out when the diameter of the screw is plainly given, with the note "standard" near it; in this case the workman is expected to determine the number of threads to the inch from the table of U. S. Standard Threads. ROGERS' DRAWING AND DESIGN. 237 The proportions of bolt heads and nuts which have been accepted in this country as a standard are as follows : The distance between the paral- lel sides of heads and nuts is equal to I Yi times the diameter of the bolt, plus y% inch=i3^ D + 5^ inch. The thickness of heads is equal to one-half of the distance between the parallel sides. i}4 D + ys inch. 2 The thickness of nuts is equal to the diameter of the bolt^ D. The same proportions are used for square heads and nuts. In all these formula; D expresses the diameter of the bolt. Fig. 351 shows the conventional method of representing a hexagonal nut for a 2" bolt, The height of the nut is equal to 2". The two views may be drawn similar to the two views of a hexagonal prism, ex- plained in the chapter on projec- tion. The curve cde is drawn first, with a radius equal to the height of the nut. When the points c and e are thus determined, a fine straight line is drawn Fia. :i45. Fia. 346, Fio. 349. 238 ROGERS' DRAWING AND DESIGN. Fig. 351. ROGERS' DRAWING AND DESIGN. 239 through these points and extended in both direc- tions so as to cut all vertical edges of the nut in both views, at the points a, g, h, k and m. Arcs are then Fig. 352. Fig. 353. drawn through the points a, b, c and through e, f, g. The same is done in the other view in passing arcs through h, i, k and k, 1 and m. These arcs are struck with compasses, after a centre is found by trial with the compasses. The chamfer at aa and g3 may be drawn by 45° lines, from the points a and g respec- tively. Fia. 354. Fig. a55. This is not the exact construction of the curves as they appear on a hexagonal nut. However, the exact curves are not of any importance on a work- ing drawing, and it will be found that this prac- 240 ROGERS' DRAWING AND DESIGN. tical shop method effects a material saving of time and trouble, particularly as the representation of heads and nuts is of very frequent occurrence in ma- chine drawing. In drawing a hexagonal nut or head, it is the general custom to show three faces of each. A square nut or bolt head is generally shown by drawing one face of each only. Fig. 352 illus- trates a bolt with hexagonal check nuts. It is more conven- ient to make both nuts of stand- ard thickness, that is equal to the diameter of the bolt, although it is often found that the inner nut is made thinner. In the illustration the outer nut is chamfered on both faces. In Fig. 353 a bolt with a square head is shown. The distances between the parallel faces of this head is equal to i y^ times the diameter of the bolt plus y% inch. The height is equal the distance between the to ^ parallel faces. The arc for the chamfer of the head is usually drawn with a radius equal to 2^ times the diameter of the bolt. A set screw is shown in Fig. 354. The figure illustrates all required proportions, as they are Fig. 356. ROGERS' DRAWING AND DESIGN. 241 commonly used. The point of the set screw is usually made with an arc having a radius equal to four times the diameter of the screw. A stud-bolt is one which is threaded at both ends. Fig. 355, one end being screwed into one of the i... 'rd- -lU- — I--- FiQ. 3oT. Fig. 358. Fig. 3.59. pieces of a machine to be connected, while the other end passing through the other piece, which is to be fastened to the first, carries an ordinary nut, as in Fig. 356, which illustrates how a stuffing box is fastened to a cylinder head. The conventional way of representing screws with square heads is shown in Figs, 357, 358. A round head screw is shown in Fig. 359. The head of the screw is slotted. In the top view the parallel lines showing the slots should be drawn at an angle of 45^^ with a horizonal line. This head is particularly adopted for countersunk work. In conclusion a few words are added concerning the strength of bolts. Tke weakest part of the bolt is the section at the bottom of the thread. The fol- lowing is a table of the tensile strength of U. S. Standard Bolts at 5,000 lbs. per sq. in. : Tensile Strength of U. S. Standard Bolts AT 5,000 LBS. PER Square Inch. Diameter of Tensile Diameter of Tensile Screw. Strength. Screw. Strength. \ 134 1-1 5,300 ■h 226 H 6,400 1 339 H 7,650 1^ 465 If 8,800 i 625 i| 10,150 ■^ 809 2 11,500 % 980 H 15,600 \ 1,500 H 18,500 \ 2,100 4 23,000 1 2,750 3 27,200 H 3,450 3^ 37,700 li 3,900 4 49,500 The figures in the second and fourth columns show the total load which can be sustained hy bolts of the above diameters. In calculating the strength of a bolt the stress to which it is subjected by the use of the wrench must be taken. 242 ROGERS' DRAWING AND DESIGN. The figures in the second and the fourth columns show the total load which can be sustained by bolts of the respective diameters. In calculating the strength of a bolt, the stress to which it is subjected resisting strength, the value of the safe stress per square inch of section must be taken comparatively low, and it is advisable for the purpose of overcom- ing all difficulties here mentioned, not to take the k- -15D Q 4 0>|o Fio. 360. 0)|b •*W by the use of the wrench must be taken into consid- eration. Small bolts frequently break because of this strain. It is also necessary to take into account the man- ner in which the load is applied. As the nature of metal of the bolt may not be known as to its safe stress higher than 5,000 lbs. per sq. in. as given in the table. Fig. 360 shows the generally adopted proportions of a wrench. The wrench may be drawn for any size of a bolt head or nut, with the proportions of the parts as given in this illustration. ROGERS' DRAWING AND DESIGN. 243 RIVETS AND RIVETED JOINTS. For fastening together two or more comparatively thin pieces of metal, rivets are generally employed ; their greatest application is found in boiler work, where the joining of plates by riveting is found to be the only practical method. This method of fastening, however, is compara- tively expensive and unsatisfactory in many ways ; the rivets form a permanent fastening and can only be removed by cutting off one of the heads ; this creates trouble and expense. The process of punching the holes in the plates for riveting also has a serious effect by reducing the tensile strength of the plates by the disturbing in- fluence of the punch on the metal near the riveted joints ; for better work the holes are now generally made by drilling ; this, again, is more expensive, especially without the use of multiple drilling ma- chines. The injury due to punching, when the plates have not been cracked by the process, may be remedied by annealing them after punching; the ill effect of punching may also be removed by punching the holes ys" smaller in diameter than the required size of the hole, which may then be completed by ream- ing. Other injurious effects of punching are, i, the difficulty of correct spacing by this method, and 2, the fact that a punched hole is always tapered, the wider end of the hole being tha't next to the die. Fig. 361. Rivets are made in different forms ; that most commonly employed being of a spherical or cup head form, as illustrated in Fig. 361 ; both parts of this rivet show the spherical head. 244 ROGERS' DRAWING AND DESIGN. The rivet shown in Fig. 362 has a conical head, the lower part showing a pan head. The right pro- portions of the parts of the above rivets are given in the illustrations. Fig. 382. Fig. 363 shows a rivet with countersunk heads. The usual proportions of this kind of rivet are marked on the figure. In all the above illustrations the diameter of the rivet is taken as the unit of all proportions. The construction of the spherical head, Fig. 361, is as follows : Fio. 36.3. With a radius equal to one-half the diameter of rivet, from the center A on the vertical center line, describe a circle cutting the center line at the points B and C. Set the compasses to the distance BC and from the point B as center, describe an arc cut- ting the outline of the upper plate in the point D. Make BE equal to the distance AD and with E as ROGERS' DRAWING AND DESIGN. 245 center and CD as radius, describe the arc which forms the outhne of the spherical head. The construction of the other kinds of rivets may be easily understood from the illustrations without special explanation. The length of the rivet required to form the head is about i^ times the diameter of the rivet. For countersunk rivet heads, a trifle more than one-half of this amount is allowed. Riveted joints may give way because of the tear- ing of the plates between the rivets, as illustrated in Fig. 364, by breaking of the plates between the 246 ROGERS" DRAWING AND DESIGN. Fig. 3fi«. Fig. 3H7 rivet holes and the edge of the plate, as shown in Fig. 365 ; by crushing of the plate or by crushing of the rivet, and by the breaking of the rivet through shearing, as indicated by Fig. 366. By the pitch of rivets is meant the distance be- tween the centers of two adjoining rivets, in a single riveted joint, that is where the seam is formed by one row of rivets, Fig. 367. When more than one row of rivets make the joint, the pitch is the distance between the center lines of rivets in the same row, Fig. 368. -; '" J=>itch Fig. 368. ROGERS' DRAWING AND DESIGN. 247 The distance between the centers of two adjoin- ing rivets, both in the same diagonal row is called the diagonal pitch, Fig. 369. The strength of a riveted joint depends upon the arrangement of the rivets and upon their pro- portions. Since a rivet may part either by shearing or by crushing, it is necessary for a given thickness of plate to find the proper diameter of a rivet having Fig. 370. T^i Fig. 3T1. f/ v^^/' ''^^'' 248 ROGERS' DRAWING AND DESIGN. equal shearing and crushing strength. The rela- tion between the thickness of the plate and the di- ameter of the rivet, calculated for single shear, is Fig. 373. expressed by the following formula;, of which the first is true for iron rivets and the second for steel rivets : d=2.o6 t for iron rivets. d=2.28 t for steel rivets. /A ^'k ^\ \^ .^ ?^ ^^ ft Fig. 374. Fig. 375. vS Fig. 376. ROGERS' DRAWING AND DESIGN. 249 d expresses the diameter of the rivet and t stands for the thickness of the plate. For plates thicker than f ^-in. the diameter of the rivet may be smaller in proportion to the thickness of the plate than is required by these formulae. The proportions commonly observed in practice for lap-joints and single-strap butt-joints is given in the following- table: Thickness of plate in inches. Diameter of rivet in inches A 1 i 1 14 i\ 1 f J_ i f 1 f 7 I I 4 H Numerous styles of riveted joints are in general use. The two classes into which the different styles may be divided are the lap-joint and the butt-joint. In the lap-joint, Fig. 370, the plates overlap each other. Figs. 371, 372 show other examples of this form of riveted joint. Fig. 373 shows a butt-joint. Here the plates are butted against each other and a cover plate or strap is placed over their junction and the rivets passed through the plates and strap. Fig. 374 shows a butt-joint with two cover plates. The examples of joints thus far illustrated differ as to the number of rows of rivets that are used for the seam. Fig. 370 is a single-riveted joint. The butt-joints shown in Figs. 373, 374, are also single- riveted. In a single-riveted joint the edge of each plate is pierced by only one row of rivets. -nSSr^ ; Fig. 377. Fig. 378. I '^ ; Si --U ^1. I .1 _. I w- ^m-i-m^ Fig. 3.9. 250 ROGERS' DRAWING AND DESIGN. ^--^ ^ Fig. 380. Figs. 371 and 372 show double-riveted joints ; here the edge of each plate is pierced by a double row of rivets. When the rivets are opposite each other, as in Fig. 372, the seam is known as chain- riveted. When the positions of the rivets in one row are opposite the spaces between the rivets in the other row, the seam is staggered. The following illustrations are examples of riveted joints taken from practice in boiler work. Fig. 375 . A double-riveted lap-joint for two y^," plates, having -j-l" rivets, 1/% holes. The pitch of the rivets in this case is equal to 2^". Fig. 381. ,~^ A similar joint for S/i' plates is also shown in Fig. 376. Here the pitch is 2^" and the rivets ir', the holes being made i ". Another joint of the same T FiG. 382. ROGERS' DRAWING AND DESIGN. 251 Fig. 383. /^Si'. im* Fici. 3M. Fig. 385. character is illustrated in Fig. 377. Here the plates and rivets are the same as in Fig. 376 ; the pitch, however, is 33^ ". In the double-riveted lap-joint shown in Fig. 378, the plate is fl ', the rivets lyV'. the holes are made i|" and the pitch is 3". A butt-joint with double cover plates is illus- trated in Fig. 379. Here the plate is |" steel and i" rivets. The inner covering strap is J^g" thick, the outside strap is equal in thickness to that of the plate. Similar joints are shown in Figs. 380, 381, 382, 383 and 384. The joint shown in Fig. 385 is not used very frequently. POWER TRANSMISSION. The oft-repeated word transmission comes from two Latin words, trans, across, or over, and mittere, to send, hence, to carry from one place to another; the illustration of a few devices for the transmission of power from its cause to its place of useful employment is the limit of this section of design. Prime movers or receivers of power, are those pieces or combination of pieces of mechanism which receive motion and force directly from some natural source of energy ; the mechanism belonging to the prime mover may be held to include all pieces which regulate or assist in regulating the transmission of energy, from the source of energy or power. Throughout this preliminary sketch, power and energy are used synonymously. The useful zvor k of the prime mover is the energy exerted by it upon that piece which it directly drives; and the ratio which this bears to the energy exerted by the source of energy is the efficiency of the prime mover; in all prime movers the loss of energy may be distinguished into two parts, i, 7ieces- sary loss ; 2, zvaste. The sources of power in practical use may be classed as follows : (a) Strength of men and ani- mals, (b) Weight of liquids, (c) Motion of fluids, (d) Heat, (e) Electricity and magnetism. The duty. of a prime mover is its useful work in some given unit of time, as a second, a minute, an hour, a day. Among the first examples of power transmission may be mentioned the case of a man hauling up weight with a rope or pushing or pulling an oar or capstan; in these instances the man is the prime mover and the duty performed is the raising of the weight and the moving of the vessel. The various combinations of mechanical 'povfers produce no force : they only apply it. They form the communication between the moving power and the body moved ; and while the power itself may be in- capable of acting except in one direction, we are able by means of cranks, levers, and gears, to direct and modify that force to suit our convenience. Every one may see examples of this in the construction of the most common pieces of machinery as well as in the most complicated. 355 256 ROGERS' DRAWING AND DESIGN. SHAFTS. When a shaft is rotated by a lever attached to it, as in Fig. 3S6, or by a pulley or a gear-wheel as in Fig. 387, and a force P is applied to the free end of the lever or to a point at the rim of the pulley or u n ?; at the pitch-circle of the gear-wheel a twisting strain is produced on the shaft, this twisting strain causes a combination of stresses within the fibres of the Fig. 38: shaft, which mainly consist of shearing stress. The shearing stress is equal to nothing at the center of the shaft and it is greatest at its circumference. The twisting strain is obtained by multiplying the length of the lever, or the perpendicular distance from the point at which the force is applied to the center of the shaft, by the force P. If this distance be equal to R, Fig. 386, then R X P = T, which is called the twisting moment and is expressed in inch pounds. It is evident that the twisting moment must be equal to the resisting moment of the shaft. For finding the diameter of a crank shaft of a stationary engine with cylinders up to 30" in diam- eter some authorities recommend the following practical rule : The diameter of the crank shaft is equal to the radius of the cylinder minus 5^ of an inch. In practice many different diameters are found performing the same work. Now let T = twisting moment on shaft in inch pounds. N = number of revolutions of the shaft per minute. H = horse power transmitted, then the horse power equals 2 X 3.1416 X T X N H : 0.00001587 OJ'"- The number 33,000 in this formula expresses 33,000 foot pounds of work performed per minute, and this amount of work is called one horse power. The above formula gives a method of finding the horse power transmitted by a shaft. ROGERS' DRAWING AND DESIGN. 257 Rule : Multiply the twisting moment in inch pounds by the number of revolutions per minute, and multiply the product by the number o.oooo- 1587 the product will be the horse power trans- mitted by the shaft. Example : Find the horse power transmitted by a shaft mak- ing 100 revolutions per minute, provided with a gear wheel 36 inches in diameter (pitch circle), the turning force being 4,000 pounds. Solution : Multiply the pitch radius of the wheel,=i8 inches by the force applied, ^4,000 pounds, and multiply the product by the number of revolutions and by the number 0.00001587: Horse power= 18 X 4000 X 100 X 0.00001587=114.264. From the above formula the following expres- sions are obtained : 12 X 33,000 X H 63025.21 X H 2 X 3.1416 X N N From the same formula the twisting moment may be determined when the horse power transmitted by the shaft and the number of its revolutions are given. The number of revolutions may also be obtained from the same formula ; thus, 12 X 33,000 X H 63025.21 X H ^ "2 X 3. 14 1 6 XT~ '^ T Example : To find the number of revolutions which a shaft must make per minute in order to transmit 114.264 horse power, when a force of 4,000 pounds acting on the pitch circle of a gear-wheel of 36" in diameter produces the twisting moment. Solution : The twisting moment in this case is equal to 18 X 4,000= 72,000-inch pounds To find the num- ber of revolutions required divide the given horse power 114.264 by 72,000 and multiply the product by the number 63025.21 thus obtaining a quotient of 100.2 or the revolutions per minute. When the twisting moment only is to be consid- ered in calculating the diameter of a round shaft, which is to transmit a given horse power at a given speed, the following formula may be used : The cube of the diameter of the shaft or. D Twisting moment 0.196 X stress in pounds per square inch. 258 ROGERS' DRAWING AND DESIGN. The stress is taken in pounds per square inch at the outer fibres of the diameter of the shaft. For steel shafts the stress may betaken at 10,000 pounds and for wrought iron at 8,000 pounds per square inch. Long shafts are subjected to combined twisting and bending actions. Let B = bending moment ; T == twisting moment ; Ti = the equivalent twisting moment. Then Ti = B + V^M^T^ In practice for long shafts in factories the follow- ing simple formula is recommended : D^ = 125 X horse power number of revolutions of shaft. The speed of the shaft depends upon the speed of the driving belt or by the diameters of the pulleys upon it. Shafts in machine shops are run from about 120 to 150 revolutions per minute; wood working machinery shafts usually run from about 200 to 250 revolutions per minute. Shafts in woolen mills run up to 400 revolutions per minute. Line shafts should, as a rule, not be less than i^" thick in diameter. The distance between the centers of the bearings should not be great enough to permit a deflection of more than ^w" per foot of length. The more pulleys are on the shaft the closer the bearings must be. The beams may be placed about 8 feet apart, and each beam to be provided with a hanger on its lower side. To prevent end motion on shafts a collar is placed on each side of one of the bearings. JOURNALS. That part of a horizontal shaft which rotates in a bearing is called z. journal. The pressure of a shaft on a journal acts in a direction perpendicular to its axis. When the shaft is placed in an inclined posi- tion the pressure acts in a direction inclined to the axis of the shaft. The pressure of a shaft placed in a vertical position acts in the direction of its axis. The journal of a vertical shaft is called a pivot. The diameter of a journal must be made as small as the required strength will permit and as long as is necessary to keep the pressure per square inch as small as possible. This pressure per square inch is not measured on a circumference of the journal but by the area of its projection. Example: A journal 3" in diameter and 6" long will have a projected area of 3 X 6=18 square inches. Now if the pressure of the journal is 300 pounds per square inch then the total pressure is equal to 18 X 300=5,400 pounds. ROGERS' DRAWING AND DESIGN. 259 Example : If the total pressure of a 3 " diameter journal equals 5,400 pounds and it is desired not to exceed a pressure of 300 pounds per square inch then the length of the journal is found thus : 5,400 , . , ^L-L = 6 inches. 300 X 3 Example : If a given journal is 3 inches in diam- eter and 6 inches long and its total pressure is known to be 5,400 pounds, then the pressure per square inch of projected area is found as follows: 5,400 X 6 300 pounds. To find the pressure per square inch of projected area for a pivot bearing, multiply the square of the diameter of the shaft by .7854 and divide the total pressure of the shaft by the product thus found. The magnitude of pressure per square inch varies greatly in different cases in practice. It is gener- ally reduced where a greater speed is required. The maximum intensity of pressure on the main journal bearings of steam engines is 600 pounds per square inch for slow running and 400 pounds for high speed engines. Wherever possible it is ad- vantageous to make long bearings, thus reducing the pressure by about 200 to 300 pounds per square inch. Some manufacturers allow a pressure of 150 pounds per square inch for cast iron journals for factory shafts. For pivot bearings the following pressures per square inch are given by a high authority, as being the most desirable. 1. Wrought iron pi^-ot on gun metal bearing, 700 pounds. 2. Cast iron pivot on gun metal bearing, 470 pounds. 3. Wrought iron bearing on lignumvitae bearing, 1,400 pounds. According to the latest practice it seems, how- ever, that, for pivots which have to run continuously, the above-mentioned pressures should be reduced to one-half. BEARINGS. The simplest form of a journal bearing for a shaft or spindle of a machine is simply a hole in the frame supporting the rotating piece. If it is necessary to increase the length of the bearing the frame must be made thicker in this particular place by casting bosses on it as shown in Figs. 388 and 389. Fig. 389 is an end view and 388 is a section of such a bear- ing. The above described form of bearing is not 260 ROGERS' DRAWING AND DESIGN. durable as it has no means of adjustment for taking up the wear, and it cannot be renewed without re- newing part of the frame of the machine. It is therefore better to use the form of solid bearing shown in Figs. 390 and 391. In this case the hole is bored much larger than the journal, and lined with a solid bushing of soft metal, which can easily be replaced when worn. This arrangement requires a screw or key to hold the bushing in place ; in most cases the bushing is driven into the hole with con- siderable force to prevent it from turning, see Figs. 390 and 391. Bearings for horizontal shafts have different names, which indicate the manner in which they are used. Fig. 390. Fig. 391. Fig. 38«. HANGERS. When a bearing, is suspended from the ceil- ing it is called a hanger. Figs. 392 to 395 show the various details of a hanger made by a leading manufacturer. Fig. 392 is a side view ; Fig. 394 the longitudinal section. This design was first introduced by Sellers, and has been reproduced and modified by different manufacturers. It has a bearing box. Fig. 393, with a spherical center which is held between the ends of two hollow stems, all these parts are made of cast iron. Fig. 3m. ROGERS' DRAWING AND DESIGN. 261 Fio. 395. Fig. 393. 262 ROGERS' DRAWING AND DESIGN. These stems, Fig, 395, are provided with screw threads at their outer ends, ordinarily shallow square thread. The bosses on the frame are also provided with a similar screw thread, into which fits the screw of the stem. By means of the thread on the stems the height of the bearing can be adjusted, and the spherical centers allow a considerable adjustment in other directions. This construction makes the setting up or lining up of shafting much easier and the hangers made as described above enjoy therefore the greatest popularity at the present time. -^■- WALL BRACKETS. When a shaft is to be supported by a bearing fixed to a wall or pillar, a wall bracket is generally used for this purpose. In Figures 396 to 398 is shown a form of a wall bracket of an elegant and most solid design. nzj -aj ..,._51'_ Figs. 393, 397 and ; ROGERS' DRAWING AND DESIGN. 263 PEDESTALS AND PILLOW-BLOCKS. The words pedestal, pillow-block, bearing and journal box are used indiscriminately. A bearing designed to support a shaft above a floor or any fixed surface is called a pedestal or pillow-block, depending upon the type of bearing, as will be seen later. A simple pillow-block is shown in Figs. 399 and 400. It consists of two parts, the box which supports the journal and the cap which is screwed down to the box by two screws or bolts, called cap-screws. Fig. 399 shows the front view of a complete pil- low-block with cap and bolts. Fig. 400 is a top view of base with the cap removed. The seats in the journal-box are usually babbit- ed, that is, lined Figs. ; with babbit, a soft metal whose composition is as fol- lows : One pound of copper, ten pounds of tin and one pound of antimony. To hold the babbit in place recesses are cast in the cap and base, extending almost across the entire width of the bearino-. The hangers as well as the wall brackets shown above have the bearings babbited in the same manner. The babbit is cast in as follows : A mandrel, having a diameter a trifle less than that of the journal, is placed in position within the bearing box which the shaft is to occupy. The babbit, in a molten state, is then poured around it and the bearing is then bored to the proper diameter. Instead of using babbit for the friction sur- faces in bearings other metals, such as brass, gun metal or oth- er alloys may be used. AND 400. 264 ROGERS' DRAWING AND DESIGN. The melting point of these metals, however, is so high, that they cannot be poured into the box and cap directly, as in the case of a babbited bearing. They consequently are made as separate pieces, called steps or brasses, and are fitted into the box and cap in different ways. In some cases, where very little, and slow motion is required, the method described in Figs. 390 and 391 may be employed. The bearing in this case is made by boring a hole through the cast- ing, and the brasses consist of a simple sleeve, which is called a bushing. This bushing is sim- ply turned off and bored, and is then forced into the hole. Bearings that are made in halves, for the taking up of wear, and for removing the shafts, are fitted with two brasses however, the bearing and the cap brass. In this case the outside of the brasses is often made square or octagonal, fitting into recesses of similar shape in the box and cap of the pillow-block. This is done to prevent them from turning with the .shaft. To prevent the brasses from sliding out endwise, they are provided with a shoulder on each end, which fits over the ends of the bearing-. Brasses of the octagonal type as well as their applica- tion are fully illustrated in Figs. 402 to 404. When a shaft is to~be supported a considera- ble distance above the floor, the pillow-block is placed on a stand- ard. This standard may be cast separate from the pillow-block, and in this case both are fastened together by bolts. Note. — Brasses are made from different alloys which vary accord- ing to the judgment of the designer. Some engineers recommend the following composition : Six pounds of copper, one pound of tin and to every hundred pounds of this mixture one-half pound of zinc and one- half pound of lead are added. ROGERS DRAWING AND DESIGN. 265 [f^-'lr--^ '%' *--^---H V^ 266 ROGERS' DRAWING AND DESIGN. Very often, however, the pillow-block and stan- dard are cast in one piece and it is then called a pedestal. A pedestal is shown in Figs. 402 to 404. Fig. 403 is a front elevation and 402 shows its top view. Fig. 404 represents a side view of the pedestal. The various parts of this pedestal are exactly the same as in the above-described pillow-block with the exception that the seats or steps in this case are of an octagonal shape. There is no established standard of proportions for the parts of a bearing ; the proportions of pillow- blocks made by different manufacturers vary con- siderably. A pedestal for supporting a very small shaft is often obtained by turning a hanger upside down, reversing, of course, the bearing. Such small pedestals are usually called Jloor stands. The main bearings of large engines with girder beds are also often called pedestals. BELTS AND PULLEYS. Belts most commonly used are made of leather ; they may be single or double ; in damp places, canvas belts, covered with rubber are sometimes used ; leather belts are usually run with the hair side on the outside or away from the pulley. Long belts when running in any other than a verti- cal direction, will work better than short belts, as their own weight holds them firmly to their work. Fig. 405 shows an open belt, and Fig. 406 a crossr belt. Pulleys connected by open belts run in the same direction, while those connected by cross-belts run in opposite directions. When two pulleys are con- nected by a belt, the motion of one, the driving pulley, is transmitted to the other pulley, the follower. If we assume that there is no stretching or slipping of the belt, every part of the circumference of the follower will have the same velocity as the driving pulley being equal to the velocity of the belt passing over them. If the pulleys are of different diameters, for in- stance, if the driver has a diameter two times greater than the diameter of the follower, the latter will make two complete revolutions for each revolution of the driver. This ratio between the speeds of two pulleys is expressed in the following ROGERS' DRAWING AND DESIGN. 267 Fig. 405. Fio. 407 Fig. 406. 268 ROGERS' DRAWING AND DESIGN. Rule : The tiumber of revolutions of two connected pulleys are inversely proportional to their diameters. This may be expressed in the following formula : Number of revol. diameter of second pulley of nrst pulley Number of revol. Y)\2.m^^^v of first pulley of second pulley Example : A pulley 40" in diameter, making 300 revolutions per minute, drives a second pulley 20" in diameter. How many revolutions per minute does the second pulley make ? 40X300 No. of revol. of second pulley == ^600 revol. ^ ^ 20 To find the revolutions of the follower : Multiply the diameter of driver by its number of revolutions, and divide the product by the diameter of the follower. Example : The follower is 20" in diameter and makes 1 50 rev- olutions. What is the size of the driver used on a driving shaft that makes 200 revolutions per minute ? Diameter of driver =-2 = i5 inches, that is, 200 the diameter of the driver is found by multiplying the diameter of the follower by its number of revo- lutions, and dividing the product by the number of revolutions of the driver. For four pulleys connected by belts, as shown in Fig. 407, the following rule is to be applied : The number of revohitions of the first pulUy, tnul- tiplied by the diameter of each of the driver.-, equals the num.ber of revolutions of the last pulley, miilti- plied by the diameter of each follower. Example : Let the diameter of the drivers be 40" and 30", the diameter of the first follower 10" and of the second follower 15". What is the number of revo- lutions of the last shaft, when the first shaft makes 100 revolutions per minute ? Here the speed of the last shaft, multiplied by the diameter of the followers, 10" and 15', must equal the speed of the first shaft, 100 multiplied by the diameter of the drivers, 40" and 30" ; that is, speed of last shaft X 10 X 15 = 100 X 40 X 30, or 1 f 1 1 r 100 X 40 X ^o „ , speed 01 last snait = ^^ = 800 revol. ^ 10 X 15 When the number of revolutions of the first and last shafts are known, and it is required to find the diameters of the pulleys, apply the following ROGERS' DRAWING AND DESIGN. 269 RULH : Divide the higher number of revolutions by the lozver. In a case where four pulleys are to be used, we find the numbers whose product is equal to the quotient resulting from the above division of the speeds. One of these numbers is taken as the ratio of the diameters of one pair of the pulleys, and the other number, of the other pair. Example : It is required to run the last shaft with a speed of 1,500 revolutions, the driving shaft making 300 revolutions per minute. What size of pulleys are required when four pulleys are to be used ? The quotient resulting from division of the two speeds, equals — ^ — -^ 5. Two numbers whose product equals 5 are 2)^ and 2. Consequently one pair of the pulleys must betaken in the ratio of 2^4 to I and the other pair as 2 to i. Therefore, the first pair may be 30" and 12" and the other pair 24" and 1 2 " To find the speed of the belt : Multiply the circumference of the pulley by the number of revolutions per minute. Example : Let the diameter of the pulley be equal to 2 ft. and the number of revolutions per minute 100. Then, 2 X 3.14 X 100 -= 628 ft. per minute, the speed of the belt. The relation between the speed of the belt in feet per minute, the width of the belt in inches, and the horse power to be transmitted, is expressed in the following practical formulae : The horse power to be transmitted is found, by multiplying the speed by the width of belt and divid- ing the above product by goo ; or To find the required width of the belt, multiply the horse power to be transmitted, by goo, and divide the product by the speed ; or To fijtd the speed in feet per minute, multiply the horse pozoer by goo, and divide the product by the width of the belt. Example : Two pulleys, each 2 ft. in diameter, connected by a belt, make 200 revolutions per minute. It is de- sired to transmit 20 H. P. What is the proper width of the belt to be used ? The speed of the belt is equal to 2 x 3.14 x 200 = 1,256 ft. per minute; consequently the width of the belt equals 2 = 14.6 inches, or a belt 14^ inches. 1.256 270 ROGERS" DRAWING AND DESIGN. The above formulae are true of a single belt. When double belts are used, made of two single belts cemented and riveted together through their entire length, they should be able to transmit twice as much power as a single belt, and even more. The above formulae may be applied to the calcu- lation of double belts, provided the number 630 is put in the formula instead of the constant number 900. This will give the required proportions for belts, when used upon small pulleys, in which case more power is required for the transmission. SPEED OF MACHINE TOOLS. I n selecting the speed of pulleys, the designer must be guided by the speed of the machine which is to be driven. The speed of different machines varies according to the work which they perform, as, for example, the cutting speed of machine tools, or the velocity of emery wheels. Grindstones in machine shops, suitable for grind- ing machinists' tools may be run with a peripheral speed of about 900 ft. per minute ; grindstones for pattern makers' use, run about 600 ft. per minute. Emery wheels may be run with a peripheral veloc- ity of 5,500 ft. per minute. Polishing wheels, such as leather-covered wooden wheels, or rag wheels, may run with a peripheral velocity of 7,000 ft. per minute. The speed of cut for cast iron is 20 to 30 ft. per minute, for tool steel about 10 ft. per minute. Cut- ters in the milling machine may be run with a per- ipheral velocity of 80 ft. per minute for gun metal ; 35 to 40 ft. for cast iron ; and for machine steel about 30 ft. per minute. Example : What is the proper number of revolutions of tKe spindle of a machine shop grindstone 24" in diameter? The usual peripheral speed is 900 ft. per minute. The circumference of the given stone is equal to 2 ft. X 3.14=6.28 ft. ^ — r=about 143 revolutions per minute 6.28 ^■^ ^ Example : Let the emery wheel in a grinder be 12" in diam- eter, and let it be required to run the wheel with a peripheral velocity of 3,600 ft. per minute. What should be the speed of the spindle of this bench grinder ? The circumference of the wheel is 3. 14 ft. Divide 3,600 by 3.14 and the speed of the spindle is found. 3,600 3-14 = about 1,150 revol. ROGERS' DRAWING AND DESIGN. 271 The following are rules recommended by practical experience for the use of belts. Pulleys of small diameter, say of less than i8", should not be used for double belts. Narrow, thick belts work better than thin ones. If wide belts are used, it is proper to increase their thickness. This, however, is only true within certain limits ; the tendency among engineers is to go to the ex- treme in this direction ; it depends largely upon the class of work the belt is to be used for, and the only wa}' anyone can claim to be expert in this line is through practical experience and good judgment. The weakest part of the belt is at the joint ; for this reason joints should be made very carefully according to the most approved methods ; the same fastening does not answer for all belt-joinings. It is not advantageous to place two pulleys con- nected by a belt too near one to the other. A distance of 15 ft. between the shafts for narrow belts running over small pulleys is a good average. Wider belts running over larger pulleys for good work require a greater distance between the shafts. 30 ft. is a good average for such cases. The distance between the shafts should not be made too great, as this may cause too much of a sag of the belt, which may pro- duce such a pressure on the journals of the shaft as to injure them. Running belts in a vertical direction should be avoided whenever possible. Machine tools driven by vertical belts require particularly good well- stretched leather belts, which must be kept very tight. In tightening belts it must be remembered, that while tightening the belt, the pressure on the bear- ing is also increased, causing greater friction and wear on the bearing, especially with overhung pulleys. The angle of the belt with a horizontal line should not exceed 45° whenever possible. Belts are not run advantageously when their speed exceeds 2,500 ft. per minute. PULLEYS. The rim of a belt pulley may be made either straight, Fig. 408, or convex, as in Fig. 409. It would seem that the belt would remain on the straight pulley more readily than on the convex one. Experience shows, however, that the belt always tends to run on the highest part of the pulley, pro- vided it does not slip, in which case the belt will fall off more readily from a convex surface than from a straight rim of the pulley. 272 ROGERS' DRAWING AND DESIGN. ^^^\\^^ kmkkkk^ Fig. 408. Fio. 4(S. Fig. 410. Fig. 411. The flat or straight rim pulley is used where it is necessary to move the belt from one side of the rim to the other, as in the case where one pulley drives a pair of fast and loose pulleys. Whenever there is frequent slipping off" the rim of a belt, through a temporary increase in resist- ance, the pulfey is provided with flanges, as shown in Fig. 410. The amount of curvature in a section of the rim, is made greater, the faster the speed at which it runs. The curve may be an arc described with a radius' equal to about . 5 times the breadth of the pulley. The breadth of the pulley is generally made a little wider than the width of the belt, Fig. 411. The thickness of the rim at the edge may be found by dividing the diameter of the pulley by 200 and add- ing ys of an inch. For a pulley 25" in diameter, the thickness of the rim should be 25 inches V^ inch == i^ inch. 200 The thickness of the walls in the central part of the pulley, called the hub. is found by a formula given by Mr. Thomas Box, as follows : Thickness of hub = 1 1- ^, where D is the 96 8 diameter of the pulley and d the diameter of the shaft. ROGERS" DRAWING AND DESIGN. 273 Prof. Unwin gives the following formuls;: For a single belt, the thickness of hub = 0.14 V"B"D~ + 14 in. For a double belt, the thickness of hub = 0.18 V B D -(- J4! in. where B indicates the breadth of the pulley. The length of the hub is made from 73 B up to B. This is true for fast pulleys only. The hubs in loose pulleys are usually longer than in fast pulleys. The hubs in loose pulleys need not be so thick, and they project about ^ inch beyond each side of the face of the pulley. Fig. 412 shows loose and fast pulleys ?^W^??^l////M; 22Z^ Fig. 412. Fig. 413. Fig. 414. 274 ROGERS' DRAWING AND DESIGN. ,m,»m;^'^7777m Fig. 415. Fig. 416. The arms of pulleys are usually straight, but sometimes they are curved, as shown in Figs. 413 Fig. 417, and 414. It is the general practice in machine shops to draw the section of a pulley, as shown in Fig. 416, no matter what shape the arms may have. The straight-armed pulley is simplest in appear- ance and construction. There is no fixed rule for the number of arms in a pulley. Usually those up to 18" in diameter have four arms, and those of larger diameters, si.x arms. The cross-section of the arms of cast-iron pulleys is generally oval-shaped and of the proportions shown in Fig. 417. The longest a.xis of the oval a, may be found from the following practical formula;: the breadth_a being taken at the center of the pulley, supposing the arm to be continued through the hub to that point. B D 4N B D 2 N for sintrle belts, and for double belts. In these formulae B is the width of the pulley, D is its diameter and N the number of revolutions per minute. The proportions of the section of the arm near the rim may be two-thirds of the proportions given in the above formulae. It will be noticed that the breadth of the oval is given in the cubic power. To find the actual breadth a, multiply B by D, divide the product by 4 N and then find the cube root of the resulting number. ROGERS' DRAWING AND DESIGN. 