uJilMBjja.j, D. VAN NOSTEAND, PUBLISHEB, 16 MURRAY STREET AND V7 WARRKN STREBT, NEW YORK. OP THE TJ&I7ERSIT7 A 'TEXT BOOK OF DRAWING; FOR THE USE OF MECHANICS AND SCHOOLS, IV WHICH TITR DEFINITIONS AND RULES OF GEOMETRY ARE FAMILIARLY EXPLAINED, TI13 PRACTICAL PROBLEMS ARK ARRANGED FROM THE MOST SIMPLE TO THE MORE COMPLEX, AND IN THEIR DESCRIPTION TECHNICALITIES ARE AVOIDED AS MUCH AS POSSIBLE; WITH ILLUSTRATIONS FOR DRAWING- PLANS, SECTIONS AND ELEVATIONS OF BUILDINGS AND MACHINERY . AN INTRODUCTION TO ISO-METRICAL DRAWING : A COURSE OF LINEAR PERSPECTIVE AND SHADOWS: AN ESSAY ON THE THEORY OF ' YV * s - ITS APPLICATION TO ARCHITECTURAL AND MECHANlCXlV I/RA vVVNGS THE WHOLE ILLUSTRATED WITH STIEIEJILi BY WM. MTNIF1E, ARCHITECT. NINTH THOUSAND, REVISED BY THE AUTHOR, NEW YORK: D. VAN NOSTEAND, PUBLISHES, 23 MURRAY STREET AND 27 WARREN STREET. 09 THB " *> v^ y V) Entered, according to the Act of Congress, in the year 1867 BY WM. M 1 N I F I E, In the Clerk's Office of the District Court of Maryland PREFACE. HAVING been for several years engaged in teaching Architectural and Me- chanical drawing, both in the High School of Baltimore and to private classes, I have endeavored without success, to procure a book that I could introduce as a text book, works on Geometry generally contain too much theory for the purpose, with an insufficient amount of practical problems ; and books on Architecture and Machinery are mostly too voluminous and costly, contain- ing much that is entirely unnecessary for the purpose. Under these circum- stances, I collected most of the useful practical problems in geometry from a variety of sources, simplified them and drew them on cards for the use of the classes, arranging them from the most easy to the more difficult, thus leading the students gradually forward ; this was followed by the drawing of plans, sections, elevations and details of Buildings and Machinery, then followed Isometrical drawing, and the course was closed by the study of Linear per- spective and shadows; the whole being illustrated by a series of short lectures to the private classes. I have been so well pleased with the results of this method of instruction, that I have endeavored to adopt its general features in the arrangement of the following work. The problems in constructive geometry have been selected with a view to their practical application in the every-day business of the Engineer, Architect and Artizan, while at the same time they afford a good series of lessons to facilitate the knowledge and use of the instruments requir- ed in mechanical drawing. The definitions and explanations have been given in as plain and simple language as the subject will admit of; many persons will no doubt think them too simple. Had the book been intended for the use of persons versed in geometry, very many of the explanations might have been dispensed with, but it is intended chiefly to be used as a first book in geometrical drawing , by persons who have not had the benefit of a mathematical education, and who in a majority of cases, have not the time or inclination to study any com- plex matter, or what is the same thing, that which may appear so to them. Arid if used in schoolsjits detailed explanations, we believe, will save time to the teacher, by permitting the scholar to obtain for himself much information that he would otherwise require to have explained to him. But it is also intended to be used for self -instruction , without the aid of a teacher, to whom the student might refer for explanation of any difficulty; under these circumstances I do not believe an explanation can be couched in too simple language. With a view of adapting the book to this class of stu- dents, the illustrations of each branch treated of, have been made progressive, commencing with the plainest diagrams; and even in the more advanced, the object has been to instil principles rather than to produce effect, as those once IV obtained, the student can either design for himself or copy from any subject at hand. It is hoped that this arrangement will induce many to study draw- ing who would not otherwise have attempted it, and thereby render them- selves much more capable of conducting any business, for it has been truly said by an eminent writer on Architecture, " that one workman is superior to another (other circumstances being the same) directly in proportion to his knowledge of drawing, and those who are ignorant of it must in many re- spects be subservient to others who have obtained that knowledge." The size of the work has imperceptibly increased far beyond my original design, which was to get it up in a cheap form with illustrations on wood, and to contain about two-thirds of the number in the present volume, but on examining some specimens of mathematical diagrams executed on wood, I was dissatisfied with their want of neatness, particularly as but few students aim to excel their copy. On determining to use steel illustrations I deemed it advisable to extend its scope until it has attained its present bulk, and even now I feel more disposed to increase than to curtail it, as it contains but few examples either in Architecture or Machinery. I trust, however, that the objector to its size will find it to contain but little that is absolutely useless to a student. In conclusion, I must warn my readers against an idea that I am sorry to find too prevalent, viz : that drawing requires but little time or study for its attainment, that it may be imbibed involuntarily as one would fragrance in a flower garden, with little or no exertion on the part of the recipient, not that the idea is expressed in so many words, but it is frequently manifested by their dissatisfaction at not being able to make a drawing in a few lessons as well is their teacher, even before they have had sufficient practice to have obtained a free use of the instruments. I have known many give up the study in con- equence, who at the same time if they should be apprenticed to a carpenter, yould be satisfied if they could use the jack plane with facility after several weeks practice, or be able to make a sash at the end of some years. Now this idea is fallacious, and calculated to do much injury; proficiency n no art can be obtained without attentive study and industrious persever- ance. Drawing is certainly not an exception : but the difficulties will soon anish if you commence with a determination to succeed ; let your motto be PERSEVERE, never say "it is too difficult;" you will not find it so difficult as you imagine if you will only give it proper attention ; and if my labors lave helped to smooth those difficulties it will be to me a source of much gratification. WM. MINIFIE. BALTIMORE, 1st March, 1849. PREFACE TO THE REVISED EDITION IN issuing this seventh edition of THE GEOMETRICAL DRAWING BOOK, the author desires to return his grateful thanks to the public, for the favor with which the previous editions have been received : more especially for the favor- able notices elicited from the press, both in the United States and Great Britain, particularly from that portion devoted to Fine Art, Architecture, En- gineering and Mechanics, as in all those pursuits a knowledge of drawing is indispensable to success, and the conductors of its literature may be fairly con- sidered as the most competent to decide on the merit of a treatise on this sub- ject ; their approval has, therefore, afforded me the more satisfaction. Many of the schools and colleges of the Union have adopted the work as a Text Book. It has also been recommended by the Department of Art of the British Government to the National and other Public Schools and Institutions throughout the kingdom. The present edition has been carefully examined and the few typographical errors corrected. The Essay on the Theory of Color and its application to Architectural and Mechanical Drawings, which was issued as an appendix to the fourth and following editions, has now been revised and arranged in the body of the work, and a full Index to the Essay added, together with a few other matters of interest, which it is hoped will be found to add to the usefulness of the book, and enable the student much more readily to refer to the informa- tion required. BALTIMORE, J/ay, 1867. ILLUSTRATIONS. PLATE Definitions of lines and angles, ....... i. Definitions of plane rectilinear superficies, ii. Definitions of the circle, iii. To erect or let fall a perpendicular, iv. Construction and division of angles, v. Construction of polygons, vi. Problems relating to the circle, . . , . . . vii. Parallel ruler, and its application, viii. Scale of chords and plane scales, ix. Protractor, its construction and application, .... x. Flat segments of circles and parabolas, xi. Oval figures composed of arcs of circles, . . . . xii. Cycloid and Epicycloid, xiii. Cube, its sections and surface, ....... xiv. Prisms, square pyramid and their coverings, . . . . . xv. Pyramid, Cylinder, Cone and their surfaces, .... xvi. Sphere and covering and coverings of the regular Polyhedrons, . . xvii. Cylinder and its sections, ........ xviii. Cone and its sections, xix. Ellipsis and Hyperbola, . . ... . . . xx. Parabola and its application to Gothic arches, xxi. To find the section of the segment of a cylinder through three given points, .......... xxii. Coverings of hemispherical domes, xxiii. Joints in circular and elliptic arches, ...... xxiv. Joints in Gothic arches, ......... xxv. Design for a Cottage ground plan and elevation, . . . xxvi. Design for a Cottage chamber plan and section, .... xxvii. Details of a Cottage -joists, roof and cornice, .... xxviii. Details of a Cottage parlor windows and plinth, . . xxix. Octagonal plan and elevation, xxx. Circular plan and elevation, ....... xxxi. Roman mouldings, .... .... xxxii. Grecian mouldings, . . . . . . . xxxiii. Plan, section and elevation of a wheel and pinion, . . . xxxiv. To proportion the teeth of wheels, ...... xxxv. Cylinder of a locomotive, plan and section, .... xxxvi. Cylinder of a locomotive, transverse section and end view, . xxxvii VI Isometrical cube, its construction, xxxviii. Isometrical figures, triangle and square, .... xxxix. Isometrical figures pierced and chamfered, . . xl. Isometrical circle, method of describing and dividing it, . . . xli. Perspective Visual angle, section of the eye, &c xlii. " Foreshortening and definitions of lines, . . . xliii. " Squares, half distance, and plan of a room, . . . xliv. " Tessellated pavements, xlv. " Square viewed diagonally. Circle xlvi. " Line of elevation, pillars and pyramids, . . . xlvii. " Arches parallel to the plane of the picture, . . . xlviii. " Arches on a vanishing plane,. xlix. " Application of the circle, . . 1. Perspective plane and vanishing points, . . li. " Cube viewed accidentally, ...... Hi. " Cottage viewed accidentally, Hii. " Frontispiece. Street parallel to the middle visual ray, liv. Shadows, rectangular and circular, lv. Shadows of steps and cylinder, ...... Ivi. DEFINITIONS Oh' LINES AN11 ANdf.KS. 0* THB TJKIVBRSIT7 PRACTICAL GEOMETRY, PLATE I. DEFINITIONS OF LINES AND ANGLES. 1. A POINT is said to have position without magnitude; and it is therefore generally represented to the eye by a small dot, as at A. 2. A LINE is considered as length without breadth or thickness, it is in fact a succession of points; its extremities therefore, are points. Lines are of three kinds ; right lines, curved lines, and mixed lines. 3. A RIGHT LINE, or as it is more commonly called, a straight line, is the shortest that can be drawn between two given points as B. 4. A CURVE or CURVED LINE is that which does not lie evenly between its terminating points, and of which no portion, how- ever small, is straight ; it is therefore longer than a straight line connecting the same points. Curved lines are either regular or irregular. 5. A REGULAR CURVED LINE, as C ; is a portion of the circum- ference of a circle, the degree of curvature being the same throughout its entire length. An irregular curved line has not the same degree of curvature throughout, but varies at dif- ferent points. 6. A WAVED LINE may be either regular or irregular ; it is com- posed of curves bent in contrary directions, j ^ Jis & regular waved line, the inflections on either side of .tbe ,dott$ bring ' equal; a waved line is also called a line contrary flexure, and a serpentine line. 7. MIXED LINES are composed of straight and curved lines, as D. 8. PARALLEL LINES are those which have no inclination to t:ich other, as F, being every where equidistant; consequently they could never meet, though produced to infinity. 8 PLATE I. If the parallel lines G were produced, they would form two concentric circles, viz: circles which have a common centre, whose boundaries are every where parallel and equidistant. 9. INCLINED LINES, as H and /, if produced, would meet in a point as at K, forming an angle of which the point K is called the vertex or angular point, and the lines H and / the legs or sides of the angle K; the point of meeting is also called the summit of an angle. 10. PERPENDICULAR LINES. Lines are perpendicular to each other when the angles on either side of the point of junction are equal; thus the lines JV. 0. P are perpendicular to the line L M. The lines N. 0. P are called also vertical lines and plumb lines, because they are parallel with any line to which a plummet is suspended ; the line L. JV/ is a horizontal or level line ; lines so called are always perpendicular to a plumb line. 11. VERTICAL and HORIZONTAL LINES are always perpendicular to each other, but perpendicular lines are not always vertical and horizontal; they may be at any inclination to the horizon pro- vided that the angles on either side of the point of intersection are equal, as for example the lines X. Y and Z. 12. ANGLES. Two right lines drawn from the same point, di- verging from each other, form an angle, as the lines S. Q. R. An angle is commonly designated by three letters, and the letter designating the point of divergence, which in this case is Q, is always placed in the middle. Angles are either acute, right or obtuse. If the legs of an angle are perpendicular to each other, they form a right angle as T. Q. R, (mechanics' squares, if true, are always right angled;) if the sides are nearer together, as S. Q. R, they form an acute angle; if the sides are wider apart, or diverge from each other more than a right angle, they form an obtuse angle, as V. Q. R. The magnitude of an angle does not depend on the length of the sides, but upon their divergence from each other ; an angle is r said r to be greater or less than another as the divergence is greater or r less; r th^s r the r r bUuse angle V. Q. R is greater, and the acute than the right angle T. Q. R. UNIVERSITY) DEFINITIONS. PLANE RECTILINEAR SUPERFICIES. TRIANGLES OR TRIGONS. C d QUADRILATERALS, QUADRANGLES OR TETRAGONS. PARALLELOGRAMS. Male .7 DEFINITIONS OF THE CIRCLE. SEMICIRCLE. SEGMENTS. c, I ' A ! 5. QUADRANT. COMPLEMENT. E 7 SUPPLE Ml-:.\-T. 9. SINE. ro-; L 9 PLATE II. PLANE RECTILINEAR SUPERFICIES, 13. A SUPERFICIES or SURFACE is considered as an extension of length and breadth without thickness. 14. A PLANE SUPERFICIES is an enclosed flat surface that will coincide in every place with a straight line. It is a succession of straight lines, or to be more explicit, if a perfectly straight edged ruler be placed on a plane superficies in any direction, it would touch it in every part of its entire length. 15. When surfaces are bounded by right lines, they are said to be RECTILINEAR or RECTILINEAL., As all the figures on plate second agree with the above definitions, they are PLANE RECTI- LINEAR SUPERFICIES. 16. Figures bounded by more than four right lines are called POLYGONS; the boundary of a polygon is called its PERIMETER. 17. When SURFACES are bounded by three right lines, they are called TRIANGLES or TRIGONS. 18. AN EQUILATERAL TRIANGLE has all its sides of equal length, and all its angles equal, as Jl. 19. AN ISOSCELES TRIANGLE has two of its sides and two of its angles equal, as B. 20. A SCALENE TRIANGLE has all its sides and angles unequal, as C. 21. AN ACUTE ANGLED TRIANGLE has all its angles acute, as Ji and B. 22. A RIGHT ANGLED TRIANGLE has one right angle; the side opposite the right angle is called the hypothenuse; the other sides are called respectively the base and perpendicular. The figures A. B. C, are each divided into two right angled triangles by the dotted lines running across them. 23. AN OBTUSE ANGLED TRIANGLE has one obtuse angle, as C. 24. If figures A and B were cut out and folded on the dotted line in the centre of each, the opposite sides would exactly coin- cide ; they are therefore, regular triangles. 25. Any of the sides of an equilateral or scalene triangle may be called its BASE, but in the Isosceles triangle the side which is 10 PLATE II. unequal is so called, the angle opposite the base is called the VERTEX. 26. THE ALTITUDE of a Triangle is the length of a perpendicular let fall from its vertex to its base, as a. A. and b. B, or to its base extended, as d. d, figure C. The superficial contents of a Triangle may be obtained by mul- tiplying the altitude by one half the base. 27. When surfaces are bounded by four right lines, they are called QUADRILATERALS, QUADRANGLES or TETRAGONS; either of the figures D. E. F. G. H and K may be called by either of those terms, which are common to all four-sided right lined figures, although each has its own proper name. 28. When a Quadrilateral has its opposite sides parallel to each other, it is called a PARALLELOGRAM ; therefore figures D. E. F and G are parallelograms. 29. When all the angles of a Tetragon are right angles, the figure is called a RECTANGLE, as figures D and E. If two opposite angles of a Tetragon are right angles, the others are necessarily right too. 30. If the sides of a Rectangle are all of equal length, the figure is called a SQUARE, as figure D. 31. If the sides of a Rectangle are not all of equal length, two of its sides being longer than the others, as figure E, it is called an OBLONG. 32. When the sides of a parallelogram are all equal, and the an- gles not right angles, two being acute and the others obtuse, as figure F, it is called a RHOMB, or RHOMBUS ; it is also called a DIAMOND, and sometimes a LOZENGE, more particularly so when the figure is used in heraldry. 33. A parallelogram whose angles are not right angles, but whose opposite sides are equal, as figure G, is called a RHOMBOID. 34. If two of the sides of a Quadrilateral are parallel to each other as the sides H and in fig. H, it is called a TRAPEZOID. 35. All other Quadrangles are called TRAPEZIUMS, the term being applied to all Tetragons that have no two sides parallel, as K. NOTE. The terms TRAPEZOID and TRAPEZIUM are applied indiscrimin- ately by some writers to either of the figures H and K; by others, fig. H is called a Trapezium and fig. K a Trapezoid, and this appears to be the more correct method; but asTrapezoid is a word of comparatively modern origin, I have used it as it is most generally applied by modern writers, more par- ticularly so in works on Architecture and Mechanics. PLATE III. 11 36. A DIAGONAL is a line running across a Quadrangle, connect- ing its opposite corners, as the dotted lines in figs. D and F. NOTE. I have often seen persons who have not studied Geometry, much confused, in consequence of the number of names given to the same figure, as for example fig. D. 1st. It is a plane Figure see paragraph 14. 2nd. It is Rectilineal, being composed of right lines. 3rd. It is a Quadrilateral, being bounded by four lines. 4th. It is a Quadrangle, having four angles. 5th. It is a Tetragon, having four sides. 6th. It is a Parallelogram, its opposite sides being parallel. 7th. It is a Rectangle, all its angles being right angles. All the above may be called common names, because they are applied to all figures having the same properties. 8th. It is a Square, which is its proper name, distinguishing it from all other figures, to which some or all of the above terms may be applied. All of them except 7 and 8, may also be applied to fig. F, w : th the same propriety as to fig. D ; besides these, fig. F has four proper names distin- guishing it from all other figures, viz: a Rhomb, Rhombus, Diamond and Lozenge. If the student will analyze all the other figures in the same manner, he will soon become perfectly familiar with them, and each term will convey to his mind a clear definite idea. PLATE III. DEFINITIONS OF THE CIRCLE. 1st. A CIRCLE is a plane figure bounded by one curve line, every where equidistant from its centre, as fig. 1. 2nd. The boundary line is called the CIRCUMFERENCE or PE- RIPHERY, it is also for convenience called a Circle. 3rd. The CENTRE of a circle is a point within the circumference, equally distant from every point in it, as C, fig. 1. 4th. The RADIUS of a circle is a line drawn from the centre to any point in the circumference, as C. Jl, O. B or C. D, fig. 1. The plural of Radius is RADII. All radii of the same circle are of equal length. 5th. The DIAMETER of a circle is any right line drawn through the centre to opposite points of the circumference, as Ji. B } fig. 1. 12 PLATE III. * The length of the diameter is equal to two radii; there may be an infinite number of diameters in the same circle, but they are all equal. 6th. A SEMICIRCLE is the half of a circle, as fig. 2; it is bounded by half the circumference and by a diameter. 7th. A SEGMENT of a circle is any part of the surface cut off by a right line, as in fig. 3. Segments may be therefore greater or less than a semicircle. 8th. An ARC of a circle is any portion of the circumference cut off, as C. G. D or E. G. F, fig. 3. 9th. A CHORD is a right line joining the extremities of an arc, as C. D and E. F, fig. 3. The diameter is the chord of a semicircle. The chord is also called the SUBTENSE. 10th. A SECTOR of a circle is a space contained between two radii and the arc which they intercept, as E. C. jD, or 0. C. H, fig. 4. llth. A QUADRANT is a sector whose area is equal to one-fourth of the circle, as fig. 5 ; the. arc D. E being equal to one-fourth of the whole circumference, and the radii at right angles to each other. 12. A DEGREE. The circumference of a circle is considered as divided into 360 equal parts called DEGREES, (marked ) each degree is divided into 60 minutes (marked ') and each minute into 60 seconds (marked "); thus if the circle be large or small, the number of divisions is always the same, a degree being equal to 1 -360th part of the whole circumference, the semicircle equal to 180, and the quadrant equal to 90. The radii drawn from the centre of a circle to the extremities of a quadrant are always at right angles to each other; a right angle is therefore called an angle of 90. If we bisect a right angle by a right line, it would divide the arc of the quadrant also into two equal parts, each part equal to one-eighth of the whole circumference containing 45; if the right angle were divided into three equal parts by straight lines, it would divide the arc into three equal parts, each containing 30. Thus the degrees of the circle are used to measure angles, and when we speak of an angle of any number of degrees, it is understood, that if a circle with any length of radius, be struck with one foot of the dividers in the angular point, the sides of the angle will intercept a portion of the circle equal to the number of degrees given. NOTE. This division of the circle is purely arbitrary, but it has existed PLATE III. 13 from the most ancient times and every where. During the revolutionary period of 1789 in France, it was proposed to adopt a decimal division, by which the circumference was reckoned at 400 grades; but this method was never extensively adopted and is now virtually abandoned. 13. The COMPLEMENT of an Jirc or of an JLngle, is the difference between that arc or angle and a quadrant; thus E. D fig. 6 is the complement of the arc D. B, and E. C. D the complement of the angle D. C. B. 14. The SUPPLEMENT of an Arc or of an Jingle, is the difference between that arc or angle and a semicircle ; thus D. Jl fig. 7, is the supplement of the arc D. By and D. C. Jl the supplement of the angle B. C. D. 15. A TANGENT is a right line, drawn without a circle touching it only at one point as B. E fig. 8; the point where it touches the circle is called the point of contact, or the tangent point. 16. A SECANT is a right line drawn from the centre of a circle cutting its circumference and prolonged to meet a tangent as C. E fig. 8. NOTE. SECANT POINT is the same as point of intersection, being the point where two lines cross or cut each other. 17. The CO-TANGENT of an arc is the tangent of the comple- ment of that arc, as H. K fig. S. NOTE. The shaded parts in these diagrams are the given angles, but if in fig. 8, D. C. H be the given angle and D. H the given arc, then H. K. would be the tangent and B. E the co-tangent. 18. The SINE of an arc is a line drawn from one extremity, per- pendicular to a radius drawn to the other extremity of the arc as D. F fig. 9. 19. The CO-SINE of an arc is the sine of the complement of that arc as L. D fig. 10. 20. The VERSED SINE of an arc is that part of the radius inter- cepted between the sine and the circumference as F. B fig. 9. 21. In figure 11, we have the whole of the foregoing definitions illustrated in one diagram. C. HC. D C. B and C. Jl are Radii; Jl. B the Diameter; B. C. D a Sector; B. C. H a Quadrant. Let B. C. D be the given Jingle, and B. D the given Jlrc, then B. D is the Chord, D. H the Complement, and D. Jl the Supplement of the arc; Z>. C. H the Complement and D. C. JL the Supplement of the given angle; B. E the Tan- gent and H. K the Co-tangent, C. E the Secant and C. K the Co-secant, F. D the Sine, L. D the Co-sine and F. B the Versed Sine. 14 PLATE IV. TO ERECT OR LET FALL A PERPENDICULAR. PROBLEM 1. FIGURE 1. To bisect the right line A. B by a perpendicular. 1st. With any radius greater than one half of the given line, and with one point of the dividers in A and B successively, draw two arcs intersecting each other, in C and D. 2nd. Through the points of intersection draw C. D, which is the perpendicular required. PROBLEM 2. FIG. 2. From the point D in the line E. F to erect a perpendicular. 1st. With one foot of the dividers placed in the given point D with any radius less than one half of the line, describe an arc, cutting the given line in B and C. 2nd. From the points B and C with any radius greater than B. D, describe two arcs, cutting each other in G. 3rd. From the point of intersection draw G. Z), which is the per- pendicular required. PROBLEM 3. FIG. 3. To erect a perpendicular when the point D is at or near the end of a line. 1st. With one foot of the dividers in the given point D with any radius, as D. E, draw an indefinite arc G. H. 2nd. With the same radius and the dividers in any point of the arc, as E, draw the arc B. D. F, cutting the line C. D in B. 3rd. From the point B through E draw a right line, cutting the arc in F. UITI7BRSITT Tit KlifiCT OR LET FALL A PERPENDICULAR. ////. 2. Fiq. 5. CONSTRUCTION AND DIVISION OF ANGLES. Fia Fiq. 6. Fiq. 5. " kSOT PLATE IV. 15 4th. From F draw F. D, which is the perpendicular required. NOTE. It will be perceived that the arc B. D. F is a semicircle, and the right line B. F a. diameter ; if from the extremities of a semicircle right lines be drawn to any point in the curve, the angle formed by them will be a right angle. This affords a ready method for forming a " square corner," and will be found useful on many occasions, as its accuracy may be de- pended on. PROBLEM 4. FIG. 4. Another method of erecting a perpendicular when at or near the end of the line. Continue the line H. D toward C, and proceed as in problem 2; the letters of reference are the same. PROBLEM 5. FIG. 5. From the point D to let fall a perpendicular to the line A. B. 1st. With any radius greater than D. G and one foot of the com- passes in D, describe an arc cutting Ji. B in E and F. 2nd. From E and F with any radius greater than E. G ; describe two arcs cutting each other as in C. 3rd. From D draw the right line D. C, then D. G is the per- pendicular required. PROBLEM 6. FIG. 6. When the point D is nearly opposite the end of the line. 1st. From the given point D, draw a right line to any point of the line Jl. B as 0. 