275 For varying the velocity of a shaft, speed cones are used, Fig. 418. As the belt will have a tendency to climb a conical pulley, special provision must be made for keeping the belt in place. It is also desir- able to have both cones alike, so that they can be cast from one pattern. Cone pulleys or speed pulleys, are frequently made in a series of steps, as shown in Fig. 419, in which case they are termed step-pulleys. It is an established fact, that when two cones are placed with their centers at a given distance, and are so related that the sum of their radii remains con- stant, an endless cross-belt, containingr both cones will not change in length in the smallest deeree during the change in the actual diameter of each cone. It is necessary to keep in mind the fact that the sum of the radii of both cones and the distance be- tween their centers remain constant. As a result of this the sum of the radii of two opposite pulleys in a series of steps must be the same for all steps, as only with this condition will a crossed belt fit any pair of pulleys in the series. Fig. 420 shows three sets of pulleys which may be arranged into a step pulley with three sets or steps. The distances between each set of pulleys is the same, and the sum of the diameters of the L*__ pulleys in each one of the three sets is also ihe same. As a result of these conditions the length of the crossed belt for all sets is the same. Fig. 418. The above statement does not hold true for open belts. The middle sections of cone pulleys for open 276 ROGERS' DRAWING AND DESIGN. belts must be larger proportionately than for crossed belts. In the pair of cone pulleys shown in Figs. 421 and 422, both are made alike, and the first one makes N revolutions per minute ; let it be required that the equal the small diameter multiplied by the square root of the quotient of m divided by n. As was remarked before, for open belts, the middle diameter of the cone pulleys must be made larger. If D and d are the large and small diameters of a Fio. 419. second pulley should have a range of speed from m to n revolutions, m beine the greater number. Then N must equal the square root of the product of m and n, thus, N ^ Vm x n. The large diameter in the cone pulleys must cone pulley, then the proper middle diameter is 1 , D + d o'o8 (D-d)= , r ■ .u A- 1 to -I- ^^?; -, where L is the dis- equal to + 2 ■ C tance between the two shafts. When the middle diameter is thus found, the ROGERS' DRAWING AND DESIGN. «vy outline of the cone is laid out by an arc of a circle passing through the ends of the diameters D and d as well as the ends oi the middle diameter. When it is desirable to substitute a step pulley for a con- tinuous cone, A B D C, Fig. 423, the cone is divided into the required number of equal parts by parallel Figs. 421 and 422. lines, like E F, etc., drawn at equal distances. These diameters are then taken as center lines for the different steps. 278 ROGERS' DRAWING AND DESIGN. GEAR WHEELS. When two wheels with parallel axes, as shown In Fig. 424, are placed firmly together so as to form a rolling contact, the motion of one wheel, if there is no slipping, will produce a motion in the other Fig. 423. wheel ; in this case a point on the rim of one wheel will travel exactly at the same rate of speed as any point on the rim of the other wheel ; a rotation of this kind is called a positive rotation. Both wheels when in positive rotation by rolling contact, will have the same ratio of velocity, or as it is generally termed a constant velocity ratio. The number of revohitions of the shafts will be inversely proportional to the diameters of the wheels, and this ratio will remain constant, provided there is no slipping. With wheels having smooth surfaces it is im.pos- sible to maintain a constant velocity ratio, hence, to Fig. 424. secure this condition, the wheels are provided with teeth which will enable them to rotate without the possibility of slipping. To avoid a separate velocity for each tooth and to obtain an equal speed velocity in all parts of the wheel, the teeth are designed with proper proportions, which will be explained and illustrated hereafter. ROGERS' DRAWING AND DESIGN. 279 / Fig. 425. r- 280 ROGERS' DRAWING AND DESIGN. The rims of two imaginary wheels which have the same axes and which would have the same velocity ratio as two given gear wheels and the same width, form what are z^\t.d, pitch surfaces ; the circles rep- resenting the section of both pitch surfaces, at right angles to the axes, are called the pitch circles. The part of a tooth in a gear wheel outside of the pitch circle is called the addendum, and the part of the outline or curve of the tooth on the addendum is called the face of the touth, as shown in Fig. 425. That part of the tooth inside of the pitch circle is called the dedendum, and the part of the surface of the tooth inside of the pitch circle forming the front or back of the dedendum is called the flank of the tooth. The point where the flank and the face meet is called the pitch point and is situated on the pitch circle. The circle passing through the tops of the teeth is called the addendum circle and is equal in diameter to the blank or disc, from which the gear is to be cut. The circle passing through the bottom of the teeth \'s,Z'A\&A\h.& dedendum circle. The distance measured on the pitch circle between the pitch points of two con- secutive teeth is called the circular pitch of the gear wheel. The circular pitch includes one thickness of tooth and one space between teeth ; the circular pitch is equal to the circumferetice of the pitch circle divided by the number of teeth in the gear wheel. Example : If the diameter of the pitch circle is equal to D. The circumference of the pitch circle is equal to 3.1416 X D. Let the number of teeth in the wheel be N. Then 3.1416 X D N is the circular pitch. ^y diametral pitch is meant the number of teeth in the gear per one inch of its pitch circle diameter. Example : If the diameter of the pitch circle is equal to D Inches, and the number of teeth equals N ; then N iy= diametral pitch. The diametral pitch expresses in a direct and sim- ple manner the ratio between the diameter of the pitch circle and the number of teeth. Usually it may be expressed by a whole, number and therefore its form is convenient for expressing the proportions of the teeth, which are usually dependent upon the pitch, for this reason nearly^all gear calculations are made in terms of the diametral pitch. Rule : To change the diametral pitch to circular pitch, divide J. I ^16 by the diametral pitch. To change the circular to diametral pitch, divide j. 1^16 by circular pitch. ROGERS' DRAWING AND DESIGN. 281 The proportions commonly adopted for gears made with precision, are as follows : The addendum equals i divided by the diametral pitch. ExAMPLii: : If the diametral pitch equals 2 then the adden- dum equals yi. The pitch circle diameter, plus tivice the addendum equals the blank diameter of the S^ear. The dedendun is equal to the addendum, plus the bottom clearance. The clearance is generally equal to To the thickness of the tooth, measured on the pitch line. The thickness of the tooth and width of space, measured on the pitch circle, are each equal to one-half the pitch, on carefully cut gears ; in practice, how- ever, it is customary to make the width of the space slighth" larger than the thickness of the tooth, in order to allow for inaccuracies of workmanship and operating, unavoidable because of the difificulty of producing theoretically correct gears. This is par- ticularly necessary in cast gears. The difference between the thickness of the tooth and the width of space is called the back lash ; the amount of back lash necessary for a gear must be left to, the best judgment of the designer ; in cast iron gears it is \ sometimes equal to yV of the circular pitch ; this is good practice only for very rough castings. The flank and the dedendum circle are joined by small arcs, to avoid sharp corners at the root of the tooth. These are called filets and are usually made with a radius equal to one-seventh of the distance between two consecutive teeth, -measured on the adden- dum circle. When two gear wheels with parallel shafts are turning one the othelv^the distance between the centers of the shafts is eq\lal to the sum of the pitch diameters of both gears divided by 2. Example. — Let D equal the pitch diameter of one wheel and d the pitch diameter of the other wheel ; then the distance between the centers in this • f u 1 ■ D + d pair 01 i^ear wheels is — The number of teeth is found by dividing the cir- cumference of the pitch circle by the circular pitch. If the diametral pitch be given, the number of teeth is found by multiplying the pitch diameter by the diametral pitch. The pitch diameter is found by multiplying the num,ber of teeth by the circular pitch and dividing the product by j. 1^16. 282 ROGERS' DRAWING AND DESIGN. If the diametral pitch is given, the pitch diame- ter can be found by dividing the number of teeth by the diametral pitch. The diameter of the blank equals the pitch diame- ter plus 2 divided by the diametral pitch. If the number of teeth and the diametral pitch are known, add 2 to the number of teeth and divide by the diametral pitch. Gears may be classified as follows : Spur Gears for shafts which are parallel ; in this case the pitch surfaces would be cylinders. A gear wheel with a comparatively small number of teeth is termed a pinion. Bevel gears for con- necting shafts which intersect when lengthened. The pitch surfaces in this case are cones. Bevel gears of the same size connecting shafts at right angles are called miter gears. If the shafts are neither intersecting nor parallel, the pitch surfaces will be hyperboloids of revolution and the gears are called hyperbolic or skew gears. In all gears enumerated up to the present, the teeth are made with rectilinear elements and the pitch surfaces touch each other along straight lines. Worm Gears are used for connecting shafts which are at right angles to each other and which do not meet when lengthened indefinitely in either direction. The pitch surfaces meet in spiral lines. When the pitch circle is made with a diameter in- definitely increased, it will become a straight line and the gear is termed a rack. There are two kinds of teeth generally used and classified according to the methods of producing them ; these are involute teeth and cycloidal teeth. The cycloidal system of gearing was, for a long time, used almost exclusively ; of late, however, the involute system is rapidly gaining in popularity and many engineers advocate its general application in all cases. Involute teeth are of greater strength and will run well with their centers at varying distances and still transmit uniform velocity. The chief objection that has been raised against involute teeth is the obliquity of action, causing increased pressure upon the bear ings. If a flexible line be wound around a circle, and the part which is off the circle is kept stretched and straight, any point in it will describe a curve which is the involute of the circle. To draw the involute to a given circle, PABCO, Fig. 426. Divide the given circle into any number of equal parts by the points P, A, B, C, D, etc., through which points draw tangents to the circle. Make the length Aa equal to the length of one ROGERS' DRAWING AND DESIGN. 283 division AP, the length Bb equal to two such divi- sions Cc to three divisions, Dd to four and so on. Through the points P, a, b, c, d, etc., draw the required curve, which is a portion of the circle's involute. The outline of an involute gear tooth is made with a single curve, the involute of an especially selected circle which is called the base circle. The center of the base circle lies in the center of the pitch circle, and the base circle is always smaller than the pitch circle. Different manufacturers make the base circle of an involute gear of different diame- ters. Brown and Sharp make the diameter of the base circle equal to 0.968 of the pitch circle. The ordinary method of finding the base circle is as follows: If H is the center of the wheel and P the pitch circle, draw the addendum and deden- dum circles, Fig. 427. Take any point O, as the pitch point on the pitch circle and draw a radial line HH through this point. Draw a line EE making an angle of 75° with the radial line HH. The base circle is found by drawing inside of the given wheel, a circle tangent to the inclined line, EE (which line is called the line of action). Let the base circle intersect the line H H at the point W. From this base circle the involute is drawn, passing through the point W and extending to the point V on the Fig. 426. addendum circle. The part of the flank between the base circle and the dedendum circle is straight and is part of the radius of the circle. No wheel having less than 12 teeth will gear cor- rectly together when the base circle is laid out in 384: ROGERS' DRAWING AND DESIGN. Fw. 427. this manner ; in practice a curve wliich approximates the required involute curve is generally employed. In the Brown & Sharp system the line of action is drawn so as to make an angle of 75 3^°, Fig. 428. This is true for gears having more than thirty teeth : for gears having a smaller number of teeth, special rules are followed. The Brown & Sharp method explained above cannot be used for involute gears having less than thirty teeth, as the space left at the root is too narrow for the free motion of the mating gear. In such cases the curve is drawn from the base circle to the addendum ; from the base circle to the dedendum circle, the flank is drawn parallel to a radius of the wheel through the middle of the space between two adjoining teeth, being joined to the dedendum circle by ah arc or fillet. In an involute rack, in many shops, the teeth are made with straight lines passing through the pitch points of the teeth, as shown in Fig. 429. The direction of the straight edges of the teeth is at right angles to the line of action, that is generally lines making angles of 75° with the pitch line. For racks which are to run with pinions having fewer than thirty teeth, the outline of the teeth on the rack near the addendum are rounded to prevent interference with the flank of the pinion tooth. In the cycloidal system of gearing the outline of a tooth is made by a double curve ; here the face is a por- ROGERS' DRAWING AND DESIGN. 285 Fig. 428. tion of an epicycloid and the flank a hypocycloid, both joined in the pitch point. If a circle is made to roll along a straight line, always remaining in the same- plane, a point in the circumference of the rolling circle will describe a cycloidal curve. The rolling circle is called the gen- erating circle or describing circle. .cf/'/^v" Fig. 429. If the generating circle rolls along a straight line it will describe a cycloid. If the generating circle rolls along the outside of a circle it will describe an epicycloid, and when rolling along the inside of a circle, it will describe a hypoc\'cloid. The construction of these curves is shown in Figs. 430, 431 and 432, To draw the cycloid in Fig. 430, draw a straight line AC. Describe the generating circle and divide it into any number of equal parts by the points i, 2, 3, 4, etc. From B, the point of contact of the gen- erating circle with the straight line, set off distances equal to the portions of the circle, so that BC will be equal in length to one-half of the circumference of thfe rolling circle and will be divided into the same number of equal parts by the points i', 2',3', etc. Through these points draw lines perpendicular to the line ABC. Through the center of the generating circle draw a line parallel to the line AC ; this line will cut the perpendicular in the points a, b, c, d, e, etc. With these points as centers, describe arcs of circles with a diameter equal to the diameter of the generating circle. These arcs will touch the line AC in the points i', 2', 3', 4', etc. Through the point 2 on the generating circle, draw a line parallel to AC, cut- ting the arc which passes through i'. 286 ROGERS' DRAWING AND DESIGN. Through point 3 in the generating circle, draw a line parallel to AC and cutting the arc which passes through the point 2'. Through the points 4, 5, 6, etc., in the generating circle, draw parallels to meet the arcs cutting the points 3', 4', 5', etc., respectively. The intersections of these lines with the arcs deter- mine the required curve. Fig. 431. To draw the epicycloid, describe the generating circle tangent to the circumference of the given circle at the point B. Divide the generating circle into any number of equal parts by the points 1, 2, 3, 4, etc. Set ofi the equal portions of the generating circle on the circumference of the given circle by the points i', 2', 3', etc., through which draw radial lines extended outside of the base circle AC and cutting them in the points a, b, c. d, etc., by a circle having one common center with the base circle and passing through the center of the gener- ating circle O. With the points a, b, c, d, etc., as centers, draw arcs with a radius equal to the radius of the gener- ating circle, these arcs touching the base circle in the points 1', 2', 3', etc. Through the points i, 2, 3, etc., in the generating circle draw arcs concentric with the base circle AC, to meet the arcs touching the points i', 2', 3', etc., respectively, and through these points of intersection draw the required epicycloid. Fig. 430. 288 ROGERS' DRAWING AND DESIGN. The hypocycloid is drawn in the same manner, as shown in Fig. 432. To lay out the outlines of a cycloidal gear, draw the pitch circle, Fig. 433, and divide it into a number of equal parts, corresponding to the number of teeth required. Each one of these parts is equal to the cir- cular pitch of the wheel. Bisect each one of these pitch distances to obtain the thickness of the tooth and the width of the space between the teeth. Wherever necessary make the space larger than the thickness, thus providing for back lash. Next, draw the addendum and dedendum circles, and select the proper describing circle. The profile of the tooth between the dedendum and the pitch circle, the face of the tooth, is made by an epicycloid gen- erated by the generating circle as it rolls along the circumference of the pitch circle. The flank, that is the outline of the tooth between the pitch circle and the dedendum circle, is a hypocycloid of a generating circle equal to the above generating circle, or, if convenient, with a generating circle having a different diameter. For two gears which are to run together, as in Fig. 434, the faces of the teeth in both wheels must be described by one generating circle. Wherever one generating circle is used for the face as well as the flank of one wheel, the same generating circle should be used for both face and flank of the teeth in the mating gear. When the outline of one tooth is found, a template of thick paper may be cut to one of its sides and by attaching this template to an arm of suitable length, which may be held to the center of the wheel by a -pin, we can swing it around and bring it in position to draw the profiles of the rest of the teeth. Since it would be too much to describe all teeth by tracing for each one of them the proper cycloidal curves, it is usual to approximate these curves by means of circular arcs. We find an arc which very closely coincides with the proper curve for the face, and the same is done for the flank. The centers of these arcs are found by trying with the compass until the proper arc is found. In Fig. 435, the point A is found to be the center of an arc which very closely coincides with the flank of the tooth ab. Draw a circle through A, concen- tric with the pitch circle. The centers of all arcs for the flanks of the rest of the teeth in this gear will lie in the circumference of this circle, and the radii of these arcs will equal in length Aa. To draw the flank d on the tooth cd, set the compasses to a radius equal to Aa, put the needle point in d and ROGERS' DRAWING AND DESIGN. ^.— . *^.~ 289 Fig. 438. 290 ROGERS' DRAWING AND DESIGN. J)e5cribit\g C/rc/e ^pjcycloid Fig. 433. "'•«, '^^ \ ■c/ ROGERS' DRAWING AND DESIGN. 291 i i 1 rxpxiFh ^ (^^] ^r- \' ^^ ^ — r~~~--. f/ O y^ """\//x f—\ /S. / \ \_3t — \ P<('"V / \ (\^ \ — -^ /^>. ^ ) cl r~^ \i }p 1 ^\/^^\\ y^^ 1 C i ^p V ) ^ I / \ V y / c h ^ s CA. \ /•"■"^^^ ^^_^ ^\y s \ \ \ A V'^ ^-^^ ^ ( ^ 'bAcf .^ / \ \^ ^7 Sf J Fig. 434. I X)^"-^ ^ yi r\A h/\fkj ^ 292 ROGERS' DRAWING AND DESIGN. cut the circle A2 by the arc at the point 2 ; this point will be the center for the arc of the flank d. In this way all flanks may be drawn. When the center of an arc, which as closely as possible coincides with the face of the tooth ab is found, the circle of centers for the faces is drawn, and all arcs for the faces of the teeth will lie in this circle The generating circle may be, within certain limits, of any diameter, so long as it is not greater than the radius of the wheel on which it is used. When the diameter of the generating circle is equal to the radius of the pitch circle, the path of any point in the circumference of the generating circle is a straig^ht line. According to the Brown & Sharp system, in cy- cloidal gearing, the diameter of the generating circle is equal to the radius of a 15-tooth gear of the pitch required, this being the base of the system. The teeth of the rack of this system have double curves, which may be traced by the base circle, rolling al- ternately on each side of the pitch line. The same generating circle is used for all gears of the same pitch. According to the prevailing practice, the flank of the 15-tooth pinion in cycloidal gearing is made radial ; accordingly the diameter of the generating circle equals one-half of the pitch diameter of a 15- tooth pinion. According to other practice a 12- tooth pinion is taken as the base. In Fig. 436, is shown a cycloidal rack and pinion. The curves of the teeth-profiles for the rack are generated by rolling the generating circle along each side of the pitch line, on which all pitch points are set off. A spur gear in which the teeth are on the inside of the rim is termed an annular or internal gear In such a gear the teeth correspond with the spaces of an external gear of the same pitch circle, as do also the other proportions of the teeth. They are consequently designed in the manner described above as involute or as cycloidal gears. One particular rule must be observed in regard to epicycloidal internal gears ; the difference between the diameters of the pitch circles must be at least as great as the sum of the diameters of the desci'ibing circles. Bevel gears are used to connect two shafts which intersect when lengthened indefinitely. In most cases the shafts are at right angles with each other. The pitch surfaces of bevel gears are cones which have a common vertex, the point of intersection of the axes of the shafts. Fig. 437. ROGERS' DRAWING AND DESIGN. 293 Fio. 435. ROGERS' DRAWING AND DESIGN. Before proceeding to draw a pair of bevel gears draw a section through the shafts of both gears, thus showing a section of one-half of each gear. Draw the two axes of the shafts, OA and OB meet- ing at O, Fig. 438 shows the two axes at right angles with each other. Determine the diameter of the largest pitch cir- cles in the bevel gears proportionate to the required velocity ratio corresponding to the circles which form the bases of the two pitch cones. ^ Let ef be the maximum pitch diameter of the larger, and gh the maximum pitch diameter of the smaller bevel gear. An indefinite distance awaj' from and parallel to OB draw the line ef ; then draw the line gh parallel to OA, each one of these lines being bisected by the axes. Through the points e and f draw lines parallel to OA and through g and h lines parallel to OB. " The lines intersect in the points E, F and H. Connect the point O by straight lines with the points E, F and H. The resulting triangles EOF and FOH are sections of the pitch cones of the bevel gear. Make FG equal to the width of the face of the teeth. From the point G draw the lines IG and GJ parallel to EF and FH respectively. Each one of the gears is then com- pleted separately with the required proportions for the teeth. The manner in which this is done is ROGERS' DRAWING AND DESIGN. 295 Fio. 438. 296 ROGERS' DRAWING AND DESIGN. illustrated In Fig. 439. In this figure, ABC is a part of the pitch cone laid out according to the principles explained with Fig. 438. This view must be drawn first. K are the out- line's of the teeth of a spur gear laid out for a pitch diameter equal to the maximum pitch of the bevel gear. The proportions of the teeth on the bevel gear are laid off from these outlines. EB and BD are the addendum and dedendum ; the line ED being drawn at right angles to the line AB of the pitch cone. BF is made equal to the length of the teeth on the bevel gear ; at F a line perpendic- ular to A B is drawn, and the addendum and deden- dum of the smallest outline of the tooth is deter- mined by the intersection of the line GH with the lines EA and DA. Next the other view is drawn. OJ is a line drawn parallel to BC. A perpen- dicular to this line dropped from the point B deter- mines the maximum pitch circle and a perpendic- ular from the point F the minimum pitch circle. In the same manner the addendum and deden- dum circles for the largest as well as for the smallest profiles of the teeth may be found by dropping per- pendiculars from the points E and G, D and H. The maximum pitch circle Is then divided Into a number of equal parts corresponding to the number of teeth required. Through each one of the divisions a line is drawn to the center O ; these lines are the center lines of the teeth. The proportions of the teeth shown at K are then set off from each center line for the purpose of forming the projection of the teeth. The distance a is set off on the largest de- dendum circle, the distance b oA the maximum pitch circle ; the distance c on the addendum circle. All these lengths are set off so as to be bisected by the center line of the tooth for which they are in- tended. The projection of the smallest profile of the tooth Is obtained by drawing radial lines from the points at the addendum of the large profile to the addendum circle to the smallest addendum cir- cle. The pitch points of the tooth for the projec- tion of the smallest profile at L, is obtained by drawing radial lines from the pitch points of the large profile to the smallest pitch circle, and the root of the smallest profile is obtained in the same manner by drawing radial lines from the dedendum points of the large profile to the smallest dedendum circle. ROGERS' DRAWING AND DESIGN, 297 -"^h^^.~. ■<^ ^^ ^ ^--feSSi^ J^^ A. 1 A \\ \ \ i \ / ;| t \ \ / p""^ W^"^--- ::---sA "^-N. ,^;''^. V 1 ^ \ \\1 \ A \ 1 : ; 1 1 : I ^^r -; ■:" - ■.. :%^^»>^ 7^^ \^ \/ 1 \L \ ' i ' i i 4 \i;;i -W;;: ^J ^v^""^^ Y--'/''' 1 ' 1 1 ' ' III X^j/ N /-v i j V ; i I i i p ' 1 [ • > ! i III' III ■ ' 1 1 ' 1 1 j ' ' ■ 1 ' III < ! ■V, ^ X \ \ / ^ 1 1 ' ' ' 1 1 ' ' ' 1 ' 1 1 X "" 1 II ', < 1 [ I I ' 1 'II 1 1 \ Ml li /ii iii' iM \ s \ i-iC'"" V-W " ^-~"^ l^~^i ' 'i ' ' i ' 1 J ^^ -'^';^2$tt 11 ' itr/t^''''-^ jii : ,Yo^3^r^'\ \\\ \ ; ; 1 \ III / T''^fy^ 1 ! \ -''''y\'\\ \\A ' ' 1 / // / / / /^/i ' ' 1 1 1 ^^4 >, > i i I I I i / / / fe ^^^ ^£^: X£M/ ^^^^Vi \ j / 1 / / / ^~N^s»;Ox^\,v^'' yC \ 1 1 / ^^^^^/ \ i ' /^\^^^^^^-^-^^^^<^^^' , J 1' / ;// '^ -^""^ — ^v -' N^fe^S--^^^'^^!!!^-- — " — "\ ' i; ; 1 // / ^ 1 >. ^\!'Vt^ — " I ■ 1 1 1 o_^ \ fe===:z^ ::-;;;;:::::--■>■• i 14 .. __ 4'! f -+ 1 ' 1 ;• \ I / / r~ f^::":::i:::::^-:iA... I Fig. 439. ^^^^^^^=5:&/ 298 ROGERS' DRAWING AND DESIGN. In speaking of the diameter of a bevel gear, the largest diameter of the pitch cone is meant. The relations between the pitch diameter of a bevel gear pitch, number of teeth and velocities are the same as for spur gears, and all calculations are made in the'same manner. In Figs. 440, 441 and 442 are shown 3 views of a completed bevel gear. For transmitting motion from one shaft to another at riofht anofles to it, when the axes of the shaft do not intersect, the worm gear and worm shown in Figs. 443 and 444 are used. The section of the worm shown in Fig. 445 is of the same outlines as a rack of corresponding pitch, and may be of either the involute or cycloidal form. The involute form is generally adopted, as its teeth are easier to pro- duce. The worm is cut in a lathe like a screw. The diameter of the worm is ordinarily taken about 5 times the pitch, although it could be made of any convenient diameter. The worm must always be the driver. It is not well adapted to the transmis- sion of heavy power, as the tooth action is purely a slidingf one. Fig. 446 shows a partial section of a worm gear made by a plane perpendicular to the axis of the worm through the axis of the gear. When made by the involute system the worm teeth will be straight. The worm may be drawn by the aid of helices like a screw. The worm may also be single, double, etc., like a screw. Figs. 443 and 444 show the outside views of a worm gear. Fig. 446 also shows tl)e outside view of a complete tooth. The drawing of the outside views of 'i worm gear involves considerable labor. Fortunately, however, it is wholly unnecessary for the purposes of machine ; shop construction, to make complete outside views of worm gears. The same is also true in the case of bevel gears. A sectional view is most generally adopted to show the wheel. The wheel is usually made to embrace about one-sixth the circumference of the worm. vVhen the diameter of the worm is increased and approaches the diameter of the wheel and when the worm is given a multiple thread and the number of teeth in the worm wheel is comparatively low, then both worm and worm wheel take the shape of spiral gears. To add strength to spur gears, the rim is made wider than the teeth, and is carried outward, as shown in Figs. 447 and 448. This is called shroud- ing of the teeth. If two wheels gearing together do not differ greatly in diameter, each may be ROGERS' DRAWING AND DESIGN. 299 Fig. 440. Fig. UL Fig. 442. 300 ROGERS' DRAWING AND DESIGN. shrouded to the pitch on both sides ; but when one is very much larger than the other, it is usual to shroud the smaller only. For light spur gears the rim is generally made as shown in Fig. 449. The section shown in Fig. 450 shows the rim of a heavy spur gear. The propor- tions marked on the sections in these figures are in terms of the circular pitch of the gears. Sections of arms for gear wheels are shown in Figs. 451, 452 and 453. For light spur gears the section shown in Fig. 451 is used. For heavy spur wheels the sections shown in Fig. 452 is adopted. The section in Fig. 453 is for bevel gears. The number of arms is approximately + 4 when D is the diameter of the pitch circle in inches. The nearest even number may be taken. The width of the arms may be calculated in the following manner. In the arm shown in Fig. 451 the greatest breadth of the arm is equal to B ; then, 1.6 X width of tooth X cir- cular pitch X pitch diam. 2 X number of teeth and for the other sections, T53 width of tooth X pitch diameter 2 X number of teeth. A practical rule for finding the thickness of the hub is The thicknes of hub = 1.6 X circular pitch X 0.2 pitch diameter The length of the hub is, in most cases, equal to the width of the tooth ; it may vary up to i}4 times the width of the tooth. The pressure on one tooth may be taken, for mqat- purposes, to equal | of the whole pressure ; that is of the driving force at the pitch line, calculated from the horse power and the speed. H X 33,000 P = V where H is the given horse power and V the veloc- ity of the spur gear. The velocity of the given wheel is found by multiplying the circumference in feet by the number of revolutions per minute. The thickness of the tooth on the pitch line may be found from the following formula, ?'. c, Square of thickness= 3 times pressure on one tooth safe stress For cast iron gears the safe stress may be taken to equal 4,000 pounds per sq. in. ROGERS' DRAWING AND DESIGN. 301 Fia. 443. Fia.444, 302 ROGERS' DRAWING AND DESIGN. ROGERS' DRAWING AND DESIGN. 303 FlO. Hi. f 047 ^M FiQ.aa. Fig. 4i8. ITiO. 450. Fig. 451. rS ^ — ■*- Fig. 452. -B 1 Fig. 453. 304 ROGERS' DRAWING AND DESIGN. Fig. 454. Example : If the whole driving force at the pitch circle is equal to 12,000 pounds, then the pressure on one tooth will be 8, coo pounds, and the square of the 1-1 r 1 1-11 1 8,000 X 3 thickness 01 the tooth wul equal =6. ^ 4,000 The thickness of the tooth must then be equal to the square root of 6, i. e., 2.449 •"> ^^Y '^% \nc\\ TRAINS OF GEAR WPiEELS. When a train of gear wheels is employed in a machine, the usual arrangement is to fasten two gear wheels of unequal size upon every axis, except the first and the last, and to make the larger wheel of any pair engage the smaller one of the next pair. If the wheel A in Fig. 454 is the driving wheel, and the wheel F the last follower, and if it be deter- mined that for each single revolution of A, the wheel F shall make 50 revolutions, then it is said that the value of the train is equal to 50. That is, the value of a train of gear wheels is equal to the number of revolutions in the last fol- lower in a given time, divided by the number of revolutions of the main driver in the same time. Suppose that the first wheel had A teeth, the second B, the third C teeth, the fourth wheel D teeth, the next E and the last F teeth ; then, -^-== the velocity ratio between the first and the B second axes upon which are fastened the wheels B and C ; -— = the velocity ratio between the second and third axes ; JE T and A B ^^ velocity ratio between third and fourth axes, X- C X ,^ == the velocity ratio between the D h two extreme axes, that is, it will equal the value of the train. Example : Let A have 120 teeth, B 15 teeth, C 100 teeth, D 50 teeth, E 150 teeth and F 30 teeth. Then, the value of the train is equal to 120 15 -X- 100 50 X 150 80 METAL WORKING MACHINERY. The discovery of metals and the means of working them are among the first stages in the develop- ment of primeval man ; the earliest evidence of a knowledge and use of metals is found in the primitive implements of the so-called Bronze and Iron Age. Attention is called to the interesting note below. The Old Testament mentions six metals — gold, silver, copper, iron, tin and lead ; the old Greeks in addition to these, and to bronze, came also to know mercury ; the same set of metals without addi- tions seem to be the only ones known until the Fifteenth Century when atitimony was discovered ; about 1730 A. D., arsenic and cobalt were discovered, nickel and manganese were, discovered in 1774; in the meantime something had become known in a general way of zinc, bismuth diXxA platinum. Since the date last mentioned the discovery of many rare metals has become frequent, aluminum being among the last most useful and interesting discoveries of metals unknown at the beginning of the Nineteenth Century. The following pages deal, in text and illustrations, with iron working machinery, as agamst those machines devised to work i7i luood, etc., and few as are the cases named they show vividly the progress made in the methods of working the metals named. In designitig machines it is well to keep in mind, i, that each machine ought to be made of as lew parts as possible, 2, as simple as possible, 3, the strength of every part should be made proportional to the stress it has to bear, 4, all superfluous weight which clogs the machine's motion should be avoided, 5, all parts should be contrived to last equally well, 6, in wheels with teeth, the number of teeth that play together ought to be so constructed that the same teeth may not meet at every revolution, but as seldom as possible. Note. — " Some recent analyses of the iron of prehistoric weapons have brought to light the interesting fact that many of the prehistoric specimens of iron manufacture contain n consiilerable percenlage of nickel. This special alloy does not occur in any known iron ores but is invariably found in meteoric iron. It thus appears that iron was manufactured from meteorolites which had fallen to the earth in an almost pure metallic state, possibly long before prehistoric man had learned how to dig for and smelt iron in any of the forms of ore which are found on this planet." — Enc. Briiannica. 307 308 ROGERS' DRAWING AND DESIGN. DES AND PRESSES. The use of dies and presses has increased in recent years to an almost marvelous extent, and a numberless variety of articles are now being pressed out easily and rapidly by the aid of dies, which in former times involved great labor as well as a long special training. The number and variety of dies are so very large that it is beyond the limits of this book to give even a partial list or classification of these useful tools. In this section we shall limit ourselves to a few examples of the most frequently employed forms of dies, so as to give the reader an opportunity to ac- quaint himself with this importamt part of modern mechanism sufficiently to understand further special literature upon this subject, if it should be the desire of the reader to make a thorough study of this branch of machine shop practice. The simplest form of a die is a blanking die. Blanking dies are made for the purpose of cutting out various pieces of metal from a comparatively thin sheet of metal, cardboard, etc., leaving the cut out piece perfectly flat ; this piece is called a blank. A set of blanking dies consists of a male die, or punch, and the lower or female die. The lower die has an opening exactly equal to the form of the punch. Fig. 455 shows a punch and die for a circular blank. The narrow part of the punch, the shank, is preferably made in one piece with the punch. Fig. 455. The shank is fastened to the ram of the press, while the die is secured to the bed of the press. The taper of the lower die gives the clearance, required for the purpose of facilitating the dropping of the blank from the die as soon as it is cut ; the clear- ance is made from to 3' for average work. ROGERS' DRAWING AND DESIGN. 309 For the purpose of greater rapidity of work and uniformity in the matter of spacing the holes, dies and. punches are grouped; that is, several punches Fig. 456. are fastened to one shank, while several separate openings are worked out in the lower die to corre- spond with the number of required separate dies, if the work were done by single dies. V Fio. 457 Fig. 458 shows such a gang die, as it is called, made for cutting out washers of the shape shown in Fig. 459. One stroke of the die produces two holes, as shown in Fig. 4^7. The metal is fed into the Fig. 458. KiG. 459. 310 ROGERS' DRAWING AND DESIGN. die from the side of the smaller hole. The small punch will cut a hole in it equal to the inside diam- eter of the washer, as shown in Fig. 456 ; the metal is then advanced and at the second stroke the large punch will cut out the complete washer, while the small punch pierces the metal for the next washer at the same stroke. The plate S, Fig. 458, is the stripper which takes the metal off the punches on Fig. 460. their upward stroke. It is evident that the metal must be fed below the stripper. Fig. 461 shows the simplest form of a drawing punch and die. The flat circular blank. Fig. 460, is placed upon the die so as to fit the set edge S, and is pushed through the die by the punch. While the punch returns upward, the finished shell is pulled off by the edge P, which is made very sharp for that purpose. The diameter of the punch is equal to the diameter of the die, minus two times the thick- ness of the blank. Fig. 461. Fig. 462 shows the shell. This simple form of a drawing die should be used only on shallow work, to avoid crimping around the edge of the shell. When the blank is held firmly while being drawn, the crimping even on deeper work may be avoided. ROGERS' DRAWING AND DESIGN. 311 O (O; tr %J' I w tr r^ \^ Fig. 466. Fig. 463. Fio. 464. Fig. 465. 312 ROGERS' DRAWING AND DESIGN. Another gang die is shown in Figs. 463 and 465. Fig. 466 is the blank, Fig. 465 the top view of the lower die and stripper ; Fig. 464, a sectional view of the same ; and Fig. 463 shows the punch in section. Fig. 469. T^G. 468. In Fig. 468 is shown a type of a die, which by its relative simplicity when compared with the work produced, will always stand as a beautiful example of mechanical ingenuity. ROGERS' DRAWING AND DESIGN. 313 It is a single-action cutting and drawing die, gen- erall)- called a single-action combination die; it is a combination of a blanking and a drawing die in one ; it cuts the blank and draws it up into the shell shown in Fig. 467 at the same stroke. In descending the blank is cut by the edge of the blanking punch B, meetinor the edg-e of the blanking die A. The blank continues until the drawing punch C is drawn down on the drawing die D, when the blank is drawn into the required shape. Fig. 469 shows a sheet iron dynamo armature disc. Fig. 470 is a section on a larger scale of a set of dies of latest construction, designed for cutting such discs. These dies are made to cut discs up to 100" Fig. 470. is then held firmly by the blank holder ring E and is forced down together with it, by continued down- ward motion of the blanking punch B. The blank holder ring is forced up by the elastic force of the rub- ber spring barrel R upon which the ring sets, through the medium of six pins passing through the bolster or die holder. The descent of the blanking punch B in diameter, and it is claimed that when the press is run at a speed of 55 revolutions per minute, nearly 6,000 sheets 20" in diameter may be produced in ten hours. Fig. 471 shows a well-known type of punching and shearing machine. It will be noticed that the ma- chine is powerfully geared. The machine is really 314 ROGERS' DRAWING AND DESIGN. a form of a press and contains all essential parts of such a mechanism. It will also be noticed that the machine is equipped with a stop clutch, operated by a foot-lever. The func- tion of a stop-clutch is, at the will of the operator, to suddenly make a driving connection between the constantly revolving gear wheel and the tempor- ary stationary main shaft. Its further purpose is to disconnect these members again automatically, after the shaft has made exactly one revolution and when the punch has reached its highest open position. DRILLING MACHINES. A mechanism of the greatest importance in a machine shop is the drilling machine. The ordinary drill press, as the larger drilling machines are gener- ally called, is a complicated machine tool, presenting a great number of interesting mechanical principles to the student. It is, however, not within the scope of this book to take up extensively the construction of this machine ; the figure on page 317 is an exam- ple of this class of machinery ; the drawing shows a bench drill, which embodying, as it does, all the parts essential to any drilling machine, will enable the student to understand the favorite types of the drilling machine. The following is a table of the speeds of drills for different sizes of drills and for different metals, as recommended by the Cleveland Twist Drill Company : Table of Drill Speeds. Diam- eter of Drill. Speed for Soft Steel. Speed for Cast Iron. Speed for Brass. Diam- eter of Drill. Speed for Soft Steel. Speed for Cast Iron. Speed for Brass. 