2nd. Bisect 0. D by problem 1 ; in E. 3rd. With one foot of the compasses in E with a radius equal to E. D or E. describe an arc cutting Jl. B in F. 4th. Draw D. F which is the perpendicular required. NOTE. The reader will perceive that we have arrived at the same result as we did by problem 3, but by a different process, the right angle being- formed within a semicircle. 16 PLATE IV. PROBLEM 7. FIG. 7. Another method of letting fall a perpendicular when the given point D is nearly opposite the end of the line. 1st. With any radius as F. D and one foot of the compasses in the line Jl. B as at F, draw an arc D. H. C. 2nd. With any other radius as E. D draw another arc D. K. C, cutting the first arc in C and D. 3rd. From D draw D. C, then D. G is the perpendicular re- quired. N OTE . The points E and F from which the arcs are drawn, should be as far apart as the line A. B will admit of, as the exact points of intersec- tion can be more easily found, for it is evident, that the nearer two lines cross each other at a right angle, the finer will be the point of contact. PROBLEM 8. FIG. 8. To erect a perpendicular at D the end of the line C. D. with a scale of equal parts. 1st. From any scale of equal parts take three in your dividers, and with one foot in D, cut the line C. D in B. 2nd. From the same scale take four parts in your dividers, and with one foot in D draw an indefinite arc toward E. 3rd. With a radius equal to five of the same parts, and one foot of the dividers in B y cut the other arc in E. 4th. From E draw E. D } which is the perpendicular required. NOTE 1st. If four parts were first taken in the dividers and laid off on the line C. D, then three parts should be used for striking the indefinite arc, at v3, and the five parts struck from the point C, which w^ould give the intersection #, and arrive at the same result. 2nd. On referring to the definitions of angles, it will be found that the side of a right angled triangle opposite the right angle is called the Hypothe- nuse; thus the line E. B is the hypothenuse of the triangle E. D. B. 3rd. The square of the hypothenuse of a right angled triangle is equal to the sum of the squares of both the other sides. 4th. The square of a number is the product of that number multiplied by itself. Example. The length of the side D. E is 4, which multiplied by 4, will give for its square 16. The length of D. B is 3, which multiplied by 3, gives for the square 9. The products of the two sides added together give 25. The length of the hypothenuse is 5, which multiplied by 5, gives also 25. 5th. The results will always be the same, but if fractional parts are used in PLATE IV. 17 the measures, the proof is not so obvious, as the multiplication would be more complicated. 6th. 3, 4 and 5 are the least whole numbers that can be used in laying down this diagram, but any multiple of these numbers may be used ; thus, if we multiply them by 2, it would give 6, 8 and 10 ; if by 3, it would give 9, 12 and 15 ; if by 4 12, 16 and 20, and so on. Tne greater the distances employed, other things being equal, the greater will be the probable accuracy of the result. 7th. We have used a scale of equal parts without designating the unit of measurement, which may be an inch, foot, yard, or any other measure. 8th. As this problem is frequently used by practical men in laying off work, we will give an illustration. Example. Suppose the line C. D to be the front of a house, and it is desired to lay oif the side at right angles to it from the corner D. 1st. Drive in a small stake at D, put the ring of a tape measure on it and lay off twelve feet toward B. 2nd. With a distance of sixteen feet, the ring remaining at D, trace a short circle on the ground at E. 3rd. Remove the ring to I?, and with a distance of twenty feet cut the first circle at E. 4th. Stretch a line from D to E, which will give the required side of the building. PLATE V. CONSTRUCTION AND DIVISION OF ANGLES. PROBLEM 9. FIG. 1. The length of the sides of a Triangle A. B., C. D. and E. F being given, to construct the Triangle, the two longest sides to be joined together at A. 1st. With the length of the line C. D for a radius and one foot in Jl y draw an arc at G. 2nd. With the length of the line E. F for a radius and one foot in B, draw an arc cutting the other arc at G. 3rd. From the point of intersection draw G. A and G. B, which complete the figure. 13 PLATE V. PROBLEM 10. FIG. 2. To construct an Jingle at K equal to the Angle H. 1st. From H with any radius, draw an arc cutting the sides of the angle as at M JV*. 2nd. From K with the same radius, describe an indefinite arc at 0. 3rd. Draw K. parallel to H. M. 4th. Take the distance from M to N and apply it from to P. 5th. Through P draw K. P, which completes the figure. PROBLEM 11. FIG. 3. To Bisect the given Jingle Q by a Right Line. 1st. With any radius and one foot of the dividers in Q draw an arc cutting the sides of the angle as in R and S. 2nd. With the same or any other radius, greater than one half R. S, from the points S and R, describe two arcs cutting each other, as at T. 3rd. Draw T. Q, which divides the angle equally. NOTZ. This problem may be very usefully applied by workmen on many occasions. Suppose, for example, the corner Q be the corner of a room, and a washboard or cornice has to be fitted around it; first, apply the bevel to the angle and lay it down on a piece of board, bisect the angle as above, then set the bevel to the centreline, and you have the exact angle for cutting the mitre. This rule will apply equally to the internal or external angle. Most good practical workmen have several means for getting the cut of the mitre, and to them this demonstration will appear unnecessary, but I have seen many men make sad blunders, for want of knowing this simple rule. PROBLEM 12. FIG. 4. To Trisect a Right Angle. 1st. From the angular point Fwith any radius, describe an arc cutting the sides of the angle, as in X and W. 2nd. With the same radius from the points X and W y cut the arc in Y and Z. 3rd. Draw Y. V and Z. V, which will divide the angle as re- quired. PLATE V. 19 PROBLEM 13. FIG. 5. In the triangle A. B. C, to describe a Circle touching all its sides. 1st. Bisect two of the angles by problem 11, as Jl and B, the dividing lines will cut each other in D, then D is the centre of the circle. 2nd. From D let fall a perpendicular to either of the sides as at F, then D. F is the radius, with which to describe the circle from the point D. PROBLEM 14. FIG. 6. On the given line A. B to construct an Equilateral Triangle, the line A. B to be one of its sides. 1st With a radius equal to the given line from the points Jl and By draw two arcs intersecting each other in C. 2nd. From (7, draw C. Jl and C. B, to complete the figure. PLATE VI. CONSTRUCTION OF POLYGONS. A figure of 3 sides is called a Trigon. " 4 " " Tetragon. polygon 5 " Pentagon. " 6 Hexagon. " 7 " Heptagon. " 8 Octagon. 9 " Enneagon or Nonagon. a 10 " Decagon. II" " Undecagon. " 12 " Dodecagon. 1st. When the sides of a polygon are all of equal length and all the angles are equal, it is called a regular polygon ; if unequal; it is called an irregular polygon. 20 PLATE VI. 2nd. It is not necessary to say a regular Hexagon, regular Octa- gon, &c.; as when either of those figures is named, it is always supposed to be regular, unless otherwise stated. PROBLEM 15. FIG. 1. On a given line A. B to construct a square whose side shall be equal to the given line. 1st. With the length A. B for a radius from the points JL and B, describe two arcs cutting each other in C. 2nd. Bisect the arc C. A or C. B in D. 3rd. From C, with a radius equal to C. D, cut the arc B. E in E and the arc A. F in F. 4th. Draw JL. E, E. F and F. B, which complete the square. PROBLEM 16. FIG. 2. In the given square G. H. K. J, to inscribe an Octagon. 1st. Draw the diagonals G. K and H. J, intersecting each other in P. 2nd. With a radius equal to half the diagonal from the corners G. H. K and J, draw arcs cutting the sides of the square in 0. 0. 0, &c. 3rd. Draw the right lines 0. 0., 0. 0, &c., and they will com- plete the octagon. This mode is used by workmen when they desire to make a piece of wood round for a roller, or any other purpose ; it is first made square, and the diagonals drawn across the end ; the dis- tance of one-half the diagonal is then set off, as from G to R in the diagram, and a guage set from H to R which run on all the corners, gives the lines for reducing the square to an octagon ; the corners are again taken off, and finally finished with a tool appropriate to the purpose. The centre of each face of the octa- gon gives a line in the circumference of the circle, running the whole length of the piece; and as there are eight of those lines equidistant from each other, the further steps in the process are rendered very simple. c<).\sTnn"mt\ o o Plate 7. <; TO riii-: r//i'r '/,/;. Fin. 2. . PLATE VI. 21 PROBLEM 17. FIG. 3. In a given circle to inscribe an Equilateral Triangle, a Hexagon and Dodecagon. 1st. For the TRIANGLE, with the radius of the given circle from any point in the circumference, as at JH } describe an arc cutting the circle in B and C. 2nd. Draw the right line B. C, and with a radius equal B. C, from the points B and C, cut the circle in D. 3rd. Draw D. B and D. C, which complete the triangle. 4th. For the HEXAGON, take the radius of the given circle and carry it round on the circumference 'six times, it will give the points Jl. B. E. D. F. C, through them, draw the sides of the hexagon. The radius of a circle is always equal to the side of an hexagon inscribed therein. 5th. For the DODECAGON, bisect the arcs between the points found for the hexagon, which will give the points for inscribing the dodecagon. PROBLEM 18. FIG. 4. In a given Circle to inscribe a Square and an Octagon. 1st. Draw a diameter Jl. B, and bisect it with a perpendicular by problem 1, giving the points C. D. 2nd.' From the points Jl. C. B. D, draw the right lines forming the sides of the square required. 3rd. For the OCTAGON, bisect the sides of the square and draw perpendiculars to the circle, or bisect the arcs between the points Jl. C. B. D, which will give the other angular points of the re- quired octagon. PROBLEM 19. FIG. 5. On the given line O. P to construct a Pentagon, O. P being the length of the side. 1st. With the length of the line 0. P from 0, describe the semi- circle P. Q, meeting the line P. 0, extended in Q. 2nd. Divide the semicircle into five equal parts and from draw lines through the divisions 1, 2 and 3. PLATE VI. 3rd. With the length of the given side from P, cut 1 in S, from j S cut 2, in R, and from Q cut 2 in R; connect the points 0. Q. R. S. P by right lines, and the pentagon will be complete. PROBLEM 20. FIG. 6. On the given line A. B to construct a Heptagon, A. B being the length of the side. 1st. From Jl with JL. B for a radius, draw the semicircle B. H. 2d. Divide the semicircle into seven equal parts, and from Jl \ through 1, 2, 3, 4 and 5, draw indefinite lines. 3rd. From B cut the line A 1 in C, from C cut Jl 2 in D, from j G cut Jl 4 in F, and from F cut ^ 3 in E, connect the points by | right lines to complete the figure. Any polygon may be constructed by this method. The rule is, to divide the semicircle into as many equal parts as there are sides in the required polygon, draw lines through all the divisions except two, and proceed as above. Considerable care is required to draw these figures accurately, on account of the difficulty of finding the exact points of inter- section They should be practised on a much larger scale. PLATE VII. PROBLEM 21. FIG. 1. To find the Centre of a Circle. 1st. Draw any chord, as Jl. B, and bisect it by a perpendicular E. D, which is a diameter of the circle. 2nd. Bisect the perpendicular E. D by problem 1, the point of intersection is the centre of the circle. FIGURE 2. Another method of finding the Centre of a Circle. 1st. Join any three points in the circumference as F. G. H. 2nd. Bisect the chords F. G and G. H by perpendiculars, their point of intersection at C is the centre required. PLATE VII. 23 PROBLEM 22. FIG. 3. To draw a Circle through any three points not in a straight line } as M. N. O. 1st. Connect the points by straight lines, which will be chords to the required circle. 2nd. Bisect the chords by perpendiculars, their point of inter- section at C is the centre of the required circle. 3rd. With one foot of the dividers at C, and a radius equal to C. M, C. JV* ; or (7. 0, describe the circle. PROBLEM 23. FIG. 4. To find the Centre for describing the Segment of a Circle. 1st. Let P. R be the chord of the segment, and P. S the rise. 2nd. Draw the chords P. Q and Q. R, and bisect them by per- pendiculars; the point of intersection at C, is the centre for describing the segment. PROBLEM 24. FIG. 5. To find a Right Line nearly equal to an Jlrc of a Circle, as H. I. R 1st. Draw the chord H. K, and extend it indefinitely toward 0. 2nd. Bisect the segment in /, and draw the chords H. I and /. K. 3rd. With one foot of the dividers in H } and a radius equal to H. I, cut H. in M } then with the same radius, and one foot in M, cut H. again in JV*. 4th. Divide the difference K. JV* into three equal parts, and extend one of them toward 0, then will the right line H. be nearly equal to the curved line H. I. K. PROBLEM 25. FIG. 6. To find a Right Line nearly equal to the Semicircumference A. F. B. 1st. Draw the diameter Jl. B, and bisect it by the perpendicular F. H; extend F. H indefinitely toward G. 2nd. Divide the radius C. H into four equal parts, and extend three of those parts to G. 3rd. At F draw an indefinite right line D. E, parallel to A. . 24 PLATE VII. 4th. From G through JL, the end of the diameter Jl. B, draw G. Jl. D, cutting the line D. E in D, and from G through B draw G. B. E, cutting D. E in , then will the line D. E be nearly equal to the semicircumference of the circle, and the triangles D. G. E and Jl. G. B will be equilateral. NOTE. The right lines found by problems 24 and 25, are not mathemati- cally equal to the respective curves, but are sufficiently correct for all practical purposes. Workmen are in the habit of using the following method for finding the length of a curved line : They open their compasses to a small distance, and commencing at one end, step off the whole curve, noting the number of steps required, and the remainder less than a step, if any ; they then step off the same number of times, with the same distance,on the article to be bent around it, and add the remainder, which gives them a length sufficiently true for their purpose : the error in this method amounts to the sum of the differences between the arc cut off by each step, and its chord. PLATE VIII. PARALLEL RULER AND APPLICATION. FIGURE 1. The parallel ruler figured in the plate consists of two bars of wood or metal Jl. B and C. D, of equal length, breadth and thickness, connected together by two arms of equal length placed diagonal- ly across the bars, both at the same angle, and moving freely on the rivets which connect them to the bars; if the bar JL. B be kept firmly in any position and the bar C. D moved, the ends of the arms connected to C. D will describe arcs of circles and recede from JL. B until the arms are at right angles to the bars, as shewn by the dotted lines; if moved farther round, the bars will again approach each other on the other side. The bars of which the ruler is composed, being parallel to each other., it follows, that if either edge of the instrument is placed parallel with a line and held in that position, another line may be drawn parallel to the first at any distance within the range of the instrument. This is its most obvious use; it is generally applied to the drawing of inclined parallel lines in mechanical drawings, vertical and horizontal lines being more easily drawn with the ^quare, when the drawing is attached to a drawing board. TJHI7BRSIT7 find //.v Application fty.4. .<;. SCALE OF CHORUS /''ft/. I . i i i | i i num.! o :>*> Ao S\<> 2\tt iV> ii CALKS 01'' l-:.' w ...a?.... i4o 36 d?. M4 3O .-?-... k~>t> FLAT SEGMU'XTS . I XI) /', / It. Ul f>LAS. ^:R^: OF THT SITY PLATE X. 31 sides in the required polygon, the quotient will be the angle at the centre; for example, let it be required to find the angle at the centre of an octagon: divide 360 by 8, the nilfnber of sides, the quotient will be 45, which is the angle formed by the octagon at the centre. 2nd. To find the angle formed by two adjoining faces of a polygon, we must subtract from 180 the number of degrees in the semi- circle, the angle formed by said polygon at the centre, the re- mainder will be the angle formed at the circumference. For ex- ample let us take the octagon ; we have found in the last paragraph that the angle formed by that figure in the centre is 45; then if we subtract 45 from 180 it will leave 135, which is the angle formed by two adjoining faces of the octagon. PLATE XT. TO DESCRIBE FLAT SEGMENTS OF CIRCLES AND PARABOLAS, Very often in practice it would be very inconvenient to find the centre for describing a flat segment of a circle, in consequence of the rise of an arch being so small compared to its span. PROBLEM 34. FIG. 1. To describe a Segment with a Triangle. NOTE. In all the diagrams in this plate Ji. B is the span of the arch, A. D the rise, and C the centre of the crown of the arch. 1st. Make a triangle with its longest side equal to the chord or span of the arch and its height equal to one-half the rise. 2nd. -Stick a nail at JL and at C, pi ace the triangle as in the diagram and move it round against the nails toward JL* a pencil kept at the apex of the triangle will describe one-half of the curve. 3rd. Stick another nail at B, and with the triangle moving against C and B describe the other half of the curve. 32 PLATE XI. PROBLEM 35. FIG. 2. To describe the same Curve tvith strips of wood, forming a Triangle. 1 st. Drive a nail at A and another at J5, place one strip against JL and bring it up to the centre of the crown at C. 2nd. Place another strip against B and crossing the first at C, nail them together at the intersection, and nail a brace across to keep them in position. 3rd. With the pencil at C and the triangle formed by the strips kept against Jl and B, describe the curve from C toward Jl, and from C toward B. PROBLEM 36. FIG. 3. To draw a Parabolic Curve by the intersection of lines forming Tangents to the Curve. 1st. Draw C. 8 perpendicular to Jl. B, and make it equal to Jl. D 2nd. Join Ji. 8 and B. 8, and divide both lines into the same number of equal parts, say 8, number them as in the figure, draw 1. 1. 2. 2. 3. 3., &c., then these lines will be tangents to the curve ; trace the curve to touch the centre oi' each of those lines between the points of intersection. PROBLEM 37. FIG. 4. To draw the same Curve by another method. 1st. Divide Jl. D and B. E, into any number of equal parts, and C. D and C. E into a similar number. 2nd. Draw 1. 1. 2. 2. &c., parallel to Jl. D, and from the points of division in Jl. D and B. E, draw lines to C. The points of intersection of the respective lines, are points in the curve. NOTE. The curves found, as in figs. 3 and 4, are quicker at the crown than a true circular segment ; but, where the rise of the arch is not more than one-tenth of the span, the variation cannot be perceived. Hate 12 OVAL h'K'.riiKX rai//Y/S'/iV; OF AIICS OP flatelS. CYCLOID AXD EPICYCLOID. PLATE XI. 33 PROBLEM 38. FIG. 5. To describe a True Segment of a Circle by Intersections. 1st. Draw the chords Jl. C and B. C, and Jl. and B. 0', per- pendicular to them. 2nd. Prolong D. E in each direction to O. 0' ; divide 0. C, C. O' y Jl. D y Jl. 6, B. 6, and B. E into the same number of equal parts. 3rd. Join the points 1. L 2. 2. &c., in Jl. B and 0. O f . 4th. From the divisions in Jl. D, and B. E, draw lines to C. The points of intersection of these lines with the former, are points in the curve. A semicircle may be described by this method. PLATE XII. TO DESCRIBE OVAL FIGURES COMPOSED OF ARCS OF CIRCLES. PROBLEM 39. FIG. 1. The length of the Oval A. B. being given, to describe an Oval upon it. 1st. Divide Jl. B the given length, into three equal parts, in E and F. 2nd. With one of those parts for a radius, and the compasses in E and F successively, describe two circles cutting each other in and 0'. 3rd. From the points of intersection in and 0', draw lines through E and F, cutting the circles in V. V," and V.' VJ" 4th. With one foot of the compasses in 0, and 0' successively, and with a radius equal to 0. V," or 0.' V, describe the arcs between V. V} and V! 1 V", to complete the figure. PROBLEM 40. FIG. 2. To describe the Oval, the length A. B, and breadth C. D, being given. 1st. With half the breadth for a radius, and one foot in F, de- scribe the arc C. E, cutting Jl. B in E. 34 PLATE XII. 2nd. Divide the difference E. B between the semiaxes into three equal parts, and carry one of those divisions toward 4. 3d. Take the distance B 4, and set off on each side of the centre F at H and H. 1 4th. With the radius H. HJ describe from H and H' as centres, arcs cutting each other in K and KJ 5th. From K and K] through H and JET/ draw indefinite right lines. 6th. With the dividers in H, and the radius H. Jl, describe the curve V. JL. V" and with the dividers in H/ describe the curve VJ B. VJ" 7th. From K and K,' with a radius equal to K. C, describe the curves V. C. VJ and VJ' D. VJ" to complete the figure. PROBLEM 41. FIG. 3. Jlnother method for describing the Oval, the length A. B, and breadth C. D, being given. 1st. Draw C. B, and from B, with half the transverse axis, B. F, cut B. C in 0. 2nd. Bisect B. b j a perpendicular, cutting Jl. B in P, and (7. D in Q. 3rd. From F, set off the distance F. P to R, and the distance F. Q to S. 4th. From S, through R and P, and from Q through 7?, draw indefinite lines. 5th. From P and R, and from S and Q, describe the arcs, com- pleting the figure as in the preceding problem. NOTE. In all these diagrams, the result is nearly the same. Figs 1 and 2 are similar figures, although each is produced by a different process. The proportions of an oval, drawn as figure 1, must always be the same as in the diagram ; but, in figs. 2 and 3, the proportions may be varied ; but, when the difference in the length of the axes, exceeds one-third of the longer one, the curves have a very unsightly appearance, as the change of curva- ture is too abrupt. These figures are often improperly called ellipses, and sometimes false ellipses. Ovals are frequently used for bridges. When the arch is flat, the curve is described from more than two centres, but it is never so graceful as the true ellipsis. 35 PLATE XIII. TO DESCRIBE THE CYCLOID AND EPICYCLOID. The Cycloid is a curve formed by a point in the circumference of a circle, revolving on a level line ; this curve is described by any point in the wheel of a carriage when rolling on the ground. PROBLEM 42. FIG. 1. To find any number of Points in the Cycloid Curve by the inter- section of lines. 1st. Let G. H be the edge of a straight ruler, and C the centre of the generating circle. 2nd. Through C draw the diameter Jl. B perpendicular to G. H, and E. F parallel to G. H; then A. B is the height of the curve, and E. F is the place of the centre of the generating circle at every point of its progress. 3rd. Divide the semicircumference from B to Jl into any number of equal parts, say 8, and from Jl draw chords to the points of di- vision. 4th. From C, with a space in the dividers equal to one of the di- visions on the circle, step off on each side the same number of spaces as the semicircumference is divided into, and through the points draw perpendiculars to G. H: number them as in the dia- gram. 5th. From the points of division in E. F, with the radius of the generating circle, describe indefinite arcs as shewn by the dotted lines. 6th. Take the chord Jl 1 in the dividers, and with the foot at 1 and 1 on the line G. H } cut the indefinite arcs described from 1 and 1 respectively at D and D f , then D and D' are points in the curve. 7th. With the chord Jl 2, from 2 and 2 in G. H, cut the indefinite arcs in J and J 7 , with the chord Jl 3, from 3 and 3, cut the arcs in K and K and apply the other chords in the same manner, cutting the arcs in L. M, &LC. 8th. Through the points so found trace the curve. 36 PLATE XIII. NOTE. The indefinite arcs in the diagram represent the circle at that point of its revolution, and the points D. J. K, &c., the position of the genera- ting point B at each place. This curve is frequently used for the arches of bridges, its proportions are always constant, viz : the span is equal to the circumference of the. generating circle and the rise equal to its diameter. Cycloidal arches are frequently constructed which are not true cycloids, but approach that curve in a greater or less degree. FIGURE 2. THE EPICYCLOID. This curve is formed by the revolution of a circle around a circle, either within or without its circumference., and described by a point B in the circumference of the revolving circle. P is the centre of the revolving circle, and Q of the stationary circle. PROBLEM 43. To find Points in the Curve. 1st. Draw the diameter 8. 8, and from Q the centre, draw Q. B at right angles to 8. 8. 2nd. With the distance Q. P from Q, describe an arc 0. repre- senting the position of the centre P throughout its entire progress. 3rd. Divide the semicircle B. D and the quadrants D. 8 into the same number of equal parts, draw chords from D to 1, 2, 3, &,c., and from Q draw lines through the divisions in D. 8 to intersect the curve 0. in 1, 2, 3, &c. 4th. With the radius of P from 1, 2, 3, &c., in 0. describe in- definite arcs, apply the chords D 1, D 2, &,c., from 1, 2, 3, &c., in the circumference of Q, cutting the indefinite arcs in Jl. C. E. F, &,c., which are points in the curve. PLATE XIV- DEFINITIONS OF SOLIDS On referring back to our definitions, we find that a point has posi- tion without magnitude. A Line has length, without breadth or thickness, consequently has but one dimension. UNIVERSITY; 15. SO/JDS AXD THE IK COVERINGS. ^Hfe' ^^^ Fig. 4. PLATE XIV. 37 A Surface has length and breadth, without thickness, conse- quently has two dimensions, which, multiplied together, give the content of its surface. A Solid has length, breadth and thickness. These three dimen- sions multiplied together, give its solid content. Lineal Measure, is the measure of lines. Superficial or Square Measure, the measure of surfaces, Cubic Measure, is the measure of solids. For Example. If we take a cube whose edge measures two feet, then two feet is the lineal dimension of that line. If the edge is two feet long, the adjoining edge is also two feet long; then, two feet multiplied by two, gives four feet, which is the superfi- cial content of a face of the cube. Then, if we multiply the square or superficial content, by two feet, which is the thickness of the cube, it will give eight feet, which is its solid content. Then, two lineal feet is the length of the edge. " four square " the surface of one side. And eight cubic " the solid content of the cube. THE CUBE OR HEXAHEDRON, ITS SECTIONS AND SURFACE. FIGURE 1. 