1 1,824 2,128 3,648 ItV 108 125 "215 * 912 1,064 1,824 H 102 118 203 A 608 710 1,216 lA 96 112 192 1 456 532 912 li 91 106 182 5 365 425 730 ItV 87 101 174 1- 304 355 608 If 83 97 165 t'h 260 304 520 lA 80 93 159 1 228 266 456 U 76 89 152 tV 5 203 182 236 213 405 365 lA 78 70 85 82 146 140 w 166 194 332 ^\ 68 79 135 f 152 177 304 If 65 76 130 1 3 fF 140 164 280 m 63 73 125 I 130 152 260 n 60 71 122 If 122 142 243 in 59 69 118 1 114 133 228 2 57 67 114 ROGERS' DRAWING AND DESIGN. 316 ROGERS' DRAWING AND DESIGN, Fig. 472 shows a side elevation of a bench drill of a neat and practical design, suitable for drilling small holes. A front view of this machine is illus- trated in Fig. 473, while Fig. 474 exhibits a top view of the same. The principal parts of the machine are the vertical spindle, holding the drilling tool, a table upon which the work to be drilled is held, and a rigid frame to which all parts of the machine are fastened. The driving cone pulley, as well as the fast and loose pulley are mounted upon one horizontal shaft, at the back of the frame near its lower end. The driving belt is led upward from the cone pulley over two horizontal guide pulleys, and then in a horizontal direction to the cone pulley, which is mounted upon the vertical spindle near its highest point. The lower end of the spindle is provided with a thread for the purpose of holding a chuck which is to receive the drilling tool. The weight of drill, chuck, spindle and other parts which move down ward or upward together with these parts are counterbalanced by a weight which is hidden in the hollow frame. For the upward and downward motion of the spindle a pinion and rack motion is provided. The table m.iy also be lowered or raised according to the requirement of the work. The number of varieties of drilling machines is growing rapidly. Large drilling machines are used also for tapping holes and are generally provided with automatic feed. A class of drill presses which is particularly adapted for larger work is known as the radial drill ; in this type the work is not shifted, after drilling a hole, if there are more holes to be drilled into the same surface ; the drill spindle, with its entire mechanism is mounted upon a heavy cast iron ai^m, which swings horizontally upon the frame of the machine ; the arm may be lowered or raised to suit the work, and the spindle carriage can be moved in or out on the arm, to suit conditions. THE MILLING MACHINE. The Milling Machine may be classed as a combi- nation of several other machine tools used for cutting metals ; the work that can be done on this machine is not limited to either straight or curved surfaces, or drilling of holes ; in general construction this machine does not differ greatly from an ordinary drill press, in fact it can often be used in its place. In this machine the table upon which the work is held is movable in all directions, without disturbing ROGERS' DRAWING AND DESIGN. 317 t 3ji' i Fio. 474. Fig. 472. Fig. 473. 318 ROGERS' DRAWING AND DESIGN. the adjustment of the work, which is fed either auto- matically or by hand feed, while the rotating cutter removes the superfluous metal. In illustrations Nos. 475 and 476 are illustrated an approved form of a vertical spindle milling machine. Fig. 475 shows the front view of this machine, and Fig. 476 shows the side elevation. The whole machine is an advanced type of a modern milling machine and produces an impression of strength and neatness of design. The vertical spindle of this machine is made fully 3" in diameter ; the lower end of the spindle is provided with a thread for large mills, working in a horizontal plane. 77^1? platen as well as the saddle of this machine are 5 1]/^' long. All feed screws are provided with dials, thus enabling accurate work in a most conven- ient manner. The largest distance between the spindle and the platen is 2ii/^". The extreme distance between the rotary table and the vertical spindle is 16". There are fully eight changes of feed for the table and sixteen changes for the rotary attachment. The dimensions of this machine over all, are as follows: Height 81", width 65" and depth 88}4"- ROGERS' DRAWING AND DESIGN. 319 a Fig. 476. 320 ROGERS' DRAWING AND DESIGN. i^S > ^ rsr \ -en Fig. 4TT. FlO. 478, DRAWING AND DESIGN. 321 THE LATHE. The most important machine tool in a shop is undoubtedly the lathe. It is used for a great variety of purposes and for this reason it is made in many different special forms and designs. The simplest kind of a lathe is shown in Figs. 477 to 479. It is called a speed lathe and is used for small work which can be run at a high speed. The lathe is composed of the following principal parts : The bed. The legs or supports. The head stock. The tail stock. The tool rest. I. 2. 3- 4- 5- By means of the steps in the cofie pulley on the head stock different changes of speed of the spindle can be obtained. The tool rest, V\^^. \%o and 481, is adjustable in all directions, but it is not provided with automatic feed connections. The ordinary engine lathe, used for heavier and more accurate work, has the same main parts as the speed lathe. In this lathe, however, the carriage with its tool support is moved over the shears of the bed by the lead screw and its connections. The lead screw is splined and the feed mechanism is driven 322 ROGERS' DRAWING AND DESIGN. from a collar which has 2^. feather engaging the spline and slides over the lead screw. The form of thread used on lead screws is somewhat similar to a square thread with sides forming an angle of 14^ degrees. The lead screw is driven from the spindle of the head stock by gear wheel connections. The head stock of an engine lathe, in two views, is shown in Figs. 482 and 483. Fig. 482 shows an ele- vation of the head stock, and Fig. 483 represents its plan or top view, the back gears being plainly shown. Large engine lathes are also provided with a separate feed shaft besides the lead screw ; this shaft is driven by a belt and cone pulleys, from the stud, and is splined lengthwise ; a splined worm is fitted upon this shaft in such a manner that it can slide on it lengthwise, but is held by two projections on the apron of the carriage, so that it will slide with the carriage and at the same time turn with the feed shaft. This worm engages in a worm-wheel, connected by a clutch to a gear, which meshes with a rack under the front edge of the lathe-bed. By means of this clutch the feed can be engaged or disen- gaged. The worm-wheel also connects with a clutch, which will operate the cross-feed of the tool. Both clutches are operated by knobs at the front of the apron. Fig. 484 shows the longitudinal section and Fig. 485 is a cross or lateral section of the tail-stock of a lathe of the usual form. All back-geared lathes can be run with the back gears idle, by locking the cone with the spindle gear; in this way the spindle has only the changes in spindle speed depending upon the steps of the cone pulley as in the speed lathe. The gear wheels at the back of the head stock reduce the speed of the spindle and a double number of changes in speed can thus be obtained. Figs. 486 and 487 show the mannfer in which the spindle gear may be connected with the lead screw gear, for producing different feeds. CHANGING GEARS FOR SCREW CUTTING. The problem of cutting a screw on a lathe resolves itself into connecting the spindle of the lathe with the lead screw by a number of gears in such a man- ner that the carriage, moved by the lead screw, advances exactly one inch during the lapse of time required for the lathe spindle to make a number of revolutions equal to the number of threads to the inch in the desired screw. The lead screw has, nearly always, a single thread, and, therefore, to move the carriage for- ROGERS" DRAWING AND DESIGN. 323 CenferLine 3" t-d.m S/ideTTest- I Ufi ±^- Fig. 481. 324 ROGERS' DRAWING AND DESIGN. Fto. 482. ROGERS' DRAWING AND DESIGN. 325 Fio. 483. 326 ROGERS' DRAWING AND DESIGN. ward just one inch it must make a number of revolutions equal to its own number of threads per inch. It is, consequently, first of all necessary to know the number of threads per inch on the lead screw. The spindle of the lathe is provided with a gear which transmits the rotary motion of the spindle to the stud gear, below the spindle, by means of inter- mediate gears, situated within the head stock. There are two of these intermediate gears, one being an idle gear, for the purpose of changing the direction of the motion of the stud and through this the lead screw. The connection of the stud with the lead screw may be accomplished by simple or cotnpoujid gearing. In simple gearing the motion of the stud gear is transmitted either direct or by means of an inter- mediate gear to the gear on the lead screw. One or more intermediate gears, which simply transmit the motion received from one gear to another, do not affect the resulting ratio of a train of gears. Consequently, the intermediate gears in simple gear- ing will be disregarded in all calculations for screw cutting. The stud gear is usually equal to the driving gear on the spindle; it may, however, be of a different size and in the following problem it will be assumed that the gear on the spindle has double the number of teeth than that on the stud. The following formula will give the required ratio for the gears on the stud and on the lead screw : Number of teeth on stud gear Divided by number of teeth on lead screw gear Number of turns of spindle multipled by Number of turns of stud Number of threads on the lead screw Divided by number of threads per inch on required screw. Problem : It is required to cut a screw with i6 threads to the inch ; the lead screw has 8 threads to the inch and the spindle makes 20 turns to 40 turns of the stud. Solution : Number of teeth on stud gear 20 8 i Number of teeth on lead screw gear 40 16 4 The required ratio is one to four, i.e., when the stud gear will have 16 teeth the lead screw gear will have 16 X 4^64 teeth; now if the stud gear will have 20 teeth the lead screw gear will have 20X4 = 80 teeth and so on. ROGERS' DRAWING AND DESIGN. 327 ■,■ •.ftfN}}WW?^.V/?f Fig. 484. 328 ROGERS' DRAWING AND DESIGN. In compound gearing, as in Fig. 487, the motion of the stud gear is transmitted to the lead screw by two gears keyed together on an intermediate stud. In this case there are four changeable gears and consequently a wider range of changes than in simple gearing. Of the two gears working to- gether on the intermediate stud that one which works with the spindle stud is called the first gear and the other working with the lead screw g-ear is termed the sec- ond gear. Now assuming that the spindle gear makes 20 revolutions to 40 revolutions of the stud gear and that the lead screw has 4 threads to the inch it will be necessary to find the velocity ratio between the stud gear and the lead screw gear for cutting a screw with 50 threads to the inch. Fig. ^imhla Qearing. ROGERS' DRAWING AND DESIGN. 329 Number of teeth in stud _a;ear 20 4 40^50 I 25 Fio. 487. Comhound (^eartny. Number of teeth in lead screw gear that is, if the gear on the stud should have 16 teeth then tlie gear on the lead screw would have 400 teeth, or the required ratio for simple gearing. For compound gearing this ratio can be divided into fac- tors, for instance, iX|=5V' that is the velocity ratio for the spindle stud gear and the first intermediate stud gear could be made equal to i, and the same velocity ratio for the two other gears. For in- stance, if the stud gear will have 16 teeth the first intermediate stud gear must have 5 X 16^80 teeth; the second intermedi- ate stud gear could have 16 teeth and the lead screw gear 80 teeth. Or 15 and 75 could be taken for the first pair and 16 and 80 for the second pair, or in fact any pair of gears having the desired velocity ratio. Figure 488 represents a modern shaft- ing lathe, built by the Springfield Tool Company, and which can be used both for ordinary lathe work and especially for turning shafting. 330 ROGERS' DRAWING AND DESIGN. m im I ffitf rm fq| £ (ml ■Q_s z T rk. Fig. 488a. ROGERS' DRAWING AND DESIGN. 331 no. laaB. Fig, 489. Fig. 490. 332 ROGERS' DRAWING AND DESIGN. This lathe may be fed by a friction feed or by the lead screw. Slightly below and midway between the shears in the bed, passes a splined shaft, which drives the tail stock face plate by means of an inter- mediate gear, shown in Fig. 490. This small gear can be thrown out of action by the worm-wheel sec- tion shown in Fig. 489. The tail stock face plate is made for the purpose of driving the shaft by a dog, exactly as is done by the head stock. It is very convenient for turning the end of the shaft for which the head stock dog has to be removed. The long centers are a necessity in this lathe in view of the fact that they must reach through a bushing in the rest. This bushing is depended upon to support the shaft during the cutting ; it is made to fit the shaft exactly. The special rest for shaft- ing slides into place on the saddle when the com- pound rest is removed. The pump and tanks for lubricants are carried by this rest. The pump is driven by a gear wheel seen in Fig. 491, which engages with a pinion sliding upon a shaft ; this latter extends the entire length at the back of the lathe. Fig. 491 also shows the lower tank into which all the lubricant collects, from where it flows by gravity to the pump. ENGINES AND BOILERS. The study of the steam engine involves an acquaintance with the sciences of heat, of chemistry, and of pure and applied mechanics, as well as a knowledge of the theory of mechanism and the strength of materials ; many other things are needed to be known, as the student will find as he progresses in his researches, the first of which should relate to the safe and economical production of the steam itself. Nearly the whole of the Eighteenth Century passed in experiments made to reduce the energy, latent in coal and other fuels, to the service of mankind ; at its earliest point of progression the boiler and the engine were substantially one and the combined engine and boiler were known as the fire engine. At a little later period when scientific research had shown clearly the source of the power which gave vitality to the newly invented mechanism the name changed to the heat engine, it having become known that heat accomplishes work only by being let down from a higher to a lower temperature, a certain amount of heat disappearing when changed into work. The modern Steam Engine is now considered as apart from the Steam Boiler and the classification and variety of each and the successive steps of advancement, while full of interest are too voluminous to consider in this volume, but some account of their early history is given in the note below. Note. — About the year 1710, Thomas Newcomen, ironmonger, and John Cawley, glazier, of Dartmouth, in the county of Devonshire, made several experiments in private, and in the year 1712 put up an engine, operated by steam, which acted successfully. The progre.ss made was very rapid and it is recorded that in the year 1737 there was a pumping engine of the Newcomen construction working a succession of pumps each 7 inches in diameter and 24 feet apart, and making 6-feet strokes at the rate of 15 per minute, whereby water was pumped from cis- tern to cistern throughout the whole length of a shaft 267 feet deep, by steam at or near the atmospheric pressure. The construction of the Newcomen engine was greatly improved by Smeaton, who designed and erected an engine for the Chase-Water mine, in Cornwall, which had a cylinder of 72 inches in diameter, with a 9-feet stroke, and worked up to 76 horse power. There were three boilers, each fifteen feet in diameter. This was the last effort on a system then about to pass away, for the engine was set up in 1775 no less than six years after the date of Watts' patent, and we are told that when " erected it was the most powerful machine in existence." 336 ROGERS' DRAWING AND DESIGN. STEAM BOILERS. A closed vessel in which water may be heated for the purpose of generating steam is called a steam boiler ; the boiler is partially filled with water for this purpose, the level of the water in the boiler be- ing called its water line ; the space above the water line is termed the steam space. That part of the surface of the boiler, exposed to the heat of the fire and of the hot gases, is called the heating surface of the boiler, and its measure- ment is usually given in square feet. The space in which the heat is generated is called the furnace ; the surface upon which the coal is laid is called the grate surface, and its dimensions are also given in square feet. Steam boilers may be classified according to their construction and form, or according to their appli- cation. Thus, we have horizontal and vertical boilers, externally and internally fired boilers, plain cylin- drical shell boilers, fire-tube and water-tube boilers. Boilers may be stationary o'c portable ; there are locomotive boilers and marine boilers semi-portable, etc. The plain cylindrical boiler, Fig. 492, consists of a long cylinder called a shell, made of iron or steel plates riveted together, the ends of the cylinder be- ing closed by flat plates called the heads of the boiler. The furnace is arranged at the front end of the boiler, the fuel being placed on the grate through the furnace door, the ashes falling through the grate into the ash pit below. Behind the furnace is built a brick wall, called the bridge wall, which keeps the hot gases in close contact with the under- side of the boiler. ^ For long boilers of this class a second bridge-wall often is built. The hot gases flow from the furnace over the bridge walls into the chim,ney. Within the chimney is placed a damper regulating the flow of these gases. All portions of the brick work exposed to the action of the gases, are made of fire-brick. It is not desir- able to allow the upper portion of the boiler to come in contact with the hot gases, and for this reason the boiler above the water line is lined with fire brick. Water is forced into the boiler through the feed pipe leading to the lower end of the boiler, by the aid of an injector or a pump. To prevent the steam from rising above a certain pressure, a safety valve is placed at the top of the boiler. From the highest part of the boiler also, the main steam pipe leads the steam to the engine or any apparatus for which the steam is to be used. ROGERS' DRAWING AND DESIGN. 337 V//^^/////////////////////7^^ '-'- '^^'':^:- ^-^X' : '^< t'. '-'<■ /.y/,/ ^,,^ ///,,/\i\ 346 ROGERS' DRAWING AND DESIGN. Since the flat sides of the furnace and shell are liable to bulge underpressure, they must be securely braced or stayed ; the illustration shows the s^aj/ bolts which are there for that purpose. The top of the fire box is strengthened in a similar manner, as is seen in the longitudinal section of this boiler, Fig. 510; a large number of tubes pass through the boiler and are secured to a tube plate at the rear end Fig. tOo. of the boiler and to the fire box at the front end ; a cylindrical smoke box is fastened to the rear end of the boiler ; the gases of combustion pass directly from the furnace through the tubes to the smoke box and thence to the smoke stack. Vertical boilers have the advantage of taking up comparatively small floor space. They are made in a great many varieties of designs and are used par- ticularly for fire engines, hoisting engines, etc., and wherever space is limited. An interesting example of this type of boiler con- struction is shown in Figs. 511, 512 and 513; this design has given excellent economical results. It may be observed that there are no flat surfaces 000 000 000 FiQ. 506, which require staying, the top of the shell, as well as the upper plate of the fire box, are of hemispherical shape, giving the maximum strength for a given weight of material. ROGERS' DRAWING AND DESIGN. 347 The products of combustion pass from the fire box through the inclined oval-shaped flue, into a com- bustion chamber and thence through a very large number of horizontal tubes to the smoke box and thence to the chimney. The water is contained in a large number of small lap-welded tubes, connected in various ways to each other, as well as to the cylindrical drum above them. The water fills all the tubes and a part of the drum. A furnace of the usual form is placed under the front Fig. 507. As may be seen in the illustration, the combustion chamber is lined with fire brick. Whenever the generation of a large quantity of steam in a comparatively short time is required, water tube boilers are now extensively employed. In these boilers the steam space is limited to a cylin- drical shell which forms only a part of the boiler. end of the tubes, the products of combustion circu- lating around the tubes and the under side of the drum. An illustration of this type of boiler is shown in Figs. 514, 515. In this type of boiler the drum for the steam space is made comparatively large, and is located parallel to and above the network of tubes, which are 348 ROGERS' DRAWING AND DESIGN. inclined, as well as the drum, at an angle with a hori- zontal plane, so as to bring the water level to about one-third the height of the drum in the front and about two-thirds of its height in the rear. The ends of the tubes are expanded into large water legs made of wrought iron, flanged and riveted to the shell, which is cut out for a part of its circumference to Fig. m&. receive them. The two ends of the drum are of a hemispherical form and are not braced as is the case where flat heads are used. The water legs form the n£tural support of the boiler. The boiler is entirely enclosed by a brick work setting ; the furnace is situated below the front end of the boiler and is terminated by a bridge wall ; air is admitted through a channel at the bottom of the space behind the bridge wall, and is heated in pass- ing through the wall. Fig. 509. The feed water is brought through a feed pipe leading to the front head of the drum ; within the main drum is suspended a mud drum below the water ROGERS' DRAWING AND DESIGN. 349 Z' 7/i * ^ 8>% Fig. 510. 350 ROGERS' DRAWING AND DESIGN. line. The feed water enters the mud drum first, which is submerged into the hottest part of the water ; in this manner the impurities of the feed water are largely extracted. Between the horizontal rows of tubes are placed layers of fire brick, acting as baffle plates, forcing the hot gases to circulate back and forth between the tubes and finally flow out through the chimney placed above the rear end of the boiler ; the upper part of the main drum is protected by a lining of fire brick. When several of these boilers are used together, forming a battery of boilers, an additional steam drum is usually placed at right angles and above the steam drums already described. TO DESIGN A STEAM BOILER. In designing a steam boiler an engineer has to bear in mind the following considerations: i, its required strength; 2, its durability; 3, its accessibility for inspection, cleaning and repairs ; 4, its ability to perform the work ; 5, the special laws of the locality in which the boiler is to be used, as well as the rules of insurance companies; and 6, the type best adapted to existing conditions. The following example may serve to show how the parts of a boiler may be calculated. Let it be required to design a 60 horse power horizontE I multi-tubular boiler, to carry a working pressure of 150 lbs. per square inch, and which will be capable of sustaining a test pressure of 225 lbs. Let tbt: length of the tubes be 15 ft. and each tube havi;: an internal area of 6.08 sq. in., i. e., about 3 inches outside diameter. The heating surface should b<; about 37 times the grate surface. ' j Let the area of the grate in sq. ft equal G " the lieating surface in sq. ft " H " the area of smoke passage through the tubes in sq. ft " C " the water space in cubic feet. " W " the steam space in cubic feet " S According to the standard recommended by a committee of the American Society of Mechanical Engineers, it is customary to rate boilers by their horse power, considering jo potmds of water evapor- ated from feed water at 100" F., under pressure of 70 lbs. by the steam gauge, is equivalent to one horse power ; this is equivalent to 34^ lbs. of water evap- orated from feed water at a temperature of 212° F., into steam at atmospheric pressure. Now, as 343^ lbs. are evaporated for i-horse power, for a boiler of 60 H. P. 34.5 X 60 ^ 2,070 lbs. must be evaporated to meet the conditions. ROGERS' DRAWING AND DESIGN. 351 lb ■© ® ©■©-ee©<5> ©■*©•©■© ©^S- ->§; e- @%® -®-!f}-^^>®- ©{^ ©^-sfe ■© ©•©-©-©■©-© f -©-©■©■©■©-©-©■ -C# ®-©-:@;o-S ©■© ©iS:-©-^!;-©^©'^- -S ©&©■©■ 6-^© ©-©-©■© ©oe-,,^ -;®! ©©-©■©-© ©■*■ ©■©-©- ©-©-©vfro ©■©■© ®©©©0^ ©©©■©-© "K3 © 6©®© ©a> ©-©-©-* ©■©©> ' ©-©^■©■®© ®-^ffi-©-©-&©©- ®©f«©-(&-^<3®#;- 1 -e -© © ©■©© © -® © ffi ©-©o- 3r Ftg. 512, Fio. 511. Fig. 813. 352 ROGERS' DRAWING AND DESIGN. Evaporation per i lb. of coal : i, depends on the quality of the coal ; 2, the rate of combustion, and also, 3, the construction of a boiler. The following table illustrates the effect of the rate of combustion, according to tests with a boiler, which had a ratio of 25 to I of the heating surface to the grate surface. te of combustion in ounds of coal per sq. ft. of surface. Evaporation per lb. of coal. Evaporation per sq. ft. of heat- ing surface. 6 10.5 2.52 8 10.4 3-33 10 12 14 16 10. I 9-5 8.9 8.2 4.04 4-56 4-98 5-25 The evaporation per pound of coal that takes place (n the different types of boiler, according to Prof. Hutton, is as follows : Plain Cylindrical 5 to 8 Vertical 5 to 10 Water-tube 5 to 11 Cornish 6 to 11 Multi-tubular 8 to 12 Locomotive Boiler 8 to 13 The area of heating surface of each type of boiler is nearly always in a constant ratio to the grate area Below is given a table showing ike ratio be- tween the grate surface and the heating surface, gen- erally observed in the several types of boiler : Ratio of Grate Surface Style of Boiler to Heating Surface Plain Cylindrical 12 to 15 Cornish 1 5 to 30 Cylindrical Flue 20 to 25 Cylindrical Tubular 25 to 35 Locomotive Tubular 50 to 100 For the boiler which we have taken as an exam- ple, we will assume that 12 pounds of coal can be burned per square foot of grate, and that one pound of anthracite coal will evaporate about 9 pounds of water at 212° F. As we have found, 2,070 pounds of water will have to be evaporated in this boiler to give us the re- quired 60 H. P. Now, as I pound of coal will evaporate 9 pounds of water, the coal contained in i square foot of grate surface, 12 pounds, will evaporate 12 x 9^108 pounds of water. Dividing the number 2,070 by 108 gives the reqttired area of grate. ■ — ^ = 19.16 sq. ft. of grate surface, 108 ^ ^ ^ The area of smoke passage through the tubes to the grate area, C : G, is according to good practice, made equal to i : 8 for this type of boiler. The number of tubes, each 3" in diameter may be found in the following manner : ROGERS' DRAWING AND DESIGN. 353 19.16 X 144 == about 56 tubes. 8 X 6.08 Here 6.08 is the internal area of one tube. Suppose our boiler is designed for a steam engine which is to use 40 pounds of steam at a pressure of 70 pounds per horse power per hour, and that the steam space shall hold enough steam to supply the engine 30 seconds ; the absolute pressure is nearly 85 pounds and the specific volume of steam at this pressure is 5.125. The space which will be taken by steam required for one horse power is, -12 ^X 5-125 == 1.706 cubic feet. 60 X 60 Let us assume that this amount of space is taken up by the steam required for one horse power ; then the total space required to contain the steam for the 60 H. P. will be 1.706 X 60 = 102.36 cu. ft., say 103 cu. ft. That is, the steam space, S, should hold 103 cu. ft. As the water space may be taken to equal two times the steam space, the water space, W, will equal 2 X S, or 2 X 103 == 206 cu. ft. JJoTE. — The steam space required by a given boiler depends upon the purpose for which the steam is to be used. Where the steam is under high pressure and comparatively small quantities of it are with- drawn at very frequent intervals, the steam space need not be so large as in cases where large quantities are withdrawn, even though less frequently. Where the boiler supplies a steam engine, it is the general practice to have the steam space of such dimensions that it shall contain sufficient steam to supply the engine for about a half a minute. Adding the steam space and the water space to- gether, we get 309 cu. ft., for the volume of both ; this volume would determine the capacity of the shell, were there no tubes passing through it and taking up part of the space within ; in this case, therefore, the space taken up by the tubes must be added to the above volume. We have already found that the boiler will con- tain 56 tubes. The outside area of one tube of the given size, is 7.107 sq. in., consequently 56 X 7.107 X 15 144 41.45 cu. ft. = the space taken up by the tubes. Adding this to the volume found above, 309 cu. ft. plus 41.45 = 350.45 cu. ft., the entire volume of the shell. As the shell is to be 15 feet long, the area of its head must be ^^ '^^ zz.^ 3362.3 sq. in. Divid- ing this by 0.7854, we get the square of the diameter of the shell; D^= 3362.3 _ g^_ j^^ Allowing 0.7854 for the space occupied by the stays, etc., we may take D equal to 67". One-half of the outside surface of the shell equals _Z 3-14 i^ 131.4 sq. ft. The inside surfaces 12 of the tubes equal -^ ^ ^- — 61 1.8 sq. ft, 12 354 ROGERS' DRAWING AND DESIGN. Allowing for the heating surface, one-half the sur- face of the shell, and all of the inside surfaces of the tubes, we have 131.4 + 6i 1.8 = 743.2 sq. ft. H. = 743.2 sq. ft. We have already found the grate surface to be HORSE POWER OF THE STEAM BOILER. As was stated before, the evaporation of 30 lbs. of water per hour, from a feed water temperature of 100° F. into steam at gauge pressure of 70 lbs. is equal to 19.16 sq. ft. Dividing the heating surface by the grate surface, we are to get, according to the conditions given in the problem, the ratio of 2)7- '^^" — 38.7, or very nearly as required. 1 9. 1 6 the value of a commercial horse power, adopted by the A. S. M. E. Different boilers will generate steam at different pressures, receiving also the feed water at different temperatures. In order to com- pare properlj' the performances of different boilers, ROGERS' DRAWING AND DESIGN. 355 their actual evaporation must be reduced to an equivalent evaporation from and at 212° F. The problem may be stated differently as follows : It is necessary to find what would be the evaporation if Rule : From the total heat of steam at pressure of actual evaporatioji, subtract the observed tempera- ture of the feed water and add J2 ,- multiply the result by the actual evaporation and divide by g66 , i. Fig. 515. the feed water would be at 212° F., and the deliv- Example: If a boiler generates 2,000 lbs. of ered steam at o gauge pressure. steam per hour at a pressure of 100 lbs., and if the To find the equivalent evaporation of a boiler, j temperature of the feed water is 70°, what is the proceed as follows: equivalent evaporation of the boiler? 356 ROGERS' DRAWING AND DESIGN. From the steam pressure table given below, the total heat corresponding to a pressure of lOO lbs. gauge is 1184.5; consequently the equivalent evap- - (1184.5-70 + 32) X 2,000 oration is ^^ ^^^^^ — - — ^\ 966.1 To find the horse power of the boiler, divide the equivalent evaporation by 34.5. In this case, the horse power (1184-5-70 + 32) X2.000 _ ^g_g ^^^^jy_ g66.i X 34.5 Let W equal the actual evaporation H " total heat t " observed temperature of the feed water ; then, according to the above rule, the equiv- 1 . .- • w (H— t + 32) X W alent evaporation is equal to ^ r^ ^ or H— t + 3: 966.1 The quantity X W H- •t + 32 966.1 966.1 which changes actual evaporation to equivalent evaporation from and at 212° F. is called the factor of evaporation. The equivalent evaporation is equal to the actual evaporation of the boiler, multiplied by the factor of evaporation; knowing the actual evaporation, and having a table of factors of evaporation, we are eas- ily able to calculate the equivalent evaporation, or the horse power of the boiler. TABLE OF GAUGE PRESSURE AND TOTAL HEAT. ' The following is a table of steam pressures ; the table gives the pressure of the steam by gauge and the corresponding total heat required to generate one pound of steam from water at 32° F. under constant pressure. Pressure by Gauge Total Heat [I46. I Pressure by Gauge 85 Total Heat\ [i8u4 5 [I5O.9 90 [182.4 10 [I54.6 95 ti83-5 15 20 ] 25 1 [I57.8 [I60.5 [162.9 100 105 110 [184.5 [185.4 [186.4 30 1 35 [165.I 1167.1 115 120 1187.3 1x88.2 40 1 [ 169.0 125 [ 189.0 45 J 170.7 130 ] 189.9 50 ] 172.3 135 1 190.7 55 i 173-8 140 1 [191-5 60 1 175-2 145 ] 192.2 65 i 176.5 150 ] [192.9 70 1 177.9 155 1 193-7 75 1 80 1 179. 1 180.3 160 ] 194.4 ROGERS" DRAWING AND DESIGN. 357 SAFETY VALVE RULES. The safety valve provides for the safety of boilers, by allowing the steam to escape when its pressure exceeds a certain limit. The valve is kept in its seat, either by a weight at the end of a lever, as in Fig. 516, or by a heavy weight placed directly over Let AB represent the length of the lever in inches. AC W w P Let a " V the distance between the center of valve and A, also in inches. the weight in pounds. the weight of the lever in pounds. the pressure of the steam per sq. in. the area of the valve in sq. in. the weight of the valve in pounds. Fig. 516. the valve, or by a strong spring. A good safety valve must allow all excess of steam to escape as fast as it may be generated. The valve shown in Fig. 516 rests on a circular seat. To find the weight, W, or the length of the lever, AB, for a given pressure of steam : If the weight of the lever and valve be neglected, we have, when the steam reaches the limit of pressure, for which the valve is intended, AB a downward pressure of W X -— and at the same time an upward pressure equal to P X a. When the valve is just about to lift, these two pres- AB sures may be considered equal ; then W X ^ =^ AC P X a; from this, W: P X a X ~-~- in pounds. AB ^ Taking into consideration the weight of the valve, which should be done for accurate practice, we have a AB downward pressure of W X ~— , the pressure due to AB the weight W, plus w X — —=;, the pressure due to the 2 1\\^ 358 ROGERS' DRAWING AND DESIGN. weight of the lever, assuming that the weight of the lever acts downward in its middle, and plus Y, the weight of the valve. The upward pressure remains, as before, P X a. w X — ^^ + V = P X a. Here again, W X AC AC Example : Let it be required to find the weight, when the lever AB is equal to 36 in., AC equals 4 in., w equals 5 lbs., V. equals 3 lbs., P equals 80 lbs. and a equals 6 sq. in. The weight of valve and lever must be taken into account. According to the above formula, Wx 3^+ 5 x-2^^ 4 2X4- W X 9 + 4.5 X 5 + 3 W — 480—25-5 3 = 80 X 6, or = 480, or 50.5 lbs. THE STEAM INJECTOR. The injector is an instrument, by the aid of which, the energy of. a jet of steam from the boiler, is utilized in forcing a stream of water into the boiler. The injector has largely replaced other appliances for feeding steam boilers, for when the work of an injector is compared with that of a steam pump, we come to the conclusion that even if the injector may consume a little more steam than the pump, the heat is returned to the boiler, by being imparted to the feed water. A boiler is tapped at the highest point of the steam space, and a pipe leading downward is inserted into the opening. To the open end of this pipe is attached the injector, which again is connectec^ with the lower part of the boiler, into which it is to force the feed water which it receives through a special pipe fr^m the source of water, be it a tank, well, etc. The live steam which enters the injector and is given the shape of a pointed jet, forms a partial vacuum within a chamber in the injector just above the feed water pipe, allowing the water to enter the chamber, where it acquires a velocity equal to that of the jet of the entering steam, and being thus enabled to overcome the pressure within the boiler, by its momentum, it is forced through an opening and a check valve, into the boiler. While the pressure within the boiler may be taken to be pretty nearly equal in all its parts, the partial vacuum caused by the condensation of the jet of steam meeting the colder water in the injector, com- pels the jet of steam to rush into the injector at a much higher velocity than if it were discharged into ROGERS' DRAWING AND DESIGN. 359 Fig. 517. 360 ROGERS' DRAWING AND DESIGN. the atmosphere. Consequently the high velocity and the resulting momentum of the entering feed water. The accompanying illustrations, Figs. 517 and 518, show an outside view and a longitudinal section of an injector. Referring to the sectional view, it will be seen that the injector consists of a case with a steam inlet at its upper part, water inlet directly below the steam inlet, delivery outlet to the boiler and an overflow opening at the bottom of the injec- tor. Separate handles are provided to regulate the flow of steam, of feed water and delivery. The nozzles within the injector may be termed, according to their purpose : (i.) The steam nozzle, through which the steam enters into the chamber of the injector. It is bored out straight in the middle and slightly conical to- wards its ends. (2.) The combining nozzle is nearest to the steam nozzle ; here the steam and the feed water come together. This nozzle is placed in line with the first one. (3.) The condensing nozzle, is the next one ; it forms the vacuum, upon which is based the velocity of the feed water ; from this nozzle the water is driven into the (4) delivery nozzle, through which it enters the boiler. The delivery nozzle is usually made with the smallest bore of all the nozzles. The diameter of the bore in the delivery nozzle determ- ines the volume of water which may be forced through it into the boiler. The size of an injector is always given by the diameter of the smallest part of the bore in the delivery nozzle, expressed in mil- limeters ; thus a No. 6 injector has an opening of 6 millimeters in diameter. ^ To start the injector, open the water valve first. When the water appears in a solid stream in the overflow, open the steam valve, situated directly above the jet, and close the jet valve. The steam valve must always be opened slightly before closing the jet valve, so as not to break the vacuum of the injector. It will be noticed that the injector is put together in such a manner as to render feasible all repairs within by unscrewing the connected parts. The nozzles may have to be replaced from time to time, as they have to withstand the great velocity of the flow of water, which because of its impurities, occasions considerable wear on the nozzles. On account of this, all nozzles are made of a special hard metal. ROGERS' DRAWING AND DESIGN. 361 Fig. 518. 362 ROGERS' DRAWING AND DESIGN. STEAM ENGINE. The steam engine is a machine designed to trans- form the energy of steam, underpressure, into actual energy in the form of continuous rotation. For this purpose the steam is made to move the piston in the steam cylinder backward and forward, by bringing the steam into the cylinder, alternately from one side of the piston and then from the other, thus imparting a reciprocating motion to the piston. The mechanism which regulates the direction of the steam into the cylinder is called the valve meclianism or valve gear of the steam engine. When the piston, or the area which receives the pressure of the steam, travels in a circular path con- tinuously in one direction, the engine is termed a rotary steam engine or the steam turbine. The rcciprocatijjg steam engine^ in which the pis- ton travels back and forth, is the ordinary form of this important motor. It has been found to be the most convenient and most economical design, hence we shall take up for illustration and explana- tion this form, only, of the steam engine. The reciprocating motion of the piston may be transformed into a continuous rotary motion in various ways ; the crank motion is the most popular form of mechanism adopted for this purpose ; the motion of \.]\& piston is transmitted hy the piston rod, which is fastened firmly at one end to the piston, to the crosshead, from which the co7i7tecting rod leads to the crank pin oi the cra^ik; this forms a solid structure with the main shaft of the engine. The ?naifi shaft, receiving in this manner the rotary motion, serves as the source of rotary power, used for the many purposes of modern industry. The length of the cylinder is made equal to the travel of the piston, which itself is equal to-, twic« the effective length of the crank, phis the thickness of the piston, to which must be added the allowance for clearance at each end, so that the piston shall not strike the head of the cylinder, and at the same time will provide the necessary space for the steam to get behind the piston when the latter is at the end of its stroke. The length of the piston rod must be sufficient to permit the piston to return the full length of its stroke and still leave enough of the rod outside the cylinder to fasten it to the crosshead. To avoid leakage of steam through the hole in the cylinder head, through which the piston rod protrudes, 2. stiff- ing box is attached to the cylinder head. It is evident that the extreme length of the stuffing box must be added to the length of the piston rod. The crosshead is guided in its rectilinear path by close-fitting rods, bars or blocks, which are securely ROGERS' DRAWING AND DESIGN. 