1st. The cube is one of the regular polyhedrons, composed of six regular square faces, and bounded by twelve lines of equal length ; the opposite sides are all parallel to each other. 2nd. If a cube be cut through two of its opposite edges, and the diagonals of the faces connecting them, the section will be an oblong rectangular parallelogram, as fig. 2. 3rd. If a cube be cut through the diagonals of three adjoining faces, as in fig 3, the section will be an equilateral triangle, whose side is equal to the diagonal of a face of the cube. Two such sections may be made in a cube by cutting it again through the other three diagonals, and the second section will be parallel to the first. 4th. If a cube be cut by a plane passing through all its sides, the line of section, in each face, to be parallel with the diagonal, and midway between the diagonal and the corner of the face, as in 38 PLATE XIV. fig. 4, the section will be a regular hexagon, and will be parallel with, and exactly midway between the triangular sections de- fined in the last paragraph. 5th. If a cube be cut by any other plane passing through all its sides, the section will be an irregular hexagon. 6th. The surface of the cube fig. 1, is shewn at fig. 5, and if a piece of pasteboard be cut out, of that form, and cut half through in the lines crossing the figure, then folded together, it will form the regular solid. All the other solids may be made of pasteboard, in the same manner, if cut in the shape shewn in the coverings of the diagrams in the following plates. 7th. The measure of the surface of a cube is six times the square of one of its sides. Thus, if the side of a cube be one foot, the surface of one side will be one square foot, and its whole surface would be six square feet. Its solidity would be one cubic foot. NOTE. The cube may also, in general, be called a prism, and a parallelo- pipedon, as it answers the description given of those bodies, but the terms are seldom applied to it. PLATE XV. SOLIDS AND THEIR COVERINGS FIG. 1. Is a solid, bounded by six rectangular faces, each oppo- site pair being parallel, and equal to each other ; the sides are oblong parallelograms, and the ends are squares. It is called a right SQUARE PRISM, PARALLELOPIPED, Or PARALLELOPIPEDON. FIG. 2. Is its covering stretched out. FIG. 3. Is a triangular prism; its sides are rectangles, and its ends equal triangles. FIG. 4. Is its covering. PRISMS derive their names from the shape of their ends, and the angles of their faces, thus : Fig. 1 is a square prism, and fig. 2 a triangular prism. If the ends were pentagons the prism would be pentagonal; if the ends were hexagons, the prism PLATE XV. 39 would be hexagonal, fyc. The sides of all regular prisms are equal rectangular parallelograms. Fie. 5. Is a SQUARE PYRAMID, bounded by a square at its base, and four regular triangles, as shewn at fig. 6. Pyramids, like prisms, derive their names from the shape of their bases ; thus we may have a square pyramid, as in fig. 5, or a triangular, pentagonal, or hexagonal pyramid, &c., as the base is a triangle, pentagon, hexagon, or any other figure. The sides of a pyramid incline together, forming a point at the top. This point is called its vertex, apex, or summit. The axis of a pyramid, is a line drawn from its summit, to the centre of its base. *The length of the axis, is the altitude of the pyramid. When the base of a pyramid is perpendicular to its axis, it is called a right pyramid; if they are not perpendicular to each other, the pyramid is oblique. If the top of a pyramid be cut off, the lower portion is said to be truncated ; it is also called a frustrum of a pyramid, and the upper portion is still a pyra- mid, although only a segment of the original pyramid. A pyramid may be divided into several truncated pyramids, or frus- trums, and the upper portion remain a pyramid, as the name does not convey any idea of size, but a definite idea of form, viz :* a solid, bounded by an indefinite number of equal triangles, with their edges touching each other, forming a point at the top. A pyramid is said to be acute, right angled or obtuse, dependant on the form of its summit. An OBELISK is a pyramid whose height is very great compared to the breadth of the base. The top of an obelisk is generally truncated and cut off, so as to form a small pyramid, resting on the frustrum, which forms the lower part of the obelisk. When the polygon, forming the base of a pyramid, is irregular, the sides of the figure will be unequal, and the pyramid is called an irregular pyramid. * These definitions are applied to pyramids that are right and regular: it is not necessary to say, "aright, regular pyramid," as when a pyramid is named, it is always supposed to be right and regular, unless otherwise ex- pressed. 40 PLATE XVI. SOLIDS AND THEIR COVERINGS. FIG. 1. Is an HEXAGONAL PYRAMID; and fig. 2 its covering. FIG. 3. A RIGHT CYLINDER, is bounded by two uniform circles, parallel to each other. The line connecting their centres, is called the axis. The sides of the cylinder is one uniform surface, connecting the circumferences of the circle, and everywhere equidistant from its axis. PROBLEM 44. FIG. 4. To find the Length of the Parallelogram A. B. C. D, to form the Side of the Cylinder. 1st. Draw the ends, and divide one of them into any number of equal parts, say twelve. 2nd. With the space of one of those parts, step off the same number on Ji. B, which will give the breadth of the covering to bend around the circles. FIG. 5. Is A RIGHT CONE; its base is a circle, its sides sloping equally from the base to its summit. A line drawn from its summit to the centre of the base, is called its axis. If the axis and base are not perpendicular to each other, it forms an oblique, or scalene cone. PROBLEM 45. FIG. 6. To draw the Covering. 1st. With a radius equal to the sloping height of the cone, from E, describe an indefinite arc, and draw the radius E. F. 2nd. Draw the circle of the base, and divide its circumference into any number of equal parts, say twelve. 3rd. With one of those parts in the dividers, step off from F the same number of times to G, then draw the radius E. G, to complete the figure. 41 PLATE XVIX, COVERINGS OF SOLIDS. FIG. 1. THE SPHERE Is a solid figure presenting a circular appearance when viewed in any direction ; its surface is every where equidistant from a point within, called its centre. 1st. It may be formed by the revolution of a semicircle around its chord. 2nd. The chord around which it revolves is called the axis, the ends of the axis are called poles. 3rd. Any line passing through the centre of a sphere to opposite points, is called a diameter. 4th. Every section of a sphere cut by a plane must be a circle, if the section pass through the centre, its section will be a great cir- cle of the sphere ; any other section gives a lesser circle. 5th. When a sphere is cut into two equal parts by a plane passing through its centre, each part is called a hemisphere; any part of a sphere less than a hemisphere is called a segment; this term may be applied to the larger portion as well as to the smaller. PROBLEM 46. FIG. 2. To draw the Covering of the Sphere. 1st Divide the circumference into twelve equal parts. 2nd. Step off on the line Jl. B the same number of equal parts, and with a radius of nine of those parts, describe arcs through the points in each direction ; these arcs will intersect each other in the lines C. D and E* F, and form the covering of the sphere. FIGURE 3. Is the surface of a regular TETRAHEDRON, it is bounded by four equal equilateral triangles. 42 PLATE XVII. FIGURE 4. The regular OCTAHEDRON is bounded by eight equal equilateral triangles. FIGURE 5. The DODECAHEDRON is bounded by twelve equal pentagons. FIGURE 6. The ICOSAHEDRON is bounded by twenty equal equilateral triangles. The four last figures, together with the hexahedron delineated on Plate 14, are all the regular polyhedrons. All the faces and all the solid angles of each figure are respectively equal. These solids are called platoriic figures. PLATE XVIII. THE CYLINDER AND ITS SECTIONS. 1st. If we suppose the right angled parallelogram Jl. B. C. D, fig. 1, to revolve around the side A. B, it would describe a solid figure; the sides Jl. D and B. C would describe two circles whose diameters would be equal to twice the length of the re- volving sides; the side C. D would describe a uniform surface con- necting the opposite circles together throughout their whole cir- cumference. The solid so described would be a RIGHT CYLINDER. 2nd. The line Jl. B, around which the parallelogram revolved, is called the AXIS of the cylinder, and as it connects the centres of the circles forming the ends of the cylinder, it is every where equidistant from its sides. 3rd. If the ends of a cylinder be not at right angles to its axis, it is called an OBLIQUE CYLINDER. 4th. If a cylinder be cut by any plane parallel to its axis, the sec- tion will be a parallelogram, as E. F. G. H, fig. 1. PLATE XVIII. 43 5th. If a cylinder be cut by any plane at right angles to its axis, the section will be a circle. 6th. If a cylinder be cut by any plane not at right angles to its axis, passing through its opposite sides, as at K. L or M. JV, fig. 2, the section will be an ELLIPSIS, of which the line of section K. L or M. JV* would be the longest diameter, called the TRANS- VERSE or MAJOR DIAMETER, and the diameter of the cylinder C. D would be the shortest diameter, called the CONJUGATE or MINOR DIAMETER. PROBLEM 47. FIG. 3 To describe an Ellipsis from the Cylinder with a string and two pins. 1st. Draw the right lines JV*. M and C. D at right angles to each other, cutting each other in S. 2nd. Take in your dividers the distance P. M or P. N, fig. 2, and set it off from S to M and JV, fig. 3, which will make M. JV* equal to M. JV, fig. 2. 3d. From Jl, fig. 2, take A. D or Jl. C, and set it off from S to C and 7), making C. D equal to the diameter of the cylinder. 4th. With a distance equal to S. M or S. JV from the points D and C, cut the transverse diameter in E and F; then E and F are the FOCI for drawing the ellipsis. NOTE. E is a FOCUS, and F is a FOCUS. E and F are FOCI. 5th. In the foci, stick two pins, then pass a string around them, and tie the ends together at C. 6th. Place the point of a pencil at C, and keeping the string tight, pass it around and describe the curve. NOTE. The sum of all lines drawn from the foci, to any point in the curve, is always constant and equal to the major axis : thus, the length of the lines E. R, and F. R, added together, is equal to the length of E. C, and F. C, added together, or to two lines drawn from E and F, to any other point in the curve. 7th. Fig. 4 is the section of the cylinder, through L. K, fig. 2, and is described in the same way as fig. 3. The letters of refer- ence are the same in both diagrams, except that the transverse diameter L. K, is made equal to the line of section L. K, in fig. 2. 44 PLATE XVIII. 8th. The line JV. M, fig. 3, or L. K, fig. 4, is called the TRANS- VERSE, or MAJOR AXIS, (plural AXES,) and the line C. D, its CONJUGATE, or MINOR AXIS. They are also called the transverse and conjugate diameters, as above defined. The transverse axis is the longest line that can be drawn in an ellipsis. 9th. Any line passing through the centre S, of an ellipsis, and meeting the curve at both extremities, is called a DIAMETER : every diameter divides the ellipsis into two equal parts. The CONJUGATE of any diameter, is a line drawn through the centre, terminated by the curve, parallel to a tangent of the curve at the vertex of the said diameter. The point where the diameter meets the curve, is the vertex of that diameter. 10th. An ORDINATE to any diameter, is a line drawn parallel to its conjugate, and terminated by the curve and the said diameter. An ABSCISSA is that portion of a diameter intercepted between its vertex and ordinate. Unless otherwise expressed, ordinates are in general, referred to the axis, and taken as perpendicu- lar to it. Thus, in fig. '4, X. Y is the ordinate to, and L. X and K. X, the abscissae of the axis K. L. V. W is the ordinate to, and (7. F", and D. F, the abscissae of the axis C. D. PLATE XIX. THE CONE AND ITS SECTIONS. DEFINITIONS 1st. A CONE is a solid, generated by the revolution of a right angled triangle about one of its sides. 2nd. If both legs of the triangle are equal, as S. JV and JV. 0, fig. 2, it would generate a RIGHT ANGLED CONE ; the angle S being a right angle. 3rd. If the stationary side of the triangle be longest, as M. J the cone will be ACUTE, and if shortest, as T. JV, it will be OBTUSE angled. 4th. The BASE of a cone is a circle, from which the sides slope regularly to a point, which is called its VERTEX, APEX, or SUMMIT. rill-: ('YIJM)l'Ji AM) SKCTIUXS. //,/. I. Fig. TUK COM! JM) ITS SUCTIONS, /'/'. 5th. VOUSSOIRS is the name given to the stones forming the arch. 6th. The KEYSTONE is the centre or uppermost voussoir D, so called ; because it -is the last stone set, and wedges or keys the whole together. Keystones are frequently allowed to project from the face of the wall, and in some buildings are very elabor- ately sculptured. 7th. The INTRADOS or SOFFIT of an arch is the under side of the voussoirs forming the curve. 8th. The EXTRADOS or BACK is the upper side of the voussoirs. 9th. The THRUST of an arch is the tendency which all arches have to descend in the middle, and to overturn or thrust asunder the points of support. NOTE. The amount of the thrust of an arch depends on the proportions be- tween the rise and the span, that is to say, the span and weight to be sup- ported being definite; the thrust will be diminished in proportion as the rise of the arch is increased, and the thrust will be increased in proportion as the crown of the arch is lowered. 10th. The JOINTS of an arch are the lines formed by the adjoining faces of the voussoirs; these should generally radiate to some de- finite point, and each should be perpendicular to a tangent to the curve at each joint. In all curves composed of arcs of circles, a tangent to the curve at any point would be perpendicular to a radius drawn from the centre of the circle through that point, consequently the joints in all such arches should radiate to the centre of the circle of which the curve forms a part. llth. The BED of an arch is the top of the abutment; the shape of the bed depends on the quality of the curve, and will be ex- plained in the diagrams. 12th. A RAMPANT ARCH is one in which the springing points are not in the same level. 13th. A STRAIGHT ARCH, or as it is more properly called, a PLAT BAND, is formed of a row of wedge-shaped stones of equal depth placed in a horizontal line, the upper ends of the stones being broader than the lower, prevents them from falling into the void below. 14th. Arches are named from the shape of the curve of the under side, and are either simple or complex. I would define simple curves to be those that are struck from one centre, as any segment of a circle, or by continued motion, as the ellipsis, parabola, hy- perbola, cycloid and epicycloid; and COMPLEX ARCHES to be 56 PLATE XXIV. those described from two or more fixed centres, as many of the Gothic or pointed arches. The simple curves have all been de- scribed in our problems of practical Geometry; we shall however repeat some of them for the purpose of showing the method of drawing the joints. PROBLEM 60. FIG. 1. To describe a Segment or Scheme Jlrch, and to draw the Joints. 1st. Let E and F be the abutments, and the centre for describ- ing the curve. 2nd. With one foot of the dividers in 0, and the distance 0. F, describe the line of the intrados. 3rd. Set off the depth of the voussoirs, and with the dividers at 0, describe the line of the extrados. 4th. From E and F draw lines radiating to 0, which gives the line of the beds of the arch. This line is often called by masons a skew-back. 5th. Divide the intrados or extrados, into as many parts as you design to have stones in the arch, and radiate all the lines to 0, which will give the proper direction of the joints. 6th. If the point should be at too great a distance to strike the curve conveniently, it may be struck by Problem 34 or 35, Plate 1 1 ; and the joints may be found as follows : Let it be de- sired to draw a joint at 2, on the line of the extrados ; from 2 set off any distance on either side, as at 1 and 3; and from 1 and 3, with any radius, draw two arcs intersecting each other at 4 then from 4 through 2 draw the joint which will be perpendicular to a tangent, touching the curve at 2. This process must be re- peated for each joint. The keystone projects a little above and below the lines of the arch. PROB. 61. FIG. 2. THE SEMICIRCULAR ARCH. This requires but little explanation. Ji. B is the span and C the centre, from which the curves are struck, and to which the lines of all the joints radiate. The centre C being in the springing line of the arch the beds are horizontal. UFI7EESIT7 . : ' Plate : J / JOISTS AY .1/1 C//KS /'/,/. V /'///. .y. PLATE XXIV. 57 PROB. 62. FIG. 3. THE HORSE SHOE ARCH Is an arc of a circle greater than a semicircle, the centre being above the springing line. This arch is also called the SARACENIC or MORESCO arch, because of its frequent use in these styles of architecture. The joints radiate to the centre, as in fig. 2. The joint at 5, below the horizontal line, also radiates to 0. This may do very well for a mere ornamental arch, that has no weight to sustain; but if, as in the diagram, the first stone rests on a horizontal bed, it would be larger on the inside than on the outside, and would be liable to be forced out of its position by a slight pressure, much more so than if the joint were made horizontal, as at 6. These remarks will also apply to fig. 4, Plate 25. PROBLEM 63. FIG. 4. To describe an Ogee Jlrch, or an Jlrch of Contrary Flexure. NOTE. This arch is seldom used over a large opening, but occurs frequently in canopies and tracery in Gothic architecture, the rib of the arch being moulded. 1st. Let JL. E be the outside width of the arch, and C. D the height, and let JL. E be the breadth of the rib. 2nd. Bisect Jl. B in C, and erect the perpendicular C. D; bisect Jl. C in F, and draw F. J parallel to C. D. 3rd. Through D draw J". K parallel to Jl. B, and make D. K equal to D. J. 4th. From F set off F. G, equal to Jl. E the breadth of the rib, and make C. H equal to C. G. 5th. Join G. J and H. K; then G and H will be the centres for drawing the lower portion of the arch, and J and K will be the centres for describing the upper portion, and the contrary curves will meet in the lines G. J and H. K. PROBLEM 64. FIG. 5. To draw the Joints in an Elliptic Jlrch. Let Jl. B be the span of the arch, C. D the rise, and F. F the foci, from which the line of the intrados may be described. 58 PLATE XXIV. The voussoirs near the spring of the arch are increased in depth, as they have to bear more strain than those nearer the crown ; the outer curve is also an ellipsis, of which Hand If are the foci. To draw a joint in any part of the curve, say at 5. 1st. From F and F the foci, draw lines cutting each other in the given point 5, and continue them out indefinitely. 2nd. Bisect the angle 5 by Problem 11, Page 18, the line of bisection will be the line of the joint. The joints are found at the points 1 and 3 in the same manner. 3d. If we bisect the internal angle, as for the joints 2 and 4, the result will be the same. 4th. To draw the corresponding joints on the opposite side of the arch, proceed as follows : 5th. Prolong the line C. D indefinitely toward E, and prolong the lines of bisection 1, 2, 3, 4 and 5, to intersect C. E in 1, 2, 3, &c., and from those points draw the corresponding joints be- tween JL and D. PLATE XXV. TO DESCRIBE GOTHIC ARCHES AND TO DRAW THE JOINTS, The most simple form of Pointed or Gothic arches are those com- posed of two arcs of circles, whose centres are in the springing line. FIGURE 1. THE LANCET ARCH. When the length of the span Jl. B is much less than the length of the chord Jl. C, as in the diagram, the centres for striking the curves will be some distance beyond the base, as shewn by the rods; the joints all radiate to the opposite centres. FIG. 2. THE EQUILATERAL ARCH. When the span D. E, and the chords D. F and E. F form an equilateral triangle, the arch is said to be equilateral, and the PLATE XXV. 59 centres are the points D and E in the base of the arch, to which all the joints radiate. FIG. 3. THE DEPRESSED ARCH Has its centres within the base of the arch, the chords being shorter than the span ; the joints radiate to the centres respectively. NOTE. There are no definite proportions for Gothic arches, except for the equilateral ; they vary from the most acute to those whose centres nearly touch, and which deviate but little from a semicircle. FIG. 4. THE POINTED HORSESHOE ARCH. This diagram requires no explanation ; the centres are above the springing line. See fig. 3 ; plate xxiv ; page 57. FIGURE 5. To describe the Four Centred Pointed Jlrch. 1st. Let JL. B be the springing line, and E. C the height of the arch. 2nd. Draw B. D parallel to E. C, and make it equal to two- thirds of the height of E. C. 3rd. Join D. C, and from C draw C. L perpendicular to C. D. 4th. Make C. G and B. F both equal to B. D. 5th. Join G. F, and bisect it in H, then through H draw H. L perpendicular to G. F meeting C. L in L. 6th. Join L. F, and continue the line to N. Then L and F are the centres for describing one-half of the arch, and the curves will meet in the line L. F. JY. 7th. Draw L. M parallel to Jl. B, make 0. M equal to 0. L, and E. K equal to E. F. Then K and M are the centres for completing the arch, and the curves will meet in the line M. K. P. 8th. The joints from P to C will radiate to M; from C to JVthey will radiate to L ; from JY to B they will radiate to F, and from P to Jl they will radiate to K. NOTE 1. As the joint at P radiates to both the centres K and Jlf, and the joint at JV" radiates both to F and I/, the change of direction of the lower joints is easy and pleasing to the eye, so much so that we should be uncon- scious of the change, if the constructive lines were removed. 60 PLATE XXV. NOTE 2. When the centres for striking the two centred arch are in the springing line, as in diagrams 1, 2 and 3, the vertical side , of the opening joins the curve, without forming an unpleasant angle, as it would do if the vertical lines were continued up above the line of the centres ; it is true that examples of this character may be cited in Gothic buildings, but its ungrace- ful appearance should lead us to avoid it. PLATES XXVI AND XXVII DESIGN FOR A COTTAGE. Fig. 1. Is the elevation of the south-east front. Fig. 2. Plan of the ground floor. Fig. 3. Section through the line E. F on the plan fig. 2, the front part of the house supposed to be taken away. Fig. 4. Plan of the chamber floor. This simple design is given for the purpose of shewing the method of drawing the plans., section, elevation and details of a building ; it is not offered as a " model cottage/' although it would make a very comfortable residence for a small family. The PLAN of a building is a horizontal section; if we suppose the house cut off just above the sills of the windows of the second floor and the upper portion taken away, it would expose to view the whole interior arrangement, shewing the thickness of the walls, the situation and thickness of the partitions and the position of doors, windows, &c.; all these interior arrangements are in- tended to be represented by fig. 4. If we perform the same operation above the sills of the first floor windows, the arrangements of that floor, including the stairs and piazzas, would appear as in fig. 2. A SECTION of a building is a vertical plan in which the thicknesses of the walls, sections of the fire-places and flues, sizes and direc- tion of the timbers for the floors and roof, depth of the foundations and heights of the stories are shewn, all drawn to a uniform scale. If the front of the building is supposed to be removed, as in fig. 3, the whole of the inside of the back wall will be seen in elevation, shewing the size and finish of the doors and windows., the height of the washboards, and the stucco cornice in the parlor. In looking PLATES XXVI AND XXVII. 61 through the door at K, the first flight of stairs in the back build- ing is seen in elevation. If we suppose the spectator to be looking in the opposite direction, the back part of the house removed, he would see the inside of the front windows &c. instead of the back. An ELEVATION of a building is a drawing of the front, side or back, in which every part is laid down to a scale, and from which the size of every object may be measured. A PERSPECTIVE VIEW of a building, is a drawing representing it as it would appear to a spectator from some definite point of view, and in which, all objects are diminished as they recede from the eye. The PLANS, SECTIONS and ELEVATIONS, give the true size and arrangements of the building drawn to a scale, and shew the whole construction. The PERSPECTIVE VIEW should shew the building complete, in connection with the surrounding objects, which would enable the proprietor to judge of the effect of his intended improvement. The whole constitutes the DESIGN, which for a country house can- not be considered complete without a perspective view. To make a design for a dwelling house, or other building, it is ne- cessary before we commence the drawing, that we should know the site on which it is to be erected, and the amount of accommo- dation required. In choosing a site for a country residence many subjects should be taken into consideration ; for example, it should be easy of access, have a good supply of pure water, be on elevated ground to allow the rain and other water to flow freely from it, but not so high as to be exposed to the full blasts of the winter storms; it should have a good prospect of the surrounding country, and above all, it should be in a salubrious locality, free from the malaria arising from the vicinity of low or marshy grounds, with free scope to allow the house to front toward the most eligible point of the compass. The ASPECT of a country house is of much importance; for if the site commands an extensive view, or pleasant prospect in any di- rection, the windows of the sitting and principal sleeping rooms, should front in that direction : provided it does not also face the point from which blow the prevailing storms of the climate, this should be particularly considered in choosing the site. Rooms to be cheerful and pleasant, should front south of due east or west; at 62 PLATES XXVI AND XXVII. the same time it is desirable that the view of disagreeable or un- sightly objects should be excluded, and as many as possible of the agreeable and beautiful objects of the neighborhood brought into view. The design before us, is made to front the south-east, all the openings except two are excluded from the north easterly storms, which are the most disagreeable in the Atlantic States; the sun at noon would be opposite the angle Jl, and would shine equally on the front and side, consequently the front would have the sun until the middle of the afternoon, and the side of the front house and front of the back building would have the evening sun, rendering the whole dry and pleasant. The end of the back building, containing the kitchen and stairs, is placed against the middle of the back wall of the front building, to allow the back windows in the parlor, &c., to be placed in the middle of each room. These windows may be closed in stormy weather with substantial shutters; but in warm weather they will add much to the coolness of the rooms, by allowing a thorough ventilation. The broad projecting cornice of the house, and the continuous piazza, are the most important features in the elevation; besides the advantage of keeping the walls dry, and throwing the rain- water away from the foundation, they give an air of comfort, which would be entirely wanting without them ; for if we were to take away the piazza, and reduce the eave cornice to a slight projection, the appearance would be bald and meagre in the extreme. The projection of the piazza is increased on the front and rear, to give more room to the entrances. The front building is 36 feet wide from Jl to , fig. 2, and 20 feet deep from Jl to D. The back building is 16 feet wide, by 20 feet deep. The scale at the bottom of each plate must be used to get the sizes of all the minor parts. The height of the first story is 10 feet in the clear, and of the second story 8' 6"; these heights are laid off on a rod /?, to the right of fig. 1 ; so are also the heights of the windows, which shews at a glance, their posi- tion with regard to the floors and ceilings. This method should always be resorted to in drawing an elevation, as it will the better enable the draughtsman to make room for the interior finish of the windows and for the cornice of the room. In laying down a plan, the whole of the outer walls should be first drawn, and in setting off openings and party walls, the measure- PLATES XXVI AND XXVII. 