363 364 ROGERS' DRAWING AND DESIGN. fastened to the engine bed, or form a part of the bed casting, in one piece, and which are called the cross/lead gtiides ; these must be set absolutely parallel to the axis of the cylinder. The steam cylinder, as well as all other parts of the steam engine mechanism, is fastened to a heavy casting called the engiiie bed^ which is rigidly held upon a solid masonry foundation by means of anchor rods, whenever the engine is of the stationary type ; in marine engines the bed plate is fastened to extra heavy frames forming part of the hull. When the crank is in a horizontal position, in the plane of the piston rod, and the crank-pin lies in a line drawn through the center line of the cylinder, the piston being at one of its extreme positions of the stroke, the engine is then said to be on its dead center , as the pressure of the steam upon the piston will not result in rotation of the crank. The rotatingr shaft is usually supplied with a heavy fly wheel, intended to store up the energy of rotation, and one of the func- tions of the fly wheel is to carry the engine mechan- ism past the dead centers. Fly wheels are of many different constructions, varying from a solid cast iron wheel of small diame- ter, to built up wheels of over 30 ft. in diameter. In modern practice the rim of the wheel is made wide enough to carry the belt which transmits the motion from the engine to the machinery. Such wheels are usually called belt wheels m distinguishing them from the old-style narrow rimmed fly or bnlance wheel, which was constructed independent of the pulley carrying the belt. In Figs. 519 and 521 are shown two views of a belt wheel of modern construction, 18 ft., 8 in. in diameter. It is unnecessary to show the entire wheel, as it would be simply a repetition of similar parts. In most cases only a quadrant of the wheel is shown, accompanied by a partial section like that shown in Fig. 519. It will be noticed that the outline of the belt wheel hub is a regular dodecagon, its sides forming the planed surfaces, to which the arms are attached by means of flanges and bolts. The other ends of the arms are also flanged and connected to the dif- ferent rim sections as shown in Fig. 520. The hub in this case is made in two sections and the rim in six sections, to each of which latter two arms are flanged. The sections of both hub and rim are held to- gether by bolts passing through projecting flanges, as shown in Fig. 520 and in partial detail in Figs. 521, 522 and 523 . Where extra strength is re- quired, the reinforcements shown in Fig. 520 are often made use of. These consist simply of I-shaped ROGERS' DRAWING AND DESIGN 366 ROGERS' DRAWING AND DESIGN. pieces made of mild steel, which are shrunk into re- cesses of similar shape, making a rigid joint. A different type of reinforcement is illustrated in detail in Figs. 521, 522 and 523. Here, instead of the I-shaped piece, a wrought iron link is substituted, which is shrunk over bosses cast into the castings for this purpose. These bosses are cast so that they will not project beyond the surface of the rim, as shown in Fig. 522, leaving them flush so the wheel may be faced off after mounting. This method of rein- forcement is rapidly finding favor for permanently connecting parts of heavy machinery, such as large sectional engine beds, etc. It is very difficult, however, to disconnect such fastenings, for, when heating the link to expand it for removal, the lupfs will also be heated and ex- panded, necessitating in most cases cutting and destruction of the link. In Fig. 524 is shown a section through an arm of the wheel, and Fig. 526 illustrates a section through the rim, close to one of the flanges to which the arms are attached. Fig. 525 represents part of the hub, showing the face of the joint. The design of a fly wheel is one of the most diffi- cult tasks that an engineer m.ay meet and requires judgment and much practical experience. Often- times the success of an engine depends upon the fly wheel, for even a good and active governor would be unable to steady the motion of an engine with a variable load, were the fly wheel not able to carry the crank past the dead centers, as well as those points during the revolutions, where but little rotary motion is supplied by the driving parts. Great care must also be taken to have the material in the rim evenly distributed, for if the wheel is not balanced, the centrifugal force will have a tendency to bend the shaft, besides causing severe vibration. The rim speed of a cast iron fly wheel ■iVovX^ never exceed one mile, 5,280 feet per minute, for the cen- trifugal force has a tendency to burst the rim and this tendency increases as the speed increases. This danger is not overcome by increasing the thickness of the rim, for while the extra, thickness adds strength to the rim, the extra weight increases the centrifugal force. Two or more cylinders are often used in one steam engine ; when two cylinders are used, they are usually arranged so that the two cranks of the separate cylinders are at an angle of 90° to each other. When three cylinders are used the cranks will be at angles of 120° to each other. Such multi-cylinder engines do not require as heavy a fly wheel as a single cylinder engine of the ROGERS' DRAWING AND DESIGN. 367 Fig. 527. Fig. .528. same power, as the cranks will assist each other over the dead centers; at the same time they act so that one cylinder develops its maximum power, while the others are nearing completion of the stroke. The connecting rod of an engine is usu- ally made from four to six times the length of the crank. The top view of a horizontal engine is shown in Fig. 527. The same engine is shown in Fig. 528. One illustration shows a rio;ht hand and the other illustrates a left-hand engine. A rtght-hand engine is one in which the Jiy wheel is to the right of the observer as he stands at the head end of the cylinder looking towards the main bearing of the engine. The size of the engine is commercially rated by the length and diameter of the steam cylinder. When an engine produces a rotary motion of the fly wheel and crank, so that the crank when starting from its inner dead center rises above the axis line, or descends below it, upon the beginning of the stroke, the engine is said to "run over" or "run under," Fig. 529. It is, as a rule, desirable 368 ROGERS' DRAWING AND DESIGN. to have an engine run over, because the pressure of the connecting rod is then always downward, and is taken up by the guides and bedplate. An example of proportioning the main parts of an engine is given in Figs, 530 and 531. A vertical etigine for small power is shown in Fig. 532 ; it shows a section through the cylinder and Fig. 529. valve chest, while Fig. 533 exhibits a section on a plane at right angles to that of the first section. The avoidance of cylinder wear, and still more the small floor space required by the vertical engine, have made its use practically universal for crowded power plants, steamships, etc. The support of such engines, however, is commonly not sufficiently rigid to prevent undesirable vibration of the moving parts. The section of the vertical engine shown in Fig. 532 shows a type of a steam distributing valve, called the slide valve, one of the oldest and up to the present most reliable valve gears. The functions of a valve on a steam cylinder are primarily to admit the steam from the boiler^to one side of the piston, while the steam filling the other side of the cylinder is allowed to escape through the exhaust pipe, and second to stop the admission of steam at a certain point, for the purpose of producing its desired ex- pansion and finally to close the exhaust opening at such a point in the return stroke, that a certain volume of steam shall be left in the other side of the cylinder to be compressed behind the piston, to serve as an elastic cushion. Note. — The horizontal position of the engine is by far the most popu- lar for factories, power plants, etc., where there is considerable floor space. To have all the parts of an engine easily accessible and a solid support for the engine, as offered by a large bed, are the great advan- tages offered hy a horizontal engine ; while on the other hand, the ten- dency of the cylinder to wear vmequally is a disadvantage which cannot be denied. ROGERS' DRAWING AND DESIGN. 369 Diagram of HORIZONTAL ENGINES. Fig. ra). Fig. 531. Table of Dimensions Reference being had t( ) above Diagram. SIZE or ENGINE A B C D E F Q H 1 J K L M N P Q R iZ>^24- lO-4h l4-8i 16-51 9 15 /; Z4 7-fi 4-Qi 2-iOi 10 12 3-4 6i 13 8-0 7i 16 i3^24 l0-4h 14-81 16-51 9 15 \i Z4- 7-51 4-2^ 2-108 10 12 3-4 7 15 8-0 81 16 iS^Z4 lQ-7'z 15-31 18-01 13 15 20 2-4- 8-3'z 4-71 d-Z'z II 15 4-0 9 16 10-0 9^ 19 i6-2S mi n-dl 20-11 15 18 iz 2-9 9-51 5-3l\ 3-6l IZ 16 4-10 91 20 10-0 10 20 18^28 l2-5i 17-iii 20-3i 15 18 23 Z-9 IO-3i 5-m 3-91 13 18 4-10 //b' 24 10-0 Hi 22 20x30 is-e'2 19-91 22-61 20 10 15 3-4 12-Oi 7-2 4-9 15 20 6-0 // 28 12-0 121 24 22-30 I3-/2 19-llz U-8'2 20 10 27 3-4 I2-I(l'z 7-9'z 4-IOz 15 22 6-0 12 34 IZ-0 I3i 26 370 ROGERS' DRAWING AND DESIGN. It is evident that the opening or closing of the steam inlets or outlets of the cylinder must be care- fully timed to produce the required pressures in the different parts of the cylinder at the proper time. The valve is generally moved back and forth within the steam chest by the action of an eccentric, one type of which is shown in Fig. 536; this is securely fastened to the main shaft so as to turn together with it. In some rare cases, the eccentric is forged solid on the shaft, but the general practice is to fasten it by keying it on ; the eccentric is usually placed just outside of the bearing which holds the main shaft. This mechanism consists of two parts ; the eccen- tric proper or sheave, Fig. 536, and the eccentric strap. Figs. 534 and 535. The eccentric strap is made to fit in a groove in the face of the eccentric, or the eccentric fits in a groove in the strap. To the eccentric strap is attached the eccentric rod, which is connected to the valve rod, this finally con- necting to the valve. The eccentric is a form of crank, the difference be- tween them being that in the eccentric the crank pin is so large that it embraces the crank shaft. This is shown in Figs. 536 and 537. Twice the distance between the center of the crank shaft and the center of the crank pin is the length of a stroke produced by the crank. The same is true for the eccentric. The slide valve which has been used most exten- sively is the plain three-ported slide valve. Its form and action will be understood by reference to the illustrations shown in Figs. 538 and 539 ; here the slide valve is shown in its most elementary form, as a box open on its under side sliding over a plane surface on the outside of the steam cylinder. This surface is supplied with openings called steam ports, leading into the cylinder ; these ports, usually rect- angular in section, are indicated in the illustration by the letters S^ and S^. The third opening over which the slide valve moves, indicated by the letter E, is the exhaust open- ing, through which the steam escapes from the cylin- der. It will be seen that, when the valve is in its middle position, its two edges cover the two steam ports, while the hollow part of the slide valve is over the exhaust port. When the valve no more than covers the steam ports when in its middle position, the eccentric must be placed 90° in the advance of the engine crank. The illustration shows the piston at the left end of the cylinder, with the valve moving toward the right and about to open the steam port S^ to permit the steam to pass through this port into the cylinder, thus forcing the piston to the right. During this same time, the steam port Sg and the exhaust port E will be connected by the hollow part of the valve, ROGERS' DRAWING AND DESIGN. 371 e Fig. 532. Fig. 533. 372 ROGERS' DRAWING AND DESIGN. and the steam, which is contained to the right of the piston is allowed to escape into the exhaust. The valve will have moved to its extreme right position when the piston has reached the middle of the cylinder, and when the piston has reached its extreme right position, the valve will be in the mid- dle of its return stroke. Without this arrangement there would be no expansive working of the steam, as one end of the cylinder is left open for the admit- tance of live steam during the whole stroke of the piston, while the other end is open during the same time to the exhaust. To produce expansion, the valve must more than cover the steam ports when in its middle position. The amount which the valve projects outside of the steam ports, when the valve is in its middle position, is called the outside lap of the valve, and the amount which the valve projects on the inside of the steam ports in the direction of the exhaust ports, is called the inside lap. Fig. 540. The addition of laps necessitates a change in the angle of advance between the crank and eccentric, because the valve must be on the point of opening the steam port when the^ piston is at the beginning of its stroke, and therefore the valve must be away from its middle position, by a distance equal to the outside lap when the piston is at the beginning of its stroke. The angle between the eccentric and the crank must then be more than 90°. The correct position of the eccentric in relation to the crank is found by the construction illustrated in Fig. 541. Here AO represents the crank, while the circle CDE is the path of the center of the eccentric. On AO set off OB equal to the outside lap of the valve. Draw BD perpendicular to OB cutting the circle at D. Then OD is the position of the eccentric sheave, if the motion is in the direction of the arrow. If opposite, then OE is the right position of the sheave. The angle CO A i&a 90*^ angle, and the acute angle COD is called the angle of advance. When it is desirable to partly open the steam port just as the piston is beginning its stroke, the valve must be given a lead. The amount of such opening of the steam port at the beginning of the stroke, is called the lead of the valve. To produce the valve lead, the angle of advance of the eccentric must be increased, so as to make OB, in the above figure, equal to the outside lap, plus the lead. Let the circle ABCD in Fig. 542 represent the path of the center of the eccentric sheave. When the valve is in its middle position, moving toward the left, the eccentric will be in the position OC, moving in the direction indicated by the arrow. Let P be one position of the eccentric ; drop from the point P a line PM perpendicular to AB. Then ROGERS' DRAWING AND DESIGN. 373 Fios. 531 AITD 53S. 374 ROGERS' DRAWING AND DESIGN. the distance OMwill correspond to the distance the valve has moved from its middle position. Make QO equal to MO. If more positions of the eccen- tric are taken, and for each one a similar construc- tion employed, the points corresponding to the point Q, found above, will, when joined, produce a curve which will have the form of the two circles, AQON and OSBT. These circles are described on AO and OB as diameters. By means of these two circles the position of the valve may be readily found for any position of the eccentric. For instance, if OR is the position of the eccentric, the valve will be at a distance OS from its middle position. To find the position of the engine crank for any given position of the eccentric, say for the position OP, make the angle POL equal to the angle be- tween the eccentric and the crank, equal to a right angle plus the angle of advance; then OL will be the required position of the crank. If the distance OQ, equal to MO, be set off on OL instead of OP, and a similar operation gone through for a number of positions of the crank, it will be found that the different positions of the point Q, when joined, will form a curve just coincid- ing with the two circles described on OE and OF, Fig. 543, as diameters, the diameter EOF making the angle COE, equal to the angle of advance. ROGERS' DRAWING AND DESIGN. 375 From this diagram the position of the valve for any position of the crank may be found. If the crank is in position OL, Fig. 543, then the valve is a distance OQ from its middle position. We can also find any position of the crank for a given position of the valve. For instance, when the To determine whether the valves are set correctly, by means of diagrams taken of the steam pressure from each end of the cylinder and by observing and comparing the respective positions of the point:; of admission, cut-off, release and compression, the ap- paratus called the steam engine indicator is used. Fio. 538. valve is just on the point of opening or closing, it will be at a distance equal to the outside lap, from its central position. If an arc be described with the center O and a radius equal to the outside lap, 06, will be the position of the crank when the steam is admitted, and O62 its position when the steam is cut ofT. The position of the piston can easily be found from this. A description of the indicator and its use, together with a study of its diagrams, can not be classed with mechanical drawing, and would cause us to drift too far from our subject. A very interesting and thor- oueh treatise on the indicator is " Hawkins' Indica- tor Catechism," and the student who desires to take up the field of steam engineering, is respectfully referred to this work. 376 ROGERS' DRAWING AND DESIGN. Fig. 544 shows a valve diagram, as well as a theoretical form of an indicator diagram. The ra- dius of the eccentric equal to one-half the travel of the valve, is equal to the distance AO in the diagram. The outside lap equals OC, the lead equals b d, the angle of advance is equal to the angle EOC; the OuhideLaJi /r)6ide Lafi ■ i« =t Fig. 540. inside lap is equal in length to Og, the width of the steam port is given by MK, which is equal to gh. When the steam is admitted, the crank is in the position indicated by the line 06;. When the steam is cut off the position of the crank is 06,. When the steam is released the crank is in the posi- tion O63. Compression begins when the crank is in the position 06^; the port is completely open while the crank moves through the angle q o r, and it is completely open for exhaust while the crank moves through the angle tOs. To find the indicated horse power of a steam engine, fuuliiply the mean effective pressure in lbs. per sq. length piston divide P X Fig. 641. in. on the piston during one stroke.^ by the of the stroke in feet, then by the area of the and by the number of strokes per minute and the result by j^,ooo. LXAXN ^^. ^^^, — = Indicated H. P. of engine, 33000 P is the mean pressure ; L, A, the area of the piston ; where stroke ; ber of strokes per minute. The actual horse power the length of the and N, the num- ROGERS' DRAWING AND DESIGN. 377 may be taken as about ^ of the indicated horse power. To find the area of the piston multiply the square of its diameter by o.jS^-f.. C Fig. oi-Z. To find the mean effective pressure, divide an indicator diagram of the engitie into any number of equal parts, say lo, then measure the height of each part, half-way between the division lines, as shown by the dotted lines in Fig. 545. Add the length of all the dotted lines, and divide by the num- ber of divisions, in this case 10. Fig. .>t3. The mean effective pressure may also be found by m-easuring the area of the diagram, by means of a planimeter and dividing it by the length of the dia- 578 ROGERS' DRAWING AND DESIGN. Fig. 544. gram; multiply this result by the scale oj the indi- cator spring, and the product will be the mean effec- tive pressure. Example: Find the horse power of an engine when L equals 4 ft.; diameter of cylinder equals 32 inches, P equals 40 lbs. per sq. jn. and N equals 40 per minute. H. P. _4oX 4 X 804.25 X40_ 736 nearly. 33000 From the above formula, the proportions of a cylinder may be determined, when the horse power, pressure and nuihber of revolutions per minute are given. L X A ^ HP X 33000 j^^j.^ ^ jg ^^^ ^j.^^ ^f P X N piston, and L the length of stroke. The area may be expressed by the diameter of the piston in the following manner : A = .7854 X D^ ; consequently the above formula may be written L X .7854 X D= = ^^^^^"^ ' ^^ P X N If we make L equal to d, as is often done, we have VTTF from the above formula, d = 79.59 PN -from which we may find the required diameter of cylinder, for an engine which shall have a given horse power, ROGERS' DRAWING AND DESIGN. 379 mean pressure and number of strokes per minute. For the completed thickness of the walls of the steam cylinder, Prof. Reauleaux gives the following formula : Thickness ^^ Vg inch + lOO Example : If the diameter of the cylinder is 48 in., the thickness of its wall should be 48 'A 4- 100 = .125 + .48 = .605 For the thickness of the heads of cylinders of ordinary size, when the heads are not stiffened by radial ribs, the thickness = 0.003 X d X V boiler pressure per sq. in. Example : If the diameter of the cylinder equals 40 inches, and the 1 toiler pressure 150 lb';., then the thickness of head ^0.003 X 40 X 1/150 = 1.46 in. Having found a convenient thickness of the head and flange of the steam cylinder, upon which the head is to rest, the diameter of the bolts which fasten the cylinder head should be one-half the width of the flange. The number of required bolts may be found from the following formula : Number of bolts equals 0.7854 X square of diam- eter of cylinder, multiplied by the boiler pressure and divided by 5,000 times the area of a single bolt of the assumed diameter. Example : Let the diameter of the cylinder be equal to 32 inches ; the boiler pressure 81 lbs. per sq. in., and the assumed diameter of bolt equal to ^ in. The area of a 3/^ in. bolt equals. 0.442 sq. in., then Number of bolts ^-°:1 ^54 X 32 X 32 X 81 _^^^ Fig. 545. The steam chest must be made as small as the dimensions and travel of the valve will permit. It usually has the form of a square bo.x, surrounding the valve face. The steam chest cover, as well as the sides of it, are usually made of the same thick- ness as the cylinder walls. The size of the steam 380 ROGERS' DRAWING AND DESIGN. ports depends upon the quantity of steam which is to be admitted through them, and upon the speed of the piston. It is a good practice to make the area of the ports equal to yV of the piston area when the speed of the piston is about 600 feet per minute. riiickness of piston 4 / length of stroke in in. X a/ diameter of cylinder in in. Example: If the diameter of the cylinder is 30 in. then the required thickness of piston is « 4 Q^ v 30 X 30 = V 900 = 5.4 in. nearly. When the speed of the piston is higher or lower, the size of the ports is increased or diminished pro- portionately. To find the speed of the piston, multiply the length of stroke by double the number of revolutions of crank per minute. A practical formula for the thickness of the piston, shown in Fig. 546, is : The piston rod may be made of wrought iron, or still better of steel. It is generally keyed to the crosshead and fastened to the piston by a strong thread and nut or by wedge. The diameter of a wrought iron piston rod may be found by the following rule : Divide the diameter of the cylinder in inches by ROGERS' DRAWING AND DESIGN. 381 Fig. 547. -ii- Fig. 548. 382 ROGERS' DRAWING AND DESIGN. 60 and multiply the quotient by the square root of the initial steam pressure. Diameter of wrouoJit iron rod diam. of cylinder X V initial pressure. In this formula N equals the length of the con- necting rod divided by the length of the crank. The other letters represent the same values as in former examples in this chapter. Let the total area of the shoe of the crosshead Fig. 549. Fig. 550. Diameter of steel rod diam. of cylinder ^9 X \/ initial pressure. To find the pressure of the crosshead upon the guide, in pounds, the following may be used : 396000 X HP Pressure == V n''— I X L X N be equal to A square inches, and let the pressure allowed upon a sq. in. be equal to p ; then, A pressure of crosshead upon guide ~ P If the pressure of the crosshead be found to equal 6,568 pounds, and if the pressure per sq. in. of slide allowed be equal to 125 lbs., then, 384 ROGERS' DRAWING AND DESIGN. Fig. 552. ROGERS' DRAWING AND DESIGN. 385 ^"^ Fig. oo3. 386 ROGERS' DRAWING AND DESIGN. /^ f Fig. 554. ROGERS' DRAWING AND DESIGN. Bs-; Fig. 555. 388 ROGERS' DRAWING AND DESIGN. the area of shoe 6568 125 52.5 sq. in. If the width of the shoe be taken equal to 4 in., then the length of it will be 13. i in., as 4 X 13. i equals 52.4 sq. in., found above (nearly). The pin in the crosshead which holds the connect- ing rod, is best made with a diameter equal to that of the crank pin. The smallest diameter of the connecting rod is found by dividing the diameter of the cylinder in inches by 55, and multiplying the quotient by the square root of the steam pressure per sq. in. of piston. The greatest diameter is one and one-half times the smallest. The diameter of the crank pin is equal to ^ o */ HP ^ , '^ ^ L X N where 1 is the length of the crank pin journal in inches. This formula is true for a single crank and for one made of wrought iron. The length of the pin may be made equal to 0.013 X d^ where d is the diameter of the piston. To find the diameter of a crank shaft for station- ary engines, with cylinders up to 30 in. in diameter. divide the diameter of the cylinder by 2 ; then sub- tract y^ in., and the remainder will be the diameter of the crank shaft. THE CORLISS ENGINE. The illustrations represent views of the cylinder of the Fishkill Landing Corliss engines. Fig. 551 shows the valve gear of this engine. y^ A Corliss engine has four separate valves, two situated above the axis of the cylinder and intended for the admission and cut off of steam, while the other two are placed below the axis of the cylinder for the exhaust. The steam valves are rigidly con- nected with cranks seen on the outside of the cylin- der. All valves are cylindrical in form and extend across the cylinder above and below, respectively. The cranks on the outside of the valves are operated by a number of links, and in this manner the motion of the valves is actuated. The Corliss valve gear is used in a large number of steam engines. Fig. 551 shows a side elevation of the valve gear, while Fig. 552 exhibits a partial longitudinal section of the cylinder. A cross section being shown in Fig. 553. The cut-off mechanism is shown in detail in Figs. 554 and 555. In Figs. 556, 557 and 558 the crosshead is shown. ROGERS' DRAWING AND DESIGN. 389 f- --"-" — -fi Figs. 556, 557 and 558. 390 ROGERS' DRAWING AND DESIGN. The disc seen in the middle of the cylinder in Fig. 551, called the wrist plate, is made to rock upon the stud in its center by a rod leading from the eccentric on the crank shaft. The wrist plate has four valve connecting rods, which connect it with the bell cranks, which in turn operate the steam valves. These valve connecting: rods can be lenorth- ened or shortened, so that each valve may be set independent of the other three. As the wrist plate rocks backward and forward, the exhaust valves rock with it. The two other bell cranks, which are provided with disengaging links, generally called hooks, are also given a rocking motion by the wrist plate by hooking in the blocks which are rigidly fastened to the cranks on the outer ends of the steam valve stems, thus causing the valves to rotate with them, and causing them to open the steam ports for the admittance of steam. Having turned a certain distance, the disengaging links on the bell crank are unhooked by a cam operated by the governor, and the cranks of the valves are pulled back to their original position by means of the vertical rods from the vacuum air dash pots. The dash pot is a cylinder in which fits a piston nearly air-tight. As the valve is turned it lifts the piston in the dash pot and creates a partial vacuum below it. The atmospheric pressure acts as a weight forcing down the piston into the dash pot and at the same time closing the valves. The air below the piston in the dash pot prevents a sudden shock when the piston drops down. ,As a consequence of this arrangement, the valves, 'are entirely independent in their adjustment and the inlet ports may be suddenly opened full width by the quick movement of the steam valves, while the exhaust valves are nearly at rest. The advantagfes of the Corliss valve g-ear are the large port area, the little friction through the valves, short lengths of ports, quick opening and closing of valves_and easy adjustment. However, the great number of parts makes the expense of these engines greater, their operation noisy, besides which it is impossible to run them at a high speed. Corliss engines do not, as a rule, run higher than 150 revo- lutions per minute. ELECTRICITY, THE DYNAMO AND MOTOR. Electricity is a name derived from the Greek word electron — amber. It was discovered more than 2,000 years ago that amber when rubbed possessed the curious property of attracting light bodies. It was discovered afterwards that this property could be produced in jet by friction, and in A. D. 1600 or thereabouts, that glass, sealing-wax, etc., were also affected by rubbing, producing electricity. Whatever electricity is, it is impossible to say^ but for the present it is convenient to look upon it as a kind of invisible something which pervades all bodies. While the nature of electricity is a mystery, and a constant challenge to the inquirer, many things about it have become known — thus, it is positively assured that electricity never manifests itself except when there is some mechanical disturbance in ordinary matter, and every exhibition of electricity in any of its multitudinous ways may always be traced back to a mass of matter. The great forces of the world are invisible and impalpable ; we cannot grasp or handle them ; and though they are real enough, they have the appearance of being very unreal. Electricity and gravity are as subtle as they are mighty ; they elude the eye and hand of the most skillful philosopher. In view of this, it is well for the average man not to try to fathom, too deeply, the science of either. To take the machines and appliances as they are "on the market," and to acquire the skill to operate them, is the longest step toward the reason for doing it, and zvhy the desired resiilts follow. The design, manufacture and the practical applications of dynamo electric machinery is a theme far beyond the scope of this work, and beyond the limits of many volumes equal to it in size ; suffice it to say, that the subject is as inexhaustible as it /y useful to explore ; it is especially in this, as in other sections of the volume, that the aim of the author has been to suggest the field of work rather than to try to fully explain many things needed to be known. 393 394 ROGERS' DRAWING AND DESIGN. Fig. 559. K ROGERS' DRAWING AND DESIGN. 395 Fia. 560. xn m. n rn r :i I I I J l_l L r^ ni m 396 ROGERS' DRAWING AND DESIGN. Electricity, it is conceded, is without weight, and, while electricity is, without doubt, one and the same thing, it is for convenience sometimes classified according to its motion, as — /. Static electricity, or electricity at j'esc. 2. Citrrent electricity, or electricity in motion. J. Magnetism, or electricity iti rotation, f. Electricity in vibration. Other useful divisions are into — /. Positive and 2. Negative electricity. And into — /. Static, as the opposite of 2. Dynamic electricity. There are still other definitions or divisions which are in every-day use, such as " frictional " electricity, "atmospheric" electricity, "resinous" electricity, "vitreous" electricity, etc. Static electricity. — This is a term employed to define electricity produced by friction. It is properly employed in the sense of a static charge which shows itself by the attraction or repulsion between charged bodies. When static electricity is discharged, it causes more or less of a current, which shows itself Note. — Statics is that branch of mechanics which treats of the forces which keep bodies at rest or in equilibrium. Dynamics treats of bodies in motion. Hence static electricity is electricity at rest. The earth's great store of electricity is at rest or in equilibrium. by the passage of sparks or a brush discharge ; by a peculiar prickling sensation ; by a peculiar smell due to its chemical effects ; by heating the air or other substances in its path and sometimes in other ways. Current electricity. — This may be defined as the quantity of electricity which passes through a conductor in a given time — or, electricity in the act of being discharged, or electricity in motion. An electric current manifests itself by heating the wire or conductor, by causing a magnetic field around the conductor and by causing chemical charges in a liquid through which it may pass. Radiated electricity is electricity in vibration. Where the current oscillates or vibrates back and forth with extreme rapidity, it takes the form of waves which are similar to waves of light. Positive electricity. — This term expresses the condition of the point of an electrified body having the higher energy from which it flows to a lower level. The sign which denotes this phase of electric excitement is + ; all electricity is either positive or, — , negative. Negative electricity. — This is the reverse con- dition lo the above and is expressed by the sign or symbol — . These two terms are used in the same sense as hot and cold. ROGERS' DRAWING AND DESIGN. 3f7 Atmospheric electricity^ is the free electricity of the air which is almost always present in the atmosphere. Its exact cause is unknown. The phenomena of atmospheric electricity are of two kinds ; there are the well-known manifestations of thunder-storms ; and there are the phenomena of continual slight electrification in the air.best observed when the weather is fine ; ihe aurora constitutes a third branch of the subject. Dynamic electricity. — This term is used to define current electricity to distinguish it from static electricity. This is the electricity produced by the dynamo. Frictional electricity is that produced by the friction of one substance against another. Resinous electricity. — This is a term formerly used, in place of negative electricity. This phrase originated in the well-known fact that a certain (negative) kind of electricity was produced by rub- bing rosin. Vitreous electricity is a term, formerly used to describe that kind of electricity (positive) produced by rubbing glass. Magneto-electricity is electricity in the form of currents flowing along wires ; it is electricity derived from the motion of maonets — hence the name. o Voltaic electricity. — This is electricity pro- duced by the action of the voltaic cell or battery. Electricity itself is the same thing, or phase oj efiergy, by whatever source it is produced^ and the foregoing definitions are given only as a matter of convenience. ELECTRO-MOTIVE FORCE. The term is employed to denote that which moves or tends to move electricity from one place to an- other. For brevity it is written E. M. F.; it is the result of the difference of potential, and propor- tional to it. Just as in water pipes, a difference of level produces a pressure, and the pressure pro- duces a flow as soon as the tap is turned on, so dif- ference of potential produces electro-motive force, and electro-motive force sets up a current as soon as a circuit is completed for the electricity to floiv through. Electro-motive force, therefore, may often be con- veniently expressed as a difference of potential, and vice versa ; but the reader must not forget the dis tinction. In ordinary acceptance among engineers and prac- tical workine electricians, electro-motive force is thought of as pressure, and it is measured in units called volts. The usual standard for testing and 398 ROGERS' DRAWING AND DESIGN. comparison is a special form of voltaic cell called the Clark cell. This is made with great care and composed of pure chemicals. The term positive expresses the condition of the point having the higher electric energy or pressure, and, negative, the lower relative condition of the other point, the current is forced through the circuit by the (E. M. F.) electric pressure at the generator, just as a current of steam is impelled through pipes by the generating pressure at the steam boiler. Care must be taken not to confuse electro-motive force with electric force or electric energy ; when matter is moved by a magnet, we speak rightly of magnetic force ; when electricity moves matter, we may speak of electric force. But, E. M. F. is quite a different thing, not " force " at all, for it acts not on matter but on electricity, and tends to move it. THE DYNAMO, OR GENERATOR. The word dynamo, meaning power, is one trans- ferred from the Greek to the English language, hence the primary meaning of the term signifying the electric generator is, the electric power machine. The word generator is derived from a word mean- ing birth-giving, hence also the dynamo is the ma- chine generating or giving birth to electricity. Again, the dynamo is a machine driven by power, generally steam or water power, and converting the mechanical energy expended in driving it, into elec- trical energy of the ciir rent form. Dynamos are classified as — /. Unipolar dynamos. 2. Bi-polar (or 2-pole) dynamos. J. Multipolar dynamos. This division is caused by their different construc- tion, but, whatever their shape or size or peculiarity of application, the principles upon which they work are always the same — a dynamo is always a machine for generating electric currents. It should be understood that an electric dynamo or battery does not generate electricity, for if it were only the quantity of electricity that is desired, there would be no use for machines, as the earth may be regarded as a vast reservoir of electricity, of infinite quantity. But electricity in quantity without pres- sure is useless, as in the case of air or water, we can get no power without pressure. As much air or water must flow into the pump or blower at one end, as flows out at the other. So it is with the dynamo ; for proof that the current is not generated in the machine, we can measure the current flowing out through one wire, and on through the other — it will be found to be precisely the same. ROGERS- DRAWING AND DESIGN. 399 As in mechanics a pressure is needed to produce a current of air, so in electrical phenomena an elec- tro-motive force is necesary to produce a current of electricity. A current in either case can not exist without a pressure to produce it. To summarize, the dynamo-electric generator or the dynamo-electric machine, proper, consists of five principal parts, viz.: 1. The armature or revolving portion. 2. The field magnets, which produce the magnetic field in which the armature turns. 3. The pole-pieces. 4. The commutator or colle( tor. 5. The collecting brushes that rest on the commu- tator cylinder and take off the current of electricity generated by the machine. In brief, the purpose of the dynamo is to change mechamcal motion, applied to the armature, revolving it at high speed, into electrical energy. THE ELECTRIC MOTOR. An electric motor is a machine for converting electrical energy into mechanical energy; in other words it produces mechanical poiver when supplied with an electric current ; a certain amount of energy must be expended in driving it ; the intake of the machine is the term used in defining the energy ex- pended in driving it ; the amount of power it deliv- ers to the machinery is denominated its outpiit. The difference between the output and the intake is the real efficiency o{ the machine ; it is well known that the total efficiency of an electric distribjition system, which may include several machines, usually ranges from 75 to 80 per cent, at full load, and should not under ordinary circumstances fall off more than — say 5 per cent, at one-third to half load ; the efficiency of motors varies with their size, while a one-horse power motor will, perhaps, have an effi- ciency ot 60 per cent, a loo-horse power may easily have an efficiency of 90 per cent. A dynamo, as ordinarily constructed, consists of two parts ; the stationary magnet frame and the re volving part, or the armature. In Figs. 559 and 560 is illustrated a multipolar generator, that is a dynamo having more than two poles, in this case four. The armature is generally the revolving part and to it are secured in various manners numerous loops of wire. The space occupied at the end of the mag- net poles by the armature, is called the magnetic field. The armature revolving in this magnetic field slightly charged by the influence of the magnetism naturally retained in the iron of the magnet frame, drives its numerous conductors through the magnetic 400 ROGERS' DRAWING AND DESIGN. lines, and as each of these conductors, called loops, passes through this magnetic field, it gathers a little amount of electrical energy, or as it is generally' ex- plained, a slight electric current is made to flow through it and on through the commutators and brushes to wires outside of the dynamo ready for service. The larger the number' of loops the greater the electrical energy gathered by the armature. The amount of the electrical energy gathered by the armature conductors is greatest when the magnetism in the magnet is highest. When the armature is first set in motion the elec- tric current in it is very mild as the magnetic influ- ence is also very slight at that time, but as soon as a mild current is produced, it is made to pass through insulated wires wound around the magnet cores, which at once strengthens the magnetism and thus in turn calls forth a greater electrical energy within the armature, which greater energy is again utilized to strengthen the influence of the magnet. In this Note. — There are a great variety of armatures in use ; the drutn armature has'beeu found the most popular on account of its simplicity, and comparative efficiency. One of the types is the ring armature^ whose efficiency is so low, that it never found very extensive favor, although it is very simple and easy to repair. For large machines, the multipolar armature^ is used almost exclu- sively ; still another type, disc armature. manner by increasing the magnetic influence, the electrical energy within the armature conductors is increased up to the desired limit. When electrical energy is supplied to a dynamo the armature will turn with great velocity and force, and thus the machine will transform electrical energy into mechanical motion. In this case the machine is called a motor. The armature in the dynamo shown Fig. 559'is of the drum, type ; it consists of the core, upon Which wires are wound, these wires being connected to the commutator, upon this commutator, the brushes are riding, which gather up the current as it is delivered to the commutator by the armature, whence it is led outside of the machine to the circuit. The armature core is made of sheet iron discs usually about 0.002 inch in thickness. The outer discs being unsupported at the' outer edge, are usually made of sheets of tV thickness. Three of these at each end of the core will be enough to hold the rest of the discs from spreading. These discs are usually made of the best charcoal iron. Between the discs a thin sheet of paper is laid. The circumference of these. discs is provided with apertures of various forms, for holding the ar- mature coils in place of which several types are shown in Figs. 561-564. ROGERS' DRAWING AND DESIGN. 