63 ments should be taken from both corners, to prove tha you are correct. For example, in setting off the front door, take the width 5. from the whole width of the front, which will leave 31. 0; then lay off 15'. 6" from Jl, and also from B, then if the width of 5. is left between the points so measured, you are sure the front door is laid off correctly ; as the windows C and H are midway between the front door and the corner of the build- ing, the same plan should be followed, and as a general rule that will save trouble by preventing errors, you should never de- pend on the measurements from one end or corner, if you have the means of proving them by measuring from the opposite end also. The winding steps in the stairs may be dispensed with by adding 3 steps to the bottom flight bringing it out to the kitchen door, and by adding 1 step to the top flight ; or a still better arrange- ment might be made by adding 3 steps to the bottom flight, and retaining two of the winders: this would give 17 risers instead of 16, the present number, which would reduce the height of each to 7 3-4 inches. To ascertain the number of steps required to a story, proceed as follows : Add to the clear height of the story the breadth of the joists and floor, which will give the full height from 'the top of one floor to the top of the next. In constructing the stairs this height is laid off on a rod, and then divided into the requisite number of risers; but in drawing the plan, as in the case before us, set down the height in feet, inches and parts, and divide by the height you propose for your rise: this will give you the number of risers. If there is any remainder, it may be divided and added to your proposed rise, or another step may be added, and the height of the rise reduced ; or the height of the story may be divided by the number of risers, which will give the exact height of the riser in inches and parts. For example : The clear height of the story in the design is 10'. 0." > , , x ~,, The breadth of the joist and thickness of the floor 1.0. > this multiplied by 12 would give 132 inches, and 132 inches di- vided by 16, the number of risers on fig. 1, will give 81-4 inches; or divided by 17 would give 7 3-4 inches and a fraction. As the floor of the upper story forms one step, there will be always one tread less than there are risers. The vertical front of each step is called the rise or riser, and the horizontal part is called the tread or step. When the eaves of the house are continued 64 PLATES XXVI AND XXVII. around the building in the same horizontal line as in this design, the roof is said to be hipped, and the rafter running from the cor- ner of the roof diagonally to the ridge is called the hip rafter. REFERENCES TO THE DRAWINGS. Similar letters in the plans and sections refer to the same parts : thus T the fire-place of the parlor in fig. 2, is shewn in section at T, fig. 3, and M the plan of the back parlor window in fig. 2, is shewn in elevation at M, fig. 3. Ji. By fig. 2, is the plan, and Jl. By fig. 1, the elevation of the front wall. E. Fy fig. 2, the line of the section. Gy fig. 2, the front door. Ky fig. 2, the plan, and K, fig. 3, the elevation of the door leading to the back building. Ly fig. 2, the plan, and L, fig. 3, the elevation of the first flight of stairs. M and N, fig. 2, the plans, and M and JV, fig. 3, elevation of the back first floor windows. and Py fig. 4, the plans, and and P, fig. 3, elevation of cham- ber windows. Q, fig. 2, the plan, and Q, fig. 3, section of the parlor side window. Ry fig. 4, the plan, and R, fig. 3, section of chamber side window. *S>, fig. 4, the plan, and S, fig. 3, elevation of railing on the landing. T, fig. 2, the plan, and !F, fig. 3, section of parlor fire-place. U, fig. 2, the plan, and U, fig. 3, section of breakfast room fire-place. V and Wy fig. 4, the plans, and V and W 9 fig. 3, sections of cham- ber fire-place. X and Yy fig. 2, the plans, and X and F, fig. 3, elevations of side posts on piazza. Z. Z. Zy fig. 4, plans of closets. a. a. a.y fig. 4, flues from fire-places of ground floor. b. by fig. 3, section of eave cornice. c. Cy fig. 3, rafters of building. d. dy fig. 3, rafters of piazza. e. e. e. e. e. e. e. e. e } joists of the different stories; the ends of the short joists framed around the fire-places and flues are shewn in dark sections; the projection around the outside walls of fig. 4, shews the roof of the piazza. 65 PLATE XXVIII. DETAILS. Fig. 1 is an elevation of one pair of rafters, shewing also a section through the cornice and top of the wall. JL, section of the top of the wall. B, ceiling joist, the outside end notched to receive the cornice. C, collar beam. D. D, rafters. E, raising plate. F, wall plate. G, cantilever and section of cornice. FIGURE 2. Plan of First Floor Joists. Jl) foundation of kitchen chimney. B, foundation of parlor chimney ; C, of breakfast room do. D, double joist to receive the partition dividing the stairway from the kitchen. E. E, &LC. double joists resting on the walls and supporting the short joists F. F. F, forming the framing around the fire-places. The joists E. E. E and D, are called trimming joists. The short joists F. F. F are called trimmers, and the joists G. G. G, framed into the trimmers with one end resting on the wall are called tail joists. PLATE XXIX. DETAILS. Fig. 1 ? horizontal section through the parlor window, Jl) is the outside of the wall. B the inside of wall. 66 PLATE XXIX. C, the hanging stile of sash frame. D, the inside lining. E the outside lining F, the back lining. G. G the weights. H } the stile* of the outside or top sash. I, the stile of the inside or bottom sash. K y inside stop bead. L, jamb lining. My ground. f JV, plastering. 0, architrave. P, (dotted line) the projection of the plinth. * The stiles of a sash, door, or any other piece of framing, are the vertical outside pieces; the horizontal pieces are called rails, f Grounds are strips of wood nailed against the wall to regulate the thickness of the plastering, and to receive the casings or plinth. FIGURE 2. Vertical Section through the Sills. Jl, outside of the wall. Q, stone sill of the window. R, wooden sub-sill. S, bottom rail of sash. T, bondtimber Uy framing under window, called the back. V, cap of the back. K, the inside stop bead. FIGURE 3. Plinth of Parlor. M. My grounds. JV", plastering. y plinth or washboard. W, the base moulding. X, the floor. Many more detail drawings might be made of this design, and where a contract is to be entered into, many more should be made. Enough is here given to explain the method of drawing them; their use is to shew the construction of each part, and when drawn to a large scale, as in plate xxix, a workman of any in- telligence would be able to get out any part of the work required. Mate 30 XAI. /'/.IX AXD KUWATIOX Plate 31 CIRCULAR FLAN AND ELEVATION 67 PLATE XXX. OCTAGONAL PLAN AND ELEVATION. F IG . i. HALF THE PLAN. FIG. 2. ELEVATION. This plate requires but little explanation, as the dotted lines from the different points of the plan, perfectly elucidate the mode of drawing the elevation. The dotted line *#, shews the direction of the rays of light by which the shadows are projected ; the mode of their projection will be explained in Plates 55 and 56. PLATE XXXI. CIRCULAR PLAN AND ELEVATION This plate shews the mode of putting circular objects in elevation. The dotted lines from the different points of the plan, determine the widths of the jambs (sides) of the door and windows, and the projections of the sills and cornice. One window is farther from the door than the other, for the purpose of shewing the different apparent widths of openings, as they are more or less inclined from the front of the picture. This, as well as Plate 30, should be drawn to a much larger size by the learner ; he should also vary the position and width of the openings. As these designs are not intended for a particular purpose, any scale of equal parts may be used in drawing them. 68 PLATE XXXIL ROMAN MOULDINGS Roman mouldings are composed of straight lines and arcs of circles. NOTE. Each separate part of a moulding, and each moulding in an assem- blage of mouldings, is called a member. FIG. 1. A FILLET, BAND OR LISTEL Is a raised square member, with its face parallel to the surface on which it is placed. FIG. 2. BEAD. A moulding whose surface is a semicircle struck from the centre K. FIG. 3. TORUS. Composed of a semicircle and a fillet. The projection of the circle beyond the fillet, is equal to the radius of the circle, which is shewn by the dotted line passing through the centre L. The curved dotted line above the fillet, and the square dotted line be- low the circle, shew the position of those members when used as the base of a Doric column. FIG. 4. THE SCOTIA Is composed of two quadrants of circles between two fillets. B is the centre for describing the large quadrant; JL the centre for describing the small quadrant. The upper portion may be made larger or smaller than in the diagram, but the centre JL must always be in the line B. Jl. The scotia is rarely, if ever used alone ; but it forms an important member in the bases of columns rid n- 32, fiOMAN MOULDINGS. ft. rta. 8. Cvma fteyersa .\ cud tt fe s "S PLATE XXXII, 69 FIG, 5. THE OVOLO Is composed of a quadrant between two fillets. C is the centre for describing the quadrant. The upper fillet projects beyond the curve, and by its broad shadow adds much to the effect of the moulding. The ovolo is generally used as a bed moulding, or in some other position where it supports another member. FIG 6. THE CAVETTO, Like the ovolo, is composed of a quadrant and two fillets. The concave quadrant is used for the cavetto described from D ; it is consequently the reverse of the ovolo. The cavetto is frequently used in connection with the ovolo, from which it is separated by a fillet. It is also used sometimes as a crown moulding of a cor- nice; the crown moulding is the uppermost member. FIG. 7. THE CYMA RECTA Is composed of two arcs of circles forming a waved line, and two fillets. To describe the cyma, let I be the upper fillet and JV* the lower fillet. 1st. Bisect 7. JV*. in M. 2nd. With the radius M. N or M. /, and the foot of the divi- ders in JV* and M, successively describe two arcs cutting each other in F, and from M and / with the same radius, describe two arcs, cutting each other in E. 3rd. With the same radius from E and F, describe two arcs meeting each other in M. The proportions of this moulding may be varied at pleasure, by varying the projection of the upper fillet. FIG. 8. THE CYMA RE VERSA, TALON OR OGEE. Like the cyma recta, it is composed of two circular arcs and two fillets; the upper fillet projects beyond the curve, and the lower fillet recedes within it. The curves are described from G and H. The CYMA, or CYMA RECTA has the concave curve uppermost. 70 PLATE XXXII. The CYMA REVERSA has the concave curve below. The CYMA RECTA is used as the upper member of an assemblage of mouldings, for which it is well fitted from its light appearance. The CYMA REVERSA from its strong form, is like the ovolo, used to sustain other members. The dotted lines drawn at an angle of 45 to each moulding, shew the direction of the rays of light, from which the shadows are projected. NOTE. When the surface of a moulding is carved or sculptured, it is said to be ENRICHED. PLATE XXXIII. GRECIAN MOULDINGS Are composed of some of the curves formed by the sections of a cone, and are said to be elliptic, parabolic, or hyperbolic, taking their names from the curves of which they are formed. FIGURES 1 AND 2 To draw the Grecian ECHINUS or OVOLO, the fillets A and B, the tangent C. B, and the point of greatest projection at D being given. 1st, Draw B. H, a continuation of the upper edge of the under fillet. 2nd. Through D, draw D. H perpendicular to B. H, cutting the tangent B. C in C. 3rd. Through , draw B. G parallel to D. H, and through D, draw D. E parallel to H. B, cutting G. B in E. 4th. Make E. G equal to E. B, and E. F equal to H. C, join D. F. 5th. Divide the lines D. Fand D. C each into the same number of equal parts 6th. From the point B, draw lines to the divisions 1, 2, 3, &,c. in D. C. PLATE XXXIII. 71 7th. From the point G> draw lines through the divisions in 'D. F, to intersect the lines drawn from B. 8th. Through the points of intersection trace the curve. NOTE. A great variety of form may be given to the echinus, by varying the projections and the inclination of the tangent B. C. NOTE 2. If H. C is less than C. D, as in fig. 1, the curve will be elliptic; if H. C and C. D are equal, as in fig. 2, the curve is parabolic ; if H. C be made greater than D. C, the curve will be hyperbolic. NOTE 3. The echinus, when enriched with carving, is generally cut into figures resembling eggs, with a dart or tongue between them. FIGS. 3 AND 4. THE GRECIAN CYMA. To describe the Cyma Recta, the perpendicular height B. D and the projection A. D being given. 1st. Draw Jl. C and B. D perpendicular to JL. D and C. B par- allel to Jl. D. 2nd. Bisect Jl. D in E, and Jl. C in G ; draw E. F and G. 0, which will divide the rectangle A. C. B. D into four equal rec tangles. 3rd. Make G. P and 0. K each equal to O. H. 4th. Divide Jl. G O. B Jl. E and B. F into a similar number of equal parts. 5th. From the divisions in Jl. E and F. B, draw lines to H ; from P draw lines through the divisions on Jl. G to intersect the lines drawn from Jl. E, and from K through the divisions in 0. By draw lines to intersect the lines drawn from F. B. 6th. Through the points of intersection draw the curve. NOTE. The curve is formed of two equal converse arcs of an ellipsis, of which E. F is the transverse axis, and P. H or H. K the conjugate. The points in the curve are found in the same manner as in fig. 1, plate 20. FIG. 5. THE GRECIAN CYMA REVERSA, TALON OR OGEE. To draw the Cyma Reversa, the fillet A, the point C, the end of the curve B, and the line B. D being given. 1st. From C, draw C. D, and from B, draw B. E perpendicular to B. D, then draw C. E parallel to B. D, which completes the rectangle. 72 PLATE XXXIII. 2nd. Divide the rectangle B. E. C. D into four equal parts, by drawing F. G and 0. P. 3rd. Find the points in the curve as in figs. 3 and 4. NOTE 1. If we turn the figure over so as to bring the line F. G vertical, G being at the top, the point B of fig. 5, to coincide with the point Jl of fig. 3, it will be perceived that the curves are similar, jP. G. being the transverse axis, and JV*. H or M. H the conjugate axis of the ellipsis. NOTE 2. The nearer the line B. D approaches to a horizontal position, the greater will be the degree of curvature, the conjugate axis of the ellipsis will be lengthened, and the curve become more like the Roman ogee. FIGURE 6. THE GRECIAN SCOTIA. To describe the Grecian Scotia, the position of the fillets A and B being given. 1st. Join Jl. .#, bisect it in C, and through C draw D. .E parallel to B. G. 2nd. Make C. D and C. E each equal to the depth intended to be given to the scotia; then JL. B will be a diameter of an ellipsis, and D. E its conjugate. 3rd. Through E, draw F. G parallel to A. B. 4th. Divide Jl. F and B. G into the same number of equal parts, and from the points of division draw lines to E. 5th. Divide Jl. C and B. C into the same number of equal parts, as Jl. F, then from D through the points of division in Jl. B, draw lines to intersect the others, which will give points in the curve. PLATE XXXIV. PLAN, SECTION AND ELEVATION OF A WHEEL AND PINION. The cross lines on Q. R, fig. 2, shewing the teeth of the wheel and pinion, are drawn from the elevation as described in Plate 31, which explains the method of drawing an elevation from a circu- lar plan. This plate is introduced to give the learner an example for draw- ing machinery ; it requires but little explanation, as the relative Plate 34 ELEVATION : Plat* 35. TEETH OF WHEELS PLATE XXXIV. 73 parts are plain and simple; the same letters refer to the same parts in each figure. Thus JL, fig. 1, is the end of the axle of the wheel. Jl. By fig. 2, the top of the axle of the wheel. Jl. By fig. 3, section through the centre of the wheel. C. Dy the axle of the pinion. E. Fy flanges of the barrel, with the rope coiled between them. G. Hy bottom piece of frame. /. K. K. N, inclined uprights of frame. Ly top of frame. M. My top cross pieces of frame. 0. Py bearings of the wheel. Q. Ry plan and elevation of wheel. Ry intersection of wheel and pinion. S. Sy bottom cross pieces of frame. When two wheels engage each other, the smallest is called a pinion. PLATE XXXV. TO DRAW THE TEETH OF WHEELS 1st. The LINE of CENTRES is the line Jl. B. D, fig. 1, passing through Jif and C, the centres of the wheel and pinion. 2nd. The PROPORTIONAL or primitive diameter of the wheel, is the line Jl. B; the proportional radius Jl. K or K. B. The true radii are the distances from the centres to the extremities of the teeth. 3rd. The PROPORTIONAL DIAMETER of the pinion is the line B. D; the proportional radius C. B. 4th. The PROPORTIONAL CIRCLES or PITCH LINES are circles de- scribed with the proportional radii touching each other in B. 5th. The PITCH of a wheel is the distance on the pitch circle in- cluding a tooth and a space, as E. F or G. H } or 0. Z), fig. 2. 6th. The DEPTH of a tooth is the distance from the pitch circle to the bottom, as L. K, fig. 1, and the height of a tooth is the dis- tance from the pitch circle to the top of the tooth, as L. M. 10 74 PLATE XXXV. To draw the Pitch Line of a Pinion to contain a definite number of Teeth of the same size as in the given wheel K. 1st. Divide the proportional diameter Ji. B of the given wheel into as many equal parts as the wheel has teeth, viz. 16. 2nd. With a distance equal to one of these parts, step off on the line B. D as many steps as the pinion is to contain teeth, which will give the proportional diameter of the pinion; the diagram contains 8. 3rd. Draw the pitch circle, and on it with the distance E. F, the pitch, lay off the teeth. 4th. Sub-divide the pitch for the tooth and space, draw the sides of the teeth below the pitch line toward the centre, and on the tops of the teeth describe epicycloids. NOTE. The circumferences of circles are directly as their diameters; if the diameter of one circle be four times greater than another, the circumference will also be four times greater. Fig. 2 is another method for drawing the teeth; Jl. B is the pitch circle on which the width of the teeth and spaces must be laid down. Then with a radius D. E or D. F y equal to a pitch and a fourth, from the middle of each tooth on the pitch circle as at D, describe the tops of the teeth E and jF, from describe the tops of the teeth G and D, and so on for the others. The sides of the teeth within the pitch circle may be drawn toward the centre, as at F and H, or from the centre 0, with a radius equal to 0. Q or 0. P > describe the lower part of the teeth G and D. CYLINDER OF A LOCOMOTIVE Scale '/8 * of full . vi/.r Srri/'/)ii through *l.H. !'/./. CYLINDER OF A LOCOMOTIVE Scate '//"'of lull size . 3. End view /''/(/. /.S redo 1 1 //ifau f//i //.'///'///./ /' 75 PLATES XXXVI AND XXXVII. PLAN, SECTIONS AND END ELEVATION OF A CYLINDER FOR A LOCOMOTIVE ENGINE. Fig. 1. Top view or Plan. Fig. 2. Longitudinal Section through Jl. , fig. 1. Fig. 3. Elevation of the end B, fig. 1. Fig. 4. Transverse Section through G. H, fig. 1. REFERENCES Jl. Stuffing box. Jl. B. Line of longitudinal section. C. Steam exhaust port, or Exhaust. D. D. Steam ports or Side openings. E. Piston rod. F. Piston shewn in elevation. G. H. Line of transverse section. H. Exhaust pipe. K. Packing. L- Gland or Follower. M. M. Heads of cylinder. JV. Valve face. The piston is represented in the drawing as descending to the bot- tom of the cylinder; the bent arrows from D to C, fig. 2, and from C to H, fig. 4 ; shew the course of the steam escaping from the cylinder through the steam port and exhaust port to the ex- haust pipe ; the other arrow at Z> ; fig. 2 ; the direction of the steam entering the cylinder. 76 PLATE XXXVIII. ISOMETRICAL DRAWING. FIGURE 1. To draw the Isometrical Cube. Let A be the centre of the proposed drawing. 1st. With one foot of the dividers in A, and any radius, describe a circle. 2nd. Through the centre A, draw a diameter B. C parallel to the sides of the paper. 3rd. With the radius from the points B and C lay off the other corners of a hexagon, D. E. F. G. 4th. Join the points and complete the hexagon 5th. From the centre A> draw lines to the alternate corners of the hexagon, which will complete the figure. The isometrical cube is a hexahedron supposed to be viewed at an infinite distance, and in the direction of the diagonal of the cube; in the diagram, the eye is supposed to be placed opposite the point A: if a wire be run through the point A to the opposite corner of the cube, the eye being in the same line, could only see the end of the wire, and this would be the case no matter how large the cube, consequently the front top corner of the cube and the bottom back corner must be represented by a dot, as at the point A. As the cube is a solid, the eye from that direction will see three of its sides and nine of its twelve edges, and as the dis- tance is infinite, all these edges will be of equal length, the edges seen are those shewn in fig. 1 by continuous lines; three of the edges and three of the sides could not be seen, these edges are shewn by dotted lines in fig. 1, but if the cube were transparent all the edges and sides could be seen. The apparent opposite angles in each side are equal, two of them being 120, and the other two 60; all the opposite boundary lines are parallel to each other, and as they are all of equal length may be measured by one common scale, and all lines parallel to any of the edges of the cube may be measured by the same scale. The lines F. G, A. C and E. D represent the vertical edges of the cube, the par- <>i<' THE ISOMKTKIC.IL Plate 39. ISQMETRICAL FIGUHKX .90" 11 Fig. 2. W^Ninifu JlbnankSons PLATE XXXVIII. 77 allelograms Jl. C. D. E and JL. C. F. G, represent the vertical faces of the cube and the parallelogram Jl. B. E. F represents the horizontal face of the cube ; consequently, vertical as well as horizontal lines and surfaces may be delineated by this method and measured by the same scale, for this reason the term ISOME- TRICAL (equally measurable) has been applied to this style of drawing. FIGURE 2 Is a cube of the same size as fig. 1, shaded to make the represen- tation more obvious; the sides of the small cube Jl, and the boundary of the square platform on which the cube rests, as well as of the joists which support the floor of the platform, are all drawn parallel to some of the edges of the cube, and forms a good illustration for the learner to practice on a larger scale. NOTF. A singular optical illusion may be witnessed while looking at this diagram, if we keep the eye fixed on the point *#, and imagine the drawing to represent the interior of a room, the point A will appear to recede; then if we again imagine it to be a cube the point will appear to advance, and this rising and falling may be continued, as you imagine the angle A to rep- resent a projecting corner, or an internal angle. PLATE XXXIX. EXAMPLES IN ISOMETRICAL DRAWING. Figs. 1 and 2 are plans of cubes with portions cut away. Figs. 3 and 4 are isometrical representations of them. To draw a part of a regular figure, as in these diagrams, it is bet- ter to draw the whole outline in pencil, as shewn by the dotted lines, and from the corners lay off the indentations. The circumscribing cube may be drawn as in fig. 1, Plate 38, with a radius equal to the side of the plan, or with a triangle having one right angle, one angle of 60, and the cither angle 30, as shewn at Jl. Proceed as follows : Let B be the tongue of a square or a straight edge applied hori- zontally across the paper, apply the hypothenuse of the triangle to the tongue or straight edge, as in the diagram, and draw the 7g PLATE XL. left hand inclined lines ; then reverse the triangle and draw the right hand inclined lines ; turn the short side of the triangle against the tongue of the square, and the vertical lines may be drawn. This instrument so simplifies isometrical drawing, that its applica- tion is but little more difficult than the drawing of flat geometrical plans or elevations. PLATE XL. EXAMPLES IN ISOMETRICAL DRAWING CONTINUED. Fig 1 is the side, and fig. 2 the end elevation of a block pierced through as shewn in fig 1, and with the top chamfered off, as shewn in figs. 1 and 2. FIGURE 3. To draw the figure Isometrically. 1st. Draw the isometrical lines Jl. B and C. D; make Jl. B equal to Jl. B fig. 1, and C. D equal to C. D fig. 2. 2nd. From Jl. B and D, draw the vertical lines, and make them equal to B. G, fig. 1. 3rd. Draw K. H and L. I parallel to Ji. B, and H. I and K. L parallel to C. D. 4th. Draw the diagonals H. D and /. (7, and through their inter- section draw a vertical line M. G. F. Make G. F equal to G. F, fig. 1. 5th. Through G, draw G. JV, intersecting L. K in JV*, and from JV' draw a vertical line J\T. E. 6th. Through F, draw F. E, intersecting JV*. E in E; then E. F represents the line E. F in fig. 1 . 7th. From E and F, lay off the distances and P, and from and P draw the edges of the chamfer 0. K 0. L P. Hand P. /, which complete the outline. 8th. On Jl. B lay off the opening shewn in fig. 1, and from R } draw a line parallel to C. D. UITIVERSIT7 ISOMETRiaiL FIGURES. K . M. emeu-:. Fig. 1. a> PLATE XL 79 NOTE 1. All the lines in this figure, except the diagonals and edges of the chamfer, can be drawn with the triangle and square, as explained in Plate 39. NOTE 2. All these lines may be measured by the same scale, except the in- clined edges of the chamfer, which will require a different scale. NOTE 3. The intelligent student will easily perceive from this figure, how to draw a house with a hipped roof, placing the doors, windows, &c., each in its proper place ; or how to draw any other rectangular figure. In- clined lines may always be found by a similar process to that we have pur- sued in drawing the edges of the chamfer. FIGURE 4 Is the elevation of the side of a cube with a large portion cut out. FIGURE 5 Is the isometrical drawing of the same, with the top of the cube also pierced through. The mode pursued is so obvious, that it requires no explanation : it is given as an illustration for drawing FURNITURE, or any other framed object. It requires but little ingenuity to convert fig. 5 into the frame of a table or a foot-stool. PLATE XLI. TO DRAW THE ISOMETRICAL CIRCLE. FIGURE 1 Is the plan of a circle inscribed in a square, with two diameters Jl. E and C. D parallel to the sides of the square. FIGURE 2. To draw the Isometrical Representation. 1st. Draw the isometrical square, M. JV*. 0. P, having its opposite angles 120 and 60 respectively. 2nd. Bisect each side and draw JL. B and C. D. 80 PLATE XLI. 3rd. From draw 0. Jl and 0. D, and from M draw M. C and M. B intersecting in Q and R. 4th. From Q, with the radius Q. .#, describe the arc Jl. C } and from R, with the same radius, describe the arc D. B. 5th. From 0, with the radius 0. Jl, draw the arc Jl. D, and from M, with the same radius, describe C. B, which completes the oval. NOTE. An ISOMETRICAL PROJECTION of a circle would be an ellipsis; bat the figure produced by the above method is so simple in its construction and approaches so near to an ellipsis, that it may be used in most cases, besides its facility of construction, its circumference is so nearly equal to the cir- cumference of the given circle, that any divisions traced on the one may be transferred to the other with sufficient accuracy for all practical purposes. FIGURE 3. To divide the Circumference of the Isometrical Circle into any number of equal parts. 1st. Draw the circle and a square around it as in fig. 2, the square may touch the circle as in fig. 2, or be drawn outside as in fig. 3. 2nd. From the middle of one of the sides as 0, erect 0. K per- pendicular to E. F, and make 0. K equal to 0. E. 3rd. Draw K. E and K. F, and from K with any radius, describe an arc P. Q, cutting K. E in P, and K. F in Q. 4th. Divide the arc P 4 into one-eighth of the number of parts re- quired in the whole circumference, and from K, through these di- visions, draw lines intersecting E. in 1, 2 and 3. 5th. From the divisions 1, 2 and 3, in E. 0, draw lines to the centre P, which will divide the arc E. into four equal parts. 6th. Transfer the divisions on E. from the corners E. F. G. H, and draw lines to the centre P, when the concentric curves will be divided into 32 equal parts. NOTE 1. If a plan of a circle divided into any number of equal parts be drawn, as that of a cog wheel, the same measures may be transferred to the isometric curve as explained in the note to fig. 2, but if the plan be not drawn, the divisions can be made as in fig. 3. NOTE 2. The term ISOMETRICAL PROJECTION has been avoided, as the pro- jection of a figure would require a smaller scale to be used than the scale to which the geometrical plans and elevations are drawn, but as the isometri- cal figure drawn with the same scale to which the plans are drawn, is in every respect proportional to the true projection, and conveys to the eye the same view of the object, it is manifestly much more convenient for practical purposes to draw both to the same scale. PLATE XLI. 81 NOTE 3. In Note 2 to fig. 3, Plate 40, allusion has been made to inclined lines requiring a different scale from any of the lines used in drawing the isometric cube : for the mode of drawing those scales as well as for the further prosecution of this branch of drawing, the student is referred to Jopling's and Sopwith's treatise on the subject, as we only propose to give an intro- duction to isometrical drawing. Sufficient, however, has been given to en- able the student to apply it to a very large class of objects, and it would ex- tend the size of this work too much (already much larger than was intend- ed) if we pursue the subject in full. PERSPECTIVE, PLATE XLIL The design of the art of perspective is to draw on a plane surface the representation of an object or objects, so that the representa- tion shall convey to the eye, the same image as the objects them- selves would do if placed in the same relative position. To elucidate this definition it will be necessary to explain the man- ner in which the image of external objects is conveyed to the eye. 1st. To enable a person to see any object, it is necessary that such object should reflect light. 2nd. Light reflected from a centre becomes weaker in a duplicate ratio of distance from its source, it being only one-fourth as in- tense at double the distance, and one-ninth at triple the distance, and so on. 3rd. A ray of light striking on any plane surface, is reflected from that surface in exactly the same angle with which it impinges; thus if a plane surface be placed at an angle of 45, to the direc- tion of rays of light, the rays will be reflected at an angle of 45 in the opposite direction. This fact is expressed as follows, viz : THE ANGLE OF REFLECTION IS EQUAL TO THE ANGLE OF _ 82 PLATE XLII. INCIDENCE. This axiom, so short and pithy, should be stored in the memory with some others that we propose to give, to be brought forward and applied whenever required. 4th. Rays of light reflected from a body proceed in straight lines until interrupted by meeting with other bodies, which by reflec- tion or refraction, change their direction. 5th. REFRACTION of LIGHT. When a ray of light passes from a rare to a more dense medium, as from a clear atmosphere through a fog or from the air into water, it is bent out of its direct course: thus if we thrust a rod into water, it appears broken or bent at the surface of the water ; objects have been seen through a fog by the bending of the rays, that could not possibly be seen in clear wea- ther; this bending of the rays of light is called refraction, and the rays are said to be refracted : this- effect, (produced however by a different cause) may often be seen by looking through common window^ glass, when in consequence of the irregularities of its surface, the view of objects without is much distorted. 6th. A portion of light is absorbed by all bodies receiving it on their surface, consequently the amount of light reflected from an object is not equal to the quantity received. 7th. The amount of absorption is not the same in all bodies, but depends on the color and quality of the reflecting surface ; if a ray falls on the bright polished surface of a looking-glass, most of it will be reflected, but if it should fall on a surface of black cloth, most of it would be absorbed. White or light colors reflect more of a given ray of light than dark colors ; polished surfaces reflect more than those which are unpolished, and smooth surfaces more than rough. 8th. As all objects absorb more or less light, it follows that at each reflection the ray will become weaker until it is no longer per- ceptible. 9th. Rays received from a luminous source are called direct, and the parts of an object receiving these direct rays are said to be in LIGHT. The portions of the surface so situated as not to receive the direct rays are said to be in SHADE ; if the object receiving the direct rays is opaque, it will prevent the rays from passing in that direction, and the outline of its illuminated parts will be pro- jected on the nearest adjoining surface : the figure so projected is called its SHADOW. 10th. The parts of an object in shade will always be lighter than the shadow, as the object receives more or less reflected light from PLATE XLII. 83 the atmosphere and adjoining objects, the quantity depending on the position of the shaded surface, and on the position and quali- ty of the surrounding objects. 1 1th. If an object were so situated as to receive only a direct ray of light, without receiving reflected light from other sources, the illuminated portion could alone be seen; but for this universal law of reflection we should be able to see nothing that is not illumi- nated by the direct rays of the sun or by some artificial means, and all beyond would be one gloomy blank. 12th. Rays of light proceeding in straight lines from the surfaces of objects, meet in the front of the eye of the spectator where they cross each other, and form an inverted image on the back of the eye, of all objects within the scope of vision. 13th. The size of the image so formed on the retina depends on the size and distance of the original ; the shape of the image de- pends on the angle at which it is seen. NOTE. The size of objects diminishes directly as the distance increases, ap- pearing at ten times the distance, only a tenth part as large ; the knowledge of,this fact has produced a system of arithmetical perspective, which enables the draughtsman to proportion the sizes of objects by calculation. 14th. The strength of the image depends on the degree of illumi- nation of the original, and on its distance from the eye, objects becoming more dim as they recede from the spectator. 15th. To give a better idea of the operation of the eye in viewing an object, let us refer to fig. 1. The circle Jl is intended to repre- sent a section of the human eye, H the pupil in front, K the crystalline lens in which the rays are all converged and cross each other, and M the concave surface of the back of the eye called the retina, on which the image is projected. 16th. Let us suppose the eye to be viewing the cross B. C, and that the parallelogram N. 0. P. Q represents a picture frame in which a pane of glass is inserted; the surface of the glass slight- ly obscured so as to allow objects to be traced on it, then rays from every part of the cross will proceed in straight lines to the eye, and form the inverted image C. B on the retina. If with a pencil we were to trace the form of the cross on the glass so as to interrupt the view of the original object, we should have a true perspective representation of the original, which would form ex- actly the same sized image on the retina; thus the point b would intercept the view of B, c of (7, d of D and e of E, and if colored the same as the original, the image formed from it would be the same in every respect as from the original. 84 PLATE XLII. 17th. If we move the cross B. C to F. G, the image formed on the retina would be much larger, as shewn at G. F y and the rep- resentation on the glass would be larger, the ray from F passing through/, and the ray from G passing through g, shewing that the same object will produce a larger or smaller image on the re- tina as it advances to or recedes from the spectator; the farther it recedes, the smaller will be the image formed, until it becomes so small as to be invisible. 18th. Fig. 2 is given to elucidate the same subject. If we suppose a person to be seated in a room, the ground outside to be on a level with the bottom of the window JL. B, the eye at S in the same level line, and a series of rods C. D. E. F of the same height of the window to be planted outside, the window to be filled' with four lights of glass of equal size, then the ray from the bot- tom of all the rods would pass through the bottom of the window; the ray from the top of C would pass through the top of the win- dow; from the top of D a little farther off, it would pass through the third light; the ray from E would pass through the middle, and F would only occupy the height of one pane. 19th. Fig. 3. Different sized and shaped objects may produce the sanie image; thus the bent rods JL and C, and the straight rods B and D would produce the same image, being placed at different distances from the eye, and all contained in the same angle D. S. E. As the bent rods JL and C are viewed edgewise they would form the same shaped image as if they were straight. The angle formed by the rays of light passing from the top and bottom of an object to the eye, as D. S. E, is called the VISUAL ANGLE, and the object is said to subtend an angle of so many degrees, measur- ing the angle formed at S. 20th. Of FORESHORTENING. When an object is viewed obliquely it appears much shorter than if its side is directly in front of the eye ; if for instance we hold a pencil sidewise at arms length op- posite the eye, we should see its entire length; then if we incline the pencil a little, the side will appear shorter, and one of the ends can also be seen, and the more the pencil is inclined the smaller will be the angle subtended by its side, until nothing but the end would be visible. Again if a wheel be placed perpendicular- ly opposite the eye, its rim and hub would shew perfect circles, arid the spokes would all appear to be of the same length, but if we incline the wheel a little, the circles will appear to be ellipses, and the spokes appear of different lengths, dependant on the an- PLATE XLI1. 85 gle at which they are viewed; the more the wheel is inclined the shorter will be the conjugate diameter of the ellipsis, until the whole would form a straight line whose length would be equal to the diameter, and its breadth equal to the thickness of the wheel. This decrease of the angle subtended by an object, when viewed obliquely, is called foreshortening. PLATE XLIII. FIGURE 1. 21st. If we suppose a person to be standing on level ground, with his eye at S, the line JL. F parallel to the surface and about five feet above it, and the surface G. E to be divided off into spaces of five feet, as at B. C. D and E, then if from S, with a radius S. G, we describe the arc Jl. G, and from the points B. C. D and E we draw lines to , cutting the arc in H. K. L and M, the distances between the lines on the arc, will represent the angle subtended in the eye by each space, and if we adopt the usual mode for measuring an angle, and divide the quadrant into 90, it will be perceived that the first space of five feet subtends an angle of 45, equal to one-half of the angle that would be sub- tended by a plane that would extend to the extreme limits of vi- sion; the next space from B to C subtends an angle of about 18 1-2, from C to D about 8, and from D to E about 4 1-2, and the angle subtended would constantly become less, until the divi- sions of the spaces would at a short distance appear to touch each other, a space of five feet subtending an angle so small, that the eye could not appreciate it. It is this foreshortening that enables us in some measure to judge of distance. 22nd. If instead of a level plane, the person at *S> be standing at the foot of a hill, the surface being less inclined would diminish less rapidly, but if on the contrary he be standing on the brow of a hill looking downward, it would diminish more rapidly; hence we derive the following axiom: THE DEGREE OF FORESHORTENING OF OBJECTS DEPENDS ON THE ANGLE AT WHICH THEY ARE VIEWED 86 PLATE XLII1. 23rd. PERSPECTIVE may be divided into two branches, LINEAR and AERIAL. 24th. LINEAR PERSPECTIVE teaches the mode of drawing the lines of a picture so as to convey to the eye the apparent SHAPE or FIGURE of each object from the point at which it is viewed. 25th. AERIAL PERSPECTIVE teaches the mode of arranging the direct and reflected LIGHTS, SHADES, and SHADOWS of a picture, so as to give to each part its requisite degree of tone and color, diminishing the strength of each tint as the objects recede, until in the extreme distance, the whole assumes a bluish gray which is the color of the atmosphere. This branch of the art is requisite to the artist who would paint a landscape, and can be better learnt by the study of nature and the paintings of good masters, than by any series of rules which would require to be constantly varied. 26th. Linear perspective, on the contrary, is capable of strict mathe- matical demonstration, and its rules must be positively followed to produce the true figure of an object. DEFINITIONS. 27th. The PERSPECTIVE PLANE is the surface on which the pic- ture is drawn, and is supposed to be placed in a vertical position between the spectator and the object thus in fig. 1, Plate 42, the parallelogram i/V*. 0. P. Q is the perspective plane. 28th. The GROUND LINE or BASE LINE of a picture is the seat of the perspective plane, as the line Q. P, fig. 1, Plate 42, and G. L, fig. 2, Plate 43. 29th. The HORIZON. The natural horizon is the line in which the earth and sky, or sea and sky appear to meet; the horizon in a perspective drawing is at the height of the eye of the spectator. If the object viewed be on level ground, the horizon will be about five feet or five and a half feet above the ground line, as it is repre- sented by V. L } fig. 2. If the spectator be viewing the object from an eminence, the horizon will be higher, and if the spectator be lower than the ground on which the object stands, the horizon will be lower; thus the horizon in perspective, means the height of the eye of the spectator, arid as an object may be viewed by a person reclining on the ground, or standing upright on the ground, or he may be elevated on a chair or table, it follows that the horizon may be made higher or lower, at the pleasure of the PLATE XLIII. 87 draughtsman; but in a mechanical or architectural view of a de- sign, it should be placed about five feet above the ground line. NOTE. The tops of all horizontal objects that are below the horizon, and the under sides of objects above the horizon, will appear more or less dis- played as they recede from or approach to the horizon. 30th. The STATION POINT, or POINT of VIEW is the position of the spectator when viewing the object or picture. 31st. The POINT of SIGHT. If the spectator standing at the sta- tion point should hold his pencil horizontally at the level of his eye in such a position that the end only could be seen, it would cover a small part of the object situated in the horizon ; this point is marked as at S, fig. 2, and called the point of sight. It must be remembered that the point of sight is not the position of a specta- tor when viewing an object; but a point in the horizon directly opposite the eye of the spectator, and from which point the spec- tator may be at a greater or less distance. 32nd. POINTS of DISTANCE are set off on the horizon on either side of the point of sight as at D. D', and represent the distance of the spectator from the perspective plane. As an object may be viewed at different distances from the perspective piane, it fol- lows that these points may be placed at any distance from the point of sight to suit the judgment of the draughtsman, but they should never be less than the base of the picture. NOTE 1. Although the height of the horizon, and the points of distance may be varied at pleasure, it is only from that distance and with the eye on a le- vel with the horizon that a picture can be viewed correctly. NOTE 2. In the following diagrams the points of distance have generally been placed within the boundary of the plates, as it is important that the learner should see the points to which the lines tend ; they should be copied with the points of distance much farther off. 33rd. VISUAL RAYS. All lines drawn from the object to the eye of the spectator are called visual rays. 34th. The MIDDLE RAY, or CENTRAL VISUAL RAY is a line pro- ceeding from the eye of the spectator to the point of sight; exter- nal visual rays are the rays proceeding from the opposite sides of an object, or from the top and bottom of an object to the eye. The angle formed in the eye by the external rays, is called the visual angle. NOTE. The perspective plane must always be perpendicular to the middle visual ray. 35th. VANISHING POINTS. It has been shewn at fig. 1 in this plate that objects of the same size subtend a constantly decreasing 88 PLATE XLIII. angle in the eye as they recede from the spectator, until they are no longer visible ; the point where level objects become invisible or appear to vanish, will always be in the horizon, and is called the vanishing point of that object. 36th. The POINT of SIGHT is called the PRINCIPAL VANISHING POINT, because all horizontal objects that are parallel to the mid- dle visual ray will vanish in that point. If we stand in the mid- dle of a street looking directly toward its opposite end as in Plate 54, (the Frontispiece,) all horizontal lines, such as the tops and bottoms of the doors and windows, eaves and cornices of the houses, tops of chimnies, &c. will tend toward that point to which the eye is directed, and if the lines were continued they would unite in that point. Again, if we stand in the middle of a room looking towards its opposite end, the joints of the floor, corners of ceiling, washboards and the sides of furniture ranged against the side walls, or placed parallel to them, would all tend to a point in the end of the room at the height of the eye. 37th. The VANISHING POINTS of horizontal objects not parallel with the middle ray will be in some point of the horizon, but not in the point of sight. These vanishing points are called acciden- tal points. 38th. DIAGONALS. Lines drawn from the perspective plane to the point of distance as JY. D f and 0. Z>, or from a ray drawn to the point of sight as E. D' and F. D, are called diagonals; all such lines represent lines drawn at an angle of 45 to the perspective plane, and form as in this figure the diagonals of a square, whose side is parallel to the perspective plane. 39th. Of VANISHING PLANES. On taking a position in the mid- dle of a street as described in paragraph 36, it is there stated that all lines will tend to a point in the distance at the height of the eye, called the point of sight, or principal vanishing point; this is equally true of horizontal or vertical planes that are parallel to the middle visual ray : for if we suppose the street between the curb stones, and the side walks of the street to be three parallel hori- zontal planes as m Plate 54, their boundaries will all tend to the vanishing point, until at a distance, depending on the breadth of the plane, they become invisible. Again, the walls of the houses on both sides of the street are vertical planes, bounded by the eaves of the roofs and by their intersection with the horizontal planes of the side walks, these boundaries would also tend to the same point, and if the rows of houses were continued to a suffi- Plate 42. PERSPECTIVE. Fig. 2. Fig. 3. WT^Minifte Plate 43. PERSPECTIVE . Fig.l. /'/>/. 2. 1,5 PLATE XLIII. 89 cient distance, these planes would vanish in the same point; if the back walls of the houses are parallel to the front, the planes formed by them will vanish in the same point, and if any other streets should be running parallel to the first, their horizontal and vertical planes would all tend to the same point. J^OTE. A BIRD'S EYE VIEW of the streets of a town laid out regularly, would fully elucidate the truth of the remarks in this paragraph. When the horizon of a picture is placed very high above the tops of the houses, as if the spec- tator were placed on some very elevated object, or if seen as a bird would see it when on the wing, the view is called a bird's eye view; in a represen- tation of this kind the tops of all objects are visible, and the tendency of all the planes and lines parallel to the middle visual ray to vanish in the point of sight, is very obvious. 40th. If we were viewing a room as described in paragraph 36, the ceiling and floor would be horizontal planes, and the walls vertical planes, and if extended would all vanish in the point of sight; or if we were viewing the section of a house of several stories in height, all the floors and ceilings would be horizontal planes, and all the parallel partitions and walls would be vertical planes, and would all vanish in the same point. 41st. When the BOUNDARIES OF INCLINED PLANES are horizontal lines parallel to the middle ray, the planes will vanish in the point of sight; thus the roofs of the houses in Plate 54, bounded by the horizontal lines of the eaves and ridge, are inclined planes vanishing in the point of sight. 42nd. PLANES PARALLEL TO THE PLANE OF THE PICTURE have no vanishing point, neither have any lines drawn on such planes. 43rd. VERTICAL OR HORIZONTAL PARALLEL PLANES running at any inclination to the middle ray or perspective plane, vanish in accidental points in the horizon, as stated in paragraph 37 ; as for example, the walls and bed of a street running diagonally to the plane of the picture, or of a single house as in Plate 53, where the opposite sides vanish in accidental points at different distances from the point of sight, because the walls form different angles with the perspective plane, as shewn by the plan of the walls B. D and D. C, fig. 1. 44th. ALL HORIZONTAL LINES DRAWN ON A PLANE, or running parallel to a plane, vanish in the same point as the plane itself. 45th. INCLINED LINES vanish in points perpendicularly above or below the vanishing point of the plane, and if they form the same angle with the horizon in different directions as the gables of the 12 90 PLATE XLIII. house in fig. 2, Plate 53, the vanishing points will be equidistant from the horizon. From what has been said we derive the following AXIOMS; their importance should induce the student to fix them well in his memory : 1st. The ANGLE OF REFLECTION OF LIGHT is equal to the angle of incidence. See paragraph No. 3, page 81. 2nd. The SHADOW of an object is always darker than the object itself. See paragraph 10, page 82. 3rd. The DEGREE OF FORESHORTENING of objects depends on the angle at which they are viewed. See paragraph 20, page 84. 4th. The APPARENT SIZE of an object decreases exactly as its dis- tance from the spectator is increased. See paragraph 35, p. 87. 5th. PARALLEL PLANES and LINES vanish to a common point. See paragraph 36, page 88. 6th. ALL PARALLEL PLANES whose boundaries are parallel to the middle visual ray, vanish in the point of sight. See paragraph 36, page 88. 7th. ALL HORIZONTAL LINES parallel to the middle ray vanish in the point of sight. 8th. HORIZONTAL LINES AT AN ANGLE OF 45 with the plane of the picture, vanish in the points of distance. See paragraph 38, page 88. 9th. PLANES AND LINES PARALLEL TO THE PLANE OF THE PIC- TURE have no vanishing point. PRACTICAL PROBLEMS. 1st. To draw the perspective representation of the square N. O. P. Q, viewed in the direction of the line W. B, with one of its sides N. O touching the perspective plane G. L, and parallel with it. 1st. Draw the horizontal line V. L at the height of the eye, 2nd. From C, the centre of the side JV*. 0, draw a perpendicular to V. L, cutting it in S. Then S is the point of sight or the prin- cipal vanishing point, and C. S the middle visual ray. 3rd. As the sides JV. P and 0. Q are parallel to the middle ray C. S, they will vanish in the point of sight. Therefore from JV and draw rays to S; these are the external visual rays. 4th. From S, set off the points of distance D. D r at. pleasure, equi- distant from S, and from JV and 0, draw the diagonals J\T. D' and PLATE XLIII. 91 0. D. Then the intersection of these diagonals with the external visual rays determine the depth of the square. 5th. Draw E. F parallel to JV*. 0. Then the trapezoid JV. 0. E. F is the perspective representation of the given square viewed at a distance from W on the line W. B, equal to S. D. 2nd. To draw the Representation of another Square of the same size immediately in the rear of E. F. 1st. From E, draw E. H, intersecting 0. S in H, and from F, draw F. D, intersecting JV. S in B. 2nd. Draw B. H parallel to E. F, which completes the second square ; and the trapezoid JV. 0. H. B is the representation of a parallelogram whose side 0. H is double the side of the given square. NOTE*. If from W on the line W. B we set off the distance S. D, extending in the example outside of the plate, (which represents the distance from which the picture is viewed,) and from JVand draw rays to the point so set off, cutting P. Q in R and T, then the lines R. T and E. F will be of equal length, and prove the correctness of -the diagram. PLATE XLIV. FIGURE 1. To draw a Perspective Plan of a Square and divide it into a given number of Squares, say sixty-four. Let G. L be the base line, V. L the horizon, S the point of sight, and JV. the given side of the square. 