401 Fig. 561. Fio. 562. Fig. 561 shows a disc with round holes to contain the conductors. Fig. 562 shows plain slots with parallel slides. Fig. 563 is a slight modification of the preceding form. In the slots shown in Fig. 564 the conductors are held securely by means of hard- wood strips, which are driven in above them. The disc is punched out of sheet iron, a hole for the shaft and key-way being also cut out. The disc is then placed under a punching press with a revolving table for the disc, which is automatically moved a certain distance between each stroke of the press. Thus all slots are punched. Fig. .563. Fig. 564, Figs. 565 and 566 show a section of the commutator as well as a partial end-view. It consists of the shell, that is the outside casting, placed directly upon the shaft. One end of the shell is provided with a cir- cular projecting lip wedge-shaped in section, to support the segments, while the other end is provided with a thread. A ring, also wedge-shaped in section is placed on the shell near its end, for the purpose of supporting the other end of the segment, and a nut is then screwed upon the threaded end of the shell, pressing the ring toward the segments, and holding them securely. These segments are insu- 402 ROGERS' DRAWING AND DESIGN. lated from each other, as well as from the shell, by strips of mica or fibre, indicated by the heavy black lines in the illustration. The segments are made of hard copper. The prolongation of the segment at the back is called the ear or lug of the segment. The purpose of the lug is to provide a means to se- cure the conductor to the commutator and for this slots are cut in the lugs, just wide enough to accom- modate the wires, which may be secured to it by small screws or by soldering. The shell, ring and Fig. 566. nut are made of bronze. Clearance is allowed between the shell and the segments to secure better insulation, t is not necessary that the shell should bear on the shaft throughout its entire length, and to save boring, it is cored out, as shown in the illus- tration, leaving about an inch at each end of the shell for the support. ROGERS' DRAWING AND DESIGN. 403 Fig. fier Fig. 508. Fig. 5«9. 404 ROGERS' DRAWING AND DESIGN. FlQ. 571. Fig. 570. The commutator shell is secured to the shaft by means of a small key not shown in the illustration. In Figs. 567 and 568 is illustrated a section of a simple self-oiling bearing for the support of the shaft of the dynamo, illustrated in Figs. 559 and 560. Within the bearing box is con- tained the cylindrical brass bushing. The upper part of the bushing is provided with a slot, and here is introduced the oil ring, which rests upon the shaft, dipping into the oil contained in the reservoir below. When the shaft is revolved, the ring takes up oil and carries it to the shaft. Larger bearings are provided with two oil rings. Provision must be made to drain off the oil and to furnish a fresh supply. The upper part of the bearing box is often made with a large opening covered by a hinged lid for the purpose of inspection, as well as for supply of oil. A convenient addition is an oil gauge, which shows the amount of oil in the reservoir. ROGERS' DRAWING AND DESIGN. 405 Fig. 573. Fig. 573. In the same illustration it can be seen that the rear end of the bearing box is turned to receive the brush frame, Fig. 569, which in this case is provided with four holes to receive four brushes. Another kind of brush frame, called a rocker arm, is shown in Fig. 5 70, made for two brushes. The holes for receiving the brush holder studs are often made square. A brush holder stud is shown in Fig. 571 ; it is made of brass and is circular in section. The section shown in black represents the insulating washers and insu- latinQ- bushinofs, made of hard rubber, insulating the brush holder stud from the rocker arm. Outside on the stud is placed a brass washer and a cable luof, which is used to connect it with the main cable or leads carrying the cur- rent to the point of distribution. The cable lug is shown separately in Figs. 572 and 573. A form of brush holder which is rapidly becoming most popular, is shown in Fig. 574 ; it is called the Reaction Brush Holder. Here the brush is wedded in between the brush holder and the commutator without any support on the outer side, the pressure of the curved lever forcing the carbon brush against the inclined face of the holder as well as against the commutator. The pressure of the lever 406 ROGERS' DRAWING AND DESIGN. is caused by a helical spring, terminating in a straight projection, which can be set into any one of Fig. 574. the notches on the lever, thus regulating the pressure of the lever to any desired degree. The magnet frame may be cast in one or more parts, together with the pole pieces, or the pole pieces may be bolted to the frame. The magnet frame must be rigidly secured to the base. The bearings or pedestals may be cast in one piece with the base, or fastened to it by bolts. The magnets may be made of cast iron, wrought iron, or mallea- ble iron, according to the requirements. Two methods of exciting the field are shown in the diagrams in Figs. 575-576. The shunt method of excitation consists of forming a separate circuit of the magnetizing coils which are connected directly between the brushes, or in shunt, to the external circuit. The diagram in Fig. 575 shows the manner in which shunt winding is accomplished. Another method for excitation of the field is the series winding. Here the entire current flowing through the armature is made to flow through the magnetizing coils. A combination of series and shunt winding, gives the compound winding, shown in the diagram, Fig. 576. This winding is very extensively used for generators, but is seldom used for motors, as either a series or a shunt winding serves for almost all con- ditions of operation. ROGERS' DRAWING AND DESIGN. 407 Fk;. .'.7.-,. Fig. 576. INSTRUMENTS AND METHODS OF USE Preceding the section of " Lettering" and beginning at page 41 much valuable matter relating to the "Drawing board, T square and triangles" may be found, with many illustrations; what follows properly belongs with the above section, but is removed to a less important part of the volume because the matter is almost too elementary ; it is inserted here "lest we forget." Good tools are necessary for a proper output of good work but it is not always the man who has the most or the better tools that does the best ; a little observation also shows that every regular draughts- man has his own select tools, gathered as he has progressed in study and practice to suit his own "handy" method of work ; the time comes when the draughtsman declines the employment of any but the regu- lar instruments, relying upon his manual dexterity to execute all necessary drawings. There is an old adage to the effect that " an ounce of showing is worth a pound of telling " ; the kindly assistance of an experienced draughtsman at the beginning of one's efforts is invaluable and worth the fee that migrht be charcred. Euoene C. Peck, M. E., has written an account of the method he employed in teaching a class of the employees of the Cleveland Twist Drill Co.; it is quoted almost in full in the note below : Note. — The method employed was mapped out more with a view to teaching the employees to read drawings than to make draughtsmen of them, but at the same time so that those who cared to follow the profession in the future would be able to use all the information and prac- tice to good advantage ; no originality in plan of teaching was attempted ; the class consisted of twenty pupils who had been through fractions and percentage in arithmetic; some had taken lessons previouslv in drawing, knew the use of different instruments and understood the ordi- nary geometrical problems occurring in drawing, while others were without any such previously acquired knowledge. As very little drawing could be done in one evening in a class they were instructed to do all drawing at home. Each pupil was furnished with a blueprint of instructions such as would be needed outside of class, and also a plate (blueprint) to copy from. These plates were drawn, then blueprinted, but to a scale of about lo inches to the foot, so that no copying by dividers could be done. The first four contain the ordinary geometrical problems, the next four projection, cylindrical and conical intersections and developments ; then came the simple machine parts to teach the correct placing of views, shading, etc. From this on the plates gradually get more intricate and complicated, but in all cases are taken from our own shop drawings or a machine in the factory, and more especially is a drawing of a jig or fixture used which may have given any trouble to the machinist to read. These drawings are then made at home and left in the drawing office, where they are corrected and marked, a record of the progress of the student being kept for reference. Later they were given a little algebra in the shape of simple formulas which, by the way, gave most of them some trouble until they got to handle the characters as though they had no value, or to treat them by the rules regardless of their value. A short course in the practical laying out and working of gear problems came next, which gave very little trouble, as most of the students were more or less familiar with the subject. This year's course in class closed with logarithms, and considering that I left the theory of exponents out of the question, taught them only the use of the tables and gave them rules to solve the different examples by, they handled the subject remarkably well. 413 41J: ROGERS' DRAWING AND DESiGN. COMPASSES. Compasses are instruments for describing cir- cles, measuring figures, etc.; Figs. 577—580 show a pair of compasses, a pencil, a lengthening bar and a pen point, either of which may be inserted into a socket in one leg of the instrument when a circle in pencil or in ink is to be drawn. The other leg is fitted with a needle point and acts as the cen- ter about which the circle is to be described. The compasses shown in Fig. 577 have a single socket only; the leg with the needle-like point is called a divider point ; the other leg has a stationary needle point which is placed in the center of the circle to be drawn : it will be noticed that one leg of the compasses is jointed; this is done, so xhsiX. the com- pass points may be kept perpendicular to the paper when drawing circles. Note. — The student should learn to open and close the compasses with one hand ; those provided with a cylindrical handle at the head are to be held gently between the thumb and the forefinger and those minus the handle should be held with the needle point leg resting between the thumb and fourth finger, and the other leg between the middle and forefinger. Only one hand should be used in locating the needle point at a point on the drawing about which the circle is to be drawn, unless the left hand merely serves to steady the needle point. Having placed the needle point at the desired point, and with it still resting on it, the pen or pencil may be moved in or out to any desired radius. When the lengthen ng bar is used both hands must be em- ployed. Fig. 578. Fig. 579. Fr;. ,577. Fig. 580. The joint at the head of the compasses should hold the legs firmly in any desired position and at the same time should per- mit their being opened and closed with one hand ; the joint may be tightened or loosened by means of a screwdriver or spanner, which is furnished with the instru- NoTE. — The needle point itself, in all good instruments, is a separ- ate piece of round steel wire, placed in a socket provided at the end of the leg. The wire, as a rule, has a shoulder at its lower end, below which a fine, needle-like point projects. ROGERS' DRAWING AND DESIGN. 415 ment ; compasses should not be used for circles of too large a radius, not allowing the points to be placed at a right angle with the paper. A lengthening bar. Fig. 580, is used to extend the leg carrying the pen or pencil points, as the case may be, when circles of a large radius are to be de- scribed. Circles should be drawn with a continuous motion, with an even, slight pressure on the pen ; when inking in a circle it is well to stop exactly at the end of a single revolu- tion, as the line may become uneven when going over it a second time ; when closed, the needle point and the pen or pencil point are to be set in such a manner as to be even. DIVIDERS. Dividers are used for laying off distances upon a drawing, or for dividing straight lines or circles into parts ; an instrument of this kind is shown in Fig. 581 ; the points should be thin and sharp, so that they will not puncture holes in the paper larger than is absolutely neces- sary ; when using the dividers to space a line or circle into a number of equal parts, they should be held at the top between the thumb and fore- no. 581. finger, as when using compasses ; to mark off the spaces, the instrument should be turned alternately to the right and left. The divider shown in Fig. 581 is provided with a hair spring attachment, which enables the user to make quite ^ne adjustments ; in this case one leg is made separate from the main body of the instru- ment with its upper end terminating in a spring, which tends to bring one leg toward the other. The legs of dividers, as well as those of compasses, are made triangular in section, except near the point, where the corners are ground off sufficiently to make a round point. If the point should be left triangular, the holes punctured into the paper, would be bored out to such an extent, while turning the instrument, that accurate measurements would be impossible. It being essential that the points of dividers be kept in good condition, they should never be used for anything else, except the purpose they are made for. The joint ai the head of the dividers should be kept not too tight, for unless there is a hair- spring attachment, as described above, it will be dif- ficult to adjust the dividers accurately, owing to the spring in the legs. Lost motion in the head joint is also a very objectionable feature, and should be attended to as soon as detected. 416 ROGERS' DRAWING AND DESIGN. BOW PENCIL AND BOW PEN. A bow pencil a,ndi aubowpen are shown in Figs.583 and 584; these instruments are made for describing small Fig. 582. Fio. 583. Fig. 584. circles. The two points should be adjusted evenly, that is they should be of the same length, otherwise, very small circles cannot be described. To open or close either of the above mentioned instruments; Fig. 585. Fig. i support it in a vertical position by resting the needle point on the paper, and pressing slightly at the top with the forefinger of one hand, and turn the adjusting screw or nut with the thumb of the same hand. Bow di- viders for measuring very small dis- tances are also largely in use, see Fig. 582. DRAWING PENS. , For drawing ink lines other than arcs and circles, the drawing pen or ruling pen, is used, Figs. 585 and 586; these consist of two thin steel blades, attached to a handle made of wood, ivory or light metal ; the points are made of two steel blades which open and close, as required for thickness of lines, by a regulating thumb-screw. When using the ruling pen it should be held as nearly perpen- dicular as possible, the hand bearing slightly on the tee square or the tri- angle against which the line is drawn. The pen m-ust not be pressed against the edge of the tee square or triangle ROGERS' DRAWING AND DESIGN. 417 as the blades will then close together, thus making the line uneven. The edge should only serve as a guide ; the pen should be held with the thumb- -screw on the outside. BEAM COMPASSES. For describing very large circles beam compasses are used ; these compasses are shown in Fig. 587, with a portion of the wooden rod or beam on which they are used. At A, Fig. 587, is shown a section of the beam, which has the shape of a letter T. This form has considerable strength and rigidity. Beam com- passes, as shown in Fig. 587, are provided with extra points for pencil or ink work. While the generil a Ijustment is effected by means of the clamp against the wood, minute variations are made by the screw B, shifting one of the points. Note. — A good drawing pen should be made of properly tempered steel, neither too soft nor hardened to brittleness The nibs shoxUd be accurately set, both of the same length, and both equally firm when in contact with the drawing paper. The points .should be so shaped that they are fine enough to admit absolute control of the contact of the pen in starting and ending lines, but otherwise as broad and rounded as possible, in order to hold a convenient quantity of ink without drop- ping it. The lower (under) blade should be suificienth- firm to prevent the closinp^ of the blades of the pen, when using the pen against a straight-edge. The spring of the pen which separates the two blades should be strong enough to hold the upper blade in position, but not so strong that it will interfere with easy adjustment of the thumb-screw ; the thread of the thumb-screw should be deeply and evenly cut so as not to strip. This instrument is quite delicate, and, when in good order, is very accurate. It should be used only for fine work on paper, and never for scribing in metal. Flu. 387. DRAWING INK. Liquid India ink can be procured in bottles with glass tube feeders, as in Figs. 588 and 589, or with a quill attached to the cork, by means of which the pen may be filled by drawing it through the blades; a common writing pen may a'so be used for filling the pen in the same manner as described for the glass feeder or quill. Dry ink of good quality however in sticks, Figs. 590-593, cannot be surpassed, although it requires 418 ROGERS' DRAWING AND DESIGN. skill in its preparation. In case the stick ink is used put enough clean, filtered or distilled water in a shal- low dish or "tile" for making enough ink for the drawing in hand ; place one end of the stick in the water, and grind by giving the stick a circular motion. Do not bear hard upon the stick. Test the ink occasionally to see whether it is black. Draw a fine line with a pen and hold the paper in a strong light. If it shows brown or gray grind a while longer and test again. Keep grinding until a fine Fig. ] line shows black ; the time required to obtain the desired result depends entirely on the amount of water used. The ink should be kept free of dust and prevented from evaporating by covering it with a flat plate of some kind. Fig. 589. Figs. 590-593. Note. — If stick ink is used it is very good policy to buy a stick of the very best quality, costing, say about a dollar, as, perhaps, it will last longer than several dollars' worth of liquid ink. The only reason for using liquid ink is that all lines are then sure to be of the same blackness, and time is saved in grinding. ROGERS' DRAWING AND DESIGN. 419 When trouble is caused by the ink drying between the blades and refusing to flow, especially when drawing fine lines, the only remedy is to wipe out the pen with a cloth. Do not lay the pen down for any great length of time when it contains ink ; wipe it out first. The ink may sometimes be started by moistening the end of the finger and touching it to RULES AND SCALES. The rule is used for measuring and comparing dimensions; they are divided in inches, halves, quar- ters, eighths, sixteenths, and thirty-seconds. For some purposes the rules as explained above cannot be used, as i. e., for making drawings smaller 1 II II Tip 2963 ° 6 ' ^ 5 3 4- .4 3 6 3 7 8 2 9 10 1 11 12 llllll 3 s 9 ^ 3/jlNCH HON C ?<^INCH v_ 2 t It 01 6 ill ill ill III III ill 8 /. 9 llllllllllll llllll III llllll lllllllll 3 L ill llllllllllll ill! t 9 iliilillillilllili 6 lllllllll lllllllll! I. Fig. 594. the point of the pen, or by drawing a slip of paper between the ends of the blades. Before using the pen it is well to try it first on a piece of paper to make sure that it will produce lines of the required thickness ; the border of the sheet of paper on the drawing board may be used for this purpose, according to long established custom. or larger than the actual size of the object to be drawn. Scales are then employed as shown in Figs. 594 and 595. \ mv^ Fig. 595. 420 ROGERS' DRAWING AND DESIGN. The most convenient forms are the usual flat or triangular boxwood scales, having beveled edges, each of which is graduated for a distance of twelve (12) inches. These beveled edges serve to bring the lines of division close to the paper when the scale is flat, so that the drawing may be accurately measured, or distances laid off correctly. A very convenient form of scales is shown in Fig. 595. It represents a triangztlar scale (broken). It reads on its different edges as follows : O (a) 3 inches and 13^ inches to one foot, I inch and 3^ inch to one foot, (b) Y^ inch and 2/^ inch to one foot, i^ inch and y% inch to one foot and (c) one edge reads sixteenths the whole 12 inches of its length. PROTRACTOR. A protractor x?, shown in Fig. 596 ; it is an instru- ment for laying off or measuring angles on paper, or for dividing a circle into an equal number of parts ; it is also used in connection with a scale to define the inclination of one line to another. The outer edge of the protractor is a semicircle, with center at O and, for convenience, is divided into 180 equal parts or degrees from A to B and from B to A. Protractors are often made of metal, in which case the central part is cut away to allow the drawing under it to be seen. When using the protractor it must be placed so that the line O B, Fig. 596, will coincide with the line forming one side of the an- gle to be laid off or measured, and the center O must be at the vertex of the angle. IRREGULAR CURVES AND SWEEPS. Curves are irregular lines ; a circle is a resfu- lar line. Curves other than arcs of circles are drawn with the pencil or ruling pen by means of curved or irregular-shaped rulers, called irregular curves or szueeps, Figs. 597-608. These are made of various materials, wood, hard rubber or celluloid, in a great variety of shapes. A certain number of points Note. — A whole circle contains 360 degrees, a right angle contains 90 degrees and therefore as many as a. % ol a circle. A 45 degree angle contains as many degrees as j^ of a circle. ROGERS- DRAWING AND DESIGN. 421 Figs. 597 to BOB. 422 ROGERS' DRAWING AND DESIGN. is first determined througii which the line is to pass, and said Hne should be first sketched in lightly, freehand. The irregular curve is then applied to the curved line so as to embrace as many points as possible ; only the central points of those thus em- braced should be inked in; this process is continued until the desired curve is completed. It is very difficult to draw a smooth continuous curve. In order to avoid making the line curve out too much between the points or to cause it to change its direction abruptly where the different points join, the irregular curve should be fitted so as to pass through three points at least, and, when moving it to a new position, by setting it so that it will coincide with part of the line already drawn. When neatly penciled over, after having been sketched in free-hand, little difficulty will be experienced to ink it, the pencil line showing the direction in which the curve is to be drawn. When inking with the irregular curve, the blades of the pen should be kept against it and the thumb- screw on the outside ; the inside flat surface of the blades must have the same direction as the curve at Fig. 609. Fig. 610. the point where the pen touches the paper. It will be readily understood therefore that the direction of the pen must be continually changed, PENCILS. Drawings are generally made in pencil and then inked. A hard pe^icil'x?, best for mechanical drawing. The pencil should be sharpened as shoWn in Figs. 609 and 610. Cut the wood away, about Y^ or 3/^ of an inch of the lead projecting ; then sharpen it flat by rubbing it against a fine file or apiece of fine emery cloth or sandpaper that has been fastened to a flat stick. Grind it wedge-shaped as shown in the figure. If sharpened to a round point, the point will wear off quickly and make broad lines, thus making it very difficult to draw a line exactly through a point. The pencil for the compasses should be sharpened in the same manner but should have a narrower width. The pencil line should be made as light as pos- sible ; pressing the pencil too hard will often cut the paper or leave a deep mark which cannot be erased. The presence of too much lead on the surface of the paper tends to prevent the ink passing to the paper and in rubbing out pencil lines the ink is reduced in blackness and the surface of the paper is rough- ROGERS' DRAWING AND DESIGN. 423 ened, which is a disadvantage. As little erasing or rubbing out as possible should be done. Lines are drawn with the flat side of the lead pressed lightly against the straight-edge, as close to it as possible, the pencil being held almost vertically. DRAWING PAPER. The first thine to be considered in select'mir draza- ing paper is the kind most suitable for the proposed plan. The qualities that constitute good paper are strength, uniformity of thickness and surface, neither repelling nor absorbing liquids, admitting of consid- erable erasing without destroying the surfaces, not becoming brittle nor discolored by reasonable ex- posure or age, and not buckling when stretched or when ink or color is applied. The sizes and names of commercial drawing paper made in sheets is as follows : Cap 13x17 ins. Demy 1 5x20 Medium 17x22 Royal 1 9x24 Super Royal 19x27 Imperial 22x30 Atlas 26x34 Double Elephant 27x40 Antiquarian 30x53 For large drawings paper is made in rolls. " Detail paper" is especially made for marking out new designs ; it is made in rolls 36, 42, 44, 48 and 54 inches wide ; the size of detail drawings for shop use, of course, are dependent upon the type of the drawing, the size of the parts detailed and the scale to which they are drawn ; the following sizes are good average ones as they can be cut very economic- ally from the rolls sold in print shops : 6 x 9, 9 x 1 2, 12 X 18, i8x 24, 24 X 36, 36 X 48 and 48 x 72 inches. PREPARING FOR WORK. The paper is first secured to the drawing board by means of thumb tacks, one at each corner of the sheet. It should be stretched flat and smooth ; to obtain this result proceed as follows : press a thumb- tack through one of the corners about y^ inch or ^ inch from the edge. Place the tee square in position Note. — Border lines such as are used throughout the pages of this book are frequently of considerable service to the draughtsman but they must be used with a sense of "the fitness of things." Thus: border lines are out of place in working drawings, etc., but where a set of drawings are to be inspected and important contracts decided upon by non-technical business men or capitalists, a neat border line is often the one thing that attracts attention, to the advantage of the exhibitor of the plans and specifications used in the competition bids. The Patent Office rules also call for a border line. The size of the sheet of pure white paper on which a drawing is made must be exactly lox 15 inches. One inch from its edge a single marginal line is to be drawn leaving " the sight " precisely 8x 13 inches. 424 ROGERS' DRAWING AND DESIGN. as in drawingf a horizontal line, and straig^hten the paper so that its upper edge will be paralUl to the edge of the tee square blade. Pull the corner diag- onally opposite that in which the thumb-tack was placed, so as to stretch the paper slightly and push in another thumb-tack. Proceed in the same man- ner for the remaining two corners. The thumb-tacks or drawing-pins should have a head as thin as possible without cutting at its edgesj slightly concave on the underside next to the paper, and should be only so much convex on its upper side as will give it sufficient thickness to enable the pin to be secured to it; it is better to use four or more small pins along the edge of a sheet of paper than use one much larger pin at each corner. For particular work it is necessary to stretch the paper while it is damp. For stretching the paper in this way moisten the whole sheet on the under side, with the exception of a margin all around the sheet, of about half an inch and paste the dry border to the drawing board. To do this properly requires a certain amount of skill, and paper thus stretched gives undoubtedly a smoother surface than can be obtained when using thumb-tacks, but there are objections to this process as the paper stretched in this way is under a certain strain and may have some effect on the various dimensions of the drawing, when cut off the board. Once the drawing completed, cut the paper from the board with a knife, by following the lines previ- ously drawn all around the sheet for trimming. Make a continuous cut all around ; if one of the longer sides is cut first and then the opposite side there is danger of tearing the paper when cutting the remaining sides. PENCILING. The pencil drawing should look as nearly like the ink drawing as possible. A good draughtsman leaves his work in such a state that any competent person can without difficulty ink in what he has drawn. The pencil should always be drawn, not pushed. Lines are generally drawn from left to right and from the bottom to the top or upwards. Pencil lines should not be any longer than the proposed ink lines. By keeping a drawing in a neat, clean condi- tion when penciling, the use of the rubber upon the finished inked drawing will be greatly diminished. INKING. A drawing should be inked in only after the pen- ciling is entirely completed. Always begin at the top of the paper,first inking in all small circles and curves, then the larger circles and curves, next all horizontal lines, commencing again at the top of the drawing II ROGERS' DRAWING AND DESIGN. 425 and working downward. Then ink in all vertical lines, startinof on the left and moving- toward the right ; finally draw all oblique lines. Irregular curves, small circles and arcs are inked in first, because it is easier to draw a straight line up to a curve than it is to take a curve up to a straight line. DRAWING TO SCALE. The meaning of this is, that the drawing when done bears a definite proportion to the full size of the particular part, or, in other words, is precisely the same as it would appear if viewed through a diminishinof orlass. When it is required to make a dravving to a re- duced scale, that is, of a smaller size than the actual size of the object, say for instance, J^ full size, every dimension of the object in the drawing must be one- half the actual size ; in this case one inch on the object would be represented by 3^ inch. Such a reduced drawing could be made with an ordinary rule, this, however, would require every size of the object to be divided by the proportion of the scale, which would entail a very great loss of time in cal- culations. This can be avoided by simply dividing the rule itself by 2, from the beginning. Such a rule, or scale as it is generally called, will be divided in yi inches, each half inch representing one full inch divided into yi, j^, yi, ■^, each of these representing the same proportions of the actual sizes of the object to be drawn. From this contracted scale the dimen- sions and measurements are laid off on the drawing. A quarter size scale is made by taking three inches to represent one foot. Each of the three inches will be divided into 12 parts representing inches, each one of these again will be divided in i^, y%, ■j?^, etc.; each one of these representing to a quarter size scale the actual sizes of ^, %,yi,~^ of an inch. It must be mentioned that in several instances, in this work, distances in one figure are said to be equal to corresponding distances in the same object in another view, while by actual measurement they are somewhat different; this is owing to the use of different scales — each scale separate should be marked on the drawing. Paper scales for large drawings are extremely use- ful and remarkably accurate. The advantage they possess over other kinds is that they expand and contract equally with the drawing paper during the various changes of the weather. The nickel-plated sheet-metal steel scale which has two graduated edges conduces to most accurate work; this instrument having only two scales the annoyance experienced of frequently turning it, is greatly reduced. 426 ROGERS' DRAWING AND DESIGN. A fiat boxwood scale with beveled edges has less pitch on its side and for that reason can be more quickly and easily read than others. SELECTION OF INSTRUMENTS. The choice of drawing tools is one of the most difficult points to settle that can present itself. Suc- cess or failure may hang upon the getting the most suitable tools, hence it is well to follow the advice of some professional draughtsman, and in buying, procure such tools as are immediately needed and to add others as occasion demands. The best quality of instruments last longer and in the end are the cheapest. German silver is the best metal used, much better than brass ; the use of pocket or folding instruments is to be avoided ; if it is necessary to carry the instruments nothing is better than to fold them up in a piece of chamois leather, or to have a little satchel or grip which will also accommodate the triangles, ink, colors, etc. Louis Rouillon, B. S., Instructor of Drawing in Pratt Institute, New York, recommends the follow- ing set of tools for the beginner : Compasses, 5^ inches, with needle point; pen, pencil and lengthening bar. Drawing pen, 4^ inches. T square, 24-inch blade. 45-degree triangle, 9 inches. 30 and 60-degree triangle, g inches. I Scroll. Dixon's V. H. pencil. 12-inch boxwood scale, flat, graduated 1-16 inch the entire length. Bottle of liquid India ink, four thumb-tacks, pen and ink eraser. 20 sheets drawing paper, 11 X 15 inches, and a drawing-board about 16 x 23 inches will also be necessary. Henry Raabe, M. E., is entitled to credit for the following list of instruments : I Pair of compasses, with pencil, pen, needle point, and lengthening bar ; i Pair of dividers ; i Draw- ing pen ; 1 Bow pen ; i Bow pencil ; i Bow divid- ers ; I 45-degree triangle ; i 60-degree by 30-degree triangle ; i Tee square ; i Drawing board ; i Pro- tractor; I Scale from 1" to the foot to /^ " to the foot ; I Scale from 3" to the foot to ^" to the foot; I Pencil rubber; i Ink eraser; i Pen holder with pens ; i Pencil holder for short pencils ; Compass pencils ; Pencils from 6 H. to 3 H. (drawing pen- cils) ; Pencil pointer ; Drawing ink ; Sketch pads ; Sketch pencils (soft); Thumb tacks, paper and trac- ing cloth. PRACTICAL RULES AND USEFUL DATA, For mental drill there is nothing better than the solution of mathematical problems. It is not necessary that these problems be intricate and in the higher branches, but only not so easy as to be readily understood without active and sustained brain work. Accuracy, first of all, rapidity and a familiarity with the elements of numbers and their application to the problems immediately surrounding one, — these are the foundations of many successful lives ; to most minds the study of mathematics is dry and uninteresting ; to make the subject acceptable it must be presented in such a form as to immediately appeal to the student as of great practical value. This value is proven when applications are made to problems that confront the draughtsman and engineer in his daily routine. There is no more interesting subject for one who is disposed to study than that of useful numbers. It literally opens a new world to the student. It gives him his first idea of what it means to really /rciz;^ anything, for the demonstrations of figures and geometry prove absolutely and completely the propositions with which they deal. "In the wide expanse of mathematics it has been a task of the utmost difficulty for the author to lay out a road that would not too soon weary or discourage the student ; if he had his wish he would gladly advance step by step with his pupil, and much better explain, byword and gesture and emphasis, the great principles which underlie the operations of mechanics ; to do this would be impossible, so he writes his admonition ia two short words: In case of obstacles, 'go on.' If some rule or process seetUS too hard to learn, go around the difficulty, always advancing, and, in time, retu7'7z and conquer." The foregoing paragraphs are simply to emphasize a few words explaining the value of the tallies which are printed in the following pages ; tables of the results of mathematical calculations are of immense economy in time, in guaranteeing accuracy and the saving of much drudgery To thoroughly understand the easy and helpful uspof the tables which follow should be the pleasant task of the student ; the value of a teacher or instructor at this point cannot be over estimated ; men are not made to do their work alone, to help and to be helped is the universal law ; when assistance is to be had whether it is for pay or favor the student should avail himself of it with many thanks. 429 430 ROGERS' DRAWING AND DESIGN. ELEMENTS OF ALGEBRA. Algebra Is a mathematical science which teaches the art of making calculations by letters and signs instead of figures. The name comes from two Arabic words, algabron, reduction of parts to a whole. The letters and signs are called Symbols. Quantities in algebra are expressed by letters, or by a combination of letters and figures ; as a, b, c, 2.^, Zy, S'S, etc. The first letters of the alphabet are used to ex- press known quantities ; the last letters, those which are unknown. The Letters employed have no fixed numerical value of themselves. Any letter may represent any number, and the same letter may represent different numbers, but in each sum the same letter must always stand for the same amount. The operations to be performed are expressed by the same signs as in Arithmetic; thus + means Addition, — expresses Subtraction, and X stands for Multiplication. Thus « + /; denotes the sum of a and b and is read a plus b ; a — b means a less b; and aV^b shows that a and b are to be multiplied together. Multiplication is also denoted by a period between the factors as a.(5. But the multiplication of letters is more commonly expressed by writing them to- gether, the signs being omitted. Example : 7 abc is the same as 'j'XaXbXc. The sign of Division is -^, thus a^k' is read a divided by b ; but this is also expressed -r-/ the sign of Equality is two short horizontal lines as a^^=b and is read a equals b. The Parenthesis ( ) or Vinculum , indicates that the included quantities are taken collectively or as one quantity. Example : 3 {a-\-S) and 3«+<5 each denote that the sum of a and b is multiplied by 3. The character . " . denotes hence, therefore. A Coefficient is a number or letter prefixed to a quantity, to show how many times the quantity is to be taken. Hence a coefficient is a multiplier or factor ; thus in 5«, 5 is a numeral coefficient of a. When no numeral coefficient is expressed, i is always understood. Thus xy means \xy. ROGERS' DRAWING AND DESIGN. 431 DEFINITIONS. An Algebraic Operation is combining quantities according to the principles of algebra. A Theorem is a statement of a principle to be proved. A Problem is something proposed to be done, as a question to be solved. The Expression of Equality between two quanti- ties is called an Equation. An Algebraic Expression is any quantity expressed in algebraic language, as yi, ^a — ']a, etc. The Terms of an algebraic expression are those parts which are connected by the signs + and — . Thus in aArb there are two terms ; \nx, y and z — a there are three. A Positive Quantity is one that is to be added and has the sign + prefixed to it, as 4^ + 33. A Negative Quantity is one that is to be subtracted and has the sign — prefixed to it, as \a — 3^. A Simple Quantity is a single letter, or several letters written together without the sign + or — , as a, ab, 3Xji'. A Compound Quantity is two or more simple quantities connected by the sign + or — , as 2,a-\- i,b, ix — y. The Axioms in algebra are self-evident truths as exemplified on pages 85 and 86. ADVANTAGES OF ALGEBRA. In algebra numbers are expressed by the letters of the alphabet ; the advantage of the substitu- tion is that we are enabled to pursue our investiga- tions without being embarrassed by the necessity of performing arithmetical operations at every step. Thus, if a given number be represented by the letter a, we know that 2a will represent twice that number, and ^^ the half of that number, whatever the value of a may be. In like manner if a be taken from a there will be nothing left and this result will equally hold whether a be 5, or 7, or 1000, or any other number whatever. By the aid of algebra, therefore, we are enabled to analyze and determine the abstract properties of numbers, and we are also enabled to resolve many questions that by simple arithmetic would either be difficult or impossible. A draughtsman or engineer has but little practical use for a too extended acquaintance with algebra, as nearly all the algebraic rules have been transferred to ordinary arithmetical computation, but as the algebraic system is so inwoven into the school and college course of instruction it is well for every one to know something- of the elements of the science. Arithmeticians for very many years have made a study of the use of formulce (this is Latin for the 432 ROGERS' DRAWING AND DESIGN. word forms) in stating problems and rules ; these forms are nearly all expressed in algebraic terms, The advantage to be derived from the use of these is that it puts into a short space what otherwise might necessitate the use of a long verbal or written explanation. Another advantage is that the memory retains the form of the expression much easier ?.nd longer than the longer method of expression, and it may be re- marked that those who once become accustomed to the use of formulae seldom abandon their employment. Examples Explaining the Solvng of Formul.k 1. \{ X ^ a + b — ^ + d — f ; what must be the value of X when «= lo, <5=7, ^==9, «?^4, and /=6? First substitute the figures for the letters, thus : — A' = 10 + 7 — 9+4 — 6, then proceed as in the Arithmetical part. X ^ 21 — 15 = 6 Answer. 2. If .r == 4 g-\- 2 m — 7 11 — p + 3 ^ / find the = 6 ; /> == I ; and J ' value of X when ^= 5 ; m- Here 4_^= 4 times 5 ^ 20 ; 2 m -^ twice 3^6; ■J n= y times 6 == 42 ; and 2) ^ '^ i times 8 = 24; Hence, ^ = 20 + 6 — 42 — i +24 = 50—43 = 7 Answer. U X- %d+\c- Va./ ; find the value of .;ir when a = 10 ; d=2a^; ^^25; and/"^ 12. As «= 10, then ^ «=^5 ; as d=^ 24, then y^d=^ 6 ; as ^ = 25, then \ c =5 ; and as f^^ 1 2, then ^ Hence, .1-^ 5 — 6 + 5 — 9. = 10 — 15. = — 5 Answer. 4. If X = c — (I — f) ; find the value of jirwhen ^^^8. -y = 3/^ and / ^ I ^ x^^ — (^ — i^); here 3!/^ is divided by 2= i^. = 8-(i3^-i>^) = 8-i< = 7^ Answer. ^. X ^ a b + c d — e f ; where « ^ 2, <5 = 3, c = 4, d = s, e = 6, and/= 7. ;t: = 2X3 + 4X5 — 6X7 = 6 + 20 — 42 = 26 — 42 = — 16 Answer. AB 6. Then {{ x = j-. ^ ; what is the value of x when A^6; B = 7; C=io; and D = 16 ? 6X7 42 x^ -? = ~7- = 7 Answer. 16 — 10 6 ' ROGERS' DRAWING AND DESIGN. 438 LOGARITHMS. This word is composed of two Greek nouns' meaning reason and mimber ; a logarithm is an artificial number so related to the natural numbers that the multiplication and division of the latter may be performed by addition and subtraction and by their use the much more difificult operations of raising to powers and the extraction of roots are effected by easy cases of multiplication and division. The early computers of logarithms carried them to ten places of decimals, but it was soon found that five and seven places were sufficient for most purposes ; those given in this book are carried to six places. Naperian logarithms are called natural and also Hyperbolic logarithms ; common logarithms are called the decimal, and also the Briggsian System. In the Table, letter N over the first column stands for "number" ; after loo (see page 436) the num- bers at the top of the columns express the tenth parts of N. Note. — Logarithms were invented and a table published in 1614 by John Napier, of Scotland ; but the kind now chiefly in use were pro- posed by his contemporary Henry Briggs, of London. The first extended table of common logarithms were calculated by Adrian Vlacq in 1628, and have been the basis of every one since published ; when logarithms are spoken of without any qualifications common logarithms are to be understood. The labor of the operation incurred in the ordinary pro- cesses of arithmetic is often enormous ; by the use of logarithms this labor is greatly lessened ; logarithms are of inestimable value in the so- called higher mathematics, in navigation, in surveying, and in the inves- tigation of many problems in physics. LOGARITHMIC TABLE. When the engineer or draughtsman is required to make long and difficult calculations, consisting of the multiplication, division, squaring, etc., of num- bers, the logarithmic table will, as explained in the note, be of such assistance as may amply repay the study of the subject and the acquirement of rapid and accurate use of the table. It must be understood that but an outline only of this interesting study is here presented and that the columns of figures given in the tables beginning on page 435 are but a very small part of those published in advanced works on mathematics ; hence the ex- amples given of the use of the table are necessarily confined to very small numbers. To use the table, fi.nd the number in the first column marked N, and in the next column the corre- sponding logarithm, will be found. The figures given in the column are only the decimal part of the logarithm. The rules and ex- amples for the application of logarithms are as fol- lows : Rule : To m.ultiply tzvo numbers, add their log- arithms, a7id the result will be the logarithm of the product. 434 ROGERS' DRAWING AND DESIGN. Example: Mul tiply 25 by 6 log. 25 log. 6 = I -397940 ■778151 2. 1 7609 1 Proof: 25 6 log. 150 • 50 Rule : To divide one number by another one, sub- tract tlie logarithm of the divisor from the logarithm of the dividend. Example : Divide 1 75 by 7 log. 175 =" 2,243038 log. 7 = -845098 1.397940-= log. 25 Proof : 175 -^ 7 ^= 25 Rule . To find aiiy power of any number, multi- ply the index of the power with the logarithm of the number ; the product is the logarithm of the power. Example : Find the value of 3* log. 3 = .477121 4 Proof : 3 X 1.908484 = log. 81 9X3 = 27 X 3 F-XAMPLE : Find the value of 14^ log. 14 = 1. 146 1 28 2.292256 = log. 196 Rule : To find any root of a number, divide the logarithm of the num,ber by the index of the root. Example : Find the value of V64 log. 64= 1.806 1 80 -4- 2 =.903090)'== log. 8. Example: Find the value of V 128 log. 128 = 2.107210 -H 7 = .301030 = log. 2. To find the charactistic or whole number to be placed before the mantissa, or decimal part of the logarithm, proceed as follows : Rule : // the nmnber is between i and 10 the logarithm is only a fraction. The logarithm 0/ JO is I, between 10 and 100, a i has to be placed in front of the fractional part found in the table ; be- tween J 00 and 1,000, a 2 forms the whole number ; between 1,000 and 10,000 the figure is j, and so on. Example: What is the logarithm of 123? — by looking in the table we find log. 123 = ,089905 and placing a 2 in front of the decimal point, we have for the true log. of 123 ^ 2.089905. ROGERS' DRAWING AND DESIGN. 435 TABLE OF LOGA- RITHMS. There are two different tables of logarithms in use, one is called the Napierian system, named after its in- ventor, and the common system of which the base' is lo; the accompanying ta- bles are common loga- rithms. The logarithm of a num- ber usually consists of two parts, the integral, or whole part, and a fractional part ; the integral part is called the characteristic or index, and the fractional part the mantissa. The last word is from the Latin and means an addition. The abbrviviation of the words "logarithm of" is log. or Log., thus: log. 136 = 2-133539, the characteristic of the logarithm i ^6 beine 2, and the mantissa .133539. See Table, page 437. I n the tables the mantissas only are given. N Log. N Log. N Log. N Log. 1 000000 26 414973 51 707570 76 880814 2 301030 27 431364 52 716003 77 886491 3 477121 28 447158 53 724276 78 892095 4 602060 29 462398 54 732394 79 897627 5 698970 ^30 477121 55 740363 80 903090 6 778151 31 491362 56 748188 81 908485 7 845098 32 505150 57 755875 82 913814 8 903090 33 518514 58 763428 83 919078 9 954243 34 531479 59 770852 84 924279 10 000000 35 544068 60 778151 85 929419 11 041393 36 556303 61 785330 86 934498 12 079181 37 568202 62 792392 87 939519 13 113943 38 579784 63 799341 88 944483 14 146128 39 591065 64 806180 89 949390 15 176091 40 602060 65 812913 90 954243 16 204120 41 612784 66 ai9544 91 959041 17 230449 42 623249 67 826075 92 963788 18 255273 43 633468 68 832509 93 968483 19 278754 44 643453 69 838849 94 973128 20 301030 45 653213 70 845098 95 977724 21 322219 46 662758 71 851258 96 982271 22 342423 47 672098 72 857332 97 986772 23 361728 48 681241 73 863323 98 991226 24 380211 49 690196 74 869232 99 995635 25 397940 50 698970 75 875061 100 000000 436 ROGERS- DRAWING AND DESIGN. . =1 TABLE OF LOGARITHMS— Continued. N 1 2 3 4 5 6 7 8 9 100 000000 0U0434 000868 001301 001734 002166 002598 003029 003461' 003891 101 004321 004751 005181 005609 006038 006466 006894 007321 007748 008174 102 008600 009026 009451 009876 010300 010724 011147 011570 011993 012415 103 012837 013259 013680 014100 014521 014940 015360 015779 016197 016616 1 104 017033 017451 017868 018284 018700 019116 019532 019947 020361 020775 105 02.1189 021603 022016 022428 022841 023252 023664 024075 024486 ! 024896 106 025306 025715 026125 026533 026942 027350 027757 028164 028571 ^ \ 028978 107 029384 029789 030195 030600 031004 031408 031812 032216 032619 033021 1 108 033424 033826 034227 034628 035029 035430 035830 036230 036629 037028 109 037426 037825 038223 038620 039017 039414 039811 040207 040602 040998 110 041393 041787 042182 042576 042969 043362 043755 044148 044540 044932 111 045323 045714 046105 046495 046885 047275 047664 048053 048442 048830 112 049218 049606 049993 050380 050766 051153 051538 051924 052309 052694 113 053078 053463 053846 054230 054613 054996 055378 055760 056142 056524 114 056905 057286 057666 058046 058426 058805 059185 059568 059942 060320 115 060698 061075 061452 061829 062206 062582 062958 063333 063709 064088 116 064458 064832 065206 065580 065953 066326 066699 067071 067443 067815 117 068186 068557 068928 069298 069668 070038 070407 070776 071145 071514 118 071882 072250 072617 072985 073352 073718 074085 074451 074816 075182 119 075547 075912 076276 076640 077004 077368 077731 078094 078457 078819 120 079181 079543 079904 080266 080626 080987 081347 081707 082067 082426 N 1 2 3 4 5 6 7 8 9 ROGERS' DRAWING AND DESIGN. 437 TABLE OF LOGARITHMS— Continued. N 1 2 3 4 6 6 7 8 9 121 082785 083144 083503 083861 084219 084576 084934 085291 085647 086004 122 086360 086716 087071 087426 087781 088136 088490 088845 089198 089552 123 089905 090258 090611 090963 091315 091.667 092018 092370 092721 093071 121 093422 093772 094122 094471 094820 095169 095518 095866 096215 096562 125 096910 097257 097604 097951 098298 098644 098990 099335 099681 100026 126 100371 100715 101059 101403 101747 102091 102434 102777 103119 103462 127 103804 104146 104487 104828 105169 105510 105851 106191 106531 106871 128 107210 107549 107888 108227 108565 108903 109241 109579 109916 110253 129 110590 110926 111263 111599 111934 112270 112605 112940 113275 113609 130 113943 114277 114611 114944 115278 115611 115943 116276 116608 116940 131 117271 117603 117934 118265 118595 118926 119256 119586 119915 120245 132 120574 120903 121231 121560 121888 122216 122544 122871 123198 123525 133 123852 124178 124504 124830 125156 125481 125806 126131 126456 126781 134 127105 127429 127753 128076 128399 128722 129045 129368 129690 130012 136 130334 130655 130977 131298 J31619 131939 132260 132580 132900 133219 136 i 133539 133858 134177 134496 134814 135133 135451 135769 136086 136403 137 1 136721 137037 137354 137671 137987 138303 138618 138934 139249 139564 138 1 139879 140194 140508 140822 141136 141450 141763 142076 142389 142702 139 143015 143327 143639 143951 144263 144574 144885 145196 145507 145818 140 146128 146438 146748 147058 147367 147676 147985 148294 148603 148911 N 1 1 2 3 4 6 6 7 8 9 438 ROGERS' DRAWING AND DESIGN. TABLE OF LOGARITHMS-Continued. N 1 2 3 4 5 6 f S 9 141 149219 149527 149835 150142 150449 150756 151063 151370 151676 151982 142 152288 152594 152900 153205 153510 153815 154120 154424 154728 155032 143 155336 155640 155943 156246 156549 156852 157154 157457 157759 158061 144 158362 158664 158965 159266 159567 159868 160168 160469 160769 161068 145 161368 161667 161967 162266 162564 162863 163161 163460 163758 164055 146 164353 164650 164947 165244 165541 165838 166134 166430 166726 ' 167022 147 167317 167613 167908 168203 168497 168792 169086 169380 169674 169968 148 170262 170555 170848 171141 171434 171726 172019 172311 172603 172895 149 173186 173478 173769 174060 174351 174641 174932 175222 175512 175802 160 176091 176381 176670 176959 177248 177536 177825 178113 178401 178689 151 178977 179264 179552 179839 180126 180413 180699 180986 181272 181558 152 181844 182129 182415 182700 182985 183270 183555 183839 184123 184407 153 184691 184975 185259 185542 185825 186108 186391 186674 186956 187239 154 187521 187803 188084 188366 188647 188928 189209 189490 189771 190051 165 190332 190612 190892 191171 191451 191730 192010 192289 192567 192846 156 193125 193403 193681 193959 194237 194514 194792 195069 195346 195623 157 195900 196176 196453 196729 197005 197281 197556 197832 198107 198382 158 198657 198932 199206 199481 199755 200029 200303 200577 200850 201124 169 201397 201670 201943 202216 202488 202761 203033 203305 203577 203848 160 2041^0 204391 204663 204934 205204 205475 205746 206016 206286 206556 N 1 2 3 4 6 6 7 8 9 ROGERS' DRAWING AND DESIGN. 439 TABLE OF LOGARITHMS— Continued. N 1 2 3 4 5 6 7 8 9 161 206826 207096 207365 207634 207904 208173 208441 208710 208979 209247 162 209515 209783 210051 210319 210586 210853 211121 211388 211654 211921 163 212188 212454 212720 212986 213252 213518 213783 214049 214314 214579 164 214844 215109 215373 215638 215902 216166 216430 216694 216957 217221 i 165 217484 217747 218010 218273 218536 218798 219060 219323 219585 219846 166 220108 220370 220631 220892 221153 221414 221675 221936 222196 222456 167 222716 222976 223236 223496 223755 224015 224274 224533 224792 225051 168 225309 225568 225826 226084 226342 226600 226858 227115 227372 227630 1 169 227887 228144 228400 228657 228913 229170 229426 229682 229938 230193 1 170 230449 230704 230960 231215 231470 231724 231979 232234 232488 232742 171 232996 233250 233504 283757 234011 234264 234517 234770 235023 235276 172 235528 235781 236033 236285 236537 236789 237041 237292 237544 237795 173 238046 238297 238548 238799 239049 239299 239550 239800 240050 240300 ! 174 240549 240799 241048 241297 241546 241795 242044 242293 242541 242790 175 243038 243286 243534 243782 244030 244277 244525 244772 245019 245266 1 176 245513 245759 246006 246252 246499 246745 246991 247237 247482 247728 177 247973 248219 248464 248709 248954 249198 249443 249687 249932 250176 176 250420 250664 250908 251151 251395 251638 251881 252125 252368 252610 1 179 252853 253096 253338 253580 253822 254064 254306 254548 254790 255031 i 180 255273 255514 255755 255996 256237 256477 256718 256958 257198 257439 N 1 2 3 4 5 6 7 8 9 440 ROGERS' DRAWING AND DESIGN. - TABLE OF LOGARITHMS— Continued. N 1 2 3 4 5 6 7 8 9 181 257679 257918 258158 258398 258637 258877 259116 259355 259594 259833 182 260071 260310 260548 260787 261025 261263 261501 261739 261976 262214 183 262451 262688 262925 263162 263399 263636 263873 264109 264346 264582 184 264818 265054 265290 265525 265761 265996 266232 266467 266702 266937 185 267172 267406 267641 267875 268110 268344 268578 268812 269046 269279 186 269513 269746 269980 270213 270446 270679 270912 271144 271377 271609 187 271842 272074 272306 272538 272770 273001 273233 273464 273696 273927 188 274158 274389 274620 274850 275081 275311 275542 275772 276002 276232 189 276462 276692 276921 277151 277380 277609 277838 278067 278296 278525 190 278754 278982 279211 279439 279667 279895 280123 280351 280578 280806 191 281033 281261 281488 281715 281942 282169 282396 282622 282849 283075 192 283301 283527 283753 283979 284205 284431 284656 284882 285107 285332 193 285557 285782 286007 286232 286456 286681 286905 287130 287354 287578 194 287802 288026 288249 288473 288696 288920 289143 289366 289589 289812 195 290035 290257 290480 290702 290925 291147 291369 291591 291813 292034 196 292256 292478 292699 292920 293141 293363 293584 293804 294025 294246 197 294466 294687 294907 295127 295347 295567 295787 296007 296226 296446 198 296665 296884 297104 297323 297542 297761 297979 298198 298416 298635 1 199 298853 299071 299289 299507 299725 299943 300161 300378 300595 300813 1 200 301030 301247 301464 301681 301898 302114 302331 302547 302764 302980 1 — 1 N 1 2 3 4 5 6 7 8 » ROGERS' DRAWING AND DESIGN. 441 TABLE OF LOGARITHMS— Continued. N 1 2 3 4 5 6 7 1 8 9 201 303196 303412 303628 303844 304059 304275 304491 304706 304921 305136 202 305351 305566 305781 305996 306211 306425 306639 306854 307068 307282 203 307496 307710 307924 308137 308351 308564 308778 308991 309204 309417 204 309630 309S43 310056 310268 310481 310693 310906 311118 311330 311542 205 311754 311966 312177 312389 312600 312812 313023 313234 313445 313656 206 313867 314078 314289 314499 314710 314920 315130 315340 315551 315760 207 315970 316180 316390 316599 316809 317018 317227 317436 317646 317854 ; 208 318063 318272 318481 318689 318898 319106 319314 319522 319730 319938 209 320146 320354 320562 320769 320977 321184 321391 321598 321805 322012 210 322219 322426 322633 322839 323046 323252 323458 323665 323871 324077 211 324282 324488 324694 324899 325105 325310 325516 325721 325926 326131 212 326336 326541 326745 326950 327155 327359 327563 327767 327972 328176 213 328380 328583 328787 328991 329194 329398 329601 329805 330008 330211 214 330414 330617 330819 331022 331225 331427 331630 331832 332034 332236 215 332438 332640 332842 333044 333246 333447 333649 333850 334051 334253 216 334454 334655 334856 335057 335257 335458 335658 335859 336059 336260 217 336460 336660 336860 337060 337260 337459 337659 337858 338058 338257 218 338456 338656 338855 339034 339253 339451 339650 339849 340047 340246 219 340444 340642 340841 341039 341237 341435 341632 341830 342028 342225 220 342423 342620 342817 343014 343212 343409 343606 343802 343999 344196 9 N 1 2 3 4 5 6 7 8 442 ROGERS' DRAWING AND DESIGN. - TABLE OF LOGARITHMS-Continoed. N 1 2 3 4 5 6 7 8 9 221 344392 344589 344785 344981 345178 345374 345570 345766 345962 346157 222 346353 346549 346744 346939 347135 847330 347525 347720 347915 348110 ^ 223 348305 348500 348694 348889 349083 349278 349472 349666 349860 350054 224 350248 350442 350636 350829 351023 351216 351410 351603 351796 351989 225 352183 352375 352568 352761 352954 353147 353339 353532 353724 353916 226 354108 354301 354493 354685 354876 355068 355260 355452 355643 355834 227 356026 356217 356408 356599 356790 356981 357172 357363 357554 357744 228 357935 358125 358316 358506 358696 358886 359076 359266 359456 359646 229 359835 360025 360215 360404 360593 360783 360972 361161 361350 361539 230 361728 361917 362105 362294 362482 362671 362859 363048 363236 363424 231 363612 363800 363988 364176 364363 364551 364739 364926 365113 365301 232 365488 , 365675 365862 366049 366236 366423 366610 366796 366983 367169 233 367356 367542 367729 367915 368101 368287 368473 368659 368845 369030 234 369216 369401 369587 369772 369958 370143 370328 370513 370698 370883 235 371068 371253 371437 371622 371806 371991 372175 372360 372544 372728 236 372912 373096 373280 373464 373647 373831 374015 374198 374382 374565 237 374748 374932 375115 375298 375481 375664 375846 376029 376212 376394 238 376577 376759 376942 377124 377306 377488 377670 377852 378034 378216 239 378398 378580 378761 378943 379124 379306 379487 379668 379849 380030 240 380211 380392 380573 380754 380934 381115 381296 381476 381656 381837 N 1 2 3 4 5 6 7 8 9 ROGERS' DRAWING AND DESIGN. 443 TABLE OF LOGARITHMS— Continued. N 1 2 3 4 5 6 7 8 9 241 382017 382197 382377 382557 382737 382917 383097 383277 383456 383636 242 383815 383995 384174 384353 384533 384712 384891 385070 385249 385428 243 385606 385785 385964 386142 386321 386499 386677 386856 387034 387212 244 387390 387568 387746 387923 388101 388279 388456 388634 388811 388989 245 389166 389343 389520 389698 389875 390051 390228 390405 390582 390759 246 390935 391112 391288 391464 391641 391817 391993 392169 392345 392521 ] 247 392697 392873 393048 393224 393400 393575 393751 393926 394101 394277 248 394452 394627 394802 394977 395152 395326 395501 395676 395850 396025 249 396199 396374 396548 396722 396896 397071 397245 397419 397592 397766 260 397940 398114 398287 398461 398634 398808 398981 399154 399328 399501 261 399674 399847 400020 400192 400365 400538 400711 400883 401056 401228 262 401401 401573 401745 401917 402089 402261 402433 402605 402777 402949 1 253 403121 403292 403464 403635 403807 403978 404149 404320 404492 404663 264 404834 405005 405176 405346 405517 405688 405858 406029 406199 406370 255 406540 406710 406881 407051 407221 407391 407561 407731 407901 408070 256 408240 408410 408579 408749 408918 409087 409257 409426 409595 409764 ) 257 409933 410102 410271 410440 410609 410777 410946 411114 411283 411451 258 411620 411788 411956 412124 412293 412461 412629 412796 412964 413132 259 413300 413467 413635 413803 413970 414137 414305 414472 414639 414806 260 414973 415140 415307 415474 415641 415808 415974 416141 416308 416474 N 1 2 3 4 6 6 7 8 9 I 444 ROGERS' DRAWING AND DESIGN. - TABLE OF LOGARITHMS— Continued. N 1 2 3 4 5 6 7 8 9 261 416641 416807 416973 417139 417306 417472 417638 417804 417970 418135 262 418301 418467 418633 418798 418964 419129 419295 419460 419625 419791 263 419956 420121 420286 420451 420616 420781 420945 421110 421275 421439 264 421604 421768 421933 422097 422261 422426 422590 422754 422918 423082 265 423246 423410 423574 423737 423901 424065 424228 424392 424555 424718 i 266 424882 425045 425208 425371 425534 425697 425860 426023 426186 426349 i 267 426511 426674 426836 426999 427161 427324 427486 427648 427811 427973 268 428135 428297 428459 428621 428783 428944 429106 429268 429429 429591 269 429752 429914 430075 430236 430398 430559 430720 430881 431042 431203 270 431364 431525 431685 431846 432007 432167 432328 432488 432649 432809 271 432969 433130 433290 433450 433610 433770 433930 434090 434249 434409 272 434569 434729 434888 435048 435207 435367 435526 435685 435844 436004 273 436163 436322 436481 436640 436799 436957 437116 437275 437433 437592 274 437751 437909 438067 438226 438384 438542 438701 438859 439017 439175 275 439333 439491 439648 439806 439964 440122 440279 440437 440594 440752 276 440909 441066 441224 441381 441538 441695 441852 442009 442166 442323 277 442480 442637 442793 442950 443106 443263 443419 443576 443732 443889 278 444045 444201 444357 444513 444669 444825 444981 445137 445293 445449 279 445604 445760 445915 446071 446226 446382 446537 446692 446848 447003 280 447158 447313 447468 447623 447778 447933 448088 448242 448397 448552 N 1 2 3 4 5 6 7 8 9 ROGERS' DRAWING AND DESIGN. 445 TABLE OF ] LOGARITHMS-Contlnued. N 1 2 3 4 5 6 7 8 9 281 448706 448861 449015 449170 449324 449478 449633 449787 449941 450095 282 450249 450403 450557 450711 450865 451018 451172 451326 451479 451633 283 451786 451940 452093 452247 452400 452553 452706 452859 453012 453165 284 453318 453471 453624 453777 453930 454082 454235 454387 454540 454692 285 454845 454997 455150 455302 455454 455606 455758 455910 456062 456214 286 456366 456518 456670 456821 456973 457125 457276 457428 457579 457731 287 457882 458033 458184 458336 458487 458638 458789 458940 459091 459242 288 459392 459543 459694 459845 459995 460146 460296 460447 460597 460748 289 460898 461048 461198 461348 461499 461649 461799 461948 462098 462248 290 462398 462548 462697 462847 462997 463146 463296 463445 463594 463744 291 463893 464042 464191 464340 464490 464639 464788 464936 465085 465234 292 465383 465532 465680 465829 465977 466126 466274 466423 466571 466719 293 466868 467016 467164 467312 467460 467608 467756 467904 468052 468200 294 468347 468495 468643 468790 468938 469085 469233 469380 469527 469675 295 469822 469969 47011G 470263 470410 470557 470704 470851 470998 471145 296 471292 471438 471585 471732 471878 472025 472171 472318 472464 472610 297 472756 472903 473049 473195 473341 473487 473633 473779 473925 474071 298 474216 474362 474508 474653 474799 474944 475090 475235 475381 475526 i 299 475671 475816 475962 476107 476252 476397 476542 476687 476832 476976 300 477121 477266 477411 477555 477700 477844 477989 478133 478278 478422 N 1 2 3 4 5 6 7 8 9 ' 446 ROGERS' DRAWING AND DESIGN. TABLE OF LOGARITHMS-Continoed. m ■ N 1 2 3 4 5 6 7 8 9 301 478566 478711 478855 478999 479143 479287 479431 479575 479719 479863 302 480007 480151 480294 480438 480582 480725 480869 481012 481156 481299 303 481443 481586 481729 481872 482016 482159 482302 482445 482588 482731 304 482874 483016 483159 483302 483445 483587 483730 483872 484015 484157 305 484300 484442 484585 484727 484869 485011 485153 485295 485437 485579 306 485721 485863 . 486005 486147 486289 486430 486572 486714 486855 486997 307 487138 487280 487421 487563 487704 487845 487986 488127 488269 488410 308 488551 488692 488833 488974 489114 489255 489396 489537 489677 489818 309 489958 490099 490239 490380 490520 490661 490801 490941 491081 491222 310 491362 491502 491642 491782 491922 492062 492201 492341 492481 492621 311 492760 492900 493040 493179 493319 493458 493597 493737 493876 494015 312 494155 494294 494433 494572 494711 494850 494989 495128 495267 495406 313 495544 495683 495822 495960 496099 496238 496376 486515 496653 496791 314 496930 497068 497206 497344 497483 497621 497759 497897 498035 498173 315 498311 498448 498586 498724 498862 498999 499137 499275 499412 499550 316 499687 499824 499962 500099 500236 500374 500511 500648 500785 500922 317 501059 501196 501333 501470 501607 501744 501880 502017 502154 502291 318 502427 502564 502700 502837 502973 503109 503246 503382 503518 503655 319 503791 503927 504063 504199 504335 504471 504607 504743 504878 505014 320 505150 505286 505421 505557 505693 505828 505964 506099 506234 506370 N 1 2 3 4 5 6 7 8 9 ROGERS' DRAWING AND DESIGN. 447 TABLE OF LOGARITHMS- Continued. N 1 2 3 4 5 6 7 8 9 321 506505 506640 506776 506911 507046 507181 507316 507451 507586 507721 322 507856 507991 508126 508260 508395 508530 508664 508799 508934 509068 323 509203 509337 509471 509606 509740 509874 510009 510143 510277 510411 324 510545 510679 510813 510947 511081 511215 511349 511482 511616 511750 325 511883 512017 512151 512284 512418 512551 512684 512818 512951 513084 326 513218 513351 513484 513617 513750 513883 514016 514149 514282 514415 327 514548 514681 514813 514946 515079 515211 515344 515476 515609 515741 328 515874 516006 516139 516271 516403 516535 516668 516800 516932 517064 329 517196 517328 517460 517592 517724 517855 517987 518119 518251 518382 330 518514 518646 518777 518909 519040 519171 519303 519434 519566 519697 331 519828 519959 520090 520221 520353 520484 520615 520745 520876 521007 332 521138 521269 521400 521530 521661 521792 521922 522053 522183 522314 333 522444 522575 522705 522835 522966 523096 523226 523356 523486 523616 334 523746 523876 524006 524136 524266 524396 524526 524656 524785 524915 335 525045 525174 525304 525434 525563 525693 525822 525951 526081 526210 336 526339 526469 526598 526727 526856 526985 527114 527243 527372 527501 337 527630 527759 527888 528016 528145 528274 528402 528531 528660 528788 338 528917 529045 529174 529302 529430 529559 529687 529815 529943 530072 339 530200 530328 530456 530584 530712 530840 530968 531096 531223 531351 340 531479 531607 531734 531862 531990 532117 532245 532372 532500 532627 N 1 2 3 4 5 6 7 8 9 448 ROGERS' DRAWING AND DESIGN. TABLE OF LOGARITHMS-Continoed. N 1 2 3 4 5 6 7 8 9 341 532754 ^532882 533009 533136 533264 533391 533518 533645 533772 533899 342 534026 534153 534280 534407 534534 534661 534787 534914 535041 535167 343 535294 535421 535547 535674 535800 535927 536053 536180 536306 536432 344 536558 536685 536811 536937 537063 537189 537315 537441 537567 537693 346 537819 537945 538071 538197 538822 538448 538574 538699 538825 538951 346 539076 539202 539327 539452 539578 539703 539829 539954 540079 540204 347 540329 540455 540580 540705 540830 540955 541080 541205 541330 541454 348 541579 541704 541829 541953 542078 542203 542327 542452 542576 542701 349 542825 542950 543074 543199 543323 543447 543571 543696 543820 543944 350 544068 544192 544316 544440 544564 544688 544812 544936 545060 545183 351 545307 545431 545555 545678 545802 545925 546049 546172 546296 546419 352 546543 546666 546789 546913 547036 547159 547282 547405 547529 547652 353 547775 547898 548021 548144 548267 548389 548512 548635 548758 548881 354 549003 549126 549249 549371 549494 549616 549739 549861 549984 550106 355 550228 550351 550473 550595 550717 550840 550962 551084 551206 551328 356 551450 551572 551694 551816 551938 552060 552181 552303 552425 552547 357 552668 552790 552911 553033 553155 553276 553398 553519 553640 553762 358 553883 554004 554126 554247 554368 554489 554610 554731 554852 554973 359 555094 555215 555336 555457 555578 555699 555820 555940 556061 556182 360 556303 556423 556544 556664 556785 556905 557026 557146 557267 557387 N 1 2 3 4 5 6 7 8 9 ROGERS' DRAWING AND DESIGN. 449 TABLE OF ] LOGARITHMS-Continued. N 1 2 3 4 5 6 7 8 9 361 557507 557627 557748 557868 557988 558108 558228 558349 558469 558589 362 558709 558829 558948 559068 559188 559308 559428 559548 559667 559787 363 559907 560026 560146 560265 560385 560504 560624 560743 560863 560982 364 561101 561221 561340 561459 561578 561698 561817 561936 562055 562174 365 562293 562412 562531 562650 562769 562887 563006 563125 563244 563362 366 563481 563600 563718 563837 563955 564074 564192 564311 564429 564548 367 564666 564784 564903 565021 565139 565257 565376 565494 565612 565730 368 565848 565966 566084 566202 566320 566437 566555 566673 566791 566909 369 567026 567144 567262 567379 567497 567614 567732 567849 567967 568084 370 568202 568319 568436 568554 568671 568788 56890a 569023 569140 569257 371 569374 569491 569608 569725 569842 569959 570076 570193 570309 570426 372 570543 570660 570776 570893 571010 571126 571243 571359 571476 571592 373 571709 571825 571942 572058 572174 572291 572407 572523 572639 572755 374 572872 572988 573104 573220 573336 573452 573568 573684 573800 573915 375 574031 574147 574263 574379 574494 574610 574726 574841 574957 575072 376 575188 575303 575419 575534 575650 575765 575880 575996 576111 576226 377 576341 576457 576572 576687 576802 576917 577032 577147 577262 577377 378 577492 577607 577722 577836 577951 578066 578181 578295 578410 578525 379 578639 578754 578868 578983 579097 579212 579326 579441 579555 579669 380 579784 579898 580012 580126 580241 580355 580469 580583 580697 580811 N 1 2 3 4 6 6 7 8 9 450 ROGERS' DRAWING AND DESIGN TABLE OF ] LOGARITHMS— Continued. N 1 2 3 4 5 6 7 8 9 381 580925 581039 581153 581267 581381 581495 581608 581722 581836 581950 382 582063 582177 582291 582404 582518 582631 582745 582858 582972 583085 383 583199 583312 583426 583539 583652 583765 583879 583992 584105 584218 384 584331 584444 584557 584670 584788 584896 585009 585122 585235 585348 385 585461 585574 585686 585799 685912 586024 586137 586250 586362 586475 386 586587 586700 586812 586925 587037 587149 587262 587374 587486 587599 387 587711 587823 587935 588047 588160 588272 588384 588496 588608 588720 388 588832 588944 589056 589167 589279 589391 589503 589G15 589726 589838 389 589950 590061 590173 590284 590396 590507 690619 590730 590842 590953 390 591065 591176 591287 591399 591510 591621 591732 591843 591955 592066 391 592177 592288 592399 592510 592621 592732 592843 592954 593064 593175 392 593286 593397 593508 593618 593729 593840 593950 594061 594171 594282 393 594393 594503 594614 594724 594834 594945 595065 595165 595276 595386 394 595496 595606 595717 595827 595937 596047 596157 596267 596377 596487 395 596597 596707 596817 596927 597037 597146 597256 597366 597476 597586 396 597695 597805 597914 598024 598134 598243 598353 598462 598572 598681 397 598791 598900 599009 599119 599228 599337 599446 599556 599665 599774 398 599883 599992 600101 600210 600319 600428 600537 600646 600755 600864 399 600973 601082 601191 601299 601408 601517 601625 601734 601843 601951 400 602060 602169 602277 602386 602494 602603 G02711 602819 602928 603036 N 1 2 3 4 5 6 7 8 9 ROGERS' DRAWING AND DESIGN. 451 TABTF, OF ] LOGARITHMS— Continued. N 1 2 3 4 5 6 7 8 9 401 603144 603253 603361 603469 603577 603686 603794 603902 604010 604118 402 604226 604334 604442 604550 604658 604766 604874 604982 605089 605197 403 605305 605413 605521 605628 605736 605844 605951 606059 606166 606274 404 606381 606489 606596 606704 606811 606919 607026 607133 607241 607348 405 607455 607562 607669 607777 607884 607991 608098 608205 608312 608419 406 608526 608633 608740 608847 608954 609061 609167 609274 609381 609488 407 609594 609701 609808 609914 610021 610128 610234 610341 610447 610554 408 610660 610767 610873 610979 611086 611192 611298 611405 611511 611617 409 611723 611829 ^ 611936 612042 612148 612254 612360 612466 612572 612678 410 612784 612890 612996 613102 613207 613313 613419 613525 613630 613736 411 613842 613947 614053 614159 614264 614370 614475 614581 614686 614792 412 614897 615003 615108 615213 615319 615424 615529 615634 615740 615845 413 615950 616055 616160 616265 616370 616476 616581 616686 616790 616895 414 617000 617105 617210 617315 617420 617525 617629 617734 617839 617948 416 618048 618153 618257 618362 618466 618571 618676 618780 618884 618989 416 619093 619198 619302 619406 619511 619615 619719 619824 619928 620032 417 620136 620240 620344 620448 620552 620656 620760 620864 620968 621072 418 621176 621280 621384 621488 621592 621695 621799 621903 622007 622110 419 622214 622318 622421 622525 622628 622732 622835 622939 623042 623146 420 623249 623353 623456 623559 623663 623766 623869 623973 624076 624179 N 1 2 3 4 6 6 7 8 9 1 452 ROGERS' DRAWING AND DESIGN. TABLE OF ] LOGARITHMS- Com 4nued. N 1 2 3 4 5 6 7 8 9 421 624282 624385 624488 624591 624695 624798 624901 625004 625107 625210 422 625312 625415 625518 625621 625724 625827 625929 626032 626135 626238 423 626340 626443 626546 626648 626751 626853 626956 627058 627161 627263 424 627366 627468 627571 627673 627775 627878 627980 628082 628185 628287 425 628389 628491 628593 628695 628797 628900 629002 629104 629206 629308 426 629410 629512 629613 629715 629817 629919 630021 630123 630224 630326 427 630428 630530 630631 630733 630835 630936 631038 631139 631241 631342 428 631444 631545 631647 631748 631849 631951 632052 632153 632255 632356 429 632457 632559 632660 632761 632862 632963 633064 633165 633266 633367 430 633468 633569 633670 633771 633872 633973 634074 634175 634276 634376 431 634477 634578 634679 634779 634880 634981 635081 635182 635283 635383 432 635484 635584 635685 635785 635886 635986 636087 636187 636287 636388 433 636488 636588 636688 636789 636889 636989 637089 637189 637290 637390 434 637490 637590 637690 637790 637890 637990 638090 638190 638290 638389 435 638489 638589 638689 638789 638888 638988 639088 639188 639287 639387 436 639486 639586 639686 639785 639885 639984 640084 640183 640283 640382 437 640481 640581 640680 640779 640879 640978 641077 641177 641276 641375 438 641474 641573 641G72 641771 641871 641970 642069 642168 642267 642366 439 642465 642563 642662 G42761 642860 642959 643058 643156 643255 643354 440 643453 643551 643650 643749 643847 643946 644044 644143 644242 644340 N 1 2 2 4 5 6 7 8 9 ROGERS' DRAWING AND DESIGN. 453 TABLE OF LOGARITHMS— Continued N 1 2 3 4 5 6 7 8 9 441 644439 644537 644636 644734 644832 644931 645029 645127 645226 645324 442 645422 645521 645619 645717 645815 645913 646011 646110 646208 646306 443 646404 646502 646600 646698 646796 646894 646992 647089 647187 647285 444 647383 647481 647579 647676 647774 647872 647969 6480G7 648165 648262 445 648360 648458 648555 648653 648750 648848 648945 649043 649140 649237 446 649335 649432 649530 649627 649724 649821 649919 650016 650113 650210 447 650308 650405 650502 650599 650696 650793 650890 650987 651084 651181 448 651278 651375 651472 651569 651666 651762 651859 651956 652053 652150 449 652246 652343 652440 652536 652633 652730 652826 652923 653019 653116 460 653213 653309 653405 653502 653598 653695 653791 653888 653984 654080 451 654177 654273 654369 654465 654562 654658 654754 654850 654946 655042 452 655138 655235 655331 655427 655523 655619 655715 655810 655906 656002 463 656098 656194 656290 656386 656482 656577 656673 656769 656864 656960 454 657056 657152 657247 657343 657438 657534 657629 657725 657820 657916 456 658011 658107 . 658202 658298 658393 658488 658584 658679 658774 658870 466 658965 659060 659155 659250 659346 659441 659536 659631 659726 659821 457 659916 660011 660106 660201 660296 660391 660486 660581 660676 660771 458 660865 660960 661055 661150 661245 661339 661434 661529 661623 661718 459 661813 661907 662002 662096 662191 662286 662380 662475 662569 662663 460 662758 662852 662947 663041 663135 663230 663324 663418 663512 663607 N 1 2 3 i 6 6 7 8 9 454 ROGERS' DRAWING AND DESIGN. TABLE OF LOGAPaTHMS— Continued. IT 1 & 3 4 5 6 7 8 9 461 663701 663795 663889 663983 664078 664172 664266 664360 664454 664548 462 664642 664736 664830 664924 665018 665112 665206 665299 665393 665487 463 665581 665675 665769 665862 665956 666050 666143 666237 666331 666424 464 666518 666612 666705 666799 666892 666986 667079 667173 667266 667360 465 667453 667546 667640 667733 667826 667920 668013 668106 668199 668293 4^6 668386 668479 668572 668665 668759 668852 1 668945 669038 669131 669224 467 669317 669410 669503 669596 669689 669782 669875 669967 670060 670153 468 670246 670339 670431 670524 670617 670710 670802 670895 670988 671080 469 671173 671265 671358 671451 671543 671636 671728 671821 671913 672005 470 672098 672190 672283 672375 672467 672560 672662 672744 672836 672929 471 673021 673113 673205 673297 673390 673482 673574 673666 673758 673850 472 673942 674034 674126 674218 674310 674402 674494 674586 674677 G74769 473 674861 674953 675045 675137 675228 675320 675412 675503 675595 675687 474 675778 675870 675962 676053 676145 676236 676328 676419 676511 676602 475 676694 676785 676876 676968 677059 677151 677242 677333 677424 677516 476 677607 677698 677789 677881 677972 678063 678154 678245 678336 678427 477 678518 678609 678700 678791 678882 678973 679064 679155 679246 679337 478 679428 679519 679610 679700 679791 679882 679973 680063 680154 680245 479 680336 680426 680517 680607 680698 680789 680879 680970 681060 681151 480 681241 681332 681422 681513 681603 681693 681784 681874 681964 682055 N 1 2 3 4 5 6 7 8^ 9 ROGERS' DRAWING AND DESIGN. 455 TABLE OF ] LOGARITHMS- Continued. N 1 2 3 4 5 6 7 8 9 481 682145 682235 682326 682416 682506 682596 682686 682777 682867 682957 482 683047 683137 683227 683317 683407 683497 683587 683677 683767 683857 483 683947 684037 684127 684217 684307 684396 684486 684576 684666 684756 484 684845 684935 685025 685114 685204 685294 685383 685473 685563 685652 485 685742 685831 685921 686010 686100 686189 686279 686368 686458 686547 • 486 686636 686726 686815 686904 686994 687083 687172 687261 687351 687440 487 687529 687618 687707 687796 687886 687975 688064 688153 688242 688331 488 688420 688509 688598 688687 688776 688865 688953 689042 689131 689220 489 689309 689398 689486 689575 689664 689753 689841 689930 690019 690107 490 690196 690285 690373 690462 690550 690639 690728 690816 690905 690993 491 691081 6911-70 691258 691347 691435 691524 691612 691700 691789 691877 492 691965 692053 692142 692230 692318 692406 692494 692583 692671 692759 493 692847 692935 693023 693111 693199 693287 693375 693463 693551 693639 494 693727 693815 693903 693991 694078 694166 694254 694342 694430 694517 495 694605 694693 694781 694868 694956 695044 695131 695219 695307 695394 496 695482 695569 695657 695744 695832 695919 696007 696094 696182 696269 497 696356 696444 696531 696618 696706 696793 696880 696968 697055 697142 498 697229 697317 697404 697491 697578 697665 697752 697839 697926 698014 499 698101 698188 698275 698362 698449 698535 698622 698709 698796 698883 500 698970 699057 699144 699231 699317 699404 699491 699578 699664 699751 N 1 2 3 4 5 6 7 8 9 456 ROGERS' DRAWING AND DESIGN. TABLE OF LOGARITHMS— Continued. N 1 2 3 4 5 6 7 8 9 501 699838 699924 700011 700098 700184 700271 700358 700444 700531 700617 502 700704 700790 700877 700963 701050 701136 701222 701309 701395 701482 503 701568 701654 701741 701827 701913 701999 702086 702172 702258 702344 504 702431 702517 702603 702689 702775 702861 702947 703033 703119 703205 505 703291 703377 703463 703549 703635 703721 703807 703893 703979 704065 506 704151 704236 704322 704408 704494 704579 704665 704751 704837 704922 507 705008 705094 705179 705265 705350 705436 705522 705607 705693 705778 508 705864 705949 706035 706120 706206 706291 706376 706462 706547 706632 509 706718 706803 706888 706974 707059 707144 707229 707315 707400 707485 510 707570 707655 707740 707826 707911 707996 708081 708166 708251 708336 511 708421 708506 708591 708676 708761 708846 708931 709015 709100 709185 512 709270 709355 709440 709524 709609 709694 709779 709863 709948 710033 513 710117 710202 710287 710371 710456 710540 710625 710710 710794 710879 514 710963 711048 711132 711217 711301 711385 711470 711554 711639 711723 515 711807 711892 711976 712060 712144 712229 712313 712397 712481 712566 516 712650 712734 712818 712902 712986 713070 713154 713238 713323 713407 517 713491 713575 713659 713742 713826 713910 713994 714078 714162 714246 518 714330 714414 714497 714581 714665 714749 714833 714916 715000 715084 519 715167 715251 715335 715418 715502 715586 715669 715753 715836 715920 520 716003 716087 716170 716254 716337 716421 716504 716588 716671 716754 N 1 2 3 4 5 6 7 8 9 ROGERS' DRAWING AND DESIGN. 457 TABLE OF LOGARITHMS— Continued. N 1 2 3 4 5 6 7 8 9 521 716838 716921 717004 717088 717171 717254 717338 717421 717504 717587 522 717671 717754 717837 717920 718003 718086 718169 718253 718336 718419 523 718502 718585 718668 718751 718834 718917 719000 719083 719165 719248 524 719331 719414 719497 719580 719663 719745 719828 719911 719994 720077 625 720159 720242 720325 720407 720490 720573 720655 720738 720821 720903 526 720986 721068 721151 721233 721316 721398 721481 721563 721646 721728 527 721811 721893 721975 722058 722140 722222 722305 722387 722469 722552 528 722634 722716 722798 722881 722963 723045 723127 723209 723291 723374 529 723456 723538 723620 723702 723784 723866 723948 724030 724112 724194 530 724276 724358 724440 724522 724604 724685 724767 724849 724931 725013 531 725095 725176 725258 725340 725422 725503 725585 725667 725748 725830 632 725912 725993 726075 726156 726238 726320 726401 726483 726564 726646 533 726727 726809 726890 726972 727053 727134 727216 727297 727379 727460 534 727541 727623 727704 727785 727866 727948 728029 728110 728191 728273 535 728354 728435 728516 728597 728678 728759 728841 728922 729003 729084 536 729165 729246 729327 729408 729489 729570 729651 729732 729813 729893 537 729974 730055 730136 730217 730298 730378 730459 730540 730621 730702 538 730782 730863 730944 731024 731105 731186 731266 731347 731428 731508 539 731589 731669 731750 731830 731911 731991 732072 732152 732233 732313 540 732394 732474 732555 732635 732715 732796 732876 732956 733037 733117 N 1 2 3 4 6 6 7 8 9 458 ROGERS' DRAWING AND DESIGN. TABLE OF LOGARITHMS— Continued. N 1 2 3 4 5 6 7 8 9 641 733197 733278 733358 733438 733518 733598 733679 733759 733839 733919 542 733999 734079 734160 734240 734320 734400 734480 734560 734640 734720 543 734800 734880 734960 735040 735120 735200 735279 735359 735439 735519 544 735599 735679 735759 735838 735918 735998 736078 736157 736237 736317 645 736397 736476 736556 736635 736715 736795 736874 736954 737034 737113 546 737193 737272 737352 737431 737511 737590 737670 737749 737829 737908 647 737987 738067 738146 738225 738305 738384 738463 738543 738622 738701 548 738781 738860 738939 739018 739097 739177 739256 739335 739414 739493 549 739572 739651 739731 739810 739889 739968 740047 740126 740205 740284 550 740363 740442 740521 740600 740678 740757 740836 740915 740994 741073 551 741152 741230 741309 741388 741467 741546 741624 741703 741782 741860 552 741939 742018 742096 742175 742254 742332 742411 742489 742568 742647 553 742725 742804 742882 742961 743039 743118 743196 743275 743353 743431 554 743510 743588 743667 743745 743823 743902 743980 744058 744136 744215 555 744293 744371 744449 744528 744606 744684 744762 744840 744919 744997 556 745075 745153 745231 745309 745387 745465 745543 745621 745699 745777 557 745855 745933 746011 746089 746167 746245 746323 746401 746479 746556 558 746634 746712 746790 746868 746945 747023 747101 747179 747256 747334 559 747412 747489 747567 747645 747722 747800 747878 747955 748033 748110 660 748188 748266 748343 748421 748498 748576 748653 748731 748808 748885 N ~ - 1 2 3 4 5 6 7 8 9 ROGERS' DRAWING AND DESIGN. 