1st. From JVand 0,draw rays to S and diagonals to D. D, inter- secting each other in P and Q, draw P. Q. 2nd. Divide JV". into eight equal parts, and from the points of division draw rays to S. 3rd. Through the points of intersection formed with those rays by the diagonals, draw lines parallel to JV. 0, which will divide the square as required, and may represent a checker board or a pavement of square tiles. 92 PLATE XLIV. OF HALF DISTANCE. When the points of distance are too far off to be used convenient- ly, half the distance may be used; as for example, if we bisect S. D in 1-2 D, and JV. in C, and draw a line from C to 1-2 D, it will intersect JV. in P, being in the same point as by the diago- nal drawn from the opposite side of the square, to the whole distance at D. NOTE. Any other fraction of the distance may be used, provided that the divisions on the base line be measured proportionately. tfWWWWWVVWWMWMVtt FIGURE 2. To draw the Plan of a Room with Pilasters at its sides, the base line, horizon, point of sight, and points of distance given. NOTE. To AVOID REPETITIONS, in the following diagrams we shall suppose the base line, the horizon V. L, the point of sight S, and the points of distance D. D to be given. 1st. Let J]T. be the width of the proposed room, then draw JV". S and 0. S representing the sides of the room. 2nd. From JV* toward lay down the width of each pilaster, and the spaces between them, and draw lines to D, then through the points where these lines intersect the external visual ray JV". S, draw lines parallel with N. to the line 0. S. 3rd. From JV* and 0, set off the projection of the pilasters and draw rays to the point of sight. The shaded parts shew the posi- tion of the pilasters. 4th. If from N we lay off the distances and widths of the pilasters toward M, and draw diagonals to the opposite point of distance, N. S would be intersected in exactly the same points. NOTE. Any rectangular object may be put in perspective by this method, without the necessity of drawing a geometrical plan, as the dimensions may all be laid off on the ground line by any scale of equal parts. FIGURE 3. To shorten the depth of a perspective drawing, thereby producing the same effect as if the points of distance were removed much farther off. 1st. Let all the principal lines be given as above, and the pilasters and spaces laid off on the base line from N. U1U7BRSITT Fig.l. Fig. 3. r* Tesselated Pavements Fit/.l. Fig. 3. PLATE XLIV. 93 2nd. From the dimensions on the base line draw diagonals to the point of distance D. The diagonal from M the outside pilaster will intersect JV. S in P. 3rd. From N erect a perpendicular JV*. B to intersect the diago- nals, and from those intersections draw horizontal lines to inter- sect N. S. 4th. If from N we draw the inclined line N. E and transfer the intersections from it to V. 0, it will reduce the depth much more, as shown at 0. S. Most of the foregoing diagrams may be drawn as well with one point of distance as with two. PLATE XLV. TESSELATED PAVEMENTS. FIGURE 1. To draw a pavement of square tiles, with their sides placed diago- nally to the perspective plane. 1st. Draw the perspective square JV*. 0. P. Q. 2nd. Divide the base line JV. into spaces equal to the diagonal of the tiles. 3rd. From the divisions on JV. draw diagonals to the points of distance. 4th. Tint every alternate square to complete the diagram. FIGURE 2. To draw a pavement of square black tiles with a white border around them, the sides of the squares parallel to the perspective plane and middle visual ray. 1st. Draw the perspective square, and divide X. into alternate spaces equal to the breadth of the square and borders. 2nd. From the divisions on X. draw rays to the point of sight, and from X draw a diagonal to the point of distance. 94 PLATE XLV. 3rd. Through the intersections formed by the diagonal, with the rays drawn from the divisions on X. 0, draw lines parallel to A". 0, to complete the small squares. FIGURE 3. To draw a Pavement composed of Hexagonal and Square Blocks. 1st. Divide the diameter of one of the proposed hexagons a. b into three equal parts, and from the points of division draw rays to the point of sight. 2nd. From a, draw a diagonal to the point of distance, and through the intersections draw the parallel lines. 3rd. From 1, 2, 3 and 4, draw diagonals to the opposite points of distance, which complete the hexagon. 4th. Lay off the base line from a and b into spaces equal to one- third of the given hexagon, and draw rays from them to the point of sight; then draw diagonals as in the diagram, to complete the pavement. PLATE XLVI. FIGURE 1. To draw the Double Square E. F. G. H, viewed diagonally, with one of its corners touching the Perspective Plane. 1st. Prolong the sides of the squares as shewn by the dotted lines to intersect the perspective plane. 2nd. From the points of intersection, draw diagonals to the points of distance, their intersections form the diagonal squares. 3rd. The square Jl. B. N. is drawn around it on the plan and also in perspective, to prove that the same depth and breadth is given to objects by both methods of projection. THE UF17BRSITr /'/ /'If/./. fi'f/. 2. s Plate 4 7. LINE or ELEVATION. /vy./. Fig. 2. 5 4 3 2 1 1 PLATE XLVI. 95 FIGURE 2. To draw the Perspective Representation of a Circle mewed directly in front and touching the Perspective Plane. Find the position of any number of points in the Curve. 1st. Circumscribe the circle with a square, draw the diagonals of the square P. and JV. Q, and the diameters of the circle Jl. B and E. F, also through the intersections of said diagonals with the circumference, draw the chords R. R, R. R, continued to meet the line G. L in Fand Y! 2nd. Put the square in perspective as before shewn, draw the diagonals JV. D', and 0. D, and the radials Y. S and Y. f S. 3rd. From */Z, draw Jl. S, and through the intersection of the dia- gonals draw E. F parallel to JV. 0. 4th. Through the points of intersection thus found, viz: Jl. B. E. F. R. R. R. R trace the curve. NOTE 1. This method gives eight points through which to trace the curve, and as these points are equidistant in the plan, it follows that if the points were joined by right lines it would give the perspective representation of an OCTAGON; by drawing diameters midway between those already drawn on the plan, eight other points in the curve may be found. This would give six- teen points in the curve, and render the operation of tracing much more correct. NOTE 2. A CIRCLE in perspective may be considered as a polygon of an in- finite number of sides, or as a figure composed of an infinite number of points, and as any point in the curve may be found, it follows that every point may be found, and each be positively designated by an intersection; in practice of course this is unnecessary, but the student should remember,* that the more points he can positively designate without confusion, the more correct will be the representation. PLATE XLVII. LINE OF ELEVATION. FIGURE 2 Is the plan of a square whose side is nine feet, each side is divided into nine parts, and lines from the divisions drawn across in oppo- site directions; the surface is therefore divided into eighty-one squares. G. L, fig. 1, is the base line and I). D the horizon. 96 PLATE XLVII. 1st. To put the plan with its divisions in perspective, one of its sides N. O to coincide with the perspective plane. Transfer the measures from the side JV. 0, fig. 2, to JV. on the perspective plane fig. 1, and put the plan in perspective by the methods before described. 2nd. To erect square pillars on the squares N.- Q. W, 12 feet high and one foot diameter, equal to the size of one of the squares on the plan. 1st. Erect indefinite perpendiculars from the corners of the squares. 2nd. On JV*. JL one of the perpendiculars tliat touches the perspec- tive plane lay off the height of the column JV*. M from the ac- companying scale, then JY. M is A LINE OF HEIGHTS on which the true measures of the heights of ail objects must be set. 3rd. Two lines drawn from the top and bottom of an object on the line of heights to the point of sight, point of distance, or to any other point in the horizon, forms a scale for determining similar heights on any part of the perspective plan. To avoid confusion they are here drawn to the point B. 4th. Through M draw M. (7, parallel to N. 0, and from C draw a line to the point of sight which determines the height of the side of the column, and also of the back column erected on Q, and through the intersection of the line C. S with the front per- pendicular, draw a horizontal line forming the top of the front side of the column Q. 5th. To determine the height of the pillar at W, 1st. draw a hori- zontal line from its foot intersecting the proportional scale JV*. B in Y; 2nd. from Y draw a vertical line intersecting M. B in X ; then Y. X is the height of the front of the column W. By the same method the height of the column Q may be determined as shewn at R. T. 3rd. To draw the Caps on the Pillars. 1st. On the line C. E a continuation of the top of the front, set off the amount of projection C. E, and through E draw a ray to the point of sight. 2nd. Through C draw a diagonal to the point of distance, and through the point of intersection of the diagonal with the ray PLATE XLVII. 97 last drawn; draw the horizontal line H forming the lower edge of the front of the cap. 3rd. Through M draw a diagonal to the opposite point of dis- tance, which determines the position of the corners H and K, from H draw a ray to the point of sight. 4th. Erect perpendiculars on all the corners, lay off the height of the front, and draw the top parallel with the bottom. A ray from the corner to the point of sight, will complete the cap. The other caps can be drawn by similar means. As a pillar is a square column the terms are here used indiscrimi- nately. 4th. To erect Square Pyramids on O and P of the same height as the Pillars, with a base of four square feet, as shewn in the plan. 1st. Draw diagonals to the plan of the base, and from their inter- section at R draw the perpendicular R. T 1 . 2nd. From R draw a line to the proportional scale N. B, and draw the vertical line Z. G, which is the height of the pyramid. 3rd. Make R. T' equal to Z. G, and from the corners of the per- spective plan draw lines to T f , which complete the front pyramid. 4th. A line drawn from T to the point of sight will determine the height of the pyramid at a. NOTE 1. The point of sight S shewn in front of the column TF, must be supposed to be really a long distance behind it, but as we only see the end of a line proceeding from the eye to the point of sight, we can only represent it by a dot. NOTE 2. A part of the front column has been omitted for the purpose of shewing the perspective sections of the remaining parts, the sides of these sections are drawn toward the point of sight, the front and back lines are horizontal. The upper section is a little farther removed from the horizon, and is consequently a little wider than the lower section. This may be taken as an illustration of the note to paragraph 29 on page 87, to which the reader is referred. NOTE 3. The dotted lines on the plan shew the direction and boundaries of the shadows ; they have been projected at an angle of 45 with the plane of the picture. 13 98 PLATE XLVIII. FIGURE 1. To draw a Series of Semicircular Jlrches viewed directly in front, forming a Vaulted Passage, with projecting ribs at intervals, as shewn by the tinted plan below the ground line. 1st. From the top of the side walls JY. I and 0. K, draw the front arch from the centre H, and radiate the joints to its centre. 2nd. From the centre H and the springing lines of the arch, and from the corners Ji and M draw rays to the point of sight. 3rd. From Jl and M set off the projection of the ribs, and draw rays 'from the points so set off to the point of sight. 4th. Transfer the measurements of Jf'. B". C /f , &c., on the plan,to Jl . B . C", &,c., on the ground line, and from them draw diago- nals to the point of distance, intersecting the r.ay Ji. S in B. C. D, &c. 5th. From the points of intersection in Jl. S draw lines parallel to the base line to intersect M. S. This gives the perspective plans of the ribs. 6th. Erect perpendiculars from the corners of the plans to inter- sect the springing lines, and through these intersections draw horizontal dotted lines, then the points in which the dotted lines intersect the ray drawn from H the centre of the front arch, will | be the centres for drawing the other arches; R being the centre for describing the front of the first rib. 7th. The joints in the fronts of the projecting ribs radiate to their respective centres, and the joints in the soffit of the arch radiate to the point of sight. NOTE. No attempt is made in this diagram to project the shadows, as it would render the lines too obscure. But the front of each projection is tint- ed to make it more conspicuous. OP TBX UN 17 BE SIT 7] ARCHES IN PERSPECTIVE. ARCHES WPERSPEt TfVE. -~ : -.., s Fia.2. A B F G PLATE XLVIII. 99 FIGURE 2. To draw Semicircular or Pointed Jlrcades on either side of the spectator, running parallel to the middle visual ray. N. P and Q, O the width of the arches being given, and P. Q the space be- tiveen them. 1st. From N. P. Q and erect perpendiculars, make them all of equal length, and draw E. F and M. J. 2nd. FOR THE SEMICIRCULAR ARCHES, bisect E. F in C, and from E. C. F. and Q, draw rays to the point of sight. 3rd. From C, describe the semicircle E. F. 4th. Let the arches be the same distance apart as the width Q. 0, then from draw a diagonal to the point of distance, cutting Q. S in R, from R draw a diagonal to the opposite point of distance cut'.nig 0. S in V, from V draw a diagonal to D, cutting Q. S in W, and from W to Z>', cutting 0. S in X. 5ih. Through R. V. Fand X, draw horizontal lines to intersect the rays 0. S and Q. S, and on the intersections erect perpen- diculars to meet the rays drawn from E and F. 6th. Connect the tops of the perpendiculars by horizontal lines, and from their intersections with the ray drawn from C in 1, 2, 3 and 4, describe the retiring arches. 7th. FOR THE GOTHIC ARCHES, (let them be drawn the same dis- tance apart as the semicircular,) continue the horizontal lines across from R and F, to intersect the rays P. S and N. S, and from the points of intersection erect perpendiculars to intersect the rays drawn from M and J. 8th. From M and J successively, with a radius M. J, describe the front arch, and from .fiTthe crown^draw a ray to S; from Ji and B with the radius Ji. B, describe the second arch, and from K and L, describe the third arch. NOTE . All the arches in this plate are parallel to the plane of the picture, and although each succeeding arch is smaller than the arch in front of it, all may be described with the compasses. 100 PLATE XLIX. TO DESCRIBE ARCHES ON A VANISHING PLANE. FIGURE 1. The Front JLrch A. N. B, the, Base Line G. L, Horizon D. S, Point of Sight S, arid Point of Distance D, being given. 1st. Draw H. J across the springing line of the arch, and construct the parallelogram E. F. J. H. 2nd. Draw the diagonals H. F and J. E, and a horizontal line K. My through the points where the diagonals intersect the given arch. Then H. K. N. M and J, are, points in the curve which are required to be found in each of the lateral arches. 3rd. From F and B, draw T rays to the point of sight S. Then if we suppose the space formed by the triangle B. S. F to be a plane surface, it will represent the vanishing plane on which the arches are to be drawn. 4th. From B, set off the distance B. Jl to Z, and draw rays from Z. J and (7, to the point of sight. 5th. From Z, draw a diagonal to the point of distance, cutting B. Sin 0; through 0, draw a horizontal line cutting Z. S in P; from P, draw a diagonal intersecting B. S in Q; through Q, draw a horizontal line, cutting Z. S in /?, and so on for as many arches as may be required. 6th. From 0. Q. S. U, erect perpendiculars, cutting F. S in V. W. X. Y. 7th. Draw the diagonals /. V^ F. 7, &c. as shewn in the diagram, and from their intersection erect perpendiculars to meet F. S; through which point and the intersections of the diagonals with C. S trace the curves. FIGURE 2. To draw Receding Jlrches on the Vanishing Plane J. S. D, with Piers between them., corresponding with the given front view, the Piers to have a Square Base with a side equal to C. D. I 1st. From D on the base line, set off the distances D. C, C. B Ill - THE OJiJECT AX/> POINT OF VIEW GJl'KX TO F1XJ) THE PKItxrKCTrVK 7>LAW .LY/J VANISHING IVUXTS. UNIVERSITY PLATE XLIX. 101 and B. A to D. E, E. F and F. G, and from E. F. G, ,c. draw diagonals to the point of distance to intersect D. S. 2nd. From the intersections in D. S, erect perpendiculars; draw the parallelogram M. JY. H. I around the given front arch, the diagonals M. I and H. N, and the horizontal line L. K, prolong H. /to JandJlf. JVto V. 3rd. From B. C. D. M. f. J. K and V^ draw rays to the point of sight, put the parallelograms and diagonals in perspective at 0. P V. W and at Q. W. R. X, and draw the curves through the points as in the last diagram. 4th. From i where E. D 1 cuts D. S, draw a horizontal line cutting B. S in h, and from h erect a perpendicular cutting M. S in k. 5th. From F, the centre of the front arch, draw a ray to the point of sight, and from k, draw a horizontal line intersecting it in Z. Then Z is the centre for describing the back line of the arch with the distance Z. k for a radius. NOTE. The backs of the side arches are found by the same method as the front? of those arches. The lines are omitted to avoid confusion. The projecting cap in this diagram is constructed in the same manner as the caps of the pillars in Plate 47. PLATE L. APPLICATION OF THE CIRCLE WHEN PARALLEL TO THE PLANE OF THE PICTURE. V. L is the horizon, and S the point of sight. FIGURE 1 To draw a Semicircular Thin Band placed abov? the horizon. Let the semicircle Jl. B represent the front edge <)f : t? B the diameter, and C the centre. 1st. From JL. Cand B, draw rays to the point of sight. 2nd. From C the centre, lay off toward B, the breadth of the band C. E. 3rd. From E, draw a diagonal to the point of distance, intersect- ing C. S in F. Then F is the centre for describing the back of the band. 102 PLATE L. 4th. Through F, draw a horizontal line intersecting ./I. S in K. and B. S in L. Then F. K or F. L is the radius for describing the back of the band. FIGURE 2. To draw a Circular Hoop with its side resting on the Horizon. The front circle Jl. H. B. K, diameter JL. B, and centre C being given. 1st. From A. C and B, draw rays to the point of sight. 2nd. From C the centre, lay off the breadth of the hoop at E. 3rd. From E, draw a diagonal to D', intersecting C. Sin F, and through F, draw a horizontal line intersecting Jl. S in K, and B SmL. 4th. From F with a radius F. L or F. K, describe the back of the curve. FIGURE 3. To draw a Cylindrical Tub placed below the Horizon, whose dia- meter, depth and thickness are given. 1st. From the centre C describe the concentric circles forming the thickness of the tub, lay off the staves and radiate them toward C. 2nd. Proceed as in figs. 1 and 2 to draw rays and a diagonal to find the point F, and from F describe the back circles as before ; the hoop may be drawn from F, by extending the compasses a little. 3rd. Radiate all the lines that form the joints on the sides of the tub toward the point of sight. FIGURE 4 Is a hollow cylinder placed below the horizon, and must be drawn by the same method as the preceding figures ; the letters of re- ference are the same. NOTE. The objects in this Plate are tinted to shew the different surface more distinctly without attempting to project the shadows. 103 PLATE LI. The object and point of view given, to find the Perspective Plane and Vanishing Points. Rule 1. The PERSPECTIVE PLANE must be drawn perpendicular to the middle visual ray. Rule 2. The VANISHING POINT of a line or plane is found by drawing a line through the station point parallel with such line or plane to intersect the perspective plane. The point in the hori- zon immediately over the intersection so found, is the vanishing point of all horizontal lines in said plane, or on any plane parallel to it. 1st. Let the parallelogram E. F. G. Hbe the plan of an object to be put in perspective, and let Q be the position of the spectator viewing it, (called the point of view or station point,) with the eye directed toward K, then Q. K will be the central visual ray, and K the point of sight. Draw F. Q and H. Q, these are the external visual rays. NOTE. The student should refer to paragraphs 30 and 31, page 87, for the definitions of station point and point of sight. 2nd. Draw P. at right angles to Q. K, touching the corner of the given object at E, then P. O will be the base of the perspec- tive plane. NOTE. This position of the perspective plane, is the farthest point from the spectator at which it can be placed, as the whole of the object viewed must be behind it; but it may be placed at any intermediate point nearer the spectator parallel with P. 0. 3rd. Through Q draw Q. P parallel with E. F, intersecting the perspective plane in P, then P is the vanishing point of the lines E. F and G. H. 4th. Through Q draw Q. 0, parallel to E. H, intersecting the perspective plane in 0, then O is the vanishing point for E. H and F. G. 5th. If we suppose the station point to be removed to Jl, then A. M will be the central visual ray, Ji. F and Jl. H the external rays, and B. D the perspective plane, B the vanishing point of 104 PLATE LI. E. F and G. H, and the vanishing point of E. H and F. G will be outside the plate about ten inches distant from JL* in the direc- tion of A. C. 6th. If the station point be removed to K, it will be perceived that E. H and F. G will have no vanishing point, because they are perpendicular to the middle ray, and a line drawn through the station point parallel with the side E. H will also be parallel with the perspective plane, consequently could never intersect it. 7th. The sides E. F and G. H of the plan, would vanish in the point of sight, but if an elevation be drawn on the plan in that position which should extend above the horizon, then neither of those sides could be seen, and the drawing would very nearly approach to a geometrical elevation of the same object. NOTE. In the explanation of this plate, the intersections giving the point of sight and vanishing points, are made in the perspective, plane, which the student will remember when used in this connection, is equivalent to the base line or ground line of the picture, being the seat or position of the plane on which the drawing is to be made; but we must suppose these points to be elevated to the height of the eye of the spectator; in practice, these points must be set off on the horizontal line as described in paragraph 32, page 87. PLATE LII. To delineate the perspective appearance of a Cube viewed acci- dentally and situated beyond the Perspective Plane. FIGURE L Let Jl. B. C. D be the plan of the cube, S the station point, S. T the middle visual ray and B. L the base line, or perspective plane. 1st. Continue the sides of the plan to the perspective plane as shewn by the dotted lines, intersecting it in M. E. JVand 0. 2nd. From the corners of the plan draw rays to the station point, intersecting the perspective plane in a. d. b. c. 3rd. Through S, draw S. F parallel to Jl. D, and S. G parallel to D. C. Then F is the vanishing point for the sides A. D and B. C; and G is the vanishing point for the sides A. B and D, C. OBJECT INCLINED TO THE PLANE OF DELINEATION. W ' rl - Minifie . PLAN AND PERSPECTIVE VIEW: Fia 7 <^1P^ [BIIVBRSITT] PLATE LIT. 105 FIGURE 2. 4th. Transfer these intersections from B. , fig. 1, to B. L, fig. 2, and the vanishing points F and G to the horizon, as shewn by the dotted lines. 5th. From E and M, draw lines to the vanishing point G, and from JY and 0, draw lines to the vanishing point F. Then the trapezium A. B. C. D formed by the intersection of these lines, is the perspective view of the plan of the cube. 6th. To DRAW THE ELEVATION. At M. E. JV* and erect per- pendiculars and make them equal to the side of the cube. 7th. From the tops of these perpendiculars draw lines to the op- posite vanishing points as shewn by the dotted lines, their inter- section will form another trapezium parallel to the first, repre- senting the top of the cube. 8th. From JL. D and C, erect perpendiculars to complete the cube. NOTE. It is not necessary to erect perpendiculars from all the points of in- tersection, to draw the representation, but it is done here to prove that the height of an object may be set on any perpendicular erected at the point where the plane, or line, or a continuation of a line intersects the perspective plane; one such line of elevation is generally sufficient. 9th. To draw the figure with one line of heights, proceed as fol- lows: from A. D and C, erect indefinite perpendiculars. 10th. Make E. H equal to the side of the cube, and from //"draw a line to G, cutting the perpendiculars from D and C in K and L. 1 1th. From K, draw a line to F, cutting Jl. P in P ; from L, draw a line to F, and from P, draw a line to G, which completes the figure. NOTE.- The student should observe how the lines and horizontal planes be- come diminished as they approach toward the horizon, each successive line becoming shorter, and each plane narrower until at the height of the eye, the whole of the top would be represented by a straight line. I would here re- mark, that it would very materially aid the student in his knowledge of per- spective, if he would always make it a rule to analyze the parts of every dia- gram he draws, observe the changes which take place in the forms of ob- jects when placed in different positions on the plan, and when they are placed above or below the horizon at different distances; this 'would enable him at once to detect a false line, and would also enable him to sketch from nature with accuracy. Practice this always until it becomes a HABIT, and I can assure you it will be a source of much gratification. 14 106 PLATE LIII. TO DRAW THE PERSPECTIVE VIEW OF A ONE STORY COTTAGE, SEEN ACCIDENTALLY. FIGURE 1. Let Jl. B. C. D be the plan of the cottage, twenty feet by four- teen feet, drawn to the accompanying scale; the shaded parts shew the thickness of the walls and position of the openings, the dotted lines outside parallel with the walls, give the projec- tion of the roof, and the square E. F. G. H, the plan of the chimney above the roof. Let P. L be the perspective plane and S the station point. 1st. Continue the side B. D to intersect the perspective plane in H, to find the position for a line of heights. 2nd. From all the corners and jambs on the plan, draw rays toward the station point to intersect the perspective plane. 3rd. Through S draw a line parallel to the side of the cottage D. C, to intersect the perspective plane in L. This gives the vanish- ing point for the ends of the building and for all planes parallel to it, viz : the side of the chimney, and jambs of the door and windows. 4th. Through S draw a line parallel with B. D, to intersect the perspective plane, which it would do at some distance outside of the plate; this intersection would be the vanishing point for the sides of the cottage, for the tops and bottoms of the windows, the ridge and eaves of the roof, and for the front of the chimney. ****MWMWWfW!MMM*IVM FIGURE 2. Let us suppose the parallelogram P. L. W. X to be a separate piece of paper laid oil the other, its top edge coinciding with the per- spective plane of fig. 1, and its bottom edge W. X to be the base of the picture, then proceed as follows : 1st. Draw the horizontal line R. T parallel to W. X and five feet above it. PLATE LIII. 107 2nd. Draw H. K perpendicular to P. L for a line of heights. 