459 TABLE OF LOGARITHMS— Continued. N 1 2 3 4 5 — 6 7 8 9 561 74S9G3 749040 749118 749195 749272 749350 749427 749504 749582 749659 562 749736 749814 749891 749968 750045 750123 750200 750277 750354 750431 563 750508 750586 750663 750740 750817 750894 750971 751048 751125 751202 564 751279 751356 751433 751510 751587 751664 751741 751818 751895 751972 565 75204h 752125 752202 752279 752356 752433 752509 752586 752663 752740 566 752816 752893 752970 753047 753123 753200 753277 753353 753430 753506 567 753583 753660 753736 753813 753889 753966 754042 754119 754195 754272 568 754348 754425 754501 754578 754654 754730 754807 754883 754960 755036 569 755112 755189 755265 755341 755417 755494 755570 755646 755722 755799 570 755875 755951 756027 756103 756180 756256 756332 756408 756484 756560 571 756636 756712 756788 756864 756940 757016 757092 757168 757244 757320 572 757396 757472 757548 757624 757700 757775 757851 757927 758003 758079 673 758155 758230 758306 758382 758458 758533 758609 758685 758761 758836 574 758912 758988 759063 759139 759214 759290 759366 759441 759517 759592 57-5 759668 759743 759819 759894 759970 760045 760121 760196 760272 760347 576 760422 760498 760573 760649 760724 760799 760875 760950 761025 761101 577 761176 761251 761326 761402 761477 761552 761627 761702 761778 761853 578 761928 762003 762078 762153 762228 762303 762378 762453 762529 762604 579 762679 762754 762829 762904 762978 763053 763128 763203 763278 763353 680 763428 763503 763578 763653 763727 763802 763877 763952 764027 764101 N 1 2 3 4 5 6 7 8 9 460 ROGERS' DRAWING AND DESIGN. - TABLE OF LOGARITHMS— Continued. N 1 2 3 4 5 6 7 8 9 681 764176 764251 764326 764400 764475 764550 764624 764699 764774 764848 582 764923 764998 765072 765147 765221 765296 765370 765445 765520 765594 683 765669 765743 765818 765892 765966 766041 766115 766190 766264 766338 684 766413 766487 766562 766636 766710 766785 766859 766933 767007 767082 686 767156 767230 767304 767379 767453 767527 767601 767675 767749 767823 686 767898 767972 768046 768120 768194 768268 768342 768416 768490 768564 687 768638 768712 768786 768860 768934 769008 769082 769156 769230 769303 688 769377 769451 769525 769599 769673 769746 769820 769894 769968 770042 589 770115 770189 770263 770336 770410 770484 770557 770631 770705 770778 690 770852 770926 770999 771073 771146 771220 771293 771367 771440 771514 591 771587 771661 771734 771808 771881 771955 772028 772102 772175 772248 692 772322 772395 772468 772542 772615 772688 772762 772835 772908 772981 693 773055 773128 773201 773274 773348 773421 773494 773567 773640 773713 594 773786 773860 773933 774006 774079 774152 774225 774298 774371 774444 595 774517 774590 774663 774736 774809 774882 774955 775028 775100 775173 596 775246 775319 775392 775465 775538 775610 775683 775756 775829 775902 ' 597 775974 776047 776120 776193 776265 776338 776411 776483 776556 776629 598 776701 776774 776846 776919 776992 777064 777137 777209 777282 777354 1 599 777427 777499 777572 777644 777717 777789 777862 777934 778006 778079 1 600 778151 778224 778296 778368 778441 778513 778585 778658 778730 778802 1 1 N 1 2 3 4 6 6 7 9 9 J USEFUL TABLES FOR DRAUGHTSMEN, MACHINISTS AND ENGINEERS. TABLE OF DECIMAL EQUIVALENTS. Sths, J6ths, 32ds and 64tlis of an Inch. 8ths. 32nds. 64tbs. Il=-5i5625 J =.125 A= -03125 ,V= -015625 lt= -546875 i-.25o ^=•09375 5\=. 046875 ff= 578125 f =-375 ^=•15625 5\ = .o78i25 11= 609375 ^=.500 A= .21875 A=- 109375 ii= 640625 1= 625 A= -28125 ^=.140625 H= .671875 f =-75o M= -34375 I4=.i7i875 n= 703125 l=.875 J|=. 40625 |f=.203I25 u= 734375 i6ths. M=-46875 wX. 234375 11= 765625 f J =.0625 Ji = -53i25 H= -265625 u--= 796875 A=-i875 M=-59375 ^ = .296875 11= 828125 A=-3I25 M= -65625 li=.32Si25 11= 859375 TJ=-4375 11= -71875 ll=-359375 u= S90625 ^=•5625 If =-78125 11= .390625 M= 921S75 ii = .6875 fl=-84375 ||=.42i875 U= 953125 n=.8i25 ||=.9o625 ||=.453i25 Sf= 984375 11= -9375 M=-96875 |i=. 484375 461 d62 ROGERS' DRAWING AND DESIGN. 1— 1 m 00 D Z 1— 1 00 i o 00 Q 00 00 w Q 1 O H 1 a ri ■^ a PL, 1 W o V3 G -i a S a JO joqiun^ o 2 o ^ o S a S o n ^ 1^ m -M ,/-, rn t^ rr,' r^ o •-! e^ tn -^ ta es t^ CO OS '^ <-! v> n -ii ta to t^ a> e> o -^ e^ n -* ui to t' et> a> o ogo§o'='^'=^"'*'°'°*^** — -^ — r-^r-r- i-hS,-.cjciN«e.-g<"--t>-OCD • - ^-r-*CO?CCO(M'^4'MCN(M'M"r--.— r^t-H,— OOOOOOOOOOOOOOOOOOOOOOOOOO • • •^J-Hi-HOOOCiC50a500QOOOCOIr*t^l>'CDCDO»0»Oir5U*i'^'^'-i'^7COOirNt--'-i— 'ooooasoi • . • .c^(^^(^l(r^(^^c^l^r-^l-HI-HrH^HIHr-f-H.-HrHrM,-^rHl--^T-HI-Hr^,-HrHr-r^r^rHT-l^-HrH,^ •aSntio ojt^ luuadmi -fOSCD-ri^OCOOfM-^COOOOCqOO ^•^TfCOCCCOC^iMC^OJTJi-'i— — 1— .— f-^ocooooooooooooooooooooooooo<:> ■SM'Jaie33J0A\ '■00 -ajjvr naow 5> lunqqeEji . .00»OOiOO»OI>.CCOOOOCOOiO»ftlCOC>lOOiOOOOl^COOOO-^^fO(NOO'MOOOO"-HlC50 - - - » • .coc"^<^^oo^^o-TCcc^oocol>.o»0"^'^coro<^^'^^'^^':^l.^-^I— c^rH^^r-*,— «r-ioo - - - - • •CC00C0« UMOJa JO uvciuauiv . ■ 0'*'^c:)i>.o-t' — r^^'^oO'^'--oOl— '"^r'-c^iOOKtii— i(>D<-:cioi>.»c-^c^i— 'Ocoi'-t^'iDioiO'^^ JO jaqmnii ROGERS' DRAWING AND DESIGN. 463 RULES TABLES. Relative to the Qrcle. Area, and Qrcomfei-enccs of Circles advancing^ by tenths. To And the area of a circle — Multiply circumferencebyone-quarterofthediameter. Or multiply the square of diameter by 0.7854. Or " '' circumference " .07958. DIam. Area. CIrciun. Dloin. Area. Clrcum. 0.0 .1 .007854 .31416 3.0 .1 7.0686 7.5477 9.4248 9.7389 Or " " ^ diameter " 3.1416. '.s .031416 .070686 .62832 .94248 .2 .3 8.0425 8 5530 10.0531 10.3673 A .12566 1.2566 .4 9.0792 10.6814 To £nd circumference — .5 .19735 1.5708 .5 9.6211 10.9956 Multiply diameter by 3.1416. .6 .7 .28274 .88485 1.8850 2.1991 .6 .7 10.1788 10.7521 11.3097 11.6239 Or divide " " 0.3183. .8 .9 .50266 .63617 2.5133 2.8274 .8 .9 11.3411 11.9456 11.9381 12.2523 To And diameter — 1.0 .1 .7854 .9508 3.1418 8.4558 4.0 .1 12.5664 13.2025 12.5664 12.8805 Multiply circumference by 0.3183. .2 .3 1.1810 1.3273 3.7699 4.0841 .2 .3 13.8544 14.5220 13.1947 13.5088 Or divide " " 3.1416. .4 1.5394 4,8982 .4 15.2053 18.8230 ,5 1.7671 4.7124 .5 15.9048 14.1372 To And Radius — .6 2.0106 5.0265 .6 16.6190 14.4513 ,7 2.2698 5.3407 .7 17.3494 14.7655 Multiply circumference by 0. ISQIS- .8 2.5447 5.6549 .8 18.0956 15.0796 Or divide " " 6.28318. .9 2.8353 5.9690 .9 18.8574 15.8938 2.0 3.1416 6.2832 5.0 19.6350 15.7080 In the following tables the diameter of a given .1 .2 8.4636 3.8018 6.5973 6.9115 .1 .2 20.4282 21.2872 16.0221 16.3863 inch is to be found in the first column, the area is to .3 4.1548 7.2257 .8 22.0618 16.6504 .4 4.5239 7.6398 .4 22.9023 16.9646 be found in the second column, and the circumference in the third column. .5 .6 4.9087 5.3093 7.8540 8.1681 .5 .6 23.7588 24.6301 17.2788 17.5929 Example : A circle with a diameter of 2.7 inches .7 .8 5.7256 6.1575 8.4828 8.7965 .7 .8 25.5176 26.4208 17.9071 18.2212 has an area of 5.7256 square inches and a circumfer- ence of 8.4823 linear inches. .9 6.6052 9.1106 .9 37.8397 18.5854 464 ROGERS' DRAWING AND DESIGN. TABLES OF AREAS AND CIRCUMFERENCES OF CIRCLES— Continued. DIam. Area. Clrcum. Dlaiii. Area. CIrcuni. 6.0 28.8743 18.8496 10.0 78.5398 31.4159 .1 29.2247 19.1637 .1 80.1185 31.7301 .2 30.1907 19.4779 2 81.7128 32.0443 .3 31.1725 19.7920 .3 83.3239 33.3584 .4 32.1699 20.1062 .4 84.9487 32.6726 .5 33.1831 20.4204 .5 86.5901 32.9867 .6 34.8119 20.7345 .6 88.3473 33.3009 .7 35.2565 21.0487 .7 89.9202 33.6150 .8 36.3168 21.3628 .8 91.6088 33.9293 .9 37.3928 21.6770 .9 93.3132 34.2434 7.0 38.4845 21.9911 11.0 95.0332 34.5575 .1 39.5919 28.3053 .1 96.7689 34.8717 .2 40.7150 82.6195 .2 98.'5203 35.1858 .3 41.8539 22.9336 .3 100.2875 35.5000 .4 43.0084 23.2478 .4 102.0703 35.8143 ,5 44.1786 23.5619 .5 103.6689 36.1383 .6 45.3646 23.8761 .6 105.6833 36.4425 .7 46.5663 24.1903 .7 107.5132 36.7566 .8 47.7836 24.5044 .8 109 3588 37.0708 .9 49.0167 24.8186 .9 111.2202 37.3850 8.0 50.2655 25.1327 12.0 113.0973 37.6991 .1 51.5300 25.4469 .1 114.9901 38.0133 .2 52.8102 25.7611 .2 116.8987 38.3274 .3 54.1061 26.0753 .3 118.8229 38.6416 .4 55.4177 26.3894 .4 120.7688 38.9557 .5 56.7450 26.7035 .5 122.7185 39.2699 .6 58.0880 27.0177 .6 124.6898 39.5841 .7 59.4468 27.3319 .7 126.6769 39.8982 .8 60.8213 27.6460 .8 128.6796 40.2124 .9 62.8114 27.9602 .9 130.6981 40.5265 9.0 63.6173 28.2743 13.0 132.7323 40.8407 .1 65.0388 . 28.5885 .1 134.7822 41.1549 .3 66.4761 28.9027 .2 136.8478 41.4690 .3 67.9291 29.2168 .3 138.9291 41.7832 .4 69.3978 29.5310 .4 141.0261 42.0973 .5 70.8822 29.8451 .5 143.1383 42.4115 .6 72.3823 30.1593 .6 145.2672 42.7257 .7 73.8981 30,4734 .7 147.4114 43.0398 .8 75.4296 30.7876 .8 .9 149.. 5712 43.3540 .9 76.9769 31.1018 151.7468 43.6681 Dlam. Area. Clrcuiu. Dlam. Area. Clicuin. 14.0 .1 .2 .3 .4 153.9380 156.1450 158.3677 160.6061 162.8603 43.9823 44.2965 44.6106 44.9248 45.3389 8.0 .1 .8 .3 .4 254.4690 257.3043 260.1553 263.0820 265.9044 56..5486 56.8628 57.1770 57.4911 57.8053 .5 .6 .7 .8 .9 165.1300 167.4155 169.7167 173.03.% 174.3663 4.5.5531 45.8673 46.1814 46.4956 46.8097 .5 .6 .7 .8 .9 268.8025 271.7164 274.6459 277.5911 280.5521 58.1195 58.4336 58.7478 59.0619 59.3761 15.0 .1 .2 .3 .4 176.7)46 179.0786 181.4584 183.8539 186.2650 47.1239 47.4380 47.7522 48.0664 48.3805 19.0 .1 .2 .3 .4 283.5887 286.5211 289.5292 292.5530 295.5925 59.6903 60.0044 60.3186 60.6327 60.9469 .5 .6 .7 .8 .9 188.6919 191.1345 193.5928 196.0668 198.5565 48.6947 49.0088 49.3230 49.6372 49.9513 .5 .6 .7 .8 .9 298.6477 301.7i86 304.8058 307.9075 311.0255 61.8611 61.5752 61.8894 68.2035 62.5177 16.0 .1 .2 .3 .4 201:0619 203.5831 206.1199 208.6724 211.8407 50.2655 50.5796 50.8938 51.8080 51.6221 20.0 .1 .2 .3 .4 314.1593 317.3087 320.4739 323.6547 326.8513 62.8319 63.1460 63.4602 63.7743 64.0885 .5 .6 .7 .8 .9 213.8246 216.4243 219.0397 221.6708 234.3176 51.8363 52.1504 52.4646 52.7788 53.0929 .5 .6 .7 .8 .9 330.0636 333.2916 3a6.5353 339.7947 343.0698 64.4026 64.7168 65.0310 05.8451 65.6593 17.0 .1 .8 .3 .4 236.9801 829.6583 238 3522 235.0618 837.7871 53.4071 53.7212 54.0354 54.3496 54.6637 21.0 .1 .8 .3 .4 346.3606 349.6671 353.9894 356.3273 359.6809 65.9734 66.2876 66.6018 66.9159 67.2301 .5 .6 .7 .8 .9 240.5282 243.2849 246.0574 248.8456 251.6494 54.9779 55.2920 55.6063 55.9203 56.2345 .5 .6 .7 .8 .9 363.0503 366.4354 369.8361 373.2526 376.6848 67.5442 67.8584 68.1728 68.48W 68.8009 ROGERS' DRAWING AND DESIGN. 465 TABLES OF AREAS AND CIRCUMFERENCES OF CIRCLES— Continued. f ^-, — — DIanu' Area. Clrcum. Dlaiii. Area. Clrciini. 23.0 .380.1327 69.1150 26.0 530.9293 81.6814 .1 .<)83.5963 69.4292 .1 535.0211 81.9956 .2 SST.OTTO 69.7434 ^2 539.1287 82.3097 .3 390.5707 70.0575 .3 543.2521 82.6239 .4 394.0814 70.3717 .4 547.3911 82.9380 .5 397.6073 70.6858 .5 551.5459 83.2523 .6 401.1500 71.0000 .6 555.7163 83.5664 .7 404.7078 71.3142 .7 559.9025 83.8805 .8 408.2814 71.6283 .8 564 1014 84.1947 .9 411.870t 71.9425 .9 568.3220 84.5088 23.0 415.4756 72.2566 27.0 572.5553 84.8230 .1 419.0993 72.5708 .1 576.8043 85.1373 .2 422.7327 72.8849 2 581.0890 85.4513 .3 426.3848 73.1991 .3 585.3494 85.7655 .4 430.0526 73.5133 .4 589.6455 86.0796 .5 433.7361 73.8274 .5 593.9574 86.3938 .6 437.4354 74.1416 .6 598.2849 86.7080 .7 441.1503 74.4557 .7 602.6282 87.0231 .8 444.8809 74.7699 .8 606.9871 87.3363 .9 448.6273 75.0811 .9 611.3618 87.6504 24.0 452.3893 75.3982 28.0 615.7.522 87.9646 .1 456.1671 75.7124 .1 620.1582 88.2788 .2 459.9606 76.0265 .2 634.5800 88.5929 .3 463.7698 76.3407 .3 629.0175 88.9071 A 467.5947 76.6549 .4 633.4707 89.2313 .5 471.4352 76.9690 .5 637.9397 89.5354 .6 475.2916 77.2832 .6 642.4243 89.8495 .7 479.1636 77 ..5973 .7 646.9246 90.1637 .8 483.0513 77.9115 .8 651.4407 90.4779 .9 486.9547 78.2257 .9 055.9734 90.7920 25.0 490.8739 78.5398 29.0 660.5199 91.1063 .1 494.8087 78.8540 .1 665.0830 91.4203 .2 498.7592 79.1681 .2 669.6619 91.7343 .3 502.7255 79.48i3 .3 674.3565 9?.04>'7 .4 506.7075 79.7965 A- 678.8668 93.3628 .5 510.7052 80.1106 .5 683.4928 92.6770 .6 514.7185 80.4248 .6 688.1345 92.9911 .7 518.7476 80.7389 .7 692.7919 93.3053 .8 522.7924 81.0531 .8 697.4650 93.6195 .9 526.8529 81.3673 .9 702.1538 93.9336 nam. Area. Clrcniii. DIam. Area. Ctrcuin, 30.0 .1 .2 .3 .4 706.8583 711.5786 716.3145 721.0663 725.8336 94.2478 94.5619 94.8761 95.1903 95.5041 34.0 .1 ^2 ^3 .4 907.9203 913.2688 918.6331 934.0131 939.4088 106.8142 107.1283 107.4425 107.7566 108.0708 .5 .6 .7 .8 .0 730.6167 735.4154 740.2299 745.0601 749.9060 95.8186 96.1327 96.4469 96.7611 97.0752 .5 .6 .7 .8 .9 934.8203 940 2473 945.6901 951.1486 956.6328 108.3849 108.6991 109.0133 109.3274 109.6416 31.0 .1 .2 .3 .4 754.7676 759.6450 764.5380 769.4467 774.3712 97.3894 97.7035 98.0177 98.3319 98.6460 85.0 .1 .2 .3 .4 963.1138 967.6184 973.1397 978.6768 984.2290 109.9557 110.2699 110..5841 110.8982 111.2124 .5 .6 .7 .8 .9 779.3113 784.2672 789.2388 794.2260 799.2290 98.9602 99.2743 99 5885 99.9026 100.2108 .5 .6 .7 .8 .9 989.7980 995.3822 10009821 1006.5977 10133390 111.5265 111.8407 112.1549 113.4890 113.7832 32.0 .1 .2 .3 A 804.3477 809.2831 814 3322 819.39f0 824.4796 100.5310 100.8451 101.1593 101.4734 101.7876 36.0 .1 .2 .3 .4 1017.8760 1023.5387 1029.2172 1034.9113 1040.6212 113.0973 113.4115 113.7357 114.0398 114.3510 .5 .6 .7 .8 .9 829.5768 834.6898 839.8185 844.9628 850.1329 102.1018 102.4159 102.7301 103.0442 103.3584 .5 .6 .7 .8 .9 1046.3467 1052.0880 1057.8449 1063.6176 1069.4060 114.6681 114.9823 115.3965 115.6106 115.9348 33.0 .1 .2 .3 .4 855.3986 860.4903 865.6973 870.9202 876.15S8 103.6726 103.9867 104.3009 104.6150 104.9292 37.0 .1 .2 .3 .4 1075.2101 1081.0299 1086.8654 1092.7166 1098.5835 110.2389 116.5531 116.8672 117.1814 117.4956 .5 .6 .7 .8 .9 881.4131 886.6831 891.9688 897.3703 902.5874 105.2434 105.5575 105.8717 106.1858 106..5000 .5 .6 .7 .8 .9 1104.4662 1110.3645 1116.3786 1122.2083 1128.1538 117.8097 118.1239 118.4380 118.7523 119.0664 466 ROGERS' DRAWING AND DESIGN. TABLES OF AREAS AND CIRCUMFERENCES OF CIRCLES — Continued. Diam. Area.^ Cli'cnm. Dlain. Area. ^ Clrcuiu. Dfutn. Area. CIrcum. DIam. Area. CIrcum. 88.0 .1 .8 .3 .4 1134.1149 1140.0918 1146.0844 1152.0927 1158.1167 119.3805 119.6947 130.0088 120.3330 120.6372 42.0 .1 .3 .3 .4 1385.4434 1392.0476 1398.6685 1405.3051 1411.9574 131.9469 133.2611 132.5753 132.8894 133.2035 46.0 .1 •? .3 .4 1661.9025 1669.1360 1676.3853 1683.6502 1690.9308 144.5133 144.8274 145.1416 145.4557 145.7699 50.0 .1 '.3 A 1963.4954 1971.3572 1979.2348 1987.1280 1995.0370 157.0796 157.3938 157.7080 158 0221 158.3363 .5 .6 •7 .8 .9 1164.1564 1170.8118 1170.2830 1182.3698 1188.4724 120.9513 121.2655 121.5796 131.8938 123.3080 .5 .6 .7 .8 .9 1418.6254 1435.3093 1432.0086 1438.7238 1445.4546 133.5177 133.8318 134.1460 134.4602 134.7743 .5 .6 .7 .8 .9 1698.2272 1705.5392 1712.8670 1720.2105 1727.5697 146.0841 146.3982 146.7124 147.0265 147.3407 .5 .6 .7 .8 .9 2002.9617 2010.9020 2018.8581 2026.8299 2034.8174 158.6504 158.9646 159.2787 159.5989 159.p071 i 39.0 .1 .2 .3 A 1194.5906 1300.7246 1206.8743 1813.0396 1219.2207 122.5221 123.8363 123.1504 123.4646 123.7788 43.0 .1 .8 .3 .4 1452.2012 1458.9635 1465.7415 1472.5352 1479.3446 135.0885 135.4026 135.7168 136.0310 136.3451 47.0 .1 .2 .3 .4 1734.9445 1742.3351 1749.7414 1757.1635 1764.6012 147.6550 147.9690 148.2833 148 5973 148.9115 51.0 .1 .2 .3 .4 2042.8206 2050.8395 2058.8742 2066.9345 2074.9905 160.8813 160.5354 160.8495 161 1637 161.4779 .5 .6 .7 .8 .9 1225.4175 1231.6300 1237.8583 1244.1021 1250.3617 124.0929 134.4071 1347313 135.0354 135.3495 .5 .6 .7 .8 .9 1486.1697 1493.0105 1499.8670 1506.7393 1513.6273 136.6593 136.9734 137.2876 137.6018 137.9159 .5 .6 .7 .8 .9 1772.0546 1779. .5237 1787.0086 1794.5091 1803.0254 149.3257 149.5398 149.8540 150.1681 150.4823 .5 .6 .7 .8 .9 2083.0723 2u91.1697 2099.2829 2107.4118 2115.5563 161.7920 162.1062 162.4203 162.7345 163.0487 40.0 .1 .2 .3 .4 1256.6371 1262.9281 1269.2348 1275.5573 1281.8955 125.6637 125.9779 126.3920 126.6063 136.9303 44.0 .1 2 '.Z .4 1520.5308 1527.4503 1534.3853 1541.3360 1548.3035 138.2301 138.5443 138.8584 139.1726 139.4867 48.'0 .1 .2 .3 .4 1809.5574 1817.1050 1824.6684 18.S2.2475 1839.8423 150.7984 151.1106 151.4248 151.7389 152.0531 52.0 .1 2 !3 .4 2123.7166 2131.8926 2140.0843 8148.2917 3156.5149 163.3628 163.6770 163.9911 164.3053 164.6195 .5 .6 .7 .8 .9 1288.2493 1294.3189 1301.0043 1307.4053 1313.8219 127.2345 127.5487 137.8628 138.1770 138.4911 .5 .6 .7 .8 .9 1555.2847 1563.2826 1569.2963 1576.3255 1583.3706 139.8009 140.1153 140.4293 140.7434 141.0575 .5 .6 .7 .8 .9 1847.4528 1855.0790 1862.7210 1870.3786 1878.0519 152.3672 152.6814 152.9956 153.3097 153.6339 .5 .6 .7 .8 .9 8164.7537 2173.0082 2181.2785 2189.5644 2197.8661 164.9336 165 2479 165.5619 165.8761 166.1903 41.0 .1 .2 .3 .4 1320.2543 1326.7024 1333.1663 1339.6458 1346.1410 138.8053 139.1195 139.4336 139.7478 130.0619 45.0 .1 3 !3 .4 1590.4313 1597.5077 1604.5999 1611.7077 1618.8313 141.3717 141.6858 142.0000 142.3142 143.6283 49.0 .1 .2 .3 .4 1885.7409 1898.4457 1901.1662 1908.9024 1916.6543 153.9380 154.2522 154.5664 154.8805 155.1947 53.0 .1 .2 .3 .4 2206.1834 2214.5165 8232.8653 2231.8298 8239.6100 166.5044 166.8186 167.1327 167.4469 167.7810 .5 .6 .7 .8 .9 1352.6520 1359.1786 1365.7210 1372.2791 1378.8529 130.3761 130.6903 131.0044 131.3186 131.6327 .5 .6 .7 .8 .9 1625.9705 1633.1255 1640.^962 1647.4826 1654.6847 142.9425 143.2566 143.5708 143.8849 144.1991 .5 .6 .7 .8 .9 1924.4218 1938.2051 1940.0042 1947.8189 1955.6493 155.5088 155.8330 156.1378 156.4513 156.7655 .5 .6 .7 .8 .9 2248.0059 2256.4175 2264.8448 2273.2879 2281.7466 168.0752 168.3894 1(58.7035 169.017V 169.3318 ~^''^~~~ ROGERS' DRAWING AND DESIGN. 467 TART.RS OF ARFAS AND CTR.aiMFF.RF.NCF.S OF CIRCTP.S- Continued. DIaiQ. Area. Circum DIam. Area. CIrcuin. DIain. Area. circum. Dlam. Area. circum. 54.0 .1 . .2 .3 .4 8290.2210 2398.7112 2307.2171 2iJ15.7386 2324.2759 169.6460 169.9603 170.2743 170.5885 170.9026 58.0 .1 .2 .3 .4 2642.0794 2651.19T9 2660.3331 2669.4820 2878.6476 182.2134 182.5265 182.8407 183.1549 183.4690 62.0 .1 .2 .3 .4 3019.0705 3028.8173 3038.5798 3048.3580 3058.1520 194.7787 195.0929 195.4071 195.7212 196.0354 66.0 .1 .2 .3 .4 3421.1944 3431.5695 3441.9603 3452.3669 3463.7891 207.3451 207.6593 207.9734 208.2876 208.6017 .5 .6 .7 .8 .9 2332.8289 2341.3976 2349.9820 2258.5821 2367.1979 171.2168 171.5cl0 171.8451 J72.1593 172.4735 .5 .6 .7 .8 .9 2687.8289 2697.0259 2706.3386 2715.4670 2724.7112 183.7832 184.0973 184.4115 184.7256 185.0398 .5 .6 ,7 .8 .9 3067.9616 3077.7869 3087.6279 3097.4847 3107.3571 196.3495 196.6637 196.9779 197.2930 197.6062 .5 .6 .7 .8 .9 3473.3270 3483.6807 3494.1500 3504.6351 3515.1359 208.9159 209.2301 209..5443 209.8584 210.1725 55.0 .1 .2 .3 .4 2375.8294 2384.4767 2393.1396 2401.8183 2410.5126 172.7876 173.1017 173.4159 173.7301 174.0443 59.0 .1 .2 .3 .4 2733.9710 2743.2466 2753.5H78 2761.8448 2771.1675 185.3540 185.6681 185.9833 186.2964 186.6106 63.0 .1 .2 .3 .4 3117.2453 3127.1493 3137.0688 3147.0040 3156.9550 197.9203 198.2345 198.5487 198.8628 199.1770 67.0 .1 .3 !3 .4 3535.6.524 3536.1845 3546.7324 3557.2960 3567.8754 210.4867 210.8009 211.1150 211.4292 211.7433 .5 .6 .7 .8 .9 2419.2227 2427.9485 2436.6899 2445.4471 2454.2200 174.3584 174.6726 174.9867 175.3009 175.6150 .5 .6 .7 .8 .9 2780.5058 2789.8599 2799.2297 2808.6152 2818.0165 186.9248 187.3389 18,7.5531 187.8672 188.1814 .5 .6 .7 .8 .9 3166.9217 3176.9043 3186.9023 3196.9161 3206.9456 199.4911 199.8053 200.1195 200.4336 200.7478 .5 .6 .7 .8 .9 3578.4704 3589.0811 3599.7075 3610.3497 3621.0075 212.0.575 213.3717 212 6858 213.0000 213.3141 56.0 .1 .2 .3 A 2463.0086 2471.8130 2480.6330 2489.4687 2498.3201 175.9292 176.2433 1T6.5575 176.8717 177.18.i8 60.0 .1 .2 .3 .4 2827.4334 2836 8660 2846.3144 2855.7784 2865.2583 188.4956 188.8097 189.1239 189.4380 189.7522 64.0 .1 .8 .3 .4 3216.9909 3227.0518 3237.1285 3247.2222 3257.3289 201.0620 201.3761 201.6903 202.00'44 202.3186 68.0 .1 .2 .3 .4 3631.6811 3642.3704 3653.0754 3663.7960 3674.5324 213.628S 213.9425 214.2.J66 314.5708 214.88-19 .5 .6 .7 .8 .9 2507.1873 2516.0701 2524.9687 2533.8830 2542.8129 177.5000 177.8141 178.1283 178.4425 178.7566 .5 .6 .7 .8 .9 2874.7536 2884.2648 2893.7917 2903.3343 2912.8936 190.0664 190.3805 190.6947 191.0088 191.3230 .5 .6 .7 .8 .9 3267.4527 3277.5922 3287.7474 3297.9183 3308.1049 202.6327 202.9469 203.2610 203.5752 203.8894 .5 .6 .7 .8 .9 3685.2845 3696.0.-)23 3706.8359 3717.6351 3728.4500 215.1991 215.5133 215.8274 216.1416 216.4556 57.0 .1 .2 .3 .4 2551.7586 2560.7200 2569.6971 2578.6899 2587.6985 179.0708 179.3849 179 6991 180.0133 180.3274 61.0 .1 .2 .3 .4 2923.4666 2933.0563 2941.6617 2951.2828 2960.9197 191.6373 191.9513 192.3655 193.5796 193.8938 65.0 .1 .2 .3 .4 3318.3073 3328.5353 3338.7590 3349.0085 3359.3736 204.2035 204.5176 204.8318 205.1460 205.4602 69.0 .1 .3 ,3 .4 3739.2807 3750.1270 3760.9891 3771.8668 3782.7603 216.7699 217.0841 217.3982 217.7124 218.0365 .5 .6 .7 .8 .9 2596.7227 2605.7626 2614.8183 2623.8896 2632.9767 180.6416 180.9557 181.2699 181.5841 181.8983 .5 .6 .7 .8 .9 2970.5728 2980.2405 2989.9244 2999.6241 3009.3395 193.3079 193.5231 193.8363 191.1.504 194,4646 .5 .6 .7 .8 .9 3369.5545 3379.8510 3390.1633 3400.4913 3410.8350 205.7743 206.0885 206.4026 306.7168 307.0310 .5 .6 .7 .8 .9 3793.6695 3804.5944 3815.5350 3826.4913 3837.4633 218.3407 218.6548 218.9690 219.2832 219.5973 468 ROGERS' DRAWING AND DESIGN. TABLES OF AREAS AND CIRCUMFERENCES OF CIRCLES— Continued. Dfain. Area. CIrcuin. Diflm. Arcji. Circuin. 70.0 3848,4510 319.9115 74.0 4800.8408 232.4779 .1 3tf59.4544 220.3256 .1 4312.4721 283.7920 .3 3870.4786 230.5398 .3 43a4.1195 233.1063 .3 3881.5084 220.8540 .3 4335.7827 233.4203 A 3893.5590 331.1681 .4 4847.4616 238.7345 .5 3903.6353 331.4833 .5 4359.1563 284.0487 ' .6 8914.7073 321.7964 .6 4870.8664 234.3638 .7 8935,8049 332.1106 .7 43S3.5924 234.6770 .8 3936.9183 322.4248 .8 4394.3341 284.9911 .9 3948.0473 232.7389 .9 4406.0916 285.3053 71.0 3959.1921 333.0531 75.0 4417.8647 235.6194 .1 3970.3526 223.3672 .1 4439.6585 235.9336 .8 3981.5239 233.6814 .2 4441.4580 236.3478 .3 3992.7308 233.9956 .3 4453.2783 236.5619 A 4003.9384 334.8097 .4 4465.1142 236.8761 .5 4015.1518 224.6239 .5 4476 9659 337.1903 .6 4036.3908 334.9880 .6 4488.8332 387.5044 .7 4037.6456 255.3522 .7 4500.7163 337.8186 ,8 4048.9160 225..5664 .8 4513.6151 238.1327 .9 4060.3023 225.8805 .9 4524.5296 288.4469 73.0 4071.5041 226.1947 76.0 4536.459S 238.7610 .1 4083.8317 326.5088 .1 4548.4057 339.0752 .3 4094.1550 336.8330 .3 4560.3673 339.3894 .3 4105.5040 327.1371 .8 4572.3446 3:i9.7035 .4 4116.8687 327.4513 .4 4584.3i77 340.0177 .5 4138.3491 327.7655 .5 4596.3464 340.3318 .6 4139.6452 238.0796 .6 4608.8708 240.6460 .7 4151.0571 228.3938 .7 4620.4110 240.9602 .8 4163.4846 238.7079 .8 4632.4669 241.2743 .9 4173.9379 229.0221 .9 4344.5384 3415885 73.0 4185.3868 229.3363 77.0 4656.6357 241.9036 .1 4196.8615 329.6504 .1 4668.7287 242.2168 .3 4208.3519 339.9646 .2 4680.8474 242.5310 .8 4319.8579 330.2787 .3 4692.9818 342.H451 .4 4231.3797 330.5939 .4 4705.1319 343.1592 .5 4243.9173 230.9071 .5 4717.2977 343.4734 .6 4354.4704 231.3212 .6 4729.4793 343.7876 .7 4366.0394 381.5854 .7 4741.6765 344.1017 .8 4277.6340 331.8395 .8 4753.8894 344.4159 .9 4389.2343 232.1637 .9 4766.1181 344.7301 DIani. Area. CIrcum. Dlain. Area. CIreum. 78.0 .1 .2 .3 .4 4778.8624 4790.6335 4803.8983 4815.1897 4837.4969 245.0442 245.8584 245.6725 245 9867 246.3009 82.0 .1 .3 .3 .4 5281.0173 6393.9056 5306.8097 5319.7295 5382.6650 257.6106 257.9247 258.2389 358.5531 358.867a .5 .6 .7 .8 .9 4889.8189 4852.1584 4864.5128 4876.8838 4889.2685 346.6150 346.9293 247.2433 247.5575 247.8717 .5 .6 .7 .8 .9 5845.6163 5858.5832 5871.5658 5384.5641 5397.5782 359.1814 259.4956 359.8097 260.123? 260.4880 79.0 .1 .2 .3 .4 4901.6699 4914.0871 4936.5199 4938.9685 4951.4338 248.1858 248.5000 248.8141 349.12^3 249.4425 83.0 .1 .2 .3 .4 5410.6079 5423.6584 5486.7146 5449.7915 5462.8840 260.7522 361.0665 261.3805 361.6947 262.0088 .5 .6 .7 .8 .9 4963.9137 4976.4084 4988.9198 5001.4469 5018.9897 249.7566 350 0708 250.3850 250.6991 251.0138 .5 .6 .7 .8 .9 5475.9928 5489.1168 5502.2561 5515.4115 5528.5826 262.3230 262.6371 262.9513 263.2655 263.5796 80.0 .1 .3 .3 .4 5036.5482 5039.1335 5051.7134 5064.3180 5076.9894 251.3274 251.6416 251.9557 252.2899 252.5840 84.0 .1 .2 .3 .4 5541.7694 5554.9720 5568.1902 5581.4342 5594.6739 263.8938 264.3079 364.5221 264.8863 265.1514 .5 .6 .7 .8 .9 5089.5764 5102.3292 5114 8977 5127.5819 5140.2818 252.8983 353.3134 358.5265 253.8407 354.1548 .5 .6 .7 .8 .9 5607.9392 5621.2203 5634.5171 5647.8296 5661.1578 265.4646 265.7787 266.0939 266.4071 266.7212 81.0 .1 .3 .3 .4 5153.9973 5165.7287 5178.4757 5191.3384 5304.0168 254.4690 3.i4.7833 355.0973 255.4115 255.7256 85.0 .1 .3 .3 .4 5674.5017 5687.8614 5701.3367 5714.6277 5728.0345 267.0354 267.3495 267.6637 267.9779 268.2930 .5 .6 .7 .8 .9 5316.8110 5339.6208 5343.4463 5255.3876 .5368.1446 256.0398 256.3540 256.6681 256.9823 257.2966 .5 .6 .7 .8 .9 5741.4569 5754.8951 5768.8490 5781.8185 5795.3038 268.6062 268.9203 269.2345 269.5486 269.8628 ROGERS' DRAWING AND DESIGN. 469 TABLES OF AREAS AND CIRCUMFERENCES OF CIRCLES— Continued. »lain. Area. Clrcuni. Diam. Area. CIrcum. 86.0 .1 .2 .3 .4 6808.8048 5822.8315 5835.8539 5849.4020 5863.9659 270.2770 270.49U 370.8053 271.1194 271.4336 90.0 .1 ^2 '.3 .4 6361.7351 637.5.8701 6390.0309 6404.3073 6418.3995 282.7433 283.0575 283.3717 283.6858 284.0000 .5 .6 .7 .8 .9 5876.5454 5890.1407 5903.7516 5917.3783 5931.0206 271.7478 272.0619 272.3761 272.6902 273.0044 .5 .6 .7 .8 .9 6433.6073 6446.8309 6461.0701 6475.3351 6489.5958 284.3141 284.6283 284.9425 285.2566 285.5708 87.0 .1 .2 .3 A 5944.6787 5958.3525 5973.0430 5985.7473 5999.4681 273.3186 273.6337 273.9469 274.2610 274.5753 91.0 .1 .3 .3 .4 6503.8833 6518.1843 6532.5021 6.546.8356 6561.1848 285.8849 286.1991 286.5133 286.8274 287.1416 .5 .6 .7 .8 .9 6013.2047 6036.9570 6040 7250 6054.5088 6068.3082 274.8894 275.2035 275.5177 275.8318 276.1460 .5 .6 .7 .8 .9 657.5.5408 6589.9304 6604 3268 6618.7388 6633.1666 287.4557 287.7699 288.0840 288.3983 288.7124 88.0 .1 .2 .3 .4 6083.1384 6095.9543 6109.8008 6133.6631 6137.5411 276.4602 276.7743 277.0885 277.4026 277.7168 93.0 .1 .3 .3 .4 6647.6101 6662.0692 6676.5441 66910347 6705.5410 289.0265 289.3407 289.6548 289.9690 290.2882 .5 .6 .7 .8 .9 6151.4318 6165.3443 6179.3693 6193.2101 6307.1666 278.0309 278.3451 278.6563 278.9740 279.2876 .5 .6 .7 .8 .9 6720.0630 6734.6008 6749.1542 6763.7233 6778.3083 290.5973 390.9115 391.3256 391.5398 291.8540 89.0 .1 .2 .3 .4 6331.1389 6335.1268 6249 1304 6363.1498 6377.1849 279 6017 279.9159 280.2301 280 5442 280.8584 93.0 .1 .2 .3 .4 6792.9087 6807.5250 0822.1569 6836.8046 6851.4680 393.1681 393.4833 393.7964 393.1106 293.4348 .5 .6 .7 .b .9 6291.3356 6305.3031 6319.3843 6333.4822 6347.5958 281.1725 281.4867 281.8009 282.1150 282.4292 .5 .6 .7 .8 .9 6866.1471 6880.8419 6895.5524 6910.2786 6935.0205 293.7389 294.0531 294.3673 294.6814 294.9956 Dlain. Area. Clrcuin. DIam. 'Area. Clrcimi. 94.0 .1 .2 .3 .4 6939.7783 6954.5515 6969.3106 6984.1453 6998.9658 295.3097 295.6239 295.9380 296.2522 296.5663 97.0 .1 .2 .3 .4 7389.8113 7405.0559 7420.3162 7435.5922 7450.8839 304.7345 305.0486 305.3628 305.6770 305 9911 .5 .6 .7 .8 .9 7013.8019 7028.6538 7043.5214 7058.4047 7073.3033 296.8805 297.1947 297.5088 297.8230 298.1371 .5 .6 .7 .8 .9 7466.1913 7481.5144 7496.8532 7.521.2078 7527.5780 306.3053 306.6194 306.9336 307.3478 307.5619 95.0 .1 .2 .3 .4 7088.2184 7103.1488 7118.1950 7133.0.568 7148.0343 298.4513 298.7655 299.0796 299.3938 299.7079 98.0 .1 .2 .3 .4 7542.9640 7558.3656 7573.7830 75892161 7604.6648 307.8761 308.1903 308.5044 308.8186 309.1327 .5 .6 .7 .8 .9 7163.0276 7178.0366 7193.0612 7208.1016 7223.1577 300.0221 800.3363 300.6504 300.9646 '301.2787 .5 .6 .7 .8 .9 7620.1293 7635.6095 7651.1054 7666.6170 7682.1444 309.4469 809.7610 310.0752 310.3894 310.7035 96.0 .1 .2 .3 .4 7238.2295 7253.3170 7268.4202 7283.5391 7298.6737 801.5929 301.9071 302.2313 302.5354 302.8405 99.0 .1 .2 .3 .4 7697.6893 7713.2461 7728.8206 7744.4107 7760.0166 311.0177 311.3318 311.6460 311.9602 312.2743 .5 .6 .7 .8 .9 7313.8240 7328.9901 7344.1718 7359.3693 7374.5824 803.1637 303.4779 303.7920 304.1062 304.4203 .5 .6 .7 .8 .9 7775.6383 7791.2754 7806.9284 7823.5971 7838.2815 312.5885 312.9026 313.2168 313.5309 313.8451 100.0 7853.9816 314.1593 470 ROGERS' DRAWING AND DESIGN. CIRCULAR MEASURE. 60 seconds (") make i minute ('). 60 minutes " 1 degree (°). 360 degrees " i circum. (C). The circumference of every circle whatever, is supposed to be divided into 360 equal parts, called degrees. A degree is -3^^ of the circumference of any circle, small or large. A quadrant is a fourth of a circumference, or an arc of 90 degrees. A degree is divided into 60 parts called minutes, expressed by the sign ('), and each minute is divided into 60 seconds, expressed by (") ; so that the circum- ference of any circle contains 21,600 minutes, or 1,296,000 seconds. LONG MEASURE- 12 inches = i foot. 3 feet = I yard. 55^ yards = i rod. -MEASURES OF LENGTH. 40 rods = I furlong. 8 furlongs = i common mile. 3 miles = I league. The mile (5,280 feet) of the above table is the legal mile of the United States and England, and is called the statute mile. ROMAN TABLE. I. den otes One. XVII. denotes Seventeen. II. ' Two. XVIII. ' Eighteen. III. ' Three. XIX. ' Nineteen. IV. ' Four. XX. ' Twenty. V. ' Five. XXX. ' Thirty. VI. ' Six. XL. ' Forty. VII. . ' ' Seven. L. Fifty. VIII. Eight. LX. ' Sixty. IX. ' Nine. LXX. ' Seventy. X. ' Ten. LXXX. ' Eighty. XI. ' Eleven'. XC. ' Nfnety- XII. ' Twelve. C. ' One hundred. XIII. ' Thirteen. D. ' Five hundred. XIV. ' Fourteen. M. ' One thousand XV. ' Fifteen. X. ' Ten thousand XVI. ' Sixteen. M. ' One million. GREEK ALPHABET. A a alpha I I iota P P rho B ^ . beta K K kappa 2 (T . sigma r y . gamma A A . lambda T T . tau A S delta M 11 mu Y V . upsilon E € epsilon N V nu 4> . phi i^ ^ zeta ^ $ . xi X X . cbi H ^ eta omicron ^ 'P psi e Q theta n TT . pi fl b) . omega Note. — The letters of the Greek alphabet are used sometimes as arbitrary signs, and the letter tt (pi) is used almost universally to represent the ratio of the circumference to the diameter of the circle. ROGERS' DRAWING AND DESIGN. 471 TABLES Of Squares and Cubes, and Square and Cube Roots of numbers from i to 200. (See opposite column.) RULES. To And side of an inscribed square- Multiply diameter by 0.7071. Or multiply circumference " 0.2251. Or divide " " 4.4428. To £nd side of an equal square — Multiply diameter Or divide Or multiply circumference Or divide by 0.8862. " 1. 1 264. " 0.2821. " 3.545- Square — A side multiplied by 1.4142 equals diameter of its circumscribing circle. A side multiplied by 4.443 equals circumference of its circumscribing circle. A side multiplied by 1.128 equals diameter of an equal circle. A side multiplied by 3.544 equals circumference of an equal circle. A side multiplied by 1.273 equals circle inches of an equal circle. SQUARES, CUBES AND ROOTS. Number. Square. Cube. Square Root. Cube Root. 1 1 1 1.0 1.0 2 4 8 1.414213 1.25992 8 9 27 1.732050 1.44225 4 16 64 2.0 1.58740 5 25 125 2.236068 1.70997 6 36 216 2.449489 1.81712 7 49 343 2.645751 1.91293 8 64 512 2828427 2.0 81 729 3.0 2.08008 10 100 1000 3.162277 2.15443 11 121 1331 3.316624 2.22398 12 144 1728 3.464101 2.28942 13 169 2197 3.6055.M 2.35133 14 196 2744 3.741657 2.41014 15 225 3375 3.872983 2.46621 16 256 4096 4.0 2.51984 17 289 4913 4.123105 2.57128 18 324 5832 4.242640 2.62074 19 361 6859 4.358898 2.66840 20 400 8000 4.472136 2.71441 21 441 9261 4.582575 2.75892 22 484 10648 4.690415 2.80203 23 529 12167 4.795831 2.84386 24 576 13824 4.898979 2.88449 25 625 15625 5.0 2.92401 26 676 17576 5.099019 2.962i9 27 729 19683 .5.196152 3.0 28 784 21952 5.291502 3.03658 29 841 24389 5.385164 3.07231 30 900 27000 5.477225 3.10723 81 961 29791 5.567764 3.14138 32 1024 32768 5.656854 3.17480 33 1089 35937 5.744563 3.20753 34 1156 39304 5.830951 3.23961 35 1225 42875 5.916079 3.27106 36 1298 46656 6.0 3.30192 37 1369 50653 6.082763 3.33333 38 1444 54872 6.164414 3.36197 39 1621 59319 6.244998 3,39121 40 1 1600 64000 6.324555 3.41995 473 ROGERS' DRAWING AND DESIGN. TABLES OF SQUARES, CUBES Nuirflier. Square. Cube. Square lioot. Cube Root. 41 1681 68921 6.403134 3.44821 42 1764 74088 6.480740 3.47603 43 1849 79507 6.557438 3.50339 44 1936 85184 6.6J3349 3 53034 45 2025 91125 6.708303 3.55689 46 2116 97336 6.783330 3.58304 47 2209 103823 6.855654 3.60882 48 2304 110592 6.938303 3.63424 49 2401 117649 7.0 3.65930 50 2500 125000 7.071067 3.68403 51 2601 132651 7.141428 3.70843 53 2704 14060W 7.311102 3.73251 53 3809 148877 7.280109 3.75628 54 2916 157464 7.348469 3.77976 55 3025 166375 7.416198 3.80295 56 3136 175616 7.483314 3.82586 57 3349 185193 7.549834 3.84850 58 3364 195112 7.615773 3.87087 59 3481 205379 7.681145 3.89299 60 3600 216000 7.745966 3.91486 61 8T21 226981 7.810249 3.93649 62 3844 238328 7.874007 3.95789 63 3989 250047 7.937253 3.97905 64 4096 263144 8.0 4.0 65 4225 274625 8.063357 4.02072 66 4356 287496 8.124038 4.04124 67 4489 300763 8.185352 4.06154 68 4624 314432 8.216211 4.08165 69 4761 328509 8.306623 4.10156 70 4900 343000 8.366600 4.12128 71 5041 357911 8.436149 4.14081 73 5184 373248 8.485281 4.16016 78 5329 389017 8.544003 4.17933 74 5476 405224 8.603335 4.19833 75 5635 421875 8.660354 4.21V16 76 5776 438976 8.717797 4.23582 77 5939 456533 8.774964 4.25432 78 6084 474552 8.831760 4.27265 79 6241 493039 8.888194 4.29084 80 6400 512000 8.944271 4.30887 AND ROOTS- -Continoed. Number. Square. Cube. Square Uoot. Cube Boot. 81 6561 531441 9.0 4.33674 83 6734 551368 9.055385 4.34448 83 6889 571787 9.110438 4.36207 84 70.56 592704 9.165151 4.37951 85 7235 614125 9.218544 4.39683 86 7396 636056 9.273618 4.41400 87 7569 658503 9.327379 4.43104 88 7744 681472 9.380831 4.44796 89 7921 704969 9.433981 4.46474 90 8100 729000 9.486833 4.48140 91 8381 753571 9.539392 4.49794 93 8464 778688 9.591663 4.51435 93 8649 804357 9.643650 4.53065 94 8836 830584 9.695359 4.54683 95 9025 857375 9.746794 4.56290 96 9316 884736 9.797959 4.57785 97 9409 913673 9.848S57 4.59470 4.61043 98 9604 941193 9.899494 99 9801 070399 9.949874 4.62606 100 10000 1000000 10.0 4.64158 101 10201 1030301 10.049875 4.65701 103 10404 1061308 10.099504 4.67233 103 10609 1093737 10.148891 4.68754 104 10816 1124864 10.198039 4.70266 105 11025 1157625 10246950 4.71769 106 11236 1191016 10.295630 4.73263 107 11449 1225043 10.344080 4.74745 108 11664 1259712 10.392304 4.76230 109 11881 1295029 10.440306 4.77685 110 13100 1331000 10.488088 4.79143 111 12321 1367631 10.535653 4 80589 112 12.544 1404928 10.583005 4.82028 113 12769 1443897 10.630145 4.83458 114 13996 1481544 10.677078 4.84880 115 13;'35 1520875 10.723805 4.86294 116 134.56 1560896 10.770329 4.87699 117 13689 1601613 10.816653 4.89097 118 13924 1643032 10.862780 4.94086 119 14161 1685159 10.90871£ 4.91868 120 14400 1728000 10.954451 4.93242 ROGERS' DRAWING AND DESIGN. 473 TA RT,RS OF Square Hoot, SQUARES, CUBES AND ROOTS— Continued. Nomber. Square. Cube. Cul)e Root. Nuinlifr. Square. Culie. Square Root. cube Root. 121 14641 1771561 11.0 4.94608 161 25921 4173281 12.688577 5.44012 122 14884 1815848 11.045361 4 95967 163 26244 4351538 12.727922 5.45136 123 15129 1860867 11.090536 4.97319 163 26569 4330747 12.767145 5.43255 124 15376 1906634 11.135528 4.98663 164 26896 4410944 12.806348 5.47370 125 15625 1S53125 11.180339 5.0 165 27225 4493125 12.845232 5.48480 1S6 15S76 2000376 11.224972 5.01329 166 27556 4574296 13.884098 5.49586 127 16129 2048383 11.269427 5.02653 167 27889 4657463 13.923848 5.50687 128 16384 2097152 11.313708 5.03968 168 28224 4741632 1'3.961481 5.51784 129 16641 2146689 11.357816 5.05377 169 28561 4826809 13.0 5.53877 130 16900 2197000 11.401754 5.06579 170 28900 4913000 13.038404 5.53965 131 17161 2348091 11.445523 5.07875 171 29241 5000211 13.076696 n.55049 132 17424 2399968 11.489125 5.09164 172 29584 5088448 13.114877 6.56139 133 17689 2352637 11.532562 5.10446 173 29929 5177717 13.153946 5.57305 134 17956 2406104 11.575836 5.11723 174 30276 5268024 13.190906 5.58377 135 18225 2460375 11.618950 5.12992 175 p0625 5359375 12.238756 5.59344 136 18496 2515456 11.661903 5.14256 176 80976 5451776 13.266499 5.60407 137 18769 2571358 11.704699 5.15513 177 31329 5545233 13.304134 5.61467 138 19044 2638072 11.747344 5.16764 178 31684 5639752 13.341664 5.62533 139 19321 2685619 11.789826 5.18010 179 32041 5735339 13.379088 5.63574 140 19600 2744000 11.832159 5.19349 180 33400 5833000 13.416407 5.64621 141 19881 2808221 11.874342 5.20482 181 32761 5939741 13.453634 5.65665 142 20164 2863288 11.916375 5.21710 182 33124 6038568 13.490737 5.66705 143 20449 2924207 11.958260 5.23932 183 33489 6138487 13.537749 5.67741 144 20736 2985984 13.0 5.24148 184 33856 6339504 13.564660 5.68773 145 21025 8048625 12.041594 5.35358 185 34225 6331625 13 601470 5.69801 146 21316 3112136 12.083046 5.36563 186 34596 6434856 13.638181 5.70826 147 21609 3176528 12.123455 5.27763 187 34969 6539303 13.674794 5.71847 148 31904 8241792 12.165525 5.38957 188 35344 6644672 13.711309 5.72865 149 22201 3307949 13.266S55 5.30145 189 35721 6751269 13.747727 5.73879 150 22500 3375000 12.347448 5.31329 190 36100 6859000 13.784048 5.74889 151 22801 3442951 13.388305 5.33507 191 36481 6967871 13 820275 5.75896 152 23104 3511808 13.338828 5.33680 192 36864 7077888 13.856406 5.76899 153 23409 3581577 13.369316 5.34848 193 37249 7189057 13.892444 5.778&9 154 23716 3653264 12.409673 5.36010 194 37636 7301384 13.928388 5.78896 155 24025 3733875 12.449899 5.37168 195 38035 7414875 13.964340 5.79889 156 24336 3796416 12.489996 S.S8323 196 38416 7529536 14.0 5.80878 157 24649 3869893 12.529964 5.39469 197 38809 7645373 14.035668 .5.81864 158 24964 3944312 13.569805 5.40613 198 39304 7763392 14.071247 5.82847 159 35281 4019679 13.609520 5.41750 199 39601 7880599 14.106736 5.83827 160 25600 4096000 12.649110 5.42883 200 4on(>n 8000000 14.142135 5.84803 474 ROGERS' DRAWING AND DESIGN. UNITED STATES STANDARD SIZES OF WROUGHT IRON WELDED PIPE. Length of pipe per Length of pipe per Lenmh of No. of Inside Actual Thick- Actual External iDteroal square square Biternal Actual pipe con. We/ght threads Lengtb outelde ness. Inside circum- clrcdm- foot of loot of area, totoraal talniQg per foot per loch perfect Dom. Diameter. Diameter. ference. fereocj. outside surface Inside surface. area; one cubic fj'-'t. or length. of scrow. screw 1 .405 .068 0.269 1.272 0.848 9.440 14.15 .129 .0572 2500. .243 27 ■> 0,19 i .54 .088 0.364 1.696 1.144 7.075 10.50 .229 .1041 1385. .422 18 0.29 t .675 .091 0.493 2-121 1.552 5.657 7.67 .358 ,1916 751.5 ,561 18 0,30 4 - .840 .109 0,622 2.652 1.957 4.502 6.13 .554 .3048 472.4 .845 14 0,39 \ ■ 1.050 .113 0.824 3.299 2.589 3.G37 4.635 .866 .5333 270,0 1.126 14 0.40 1 1.315 .134 1.047 4.1,34 3.292 2.903 3.679 1.357 .8627 166.9 1.670 lU 0.51 H 1.660 .140 1.38 5,215 4.335 2.301 2.768 2.164 1.496 96.25 2.258 \n 0.54 \h 1.90 .145 1,61 5.9G9 5.061 2.010 2.371 2.835 2.038 70.65 2.694 11} 0.55 2' 2.375 .154 2.067 7.461 6.494 1.611 1.848 4.430 3.355 42.36 3.667 Hi 0.58 2 J 2.875 .204 2.467 9.032 7,754 1.328 1.547 6.491 4.783 30.11 5.773 8 0.89 3 3.50 .217 3.066 10.996 9.636 1.091 1.245 9.621 7.388 19.40 7.547 8 0.95 3i 4.0 .226 3.548 12,566 11.146 .955 1.077 12.566 9.837 14.56 9.055 8 1.00 4 4.60 .237 4.026 14.137 12,648 .849 0.949 15.901 12.730 11.31 10.728 8 1.05 41 5.0 .247 4.506 15.708 14.153 765 0.848 1,9.635 15.939 9.03 12.492 8. 1.10 5 5.563 .259 5.045 1-7.475 15.849 629 0.757 24.299 19,990 7.20 14,564 8 1.16 6 6.625 .280 6.065 20.813 19.054 .577 0.630 .34.471 28.889 4.98 18.767 8 1.26 7 7.625 .301 7.023 23.954 22.063 .505 0.544 45.663 38.727 3.72 23.410 8 1..36 8 8.625 .322 7.981 27.096 25.076 .444 0.478 58.426 50.039 2.88 28.348 8 1.46 9 9.688 .344 9.00 30.433 28.277 .394 a425 73.716 63-633 2.26 34.677 8 1,57 10 10.750 .366 10,018 33.772 31.475 .355 0.381 90.762 78.838 1.80 40.641' 8 1.68 Thread taper three-fourths inch to one foot. All pipe belo vf lyi inches is butt-welded, and proved to 300 pou nds per sc luare inch ; i^i inch and above is lap- welded and proved || to 500 pounds per s quare inch. INDEX. Abbreviations and Conventional Signs 21-22 Acute Angle, def 29 Addendum Circle, desc. and illus 279-280 Advantages of Algebra 431 Corliss Valve Gear 390 Logarithms, desc . 433 Algebra — Advantages of 431 Elements of 430 Algebraic — def. and example 431 Alphabet, Antique, desc. and illus. ... 52 Greek 46S Altitude of a Pyramid or Cone 38 A Triangle, def 31 Altitude of a Triangle, def ... 31 Aluminum, when discovered 307 Andrews, Pres't of Nebraska Univer- sity, quotation 205 Angle, Acute, def 29 Designation of by three letters . . 29 Def 27-29 Obtuse and Oblique, def 29 Right, def 29 Of Advance of Eccentric, desc . . 372 Angle, to divide into four parts, illus. and rule 88 Of Screw Thread, desc. and illus. 234 To transfer, illus. and rule 92 To bisect an, illus. and rule. ... 87 Annular Gear, desc. and illus 292 Antimony, when discovered 307 Antique Alphabet, desc. and illus 52 Apex of an Angle, def 29 PAGE Apex of Pyramid, def 38 Applied Mechanics, def 212 " Apron " of Lathe, desc 322 Arc of a Circle, def 33 Tangent and sine of an 34 To find center of 97 Areas and Circumferences of Circles, Tables of 461-467 Armature, desc 399 Construction of, desc. and illus.. 400-401 Core, desc 400 Disc, illus 312 " Arrow Heads," vi'here placed 1S5 Ash Pit of Boiler, illus. and desc 336 Assembled Drawings, def 1S3 Atmospheric Electricity, def 397 Atom, def 212 Attraction, def 207 " Axis " used in Cabinet Projection. . . I2r Axiom, def 431 Axioms, def. and examples S5-S6 Axis of the Parabola, def 36 Dabbit, Sectioning of 79-80 How shown by colors 1S8 Baffle Plates, desc 350 " Bath " for Blue Prints, Solution for. 195 Battery of Boilers, desc , . 350 Beam Compasses, illus 417 Bearing, illus 1 27 Bearing, Self-oiling, desc. and illus. . . 403-404 Bearings — Forms of, illus, and desc. . . 259-260 PAGE " Bed " — Steam Engine, desc 364 Of a Press, desc 308 Bed Plate, how to draw 146 Bell Crank, desc 390 Belts and Pulleys, desc 266-277 Crossed, illus 267 Fly-wheel, illus, 3C3 Details of 365-366 Horse Power of, rule and ex- ample 269-270 Rules, Forniuhe ami Examples for Speed of . . 268-270 Bench Drill Press, desc. and illus 317 Bending Moments, examples of, def. . 221-226 Stresses Induced by, def 220 Bevel Gears 282 Desc. and illus 292-299 How to Construct, illus. and desc. 254-297 Bismuth, when discovered 307 Bipolar Dynamo, def 398 Bisect, an Angle, to 87 A vStraight Line, to, illus 87 Black Process Copying 192-193 Blanking Die, illus. and desc 308 Block Letters and Numerals 54 Blocks, Pillow, desc. and illus 263-264 Blow=off Pipe, desc 339 Blue Priut, colored illustration 202 Blue Printing, desc 189-192 Printing, Test Pieces for 1 91-192 " Bath " for, Solution 195 Blue Prints, Mounting of, desc 195-196 475 476 ROGERS' DRAWING AND DESIGN. PAGE Blue Prints, to make Drawings from. . 196 Blue, Prussian, for Water Colors 188 Boiler Bracket, illus. and desc 340-346 Cornish, illus. and desc. . . .340, 341, 347 Cylindrical Tubular. 340-345 Dome, Development of, illus. , . . 175-176 Fire Box, desc 344 Flue, illus. and desc 340-341 Furnace, illus. and desc 336-337 Galloway, illus 344-348 Lancashire, illus. and desc 341 Locomotive, illus. and desc 344-346 Plain Cylindrical, illus 336-337 Slope Sheet, Development of, illus 176-179 Stays, illus. and desc 340-346 Vertical, illus. desc 346-347 Water Tube, Dimensions of a.. . . 350-355 Boilers and Engines, desc 335 Battery of, desc 350 Grate Surface of, desc 336 Heating Surface of, desc 336 Horse Power of, desc 354-356 Steam Space of, desc 336 Steam, desc 336 Water Tube, desc. and illus 347-348 Water Line of, desc 336 Bolt-head, Square, desc. and illus. . . . 240 Proportions of, desc. and illus. . . 237 Bolt-sheets, desc 230 Bolts, Stay, illus. and desc 346 Stud, desc. and illus 241 Weakest Part of 241 Number of, for Cylinder Head . . 379 Table of Tensile Strength of . . . . 241 Border Lines, when to be used 423 Bore Dividers, illus 416 Pen and Pencil, desc. and illus.. 416 PAGE Brackets— Wall, desc. and illus 262 Brass, Sectioning of, illus 79-80 How shown by colors. . . 188 Brasses, Compositions for 265 For Pillow Blocks 263, 264 Drill Speed for 314 Brick, Sectioning of, illus 80-81 Bridge Wall of Steam Boiler, illus. and desc 336 " Bromide " Sensitized Paper 194 Bronze Age, Implements used in 307 Brushes, desc 399 Brush Holder Frame, desc. and illus. . 403, 405 Reaction, desc. and illus 405-406 Burlingame, L. D., quotation from address 196-197 Butt-joint, illus. and desc 249 C^abinet Projection, def 113 Desc. and illus 121-122 Problems in 122-127 " Cap " Drawing Paper, size of 423 Capital Letters, use of 53 Carmine for Water Color 188 Castings, how shown by colors 1S8 Cast Iron, desc 214-215 Sectioning of 78-80 Factors of Safety-bar 217 Center of a Circle, def 33 Line, def 27 Dead, of Steam Engine, desc. . . 364 Chain Riveting, desc. and illus 250 Changing Gears for Screw Cutting 322 Chapman, Jno. G. , Quotation Check Nut, desc. and illus 240 Chimney, desc 336 Chord of a Circle, def. and illus 33 ' ' Chrome Yellow " for Water Color. . 188 PAGE Circle, Arc of a, def 33 Concentric, def 34 Circumference of a 33 Chord of a 33 Diameter of a 33 Illus. of a 33 Segment of, def 33 Rules Relating to 4 1-465 Circles and their Properties 33 Eccentric, def 35 Sector of, def 33 To Draw through three points. . . 98 Circular Measure '.,. 468 Velocity, def 211 Pitch Circle of a Gear 279-280 Circumference of a Circle, def 33 Classification of Machines, desc 215 Of Electricity 396 " Clearance" of Die. 308 Cleveland Twist Drill— Table of Drill Speed 314 Co-abbreviation of Complement, 33 Coefficient, def 207 Of Safety, def 210 Cohesion, def 207 Coloring Drawings 18S Combination Die, illus 313 Combustion, Products of 346, 347 Commercial Rating of Engines 367 Commutator, desc 399 Commutator, desc. and illus 401-402 Compasses, desc. and illus 414-415 Beam, illus 417 Composition for Brasses, desc 264 Compound Winding, desc 406-407 Compressive Strain, def .■ 217 Concave, def 33 Concentric Circles, def 34 INDEX. 477 PAGE Cone, def . and illus 38 Pulleys, desc. and illus 275, 277 Conic Sections, def 37, 160, 161 ConicaUhead Rivet, How to Draw, desc. and illus 244 Construction — Line, def 27 Materials for, def .- 214-215 Of Armature, desc. and illus. . . . 400-401 Of Commutator 401-402 Contents, Table of xix Table of 21 Conventional Method of Showing Square Headed Screws, illus. . 241 Signs for Drawing Threads 235-237 Convex, def 33 Double, def 33 Copper, Factor of Safety for 217 How Shown in Colors 18S Copyright of Work xii Corliss Engine, Fishkill Landing, desc. and illus 388-390 Valves, desc 388 Valve Gear, illus 383-390 Releasing Gear, illus 386-387 Cornish Boiler, illus. and desc. . . .340, 341, 347 Corollary, def 85 Co-sine, Abbreviation of 34 Of an Arc, illus 33 Cotangent of a Circle, illus 33 Countersunk-rivet, How to Draw, desc. and illus 244 Coupling, Flange, illus 127 Cover Plate Joint, illus. and desc 249 Cranic, Rule for Finding Length of Stroke of 370 Bell, illus 127, desc, 390 Pin — Dimensions of 388 Shaft of Steam Engine 382 Crauk, To Draw by Isometric Projec- tion 119 To Draw by Cabinet Projection. . 125 Crosshead, desc 389 Guides of the Steam Engine. . . . 362-364 Of Steam Engine, desc. and illus. 382 Pin 388 Cube, illus. and def 37 To Draw a, by a Cabinet Projec- tion 122 Cup-head Rivet, How to Draw 243, 244 Current Electricity, def 396 Curved Line, def 27 Curves and Sweeps, desc. and illus . . . 420-422 Cycloid, The, def. and illus 2S5-286 Cycloidal Gear Teeth, def. and illus. . . 285-293 Rack , def 292 Cylinder, def. and illus 37 Of a Steam Engine, desc 362 Of Corliss Engine, illus 384-385 To Draw by Isometric Projec- tion, illus H6-11S To Draw a, by Cabinet Projection 123-125 With Square Flange, How to Draw 145 Walls, etc , of Steam Engine, Thickness of 379 Cylinders — Development of Their In- tersection, desc. and illus 169-172 How to Draw by Orthographic Projection 156-159 Cylindrical Boiler, plain 336-337 Tubular Boiler, desc. and illus. . 340, 345 Ring, How to Draw by Ortho- graphic Projection 144 Damper, Chimney, desc 336 Dash Pot, Corliss 390 PAGE Data and Rules, useful 429 "Dead Center" of Steam Engine, desc 364 Decagon, def 32 Decimal Equivalents, Table of 457 Equivalents of Millimeters and Fractions 458 " Dedendum " Circle of Gear Wheel. . 279, 280 Dedication by Author vii Definitions and Terms 27-40 Algebraic 43 1 Definitions and General Considera- tions Relating to Machine De- sign 207-2 1 1 " Demy " Drawing Paper, Size of ... . 423 " Density," def 213 Design, Machine 205-206 Designing a Steam Boiler, desc 350 Machines, six points in 307 Detailed Drawings, def 183 Development of a Boiler Dome, illus.. 175-176 A Four-part Elbow, illus 172-175 A Tee Pipe 165-169 Of the Slope Sheet of a Locomo- tive Boiler, illus 176-179 Right Elbow 162-164 Surfaces, def 113 Surfaces, illus. and desc 162-179 Surfaces, problems in 162-179 Diagonal, def 32 Stays, desc 340 Diagram, Indicator, desc. and illus. 375, 376, 378 Of Dimensions of Horizontal En- gines 369 Of Way to Read Drawings 190 Zenner's, illus. and desc 377 Diameter of a Circle, illus 33 Of Journals, example 258 478 ROGERS' DRAWING AND DESIGN. Diameter of Screw, desc 233 " Diametral Pitch," def 280 Die, Blanking, desc. and illus 308 Disc, Cutting, desc. and illus. . . 313 Male and Female, desc 308 Dies and Presses, desc. and illus 3o8-3>5 Blanking, illus. and desc 308 Drawing, desc. and illus. . . .310, 312-313 Gang of, desc. and illus 308 Punches, Groups of 308 Dimension — Line, def 27 Lines 1 85- '.86 " Dimensions " How Written on Drawings 185 Of Drawings 184-187 Of Horizontal Steam Engines . . . 369 Of Pulleys 272-273 Dimensions of Steam Boilers 352-354 Directrix, def 36 Disc Armature, note 400 Dividers, desc. and illus 415 Division, sign of 23 Bow, illus 416 Dodecagon, def 32,40 Double Threaded Screw, desc 230 Draft, Split, desc 341 Drafting-room as an Interpreter to the Shop 197 Drawing a Cup-Head Rivet 243-244 A Hexagonal-nut, desc. and illus. 237-239 Helix 228-230 Instruments 411-426 Linear, Subject of 25-81 Paper ^ 422-423 Pen, illus 416-417 Tools, Good ones Necessary 413 To Scale, Instructions for 424-425 With Relation to Shop Work. . . 196-198 PAGE Drawings, Working, General Subject. 181-199 Coloring of 188 Dimensioning of 184-187 Marking of 184 To Make from Blue Prints 196 Tracing of 189-190 Drawing-board, illus 4o-4i Class, Eugene C. Peck's method of Conducting, Note 413 Dies, desc. and illus 310, 312-313 Geometrical 85-1 10 Ink 417-418 Drilling Machines 314-317 Drill Press — Bench, desc. and illus. ... 317 Speeds — Table of 314 Driving Pulleys, desc 266 Ductility, def 207-213 Dynamic Electricity, def 397 Dynamics, def 212 Dynamo — Bipolar, def 398 Electric, Machinery, desc 393 Illus 394, 395 Meaning of word 398 Multipolar, def 398 The Electric, desc 398-399 Unipolar, def 398 Eccentric Circles, def. and illus 35 Desc. 370, illus 370, 373 Eccentric, Rule for Finding Length of Stroke of 370 Strap and Rod, desc. and illus. . . 370, 374 Eccentricity, Radius of, def 35 " Efficiency," def 207 Of Electric Motor, desc 399 " Effort," def 207 " Elasticity," def 207 Modulus of, def 209, 217, 218 PAGE Elbow, Development of a Right 162-164 Development of Four Part, illus. 172-175 Electric Motor, The, desc 399 Motor, EiEciency of, desc 399 ' Electrical Machines, desc. and illus. . . 391-407 Electricity, Classification of 396-398 Def 393 Electro-motive Force, def 397, 398 Ellipse, def 36, 160 Drawing an, illus 38, 106, 107 Produced by Cutting Cone 160 E. M. F. Abbreviation, def. . . . .Xt-.,. . . . 397 Energy, def :..,.. 207 Engine, Belt Fly Wheel for, illus. and desc 363-364 Corliss — Cylinder of, illus 384 Corliss Valve Gear, illus 383 Cylinder of Corliss, illus 3S4-385 Cylinder of Steam, desc 362 Fishkill Landing Corliss, desc . . 388 Lathe, desc ... 321-322 Left-hand, desc. and illus 367 Main Shaft of a Steam 362 Note, Relating to Position of . . . . 368 Reciprocating Steam, desc 362 Right-hand, desc. and illus 367 Rotary Steam, desc 362 Engines and Boilers, desc 335 Commercial Rating of 367 Engines, Multi-cylinder Steam, desc. . 366, 367 Newcomen, desc 335 Overrunning, illus. and desc. . . . 367, 36S Right and Left Hand, illus 367 Steam, desc 362 Table of Dimensions of Horizon- tal Steam Engines 369 The Fire and Heat, desc 335 Underruning, illus. and desc. . . . 367, 368 INDEX 479 PAGE Engines, Vertical, desc 368, illus. 371 Envelope of a Solid, def 37 Epicycloid, def. and illus 2S6, 287 Equality, sign of 25 Equation, def 431 Equilateral Triangle, def 30 To Construct 92-93 Erasing, How Best Done 189 Evaporation, Equivalent 355, 356 Of Steam Boilers 350, 352 Tensile Strength, illus 219 Example for Figuring Engine Horse Power 37S Exercises in Geometrical Drawing. ... 87 " pace" of Gear-tooth, desc. and illus. 279, 280 Factor, def 207 Of Safety, def 210 False Perspective, def 113 Fatigue of Metals, def 208 Feed-pipe, desc 336 Shaft of Lathe, desc. and illus. . 322 Field Magnet, desc 399 Fifteen Degree Lines, illus. and desc. . 45 Figures, Straight-sided, defs 30 "Finished," def 184 Fire=box Boiler, desc 344 Tubes, illus. and desc 340 Flange Coupling, illus 127 Cylinder, How to Draw 145 With Bolts, How to Draw . 127 " Flank" of Gear-tooth, illus. and desc. 279, 2S0 Floor Stands or Pedestals, desc 266 Flue Boiler, illus. and de.sc 340-341 Fly Wheels, desc. and illus 364 Rim Speed of 366 Foci of an Ellipse, def 36 Focus of the Parabola, def 36 PAGE Foot, def 46S Sign for 132 Force, def 207 Electro-motive, def 397-398 Moment of, def 220 Formula for Estimating Horse Power of Crank Shaft of Steam En- gine 256-25S For Figuring H. P. of Steam Engine 376 Lathe Gear Changes 326 Size of Connecting Rod 388 Thickness of Steam Engine Piston 3S1 Prof. Unwin's, for Pulleys 273 To Find the Pressure of Cross Head on Guide 382 Formulae for Belt Speeds 26S For Screw Cutting in Lathe. 326, 32S-329 Reading of 431-432 Forty-five Degree Line, illus. and desc 45 Foundations for Steam Engine 364 Four Part Elbow, Development of, illus. 1 72- 1 75 Fractions, How Placed in Dimension Lines 1 86 Franklin Institute Standard Table. . . . 232 Free-hand Lettering Specimen 55 Friction, def 208 Frictlonal Electricity, def 397 Function of Slide Valve, desc 368 Furlong, def 468 Furnace, Boiler, illus. and desc 33^-337 (jalloway Boiler, illus. and desc 344-348 Gamboge for Water Color 188 Gang Die, illus. and desc 308-312 Gas and Vapor, Difference Between. . . 213 Gaseous Bodies, Mechanics of 212 PAGE Gauge Cock, desc. and illus 338 Glass, desc. and illus 33S-339 Pressure and Total Heat, Table. 356 Steam, desc. and illus 338 Gear — Annular, desc. and illus 292 Teeth — Cycloidal, def. and illus. 285-293 Involute, desc. and illus 282-2S4 Wheels, desc. and illus 278-304 Dimensions for, desc. and rules. 300-303 Speeds of, rule and ex 278-280 Trains of 304 Gears, Bevel, def 2S2 How to Draw, illus 292, 294-299 For Screw Cutting in Lathe. . . . 322-329 Rules for Pitch of 280-282 Spur, def 282 Worm 298, 300-302 Def : . 2S2 How to Draw 230 " Gelatine " Sensitized Paper 194 Generator — Four Pole, desc 399 The Electric, desc 398-399 Geometrical Drawing 85-1 10 Exercises in 87 Tools L'sed in 86 Magnitudes. 27 Proportion, sign of 24 Glass Gauge, desc. and illus 338-339 Grate Surface of Boilers, desc 336 Gravity, def 208 Greek Alphabet 468 Handhole of Steam Boiler, illus 339-34° Hanger, Seller's Adjustable, illus and desc 260-262 Hangers, desc. and illus 260-262 Hawkins' Treatise on the Indicator Recommended 375 480 ROGERS' DRAWING AND DESIGN. " Head=stock " of Lathe, desc. and illus 322, 324, 325 " Heart Wheel," Drawing a, illus no Heat, Total, Table of Gauge Pressure. . 356 Heating Surface of Boilers, desc 336 Surface of Steam Boiler's Ratio to Grate Surface 352 Helix, How to Draw, desc. and illus. . . 228-230 Heptagon, def 32 Hexagon, def 32 How to Construct by Instruments 50 To Construct on a Line, illus. . . . 102 To Inscribe in a Circle loi Hexagonal Nut, How to Draw, desc. and illus 147, 237, 239 Prism, How to Draw 153, def. 37 Pyramid, How to Draw 155 Hexahedron, def. and illus 40 Horizontal Engines, Dimensions of. .. 369 Lines 42 Line, def 28 Horse Power of Belts, Rule for 269-270 Of Steam Boiler, Rating of 350 Of Steam Engine, Rule for Find- ing 376 Of Boilers, desc 354-356 Transmitted by Shafts 256 How to Read Drawings, Diagram 190 Hydraulics, def 212 Hydrodynamics, def 212 Hydrostatics, def 212 Hyperbola, def 36, 161, illus. 38 Drawing an, illus 108 Hypocycloid, def. and illus 288-289 Hypothenuse, def 31 Hypothesis, def 85 Icosahedron, def. and illus 40 PAGE Illustration, Colored, Blue Print 199 Imperial Drawing Paper, size of 423 Inch, Sign for 132 India Ink, desc 417, illus. 418 How to Prepare 418 Indicator Diagram, desc. and illus. 375, 376, 378 Hawkins' Treatise on. Recom- mended 375 " Inertia," def 208 Injector, How to Operate 360 Parts of, desc. and illus 359, 360, 361 Inking, Instruction for 424 Instruments, Drawing 411-426 List and Selection of , . 426 Intersection of Solids, def. of term.. . . 37 «' Involute " Gear Teeth, def 282-284 Iron, Cast, desc 214-215 Drill Speed for 314 Factors of Safety for 217 How Shown oy Water Color. ... 188 Meteoric, Note 307 Wrought, desc 217 Wrought, Factors of Safety for. . 217 Isometric Projection, desc. and illus. . 1 14-120 Problems in, illus 115-120 Isosceles Triangle 30 To Construct an 93 Johnson, Wm. , Quotation Joints, Riveted, illus. and desc 245-251 Riveted, illus. of 249-251 Journals, desc. and example 258-259 Diameter of, example 258 Pressure on, example 258 Kinematics, def 208 PAGE Lancashire Boiler, illus. and desc. . . . 341 Lap=joints, desc. and illus 249 Lap of Valve, Outside and Inside of , . . 372 Lathe, desc. and illus 320-322 Engine, desc 321-322 Formulae for Screw Cutting Gears 326-32S, 329 Shafting, desc. and illus 330-332 Lathe-speed 320, 321 Laws, Newton's, def 214 Of Motion, three, def 214 Lead of Valve , 372 Screw of Lathe, desc. and illus. . 322 League, def 472 Left Handed Screw, desc 230 Hand Engines, illus. and desc. . . 367 Lemma, def 85 Lettering, Examples 63 Subject Treated on 53-64 Triangle, illus 55 Letters, Block, and Numerals 54 Capital, Use of 53 Reference, When to be Used on Drawings 187 Line, Broken, def 27 Center, def 27 Cun'ed, def 27 Def 27 Dimension, def :.."..' 27 Dotted, def 27 Dot and Dash, def 27 Full, def 27 Horizontal, def. . 28 Inclined, def 28 Irregular Curved, def 27 Oblique, def 28 Plumb, def. . , , "8 Regular Curved, def 27 INDEX. 481 PAGE Line, Right, def 27 Shade, def 27 To Divide a Straight Line 91 Vertical, def 28 Waved, def 27 Linear Drawing, Subject of 25, 81 Velocity, def 211 Lines, Border, When to be Used 423 Fifteen Degree, illus. and desc. . 45 Forty-five Degree, illus. and desc. 45 Parallel, def 28 Seventy-five Degree, illus. and desc 45 Shade, desc. and illus 65-73 Sixty Degree, illus. and desc .... 45 Thirty Degree, illus. and desc. , . 45 To Draw Parallel 90 Vertical and Horizontal, def. ... 44 Liquid, def 213 Load, def 20S-209 Locomotive Boiler, illus. and desc. . . . 344-349 Logarittimic Table 433 Table— Use of 433-434 Logarithms, desc 433-434 Rules for Application 434 Tables of 435-456 Advantages of 433 Long Measure, rule 468 iVlachine Design 205 Man as a, Note 216 Modulus of a, def 209 Punching and Shearing 313-315 Tool Pullej's, speed of 270 Machines, Classification of, desc 215 Desc 215-216 Drilling 314, 316-317 Electrical, desc. and illus 391-407 Machines, Metal Working, desc. and illus 307-332 Milling, illus, and desc ....316, 318-319 Six Points in Designing 307 Magnetic Field, desc 399 Magneto Electricity, def 397 Main Shaft of the Steam Engine, desc. 362 Male and Female Die, desc 308 " Man as a Machine, " Note 216 Manganese, When Discovered 307 Manhole, desc. and illus 339 Masonry, Factors of Safety for 217 " Mass," def 213 Materials for Construction, def 214-215 Strength of, def 210 Matter, Properties of, def 212-213 Three States of, def 213 McWhinney, Quotation by Prof Mean Effective Pressure, Rule for Finding 377 Measure, Circular 46S Long 468 Mechanics, Applied, def 212 Squares, Note 29 Theoretical 205 Mechanism, Theory of 205 " Medium " Drawing Paper, Size of. . 423 Metal Working Machines, desc. and illus 307-332 Metals, Discovery of, desc 307 Fatigue of, def 208 Meteoric Iron, Note 307 Mile, Common, def 468 Milling Cutter, illus 127 Machines, illus. and desc. . .316, 318-319 Machine, Vertical Spindle, illus. 318-319 Minutes, Part of a Circle, 34 Mixed — Line, def 28 PAGE Modulus, def 209 Of Elasticitj', def 209, 217-218 Of a Machine, def 209 Of Resistance, def 209 Of Rupture, def 209 Section, def 222-223 Molecule, def 212 Moment, def 209 Difference Between Weight and. . 213 Of Force, def 220 Momentum, def 209 Motion, def 209-210 Three Laws of, def 214 Motor Electric, Efficiency of, desc. . . . 399 The Electric, desc 399 Mounting Blue Prints, desc 195-196 Mud Drum, desc. and illus 348 Multi-cylinder Engines, desc. and illus 366-367 Multiplication, Sign of 23 Multipolar Dynamo, def 398 Illus 394-395 Negative Electricity, def 396-398 Quantity, def 431 Newcomen Engine, desc 335 Newton's Laws, def 214 Laws, What they tell us 214 Nickel, Note Relating to 307 Plated Sheet Steel Scale, desc. . . 425 When Discovered 307 Nozzles of Steam Injector, desc 360 Numerals, Roman 46S Nut, Square, desc. and illus 240 Check, desc. and illus 240 Hexagonal, How to Draw 147 Nuts, Proportions of, desc. and illus. . 237 482 ROGERS' DRAWING AND DESIGN. PAGE Oblique Angle, def 29 Line, def 28 Objects, Orthographic Projection of 149-159 Obtuse Angle, def 29 Octagon, def 32 How to Construct by Instruments 51 To Describe on a Line, illus 103 To Describe in a Square, illus.-. .• - IC2 To Inscribe in a Circle, illus. . . . 105 Octahedron, def. and illus 40 Operation, Algebraic, example 431 Of Slide Valve, desc. and illus . . 370 Orthographic Projection, desc. and illus 113, 128-161 Problems in, illus - 132-159 Of Oblique Objects, illus. ..... 149-159 Oval, to Draw by Circular Arcs, illus. . 105 r aper, Size for Patent Drawings 423 Parabola, Drawing a, illus 107 Illus. 36-38 160-161 Parallel Lines, def 28 To Draw a, illus. and rule 90 Parallelogram, def .• 31 To Construct a 96-97 Patent Office, Size of Official Drawing Paper 423 Peck's, Eugene C, M. E., Method of Conducting a Drawing Class, Note 413 Pedestals and Pillow Blocks, desc. and illus ; 263-266 Pen, Drawing, illus 416-417 Made for Round Writing, illus . . 56 Pencil — Bow, illus 416 Penciling, Instruction for 424 Pencils, Hard and Soft, How to Sharpen 422 PAGE Pentagon, def 32 To Inscribe in a Circle, illus. ... 104 Pentagonal Prism, def 37 Perpendicular — Line, def 29 Line, to Draw a, illus. and rule. . 89 Perspective, False, def 113 Physics, Object of Study of 212 Def 212 Pillow — Blocks and Pedestals, desc. and illus 263, 264 Brassesfor 263, 264 Pipe — Development of Tee 165-169 How to Draw a, by Orthographic Projection 142 Sizes, Table of Standard 459 Piston — Area, Rule for Finding 377 Rod of Steam Engine 362, 3S0-3S2 vSteam, desc 3S0 Pitch Circle, illus. desc 279, 280 " Pitch " of Rivets, desc. and illus. . . . 246 Of Screw, desc . 232 Of Screw, Rule How to Find. . . . 233 Pivots and Journals, Pressure on 25S, 259 Plan of the Work xvii Plane Figure, def 30 " Platen " of the Milling Machine 318 Plumb — Line, def 28 Pneumatics, def - 212 Point, def 27 Pole=pieces, desc 399 Polygon, def 30 To Draw on a Line, illus 103 To Inscribe in a Circle, illus . . . 104 Polygons, Note Relating to. 32 Polyhedron, def. and illus 40 " Pores," def 213 Positive Electricity, def 396, 398 Quantity, def 431 PAGE Power, def 210 Sources of, desc 255 Preface xv-xviii Presses, Dies and, desc. and illus 308-315 Prime Movers, Useful Work of 255 Principle of Work -211 Printing, Blue 189-192 Frame, desc. and illus 190 Frames, Note 191 Paper, Sensitizing of 193-195 Prism, Hexagonal, def 37 How to Draw 153 How to Draw by Orthographic Projection 132, 148-149 Pentagonal, def 37 Quadrangular, def 37 Prisms, illus 37 Triangular. 37 Problem, def 85, 431 Problems in Cabinet Projection, illus. 122-125 In Development of Surfaces. . . 162-179 In Isometric Projection, illus. . . 115-120 In Orthographic Projection, illus 132-159 Projection, Cabinet, desc. and illus. . . 121-127 Isometric, desc. and illus 1 14-120 Orthographic, desc. and illus. . . 128-161 " Projections," General Subject of. . . 113-179 Properties of Circles 33 Of Matter, def 212-213 Proportion of Bolt-heads, desc. and illus ; 237 Of Nuts, desc. and illus 237 Proportions for Arms of Gear Wheels. 300-303 " Proposition," def 85 Protractor, illus. and desc 420 Prussian Blue for Water Colors 188 Pulleys, Anns of, desc. and illus 274 Cone, desc. and illus 275-277 INDEX. 483 PAGE Pulleys, Crowning of, desc 272 Dimensions of, ill us. and example 272-273 Proportions for Arms of ... . . 274 Proportions for Bulbs of 272-273 Rules, examples and illus. of . , - . 271-277 Step Cone, desc. and illus, 277 Thickness of Rims of 272 Ti£;lit and Loose, desc 273 Punch and Die, illus. and desc 308 Punching and Shearing Machine 313-315 Pyramid, Hexagonal, How to Draw. . . 155 Illus 38-39 Quadrangular Prism, def 37 Quadrant, def 34 Quadrilateral Figure, def 31 Quadrisect an .\ngle, to, illus. and rule 88 Quotation from American Machinist . . vii Jno. G. Chapman L. D. Burlingame Relating to Drafting Room and Shop .... 196-197 Opposite Title Page xi President Andrews Relating to Machine Design 205 Prof. McAMiinney Will. Johnson Raabe's, H.E. , List of Drawing Instru- ments 426 Rack — Cycloidal, desc 292 Radiated Electricity, def 396 Radii of a Circle, def 33 Radius of a Circle, def. and illus 33 Eccentricity, def 35 " Ram " of a Press, desc 30S Ratio Between Heating and Grate Sur- faces 352 Velocity, def 211 PAGE Reading of Formulae 431-432 Working Drawings 198 Reciprocating Steam Engine, desc 362 Rectangle, def 31 To Construct a, illus. and rule . 96 Rectilinear Figure, def 30 Red, Vermilion, for Water Color iSS Reference Letters, When to be I'sed on Drawings 187 Resistance, Modulus of, desc 209 Theoretical and Practical 209 Rhomboid, def 31 Rhombus, def 31 Right .\ngle, def 29 Angled triangle, def 31 Handed Screw, desc 230 Hand Engines, illus. and desc . 367 Line, def 27 To Trisect, illus. and rule 88 Ring, Cylindrical, How to Draw 144 Rivet, Length of 245 Riveted Joints, Breaking of 245 Illus. and desc . . 245-251 Strength of 247-249 Riveting, Chain, desc. and illus 250 Punching Holes for 243 Rivets and Joints 243-251 Diagonal Pitch of, desc 247 Pitch of, desc. and illus 246 Staggering of, desc. and illus. . . 250 Robinson's, A.W., Office Rules, Quota- tion from 1 87 Rod, def 468 Roebling, Statement by Chas. G 15 Roman Numerals, def. and illus 468 Root, Sign of 24 Rotary Steam Engine, desc 362 "Rotary Table" of the Milling Ma- chine 318 PAGE Rouillon's, Louis, List of Drawing Instruments 426 Round-Headed Screws, desc. and illus. 241 Writing, Specimen 56 Rule for Finding the Area of a Steam ■ Piston 377 Diameter of Piston Rod 381-382 Finding the Mean Effective Pres- sure on Piston. . 377 Horse Power of Steam Engine. . 376 Horse Power of Belts 269 How to Find " Pitch " of Screw. 233 How to Use Logarithms 433, ex. 434 To Find Length of Stroke Crank and Eccentric 370 Rules and Data, Useful 429 And Examples for Safety Valve. 357-358 And Scales, desc. and illus 419 For Application of Logarithms. . 434 For Finding Dimensions of Gear Wheels. 300-303 Horse Power of Shafts 257 Pitch of Gears 280-2S2 Proportioning Pulleys 268-269 Relating to Circle 461 Relating to Square 469 Speed of Driver and Follower Pulleys 268, 269 Ruling Pen. ilUis 416 " Running Under " Engine, desc. and illus 367-36S Rupture, Modulus of, def 209 " Saddle " of the Milling Machine. . . 318 Safety, Coefficient of, def 210 Factor of, def 210 Valve — Rules and Examples for. 357-35S Scale, illus 20 Flat Box-wood, desc 426 484 ROGERS' DRAWING AND DESIGN. PAGE Scale, Nickel Plated Sheet Steel, desc. 425 Used in Lettering 53 Scales, Rules and, desc. and illus 419 Scholium, def 85 Screw Cutting, Changing Gears for. . . 322 Diameter of, desc 233 Double Threaded , desc 230 Gears for Lathe 322-329 I^eft-handed, desc 230 " Pitch " of, desc 232 Right-handed, desc 230 Set, desc. and illus 240-241 Single-threaded, desc 230 Thread — Angle of, desc. and illus. 234 Threads — Conventional Signs for Drawing 235-237 Threads, Note 230 Threads, Table of IT. S. Standard. 232 Triple-threaded, desc 232 Screws and Bolts 228-241 Round-headed, desc. and illus. . . 241 Seconds (Part of a Circle), def 34 Section Lining, illus. and desc 77-8' Modulus, def 222-223 Sectioning Metals, etc 78-80 Sector of a Circle, illus. and def 33 Segment of a Circle, def 33 Seller's Adjustable Hanger, illus. and desc 260-262 Screw Thread, desc 232 Semi°circle, def 33 Sensitizing of Printing Paper 193-195 " Sepia " for Water Colors 18S Series Winding, desc. and illus 406, 407 Set=screw, desc. and illus 240-241 Squares, illus 43-44 Use of 45-48 Shade Line, Specimens of 65-73 Shading, Parallel Line, illus. and desc. 74-77 " Shaft-feed " of Lathe, desc 322 Shafting Lathe, desc. and illus 330-332 Proper Speed of 258 Rviles for Horse Power of 257 Shafts and Shafting 256-258 Formulae for Strength of 257-258 Horse Power Transmitted by 256 Strains Produced in, desc 256 " Shank " of a Punch, desc 308 Shearing Strain, def 217 Strength, def 220 Sheet Metal, Sectioning of, illus So, 81 Shop Work, Drawing withtRelation to . 196-198 Shunt Winding, desc. and illus 406-407 " Sienna," Raw, for Water Color 188 Signs, Conventional 21 -22 For Designing Screw Threads. . . 235-237 Sine of an Arc, def 34 Single Threaded Screw, desc 230 Sixty Degree Lines, illus. and desc ... 45 Slide Valve, Function of, desc 36S Operation of, desc. and illus. . . . 370-375 Smoke^box, desc 340 " Snail," Drawing a, by Circular .\rcs, illus 109 Solid, def 27-37 A, def 213 "Solution" for Bath Used in Blue Printing 193 For Sensitizing Paper, Recipe.. . 192-193 Specimens of Lettering 53-64 Speed Lathe, illus. and desc 320-321 Of Machine Tool Pulleys 27a Shafts, Proper, desc 25S Gear Wheels, rule and example. 278-280 Sphere, def. and illus 40 Spiral, Drawing a, illus. loS-iog PAGE Spur Gears, def 282 Square, def 31 Headed Screws — Conventional Method of Representing, illus. 241 How to Construct by Instru- ments 48-49 Nut, desc. and illus 240 Rules Relating to 469 Thread, def 230-234 To Construct a, illus. and rule. . 95 To Describe about a Circle, illus. and rule \. . . . . loi To Inscribe in a Circle,\illus. and rule 100 Squares and Cubes and Square and Cube Roots, Tables of 469-471 Mechanics', Note 29 Staggering of Rivets, desc. and illus.. 250 Standards, Brasses for, desc. and illus. 265 Static Electricity, def 396 Statics, def 212 Stay Bolts, illus. and desc : 346 Stays, Boiler, illus. and desc 340-346 Through and Diagonal 340 Steam Boilers, desc 336 Boilers, Designing a, desc 350-354 Boilers, Evaporation of 350-352 Chest, desc 379 Engine, Formula for Strength of Shaft 256 Engine, Horse Power, Example of Figuring 378 Engine, Parts of 362-364 Engines, desc 362 Gauge, desc. and illus 338 Piston, Rule for Finding Area of. 377 Piston, desc 380 Ports, desc 380 INDEX. 485 steam Rating of Horse Power 350 Space of Boilers, desc 336 Total Heat Units in, Table 356 Works, illus ix Steel, desc 215 Factors of Safety for 217 How Shown by Water Color. . . . 188 Sectioning of, illus 79, 80 Soft, Drill Speed for 314 Stone, Sectioning of, illus 80, 81 Stop=clutch, desc. and illus 314 Straight Line — to Bisect a 87 To Divide a, illus. and desc gi Strain, def 210 Strains Produced in Shafts, desc 256 Tensile, def 217 Strengtli of Materials, def 210 Riveted Joints 247-249 Shearing, def 220 Tensile, def 210 Ultimate, def 210, 218 Stresses, def 210, 217-228 Induced by Bending, def 220 •• Stripper " of a Die, illus 30S Stud Bolt, desc. and illus 241 Stuffing Box of the Steam Engine. . . 362 Surface, def 27 As a Magnitude 30 Surfaces, Development of, illus. and desc 162-179 1 able. Logarithmic, begin at page 435 Example of Use of 433 Of Contents. xix Of Contents 21 Of Decimal Equivalents of Milli- meters and Fractions 458 Of Decimal Equivalents, J" iV ■ 457 PAGE Table of Standard Wire Gauges 464 Of Standard Pipe Sizes ....... 459 Diameter of Rivets and Thickness • of Plates 249 Drill Speeds 314 Evaporation of Coal 352 Dimensions of Horizontal Steam Engines 369 Tensile Strength of Bolts 241 Total Heat Units in Steam 356 U. S. Standard Threads 232 Tables of Logarithms 435-460 Of Squares and Cubes and Square and Cube Roots 469-471 And Index 427-485 Of Areas and Circumferences of Circles 461-467 Tail-stock of Lathe 322-327 Tangent, Abbreviation of 34 Of an Arc, def 34 To Draw to a Circle, illus. and rules 98-100 Of a Circle, illus 33 Tee-square, illus. and desc 42 1 enacity, def 210 Tensile-strength — Table for Bolts. . . 241 Illus 219 Def.. . . 210 Tensile Strain, def 217 Terms and Definitions 27-40 Test Pieces for Blue Printing 191-192 Tetrahedron, def. and illus 40 Theorem, def 85, 431 Theoretical Mechanics, def 205 Resistance, def 210 Theory of Mechanism, def 205 Thread — Square, desc 234 V desc. 230, illus. 231, 230 Thirty Degree Lines, illus. and desc. . 45 PAGE Through Stays, desc 340 Thumb=Tacks, How Secured to Board 423 Tight and Loose Pulleys, desc 273 Timber, Factors of Safety for 217 Tints and Colors 188 Tool Chest, to Draw by Isometric Pro- jection 119 Cabinet Projection 126 TooURest of Lathe, desc. and illus. . . 323 Tools Used in Geometrical Drawing. . . 86 Tracing Cloth, Smooth and Dull Side of 189 Of Drawings 189-190 Tracings, Order to be Followed in Making Lines i S9 Trains of Gear Wheels 304 Transferring an Angle, illus. and rule 92 Transmission, def 255 Trapezium, def 31 Trapezoid, def 31 Triangle, Altitude of a 31 Def 30 How to Construct by Instruments 50 Illus 43-44 To Construct a, rule and illus. . . 94 Triangles 30 Triangular Prism 37 Triple Threaded Screw, de.sc 232 Trisect a Right Angle, to, illus. and rule 88 Truncated Pyramid, illus 38-40 Tubes, Fire, illus. and desc. ........ 340 Ultimate Strength, def 210-218 Unipolar Dynamo, def 398 I'nit Stress and Strain, def 21S Un-vWn's, Prof., Formula for Belts. . . . 273 Use of Logarithmic Table 433-434 U. S. Standard Screw Threads 232 486 ROGERS' DRAWING AND DESIGN. PAGE Valve, Functions of, desc 368 Valves, Corliss, desc 388 Gear, Corliss, desc. and illus. . . . 388-390 Corliss, desc. and illus 3S3 Mechanism of the Steam Engine 362 Steam Engine, illus. and desc . . . 376 Safety, rule 357-358 Vapor, Difference Between Gas and. . . 213 Velocity, def. and ratio 210, 211 Circular, def 211 Linear, def 211 Ratio, def 211 Vermilion, for Water Color 188 Vertex of an Angle, def 29 Vertical Boiler, illus. and desc. . . .346, 347, 351 Engines, desc 36S, illus. 371 Lines, def 28, 44 Spindle Milling Machine, illus. . 318, 319 "View" in Drawing to be Drawn First. 153 Vis=viva, def 211 Voltaic Electricity, def 397 PAGE " Volume," def 213 V — Thread, desc. 230, illus. 231 230 ^Vater — Column, desc. and illus 339 Leg of Boiler, desc 344 Line of Boilers, desc 336 Tube Boiler, illus 354-355 Boiler, Dimensions of 350 Boilers, desc. and illus 347- 348 Wall-brackets, desc. and illus 262 Waved Line, def 27 Wedge, How to Draw 147 How to Draw by Orthographic Projection 132 Weight and Moment, Difference Be- tween 213 Wheel, Belt, Fly, illus 363 Winding Series, desc 406, 407 Shunt, desc. and illus 406-407 Wire Gauges, Table of Standard 460 Wood, Sectioning of, illus 80 PAGE Work, def 211 Explanatory Note 211 Preparing for 423-424 Principle of 211 Working Drawings, General Subject.. 181-199 Reading of 198 Worm Gears 298, 300-302 Def 282 Wrench — Proportioning of, desc. and illus 242 Use of, for Tightening Bolts, desc. 242 Wrist— Plate, desc 390 Wrought Iron, desc ip 215 Factors of Safety for. 217 Sectioning of, illus .".... 78-80 Yard, def 468 Yellow, Chrome, for Water Color 188 Aeuner's Diagram, illus. and desc. . . 377-378 " "W 'W l^F "W ■^^♦" ■"♦ ♦ ♦ ♦ "Knowledge is power^ and the price of Jcnowledge is continued study * SELF-HELP MECHANICAL BOOKS FOR HOME STUDY AND REFERENCE ♦I* l'^ THEO. AUDEL & COMPANY M EDUCATIONAL PUBLISHERS ♦*♦ 72 FIFTH AVE. .-. .-. NEW YORK A GOOD BOOK |S A GOOD FRIEND To Our Readers : The good books, here described, de- serve more than a passing notice, consider- ing that the brief description under each title indicates only their wide scope, and is merely suggestive of the mine of useful infor- mation contained in each of the volumes. Written so they can be easily under- stood, and covering the fundamental prin- ciples of engineering, presenting the latest developments and the accepted practice, giv- ing a working knowledge of practical things, with reliable and helpful information for ready reference. These books are self-educators, and "he who runs may read" and improve his pres- ent knowledge in the wide field of modern engineering practice. Sincerely, Theo. Audel & Co. Publishers 72 5th Ave., N. Y. BOOKS THAT WILL ANSWER YOUR QUESTIONS ^ '♦ ♦ ♦ ♦•«►♦♦ ■^ ♦ V ♦ ,♦ ♦ «► .♦ - Hawkins' Mechanical Dictionary. If the reader often encounters words and allusions whose meaning is not clear, or is a busy man, without time to wade through a whole volume on any subject about which in- formation is desired, or is a student or a profes- sional man, feeling the frequent need of a first class reference work, he will readily appreciate the qualities that are to be found in this most useful book. It is a cyclopaedia of words, terms and phrases in use in the Mechanic Arts, Trades and Scien- ces — "many books in one." It is the one book of reference no student or expert can dispense with. 704 pageSj 6^x8^ inches, handsomely bound. Hawkins' Electrical Dictionary. A reliable guide for Engineers, Contractor';, Superintendents, Draftsmen, Telegraph and Telephone Engineers, Wire and Linemen. Contains many books in one, and is an en- tirely new and original work. Clearly and plainly defining the full use and meaning of the thousands upon thousands of words, terms and phrases used in the various branches and de- partments of Electrical Science. No Dictionary has to the knowledge of the publishers been printed to date that has kept pace with the rapid development of Electrical Engineering. It measures 6^x8^ inches, is over one and one-half inches thick; the book weighs about two and one-half pounds, giving a finish to the book which is charming. $3^ $3^ Rogers' Erecting and Operating> Provides full information on methods success- fully proved in modern practice to be the best for use in erecting and installing heavy machin- ery of all kinds. It includes a systematic course of study and instruction in Mill Engi- neering and Millwrightingy together with time saving tables, diagrams and a quick refer- ence index. Rogers' Drawing and Design. This volume is arranged as a comprehensive, self-instruction course for both shop and draughting room. Contains 506 pages, illustrated by over 600 cuts and diagrams, very many of them full page drawings; the book is printed on a very fine grade of paper; it measu-res S^^xioJ^ inches and weighs over 3 pounds; it is in every way completely up-to-date. Rogers' Machinists Guide. Is a manual for Engineers and Machinists on modern machine shop practice, management, grinding, punching, cutting, shearing, bench, lathe and vise work, gearing, turning, and all the subjects necessary to advance in shop practice, and the working and handling of machine tools. The most complete book on these subjects. Fully illustrated. Audei's Answers on Practical Engineering. Gives you the everyday practice and simple laws relating to the care and management of a steam plant. Its 250 pages make jou familiar with steam boi'ers, steam, fuel, heat, steam gauge, installation and management of boilers, pumps, elevators, heating, refrigeration, engi- neers' and firemen's law, turbines, injectors, valves, steam traps, bells, gears, pulleys, electricity, etc., etc. $2 $2 $2 $1 Homans^ Automobiles. "Homans' Self Propelled Vehicles'* gives full details on successful care, handling, and how to locate trouble. Beginning at the first principles necessary to be known, and then forward to the principles used in every part of a Motor Car. It is a thorough course in the Science of Automobiles, highly approved by manufacturers, owners, operators and repairmen. Contains over 400 illustrations and diagrams, making every detail clear, written in plain language. Handsomely bound. Audels Answers on Automobiles. Written so you can understand all about the construction, care and management of motor cars. The work answers every question that may come up in automobile work. The book is well illustrated and convenient in size for the pocket. Audels Gas Engine Manual. This volume gives the latest and most helpful information respecting the construction, care, management and uses of Gas, Gasoline and Oil Engines, Mari-ne Motors aftd Automcbile Engines, including chapters on Producer Gas Plants and the Alcohol Motor. The book is a practical educator from cover to cover and is worth many times its price to any one using these motive powers. Audels Answers on Refrigeration (2VoIs.) Gives in detail all necessary information on the practical handling of machines and appli- ances used in Ice Making and Refrigeration. Contains 704 pages, 250 illustrations, and writ- ten in question and answer form; gives the last word on refrigerating machinery. Well bound and printed in two volumes ; complete. $2 $|--55 $2 $4 Homans' Telephone Engineering. Is a book valuable to all persons interested in this ever-increasing industry. No expense has been spared by the publishers, or pains by the author, in making it the most comprehen- sive hand-book ever brought out relating to the telephone, its construction, installation and suc- cessful maintenance. Fully illustrated with diagrams and drawings. Rogers^ Pumps and Hydraulics (2 Vols.) This complete and practical work treats ex- haustively on the construction, operation, care and management of all types of Pumping Ma- chinery. The basic principles of Hydraulics are minutely and thoroughly explained. It is illustrated with cuts, plans, diagrams and draw- ings of work actually constructed and in opera- tion, and wll rules and explanations given are those of the most modern practice in successful daily use. Issued in two volumes. Hawkins^-Lucas* Marine Engineering. This treatise is the most complete published for the practical engineer, covering as it does a course in mathematics, the management of ma- rine engines, boilers, pumps, and all auxiliary apparatus, the accepted rules for fguriruj tlie safety-valve. More than looo ready references, 807 Questions on practical ma- rine engineering are fully answered atid explained, thus forming a ready guide in solv- ing the difficulties and problems which so often arise in this profession. $1 $4 $2 Hawkins* Electricity for Engineers. The introduction of electrical machinery in almost every power plant has created a great demand for competent engineers and others having a knowledge of electricity and capable of operating or supervising the running of electrical machinery. To such persons this pocket-book will be found a great benefactor, since it con- tains just the information required, clearly ex- plained in a practical manner. It contains 550 pages with 300 illustrations of electrical appliances, and is bound in heavy red leather, size 4!^x6_J^ for the pocket. Hawkins* Engineers* Examinations , This work is an important aid to engineers of all grades, and is undoubtedly the most help- ful ever issued relating to a safe and sure preparation for examination. It presents in a question and answer form the most approved practice in the care and management of Steam Boilers, Engines, Pumps, Electrical and Refriger- ating Machines, together with much operative information useful to the student. Hawkins* Steam Engine Catechism. This work is gotten up to fill a long-felt need for a practical book. It gives directions and detailed descriptions for running the various types of steam engines in use. The book also treats generously upon Ma- rine, Locomotive and Gas Engines, and will be found valuable to all users of these motive powers, Hawkins* Steam Boiler Practice. This instructive book on Boiler Room Practice is indispensable to Firemen, Engineers and all others wishing to perfect themselves in this important branch of Steam Engineering. Besides a full descriptive treatise on Station- ary, Marine and Locomotive boilers, it contains sixty management cautions, all necessary rules and specifications for boilers, including riveting, bracing, finding pressure, strain on bolts, etc., thus being a complete hand-book on the sub- ject. $2 $2 $2 $2 $2 Hawkins* Mechanical Drawing. This work is arranged according to the cor- rect principles of the art of drawing; each theme being clearly illustrated to aid the student to ready and rapid comprehension. It contains 310 pages with oyer 300 illustra- tions, including useful diagrams and suggestions in drawings for practice. Handsomely bound in dark green cloth. Size 7x10 inches. Hawkins' Calculations for Engineers. vO The Hand Book of Calculations is a work of instruction and reference relating to the steam engine, steam boiler, etc., and has been said to contain every calculation, rule and table neces- U sary for the Engineer, Fireman and Steam ',, User. It is a complete course in Mathematics. All calculations are in plain arithmetical figures, so that they can be understood at a glance. Hawkins' Steam Engine Indicator. This work is designed for the use of erecting and operating engineers, superintendents and students of steam engineering, relating, as it does, to the economical use of steam. The work is profusely illustrated with working cards taken from every day use, and gives many plain and valuable lessons derived from the diagrams. Guarantee. These books we guarantee to be in every way as represented, and if not found satisfactory can be returned promptly and the amount paidwill be willingly refunded. All books shipped post paid. Remittances are best sent by Check, Post Office or Express Money Orders. $1 JUN ~0 !9'i2