3rd. Draw a line from K to the vanishing point without the pic- ture, which we will call Z ; this will represent the line H. B of fig. 1, continued indefinitely. 4th. From b and d draw perpendiculars to intersect the last line drawn, in o and e, which will determine the perspective length of the front of the house. 5th. On K. H set off twelve feet the height of the walls, at 0, and from draw a line to the vanishing point Z, intersecting d. e in m and b. o in n. 6th. From m and e draw vanishing lines to T, and a perpendicu- lar from c intersecting them in Fand s ; this will give the cor- ner F, and determine the depth of the building. 7th. Find the centre of the vanishing plane representing the end, by drawing the diagonals m. Y and e. s, and through their inter- section draw an indefinite perpendicular u. v, which will give the position of the gable. 8th. To FIND THE HEIGHT OF THE GABLE, Set off its proposed height, say 7' 0", from to JVon the line of heights, from JV draw a ray to Z, intersecting e. d in W, and from W draw a vanishing line to T intersecting u. v in v, then v is the peak of the gable. 9ih. Join m. v, and prolong it to meet a perpendicular drawn through the vanishing point T, which it will do in V,then N is the vanishing point for the inclined lines of the ends of the front half of the roof. The ends of the backs of the gables will vanish in a point perpendicularly below F, as much below the horizon as V is above it. 10th. FOR THE ROOF. Through v draw v. y to Z without, to form the ridge of the roof, from / let fall a perpendicular to inter- sect y. v in w, through w draw a line to the vanishing point V to form the edge of the roof. From d let fall a perpendicular to in- tersect V. w, and from the point of intersection draw a line to Z to form the front edge of the roof, from a let fall a perpendicular to de- fine the corner x, and from x draw a line to V intersecting w. y in y, which completes the front half of the roof; from w draw a line to the vanishing point below the horizon, from c let fall a perpen- dicular to intersect it in g, and through g draw a line to Z, which completes the roof. llth. FOR THE CHIMNEY. Set off its height above the ridge at JV/, from M draw a line toward the vanishing point Z, intersect- ing o. b in U, from U draw a line to the vanishing point J 1 , which 108 PLATE LIII. gives the height of the chimney, bring down perpendiculars from rays drawn from G. F and E, fig. 1, and complete the chimney by vanishing lines drawn for the front toward Z and for the side toward T. 12th FOR THE DOOR AND WINDOWS. Set off their heights at P. Q and draw lines toward Z, bring down perpendiculars from the rays as before, to intersect the lines drawn toward Z; these lines will determine the breadth of the openings. The breadth of the jambs are found by letting fall perpendiculars from the points of intersection, the top and bottom lines of the jambs are drawn toward T. NOTE 1. As the bottom of the front fence if continued, would intersect the base line at K the foot of the line of heights, and its top is in the horizon, it is therefore five feet high. NOTE 2. The whole of the lines in this diagram have been projected accord- ing to the rules, to explain to the learner the methods of doing so, and it will be necessary for him to do so until he is perfectly familiar with the subject. But if he will follow the rule laid down at the end of the description of the last plate, he will soon be enabled to complete his drawing by hand, after pro- jecting the principal lines, but it should not be attempted too early, as it will beget a careless method of drawing, and prevent him from acquiring a correct judgment of proportions. PLATE LIV. FRONTISPIECE Is a perspective view of a street 60 feet wide, as viewed by a per- son standing in the middle of the street at a distance of 134 feet from the perspective plane, and at an elevation of 20 feet from the ground to the height of the eye. The horizon is placed high for the purpose of shewing the roofs of the two story dwellings. The dimensions of the different parts are as follows : 1st. DISTANCES ACROSS THE PICTURE. Centre street between the houses 60 feet wide. Side walks, each 10 " Middle space between the lines of railway 46 " Width between the rails 49 " PLATE LIV. 109 Depth of three story warehouse 40 feet. Depth of yard in the rear of warehouse 20 " Depth of two story dwelling on the right 30 " DISTANCES FROM THE SPECTATOR, IN THE LINE OF THE MIDDLE VISUAL RAY. From spectator to plane of the picture 134 feet. From plane of picture to the corner of buildings 50 " Front of each house 20 " Front of block of 7 houses 20 feet each 140 " Breadth of street running across between the blocks 60 " Depth of second block same as the first 140 " Depth of houses on the left of the picture, behind > .Q (( the three story warehouses 3 To DRAW THE PICTURE. 1st. Let Cbe the centre of the perspective plane, H. L the hori- zon, S the point of sight. 2nd. From C on the line P. P, lay off the breadth of the street thirty feet on each side, at and 60, making sixty feet, and from those points draw rays to the point of sight; these give the lines of the fronts of the houses. 3rd. From lay off a point 50 feet on P. P y and draw a diagonal from that point to the point of distance without the picture ; the intersection of that diagonal with the ray from 0, determines the corner of the building; from the point of intersection erect a per- pendicular to B. 4th. From 50, lay off spaces of 20 feet each at 70, 90 and so on, and from the points so laid off draw diagonals to determine by their intersection with the ray from 0, the depth of each house. 5th. After the depth on 0. S is found for three houses, the depths of the others may be found by drawing diagonals to the oppo- site point of distance to intersect the ray 60 S, as shewn by the dotted lines. NOTE. As a diagonal drawn to the point of distance forms an angle of 45 with the plane of the picture, it follows that a diagonal drawn from a ray to another parallel ray, will intercept on that ray a space equal to the distance between them. Therefore as the street in the diagram is 60 feet wide and the front of each house is 20 feet, it follows that a diagonal drawn from one side of the street to the other will intercept a space equal to the fronts of three houses, as shewn in the drawing. 6th. Lay off the dimensions on the perspective plane, of the HO PLATE LIV. depth of the houses, and the position of the openings on the side of the warehouse, and draw rays to the point of sight as shewn by the dotted lines. 7th. At erect a perpendicular to D for a line of heights; on this line all the heights must be laid off to the same scale as the mea- sures on the perspective plane, and from the points so marked draw rays to the point of sight to intersect the corner of the building at B. For example, the height of the gable of the ware- house is marked at ^, from Jl draw a ray toward the point of sight intersecting the corner perpendicular at B ; then from jB, draw a horizontal line to the peak of the gable ; the dotted lines shew the position of the other heights. 8th. To find the position of the peaks of the gables on the houses in the rear of the warehouses, draw rays from the top and bottom corner of the front wall to the point of sight, draw the diagonals as shewn by the dotted lines, and from their intersection erect a perpendicular, which gives the position of the peak, the intersec- tion of diagonals in this manner will always determine the perspec- tive centre of a vanishing plane. The height may be laid off on 0. D at Z), and a ray drawn to the point of sight intersecting the corner perpendicular at (7, then a parallel be drawn from C to intersect a perpendicular from the front corner of the building at ", and from that intersection draw a ray to the point of sight. The intersection of this ray, with the indefinite perpendicular erected from the intersection of the diagonals, will determine the perspective height of the peak. 9th. The front edges of the gables will vanish in a point perpen- dicularly above the point of sight, and the back edges in a point perpendicularly below it and equidistant. 10th. As all the planes shewn in this picture except those parallel with the plane of the picture are parallel to the middle visual ray, all horizontal lines on any of them must vanish in the point of sight, and inclined lines in a perpendicular above or below it, as shewn by the gables. Ill SHADOWS, 1st. The quantity of light reflected from the surface of an object, enables us to judge of its distance, and also of its form and posi- tion. 2nd. On referring to paragraph 9, page 82, it will be found that light is generally considered in three degrees, viz : light, shade and shadow; the parts exposed to the direct rays being in light, the parts inclined from the direct rays are said to be in shade, and objects are said to be in shadow, when the direct rays of light are intercepted by some opaque substance being interposed between the source of light and the object. 3rd. THE FORM OF THE SHADOW depends on the form and posi- tion of the object from which it is cast, modified by the form and position of the surface' on which it is projected. For example, if the shadow of a cone be projected by rays perpendicular to its axis, on a plane parallel to its axis, the boundaries of the shadow will be a triangle; if the cone be turned so that its axis would be parallel with the ray, its shadow would be a circle; if the cone be retained in its position, and the plane on which it is projected be inclined in either direction, the shadow will be an ellipsis, the greater the obliquity of the plane of projection, the more elongated will be the transverse axis of the ellipsis. 4th. SHADOWS OF THE SAME FORM MAY BE CAST BY DIFFER- ENT FIGURES: for example, a sphere and a flat circular disk would each project a circle on a plane perpendicular to the rays of light, so also would a cone and a cylinder with their axes par- allel to the rays. The sphere would cast the same shadow if turned in any direction, but the flat disk if placed edgeways to the rays, would project a straight line, whose length would be equal to the diameter of the disk and its breadth equal to the thickness; the shadow of the cone if placed sideways to the rays would be a triangle, and of the cylinder would be a parallelogram. 5th. Shadows of regular figures if projected on a plane, retain in some degree the outline of the object casting them, more or 112 less distorted, according to the position of the plane; but if cast upon a broken or rough surface the shadow will be irregular. 6th. Shadows projected from angular objects are generally strong- ly defined, and the shading of such objects is strongly contrasted; thus if you refer to the cottage on Plate 53, you will perceive that the vertical walls of the front and chimney are in light, fully exposed to the direct rays of the sun, while the end of the cot- tage and side of the chimney are in shade, being turned away from the direct rays, the plane of the roof is not so bright as the vertical walls, because, although it is exposed to the direct rays of light it reflects them at a different angle, the shadow of the pro- jecting eaves of the roof on the vertical wall forms a dark un- broken line, the edge of the roof being straight and the surface of the front a smooth plane, the under side of the projecting end of the roof is lighter than the vertical wall because it is so situated as to receive a larger proportion of reflected light. 7th. Shadows projected from circular objects are also generally well defined, but the shadings instead of being marked by4>road bold lines as they are in rectangular figures, gradually increase from bright light to the darkest shade and again recede as the opposite side is modified by the reflections from surrounding ob- jects, so gradually does the change take place that it is difficult to define the exact spot where the shade commences, the lights and shades appear to melt into each other, and by its beautifully swelling contour enables us at a glance to define the shape of the object. 8th. DOUBLE SHADOWS. Objects in the interior of buildings fre- quently cast two or more shadows in opposite directions, as they receive the light from opposite sides of the building; this effect is also often produced in the open air by the reflected light thrown from some bright surface, in this case however, the shadow from the direct rays is always the strongest; in a room at night lit by artificial means, each light projects a separate shadow, the strength of each depending on the intensity of the light from which it is cast, and its distance from the object; the student may derive much information from observing the shading and shadows of objects from artificial light, as he can vary the angle, object and plane of projection at pleasure. 9th. The extent of a shadow depends on the angle of the rays of light. If we have a given object and plane on which it is pro- jected, its shadow under a clear sky will vary every hour of the ^ ' "" ' "" ' ' '" - !.! *. t^-JJ PLATE LV. 113 day, the sun's rays striking objects m a more slanting position in the morning and evening than at noon, projects much longer shadows. But in mechanical or architectural drawings made in elevation, plan or section, the shadows should always be project- ed at an angle of 45, that is to say, the depth of the shadow should always be equal to the breadth of the projection or inden- tation; if this rule is strictly followed, it will enable the work- man to apply his dividers and scale, and ascertain his projections correctly from a single drawing. NOTE. The best method for drawing lines at this angle ; is to use with the T square, a right angled triangle with equal sides, the hypothenuse will be at an angle of 45 with the sides ; with the hypothenuse placed against the edge of the square, lines may be drawn at the required angle on either side. PLATE LV. PRACTICAL EXAMPLES FOR THE PROJECTION OF SHADOWS, FIGURE 1 Is a square shelf supported by two square bearers projecting from a wall. The surface of the paper to represent the wall in all the following diagrams. 1st. Let Jl. B. C. D be the plan of the shelf; Jl. B its projection from the line of the wall W. X; B. D the length of the front of the shelf, and ,and F the plans of the rectangular bearers. 2nd. Let G. H be the elevation of the shelf shewing its edge, and J and K the ends of the bearers. 3rd. From all the projecting corners on the plan, draw lines at an angle of 45 to intersect the line of the wall W. X, and from these intersections erect indefinite perpendiculars. 4th. From all the projecting corners on the elevation, draw lines at an angle of 45 to intersect the perpendiculars from correspond- ing points in the plan ; the points and lines of intersection define the outline of the shadow as shewn in the diagram. 15 ~~ 114 PLATE LV. FIGURE 2 Is a square Shelf against a wall supported by two square Uprights. L. M. JV*. is the plan of the shelf, P and Q the plans of the up- rights, R. S the front edge of the shelf, T and V the fronts of the uprights. 1st. From the angles on the plan draw lines at an angle of 45 to intersect W. X, and from the intersections erect perpendiculars. 2nd. From R and S, draw lines at an angle of 45 to intersect the corresponding lines from the plan. FIGURE 3 Is a Frame with a semicircular head, nailed against a wall, the Frame containing a sunk Panel of the same form. 1st. Let JL. B. C. D be the section of the frame and panel across the middle, and F on the elevation of the panel, the centre from which the head of the panel and of the frame is described. 2nd. From E, draw a line to intersect the face of the panel, and from D to intersect W. X, and erect the perpendiculars as shewn by the dotted lines. 3rd. From JY and JV*/ draw lines to define the bottom shadow, and at L draw a line at the same angle to touch the curve. 4th. At the same angle draw F. G, make F. H equal to the depth of the panel, and F. G equal to the thickness of the frame. 5th. From H with the radius F. R, describe the shadow on the panel, and from G with the radius F. S, describe the shadow of the frame. NOTE. The tangent drawn at L and the curve of the shadow touch the edge of the frame in the same spot, but if the proportions were different they would not do so ; therefore it is always better to draw the tangent. FIGURE 4 Is a Circular Stud representing an enlarged view of one of the Nail Heads used in the last diagram, of which N. O. P is a sec- tion through the middle, and W. X the face of the frame. 1st. Draw tangents at an angle of 45 on each side of the curve. 2nd. Through L the centre, draw L. JV/, and make L. M equal to the thickness of the stud. |]4 PLATE LV. FIGURE 2 Is a square Shelf against a wall supported by two square Uprights. L. M. jY. is the plan of the shelf, P and Q the plans of the up- rights, R. S the front edge of the shelf, T and V the fronts of the uprights. 1st. From the angles on the plan draw lines at an angle of 45 to intersect W. X, and from the intersections erect perpendiculars. 2nd. From R and S, draw lines at an angle of 45 to intersect the corresponding lines from the plan. FIGURE 3 Is a Frame with a semicircular heady nailed against a wall, the Frame containing a sunk Panel of the same form. 1st. Let Jl. E. C. D be the section of the frame and panel across the middle, and F on the elevation of the panel, the centre from which the head of the panel and of the frame is described. 2nd. From E, draw a line to intersect the face of the panel, and from D to intersect W. X, and erect the perpendiculars as shewn by the dotted lines. 3rd. From J\T and JV"/ draw lines to define the bottom shadow, and at L draw a line at the same angle to touch the curve. 4th. At the same angle draw F. G, make F. H equal to the depth of the panel, and F. G equal to the thickness of the frame. 5th. From H with the radius F. R, describe the shadow on the panel, and from G with the radius F. S, describe the shadow of the frame. NOTE. The tangent drawn at L and the curve of the shadow touch the edge of the frame in the same spot, but if the proportions were different they would not do so ; therefore it is always better to draw the tangent. FIGURE 4 Is a Circular Stud representing an enlarged mew of one of the Nail Heads used in the last diagram, of which N. O. P is a sec- tion through the middle, and W. X the face of the frame. 1st. Draw tangents at an angle of 45 on each side of the curve. 2nd. Through L the centre, draw L. M, and make L. M equal to the thickness of the stud. Plate 55. SHADOWS. Fig. 5. Mate, 56. SHADOWS. /'/,/. 7 fflma.nkS.oi PLATE LY. 3rd. From Jkf, with the same radius as used in describing the stud, describe the circular boundary of the shadow to meet the two tangents, which completes the outline of the shadow. FIGURE 5 Is a Square Pillar standing at a short distance in front of the wall W. X. 1st. Let A. B. C. D be the plan of the pillar, and W. X the front of the wall, from JL. C. D draw lines to W. X, and from their intersections erect perpendiculars. 2nd. Let E. F. G. #be the elevation of the pillar, from F draw F. K. L to intersect the perpendiculars from C and D. 3rd. Through K, draw a horizontal line, which completes the out- line. The dotted lines shew the position of the shadow on the wall behind the pillar. PLATE LVI. SHADOW S C N T I N U E D . FIGURE 1 Is the Elevation and Fig. 2 the Plan of a Flight of Steps with rectangular Blockings at the ends, the edge of the top step even with the face of the wall. 1st. From A. B. Cand D, draw lines at an angle of 45. 2nd. From F where the ray from C intersects the edge of the front step, draw a perpendicular to JV, which defines the shadow on the first riser. 3rd. From Q where the ray from C intersects the edge of the se- cond step, draw a perpendicular to M, which defines the shadow on the second riser. 4th. From K where the ray from JL intersects the top of the third step, draw a perpendicular to 0, which defines the shadow on the top of that step. 116 PLATE LVI. 5th. From L where the ray from A intersects the top of the second step, draw a perpendicular to H intersecting the ray drawn from C in H y which defines the shadow on the top of the second step. 6th. From P where the ray from B intersects the ground line, draw a perpendicular tp intersect the ray drawn from D in E; this defines the shape of the shadow on the ground. FIGURE 3. To draw the Shadow of a Cylinder upon a Vertical Plane. Rule. Find the position of the shadow at any number of points. 1st. From Jl where the tangental ray (at an angle of 45) touches the plan, draw the ray to W. X, and from the intersection erect a perpendicular. 2nd. From Jl erect a perpendicular to B, and from B draw a ray at 45 with Jl. B to intersect the perpendicular from A in L. This defines the straight part of the shadow. 3rd. From any number of points in the plan E. H, draw rays to intersect the wall line W. X, and from these points of intersec- tion erect perpendiculars. 4th. From the same points in the plan erect perpendiculars to the top of the cylinder, and from the ends of these perpendiculars draw rays at 45 to meet the perpendiculars on the wall line; the intersections give points in the curve. NOTE 1. The outlines of shadows should be marked by faint lines, and the shadow put on by several successive coats of India ink. The student should practice at first with very thin color, always keep the camel hair pencil full, and never allow the edges to dry until the whole shadow is covered. The same rule will apply in shading circular objects ; first wash in all the shaded parts with a light tint, and deepen each part by successive layers, always taking care to cover with a tint all the parts of the object that require that tint; by this means you will avoid harsh outlines and transitions, and give your drawing a soft agreeable appearance. NOTE 2. The lightest part of a circular object is where a tangent to the curve is perpendicular to the ray as at P. The darkest part is at the point where the ray is tangental to the curve as at /?, because the surface beyond that point receives more or less reflected light from surrounding objects. 117 THEORY OF COLOR, AND ITS APPLICATION TO ARCHITECTURAL AND MECHANICAL DEALINGS. THE THEORY. WHEN we survey with attention, the beautiful coloring of the works of nature ; we cannot fail to perceive the almost infinite variety of tints and hues, of which the landscape is composed, and however its tone may be modified by the state of the atmosphere, by the changes of the seasons, or by the degree of light with which it is illuminated, we shall always find these colors blended or contrasted harmoniously; forming a glorious whole, highly satisfactory when viewed in mass, and much more so when analyzed and examined in detail. But numerous as are those hues, it has been demonstrated that all are composed of three primary colors, viz. YELLOW, RED AND BLUE. These names, however, are commonly applied to various tints of the colors, and therefore do not convey to the mind a sufficiently definite idea ; but they may be seen in their pure brilliancy, in the flowers of the Yellow Jasmine, the Red Geranium and the Blue Sage. We seldom find them used in their intensity, in nature's own paint- ing ; and chiefly on the smaller gems with which she loves to decorate her bosom, from which the above examples have been selected, and even here they are used so sparingly that few ex- amples of the pure, unmixed, primary colors can be found ; so es- pecially is this the case with blue, that many horticulturists affirm that a perfectly blue flower is unknown in nature ; but our ex- ample will give a tolerably accurate idea of the color. 118 THEOKY OF COLOR. On a cursory view, it would appear almost impossible that the blending of three simple elements can produce so great variety ; if those elements could only be used as wholes, the changes would necessarily be very limited ; but let us endeavor to realize the fact, that any given quantity of a color may be divided and subdivided, again and again, into very minute portions ; and', that any minute quantity may be blended with equal or larger proportions of both or either of the other colors, and the proportions of each or all of them changed at pleasure, and that each combination of propor- tions will modify, to some extent, the hue produced. To illustrate this position, let us suppose that we have three fluids, yellow, red and blue, four drops of each, and ascertain by calcula- tion how many different combinations of distinct hues might be produced by combining them, two or more drops together to con- stitute a hue ; the difference between each combination to be not less than a drop. The answer will be eighty-eight, and if we add the three original colors, the number will be ninety-one. The greatest disparity between the proportions in this arrangement, would be, where one drop of a color is mixed with four drops of each of the others, the smallest quantity comprising one-ninth of the whole mixture; enough sensibly to modify the hue produced. If instead of four drops we were to suppose six drops of each given ; by the same process we could produce about two hundred changes, and the greatest disproportion in any hue, would be as one to six of each of the others, comprising one-thirteenth of the whole. After this very limited illustration, let us consider that any of these distinct hues may be used of any degree of depth, from the faintest trace of color to its deepest intensity, producing innumera- ble tints, and we shall be able to form some idea of the truth, that all may be produced by three original colors. Before proceeding with this subject, and that we may do so more understandingly, it will, perhaps, be as well to define the mean- ing of some of the terms used, and first of COLOR. This term is used in a very extended sense. It may be applied to any tint or hue produced in nature by all its varied processes; and with equal propriety, to all the pigments and paints used by Artists and Painters to imitate them ; but in this branch of our subject, (the theoretical,) it is necessary to limit its application to the hues produced ; therefore when we speak of THEORY OF COLOR. 119 color, we do not mean prussian Hue, ultra-marine, or any other pigment used to produce blue in art, but simply the hue itself, and in the same manner of all the other colors. HUE. This term is synonymous with color, when it is used as re- stricted in the last paragraph, and may be applied to any tint or color produced in Nature or Art. Some writers apply these terms more restrictedly ; for example, the term color is applied to the primary colors, yellow, red and blue, and to the secondary colors produced by equal admixture of two of the primaries, viz : orange, green and purple, and the term hue is applied by them to any color in which all the prima- ries are mixed, either in equal or unequal quantities ; thus a russet hue, an olive hue, &c.; but in all cases, where any definite color is meant by either of these terms, the name of the color or hue must be added to it, to convey the proper idea to the mind. TINT. This term is used to denote the depth or strength of a color, as a dark tint, a light tint, &c. Any primary or mixed color may be used in its deepest intensity, or so faintly as to be hardly distinguished from white, or with any degree of depth between these points ; these various gradations from the deepest color, are called tints. SHADE, This term is applied to any of the gradations oi a color toward black ; as a tint is applied as above defined, to any c( its gradations toward white. Shade is also applied to all the gradations from pure white light t<> bla Many of the tints of the primary colors, and many of the hues of the secondaries and tertiaries, have also their popular names by which they are recognized, but in most cases it would be very difficult to define the strength of the tint, or composition of the hue to which these names are applied ; light tints of yellow for example, are known as cream color, straw color, primrose, &c., light tints of red are known as rose-white, carnation, coquelicot, &e.; and of blue, as pearl-white, french grays, silver grays and azure. From the combination of yellow with red, are formed gold color, giraffe and scarlet; from a mixture of red and blue, result, crim- son, pink, rose colors, lakes, peach-blossom, lilac and violet; and different hues of green are known as emerald green, grass green, invisible green, &c. Numerous other names might be added, but at best they contain but indefinite ideas of the tints or hues meant, and perhaps no two artists if called upon to compound one of them, would produce the same result; in fact, this indefiniteness of nomenclature is common to the whole scale of colors, as the tints and hues so run into each other by almost imperceptible degrees, that it is very difficult to say where any particular hue terminates and the next begins. Another element of confusion in the popular names of colors, arises from the large number of pigments of commerce made to imitate 134 THEORY OF COLOR. the different lines and tints of nature ; each bears a specific name, either derived from its hue or from the elements of which it is composed : but as they are made by different manufacturers of various degrees of purity, and each maker has his own standard of hue for each color, we need not hope to have much uniformity of result. Before closing this branch of our subject, we will recapitulate in a more succinct form, some of the facts arrived at ; that they may make more permanent impressions on the memory. 1st. A PRIMARY COLOB cannot be compounded or formed by any combination of the other colors ; they are three in number, yellow,, red and blue. 2nd. A SECONDARY COLOR is formed by admixture of any two of the primaries in equal parts, they are orange, purple and green. 3rd. A TERTIARY COLOR is formed by admixture of .two of the sec- ondaries, and consequently, contains two equivalents of one of the primaries, and one of each of the others ; these are also three in number, citron, russet and olive. 4th. BROKEN COLORS are compounded of all the primaries in indefi- nite proportions, and form three classes, brown, maroon and gray. 5th. ADVANCING COLORS are first, those which advance from a neu- tral point toward light ; and secondly, those which cause objects to appear more prominent in a composition, and consequently, to appear nearer the spectator. 6th. RETREATING COLORS are first, those which retreat from a neu- tral point toward shade ; and secondly, those which cause objects to appear more distant or to recede from the spectator. 7th. WARM COLORS are those in which red or orange predominate ; they are generally also advancing colors. 8th. COLD COLORS are generally also retreating colors, in which bine and neutral gray are the prevailing hues. 9th. POSITIVE COLORS are the three primaries, and their definite compounds as placed in the scale at page 124. All other decided compounds of two of the primaries also come under this head. 10th. NEUTRALS are compounded of the three primaries in their neutralizing proportions, in which no trace of the positive colors exist ; they are neutral black, and its shades neutral gray. More strictly speaking, we have theoretically but one neutral ; as the neutral grays are to black, what the lighter tints of a color are to its full tone, viz. the same color diluted ; but in art, the THEORY OF COLOR, 135 pigments of white and black, as well as all their intermediate shades, are termed neutral colors. llth. SEMI-NEUTRALS are the same as the broken colors, and oc- cupy a middle space between the positive colors and the neutral black. 12th. ARTIFICIAL LIGHT increases the brilliancy of all warm colors, but it deteriorates all hues of the cold colors, and entirely neutral- izes many of their lighter tints. The student who wishes further to investigate this subject, should refer to the more elaborate treatises which have been published in relation to it. The object of this short essay, is to give, in a plain and simple manner, a short synopsis of the science for the use of practical men, who have but little time to devote to its study. Among other works, "Field's Chromatics," and his "Grammar of Coloring," and "Hay on the Laws of Harmonious Coloring" will be found very useful, and I would here acknow- ledge my indebtedness to them for many of the facts herein con- tained. Numerous small books of instruction in the art of coloring and of different styles of painting have been published in England, and are easily available here at moderate cost ; the conclusion of this woik will therefore be devoted to the coloring of architectural and mechanical drawings, and to specify some of the most useful pigments for the purpose. 136 GEOMETRICAL DRAWINGS. MECHANICAL or ARCHITECTURAL DRAWINGS consist of plans,, sec- tions, elevations, details, isometrical and perspective views; these terms have already been explained at pages 60,. 61. Drawings may also bo further divided into two classes : first, the more elaborately finished drawings for the purpose of explaining the proposed construction entire and in all its parts, for the use of the proprietor and contractor, and which form in connection with accompanying specifications the basis of the contract : these should be made as full and explicit as possible, to prevent the misconceptions and disputes which often arise from the insuffi- ciency of those important requisites. The second class consists of 'outline drawings, showing the details of construction, generally drawn to a larger scale and on which the dimensions should be figured. Detail Drawings are often roughly tinted to show the materials, as well as the forms and sizes of" construction; in machine drawings where round bodies frequently occur, they are shaded by ruling parallel ink lines, almost touching each other for the darkest shades, gradually placing them wider apart as they approach the light; such drawings are generally roughly made, the workman depending more on the figured dimensions than on the scale^ For detail drawings, a plain square edged board and any strong paper is sufficient, the paper can be secured to the board by small flat headed drawing pins made for the purpose. Plain drawing boards are frequently made with clamps firmly glued and screwed to the ends, or battens made equally secure to the back of the board : both these methods are objectionable, as all wood, however well seasoned, will contract and expand with the hygrometic changes of the atmosphere, with a force that no clamps can restrain; the boards will,, consequently, warp or split under such circumstances ; a better method of constructing them,, is to sink dovetailed grooves in the back of the board, and drive in battens to fit the grooves, these require no fastening and will allow the contractions or expansions to take place freely, and at GEOMETEICAL DRAWINGS. 137 the same time retain the board straight and sound: a drawing board large enough for imperial paper, made of soft pine half an inch thick, with grooves sunk a quarter of an inch deep, the battens four inches wide by three-quarters thick, has been in use for ten years, and is now as straight and free from cracks as when it was first made. For finely finished colored drawings, the paper should be dampened and stretched smoothly on the board with its edges firmly secured, otherwise, the parts which are made wet with the color, will stretch and rise from the board, and as a natural consequence, the color will flow toward the lowest parts, and prevent your obtaining an even tint. Drawing boards for this purpose should consist of a rebated frame with a thin panel, with buttons or bars on the back to press the panel tightly against the frame. Whatman's rough surface drawing papers are best for this purpose, the sizes are as follows : Cap, 17x14 in. Elephant, 28x23 in. Demy, 18x15, Imperial, 30x22, Medium, 22x17, Columbier, 34x24, Royal, 24x19, Atlas, 33x26, Super Royal, 27x19, Double Elephant, 40x27. Antiquarian, 52x31 in. These papers have a water mark of the maker's name, and the side on which this name reads correctly, is intended as the right side to make the drawing on. To stretch the paper, wet the back of the sheet evenly and lightly with fair water, either with a flat brush or sponge, or by laying a wet cloth on it ; in this state, the sheet will continue to expand in size until the moisture has pene- trated its whole substance, which requires but a few minutes to effect ; the sheet should be about one-and-a-half inches larger than the panel in each direction ; it should then be placed on the panel projecting equally all around, its edges pressed down into the re- bate, and the panel secured firmly in the frame which clamps the paper and prevents its contraction when drying, so that the sheet remains permanently stretched, and will remain smooth and even while the color is applied. When paper is stretched on a plain board, its edges are secured to the board with strong paste ; but in this case, a damp cloth should, be kept on it until the paste is sufficiently dry, to prevent its being drawn off by the too rapid contraction of the sheet ; as soon as the paper is dry, it is ready for the drawing. 18 138 GEOMETEICAL DRAWINGS. The other requisites for geometrical drawing, are a T square, a pair of wooden right angled triangles, the one with equal &ides and angles of 45 degrees ; the other with angles of 30 and 60 degrees for isometrical drawing ; pencils, rubber, a case of drawing instru- ments, and for drawing ink lines, a cake of India ink, aporcelain slab for rubbing the ink on, and camel's hair pencils for applying the ink to the drawing pens. The construction and use of the scales and parallel ruler, have already been explained in the early part of the Drawing Book ; the other instruments in a case, require but little explanation, their uses being -almost self-evident to any intelligent student. Pencils should be of good quality, sufficiently hard to retain a good point, but not so hard as to cut the paper and it is a very impor- tant requisite in a pencil that its marks may be easily erased ; it is almost impossible, to point a pencil properly with a dull knife, you should therefore, keep a keen one, it will save your time, pen- cils and temper. Common writing ink should never be used in the drawing pens, it would soon eat away the metal and render them useless. India ink rubbed up with pure water, is the best for the purpose, as it can be made of any required shade ; a slight addition of carmine will cause the ink to flow more freely ; the pen should be set to the requisite degree of fineness, and need not be altered to add a fresh supply of ink, this should be applied to the side of the pen with a hair-pencil, which should be occasionally passsed between the blades of the pen to prevent its clogging, any surplus ink on the outside of the blades should be wiped off before applying it to the paper. In using the drawing pen either for straight or circu- lar lines, it is very important to have both the blades to rest on the paper, or the lines will be irregular. With the f square applied to the edges of the board all lines paral- lel to any of its sides may be drawn correctly ; parallel inclined lines may be drawn by a square having one of its sides moveable, and clamped by a screw to the required angle, or with the parallel ruler, or by moving a triangle along a straight edge. The outlines of the drawing should be first made in pencil, then the permanent lines ruled in India ink of a light shade, and the surplus pencil marks removed with the rubber, or with a crumb of bread, which will answer the same purpose ; the drawing is then ready for col- oring. 139 APPLICATION OF COLORS. THE student who has carefully read the first part of this essay, will readily suppose that with three good pigments representing the three primary colors, a white for lights and a black for shades, he can mix any tint or hue he may require ; and this is true to a considerable extent, but not altogether so, as unfortunately, colored pigments are not always homogeneous in their composition, and often act chemically upon each other, so as to change, or perhaps neutralize the original hues ; the knowledge of the chemical prop- erties of colors requires more study and attention than is necessary for the mechanical draftsman to devote to it, as the colorman has prepared numerous pigments of different hues to meet his wants ; these pigments are commonly called colors, and in this practical branch of our subject, we shall follow the common practice and use the terms synonymously. Colors are either transparent or opaque, a transparent color is often laid over another color to change or modify its hue, and will often produce a much more soft and agreeable tone than could be pro- duced by a single color ; this operation is technically called glazing. The draftsman will often find occasion to exercise his ingenuity in this respect as well as in the mixing of his hues, and on this sub- ject he must depend chiefly on his own taste and judgment, as it is very difficult to give any other than general rules for the pur- pose. The following are the most useful of the prepared cake colors for architectural or mechanical drawings : YELLOWS. Gamboge, Roman Ochre or Yellow Ochre, Indian Yellow. REDS. . Carmine or Crimson Lake, Vermilion, Indian Red. BLUES. Cobalt or Ultra-marine, Prussian Blue, Indigo. BROWNS. Sepia, Vandyke Brown or Burnt Umber, Raw Sienna. 140 APPLICATION OF COLOES. India ink may be used for the shades or a semi-neutral tint, corn- pounded of indigo and Indian red ; this tint may be found in cakes under the name of neutral tint. The surface of the paper should represent the lights in water color drawings, or, where very small lights are required, they may be scratched out with the point of a knife ; the last method is gener- ally resorted to where drawings are made on surface tinted or graduated paper. The secondary colors, orange, green and purple, as well as the semi-neutral grays and other broken colors, may all be compounded from the pigments in the above selection. A little practice and observation of the results of the different mixtures, will soon enable the student to combine them satis- factorily ; he will find that many of the combinations he produces, will result in a foul, cloudy mixture, these he should note par- ticularly, and avoid in future ; and those which result satis- factorily, should be still more carefully remembered for future use. As a general rule, which should be well fixed in the memory, the fewer the pigments used in the composition of a tint, the more clear and satisfactory will be the hue produced. In using any hue compounded of different colors, it should be con- stantly stirred up with a brush whilst taking a fresh supply in it, as such mixtures have always a tendency to separate, this must be more particularly attended to when vermilion enters the com- pound, as it is so much heavier than other pigments that it in- variably falls to the bottom. In mechanical drawings, the plans and sections especially, should be so colored as to show the materials and construction ; so that this information may be obtained by the mere inspection of the draw- ings, without the trouble of hunting it up in the specifications, which are seldom available at the place of construction, being gen- erally filed away in the office for particular reference. In shading such drawings, the shadows should always be projected at an angle of 45 degrees, as explained at page 113, where shadows are especially treated of. In elevations, it is not so important that the materials should be so palpably designated ; but even here it should not be altogether ne- glected, more particularly when the materials are to remain of their natural color, as the general coloring has often an important effect on the character of the design. APPLICATION OF COLOES. 141 In perspective views, this requisite is generally lost sight of, and often in architectural views, the coloring and shading are more in accordance with the practice of the landscape painter, than with that of the mechanical draughtsman. The colors used in plans and sections to denote the construction should be as nearly as possible the colors of the materials to be employed. A very excellent article on this subject has recently been published in an English translation of a popular French work on drawing; "The Practical Draughtsman." As this most likely embodies the practice adopted in both those countries, it is desirable that we in America, should assimilate our practice to theirs, so that every one may know the materials intended to be used in a construction, by an inspection of the tints, no matter where the drawing may have been made. The following extracts have been taken from this work, I would recommend their careful study, as with the few additions given below, they pretty well cover the whole ground. The colors designated, are termed "conventional colors, that is, certain colors are generally understood to indicate particular materials." India ink in these extracts, is called China ink, either name may be ap- plied to this pigment. ''Stone. This material is represented by a light dull yellow, which is obtained from Roman ochre, with a trifling addition of China ink." This color does very well for light sand-stone, but where different kinds of stone are to be used in a construction, other hues are necessary to denote them. Granite may be indicated by Prussian blue, with a little India ink added. Red sand-stone, by Indian red. White or other light colored marbles, by a light tint of yellow ochre or raw sienna. Rubble stone walls, by Prussian blue with Indian red ; these should be further designated in sections, by a few irregular lines of a differ- ent tint or hue, laid in with the hair pencil after the first tint is dry, to represent the joints and indicate the construction. "Brick. A light red is employed for this material, and may be obtained from vermilion, which may sometimes be brightened by the addition of a little carmine. A pigment found in most color- boxes and termed light red, may also be used when great purity 142 APPLICATION OF COLORS. and brightness of tint is not wanted. If it is desired to distinguish firebrick from* the ordinary kind, since the former is lighter in color and inclined to yellow, some gamboge must be mixed with the vermilion, the whole being laid on more faintly. "/Steel or Wrought Iron. The color by which these metals are ex- pressed is obtained from pure Prussian blue laid on light : being lighter and perhaps brighter for steel than for wrought-iron. "Cast-Iron. Indigo is the color employed for this metal; the ad- dition of a little carmine improves it. The colors termed Neutral Tint or Paynes Gray, are frequently used in place of the above, and need no further admixture. They are not, however, BO easy to work with, and do not produce so equable a tint. " Lead and Tin are represented by similar means, the color being rendered more dull and gray by the addition of China ink and car- mine or lake. "Copper. For this metal, pure carmine or crimson lake is proper. A more exact imitation of the reality may be obtained by the mixture with either of these colors, of a little China ink or burnt sienna the carmine or lake, of course, considerably predominating. "Brass or Bronze. These are expressed by an orange color, the former being the brighter of the two ; burnt Eoman ochre is the simplest pigment for producing this color. Where, however, a very bright tint is desired, a mixture should be made of gamboge with a little vermilion care being taken to keep it constantly agitated as before recommended. Many draughtsmen use simple gamboge or other yellow. " Wood. It will be observable from preceding examples, that the tints have been chosen with reference to the actual colors of the materials which they are intended to express carrying out the same principle, we should have a very wide range in the case of wood. The color generally used, however, is burnt umber or raw sienna ; but the depth or strength with which it is laid on, may be considerably varied. It is usual to apply a light tint first, sub- sequently showing the graining with a darker tint, or perhaps with burnt sienna. These points are susceptible of great variation, and very much must be left to the judgment of the artist. "Leather, Vulcanized India Rubber and Gutta Percha. These are all represented by very similar tints. Leather by light, and gutta percha by dark sepia, whilst vulcanized India rubber requires the addition of a little indigo to that color. APPLICATION OF COLOKS. 143 "Manipulation of the Colors. The cake of color should never be dipped in the water, as this causes the edges to crack and crumble off, wasting considerable quantities. Instead of this, a few drops of water should be first put in the saucer or on the plate, and then the required quantity of color rubbed down, the cake being wetted as little as is absolutely necessary. The strength or depth of the color is obtained by proportioning the quantity of water, the whole being well mixed, to make the tint equable throughout. When large surfaces have to be covered by one tint, which it is desired to make a perfectly even, flat tint, it is well to produce the re- quired strength by a repetition of very light washes. These washes correct each other's defects, and altogether produce a soft and pleasing effect. This method should generally be employed by the beginner, as he will thereby more rapidly obtain the art of producing equable flat tints. The washes should not be applied before each preceding one is perfectly dry. When the drawing- paper is old, partially glazed, or does not take the color well its whole surface should receive a wash of water, in which a very small quantity of gum-arabic or alum has been dissolved. In pro- ceeding to lay on the color, care should be taken not to fill the brush too full, whilst, at the same time, it must be replenished before its contents are nearly expended, to avoid the difference in tint which would otherwise result. It is also necessary first to try the color on a separate piece of paper, to be sure that it will pro- duce the desired effect. It is a very common habit with water- color artists to point the brush, and take off any superfluous color, by passing it between their lips. This is a very bad and disagree- able habit, and should be altogether shunned. Not only may the color which is thus taken into the mouth be injurious to health, but it is impossible, if this is done, to produce a fine even surface, for the least quantity of saliva which may be taken up by the brush has the effect of clouding and altogether spoiling the wash of color on the paper. In place of this uncleanly method, the artist should have a piece of blotting-paper at his side the more absorbent the better. By passing the brush over this any super- fluous color may be taken off, and as fine a point obtained as by any other means. The brush should not be passed more than once, if possible, over the same part of the drawing before it is dry ; and when the termination of a large space is nearly reached, the brush should be almost entirely freed from the color, otherwise, 144 APPLICATION OF COLOES. the tint will be left darker at that part. Care should be taken to keep exactly to,the outline; and any space contained within defi- nite outlines should be wholly covered at one operation, for if a portion is done, and then allowed to dry, or become aged, it will be almost impossible to complete the work, without leaving a dis- tinct mark at the junction of the two portions. Finally, to produce a regular and even appearance, the brush should not be over- charged, and the color should be laid on as thin as possible ; for the time employed in more frequently replenishing the brush, because of its becoming sooner exhausted, will be amply repaid by the better result of the work under the artist's hands." With this information carefully digested, in addition to the in- structions of much the same tenor, previously given for laying in shades and shadows, at page 116, the student will find but little difficulty in coloring his drawings creditably ; but he will require practice, that he may gain experience in the manipulation of hues and tints, and freedom in the use of the brush. It is important that he should first practice on drawings that re- quire but little labor to construct; so that the fear of spoiling them would not cramp his motions and prevent his acquiring the requisite freedom of hand ; if a drawing of this character be spoiled, it will be of very little consequence, as it can be easily replaced ; with an elaborate drawing the case is very different, the fear of spoiling it will often induce hesitation and awkward- ness, and produce the result feared. When a drawing is to be shaded and tinted, the shades and shadows should be laid in with India ink before applying the colors, but not so dark as required for the finished drawing, as they can be worked up to better advantage by deepening them with the local hues ; sometimes drawings are shaded with differ- ent tints of the local colors without first shading with ink ; the student should practice in both styles, so that he may be able to apply either or mix them, as his taste and judgment may direct. In shading perspective views of exteriors, the student must re- member that all his shades and hues must diminish in intensity as the distance increases ; but in limited interior views, as of a room, his shades will increase in depth toward the rear of the picture, unless such room is lighted by side windows or by other light than that obtained from the front of the picture where the section of the room is made. TO MOUNT DEAWINGS, ETC. 145 The student who endeavors to follow those instructions, will doubtlessly meet with some difficulties, but he will soon overcome them, if he will practice with the requisite attention and industry. Every obstacle encountered should be removed from the path at once, he must not pass over or around it, leaving it as a stumbling block to his future labors ; but should attack each in detail, and let every encounter act as a spur to further progress and to ulti- mate success. TO MOUNT DRAWINGS. It is often desirable for large drawings that are expected to be much handled, or to be preserved for records, to strengthen them by backing with cotton cloth, technically called "mounting" this may be done as follows : 1st. Procure a piece of new cotton cloth several inches larger each way than the drawing, and tack its edges securely down on a table, drawing-board or other flat surface ; new cloth is preferred because it shrinks considerably in drying ; that which has been washed has lost this property of shrinkage, having already gone through the process. 2nd. Prepare a strong flour paste, about the consistency of that used by paper- hangers, and with a large brush give the back of the drawing a coat of the paste ; be careful to cover the whole of the paper, that it may stretch equally. 3rd. Coat the cloth thoroughly, so that it may be saturated with paste. 4th. Lay the drawing in position on the cloth, then with two stout bone or ivory paper folders, commence at the middle of the sheet with the edges of the folders to press in every direction from the centre toward the edges, until all the surplus paste is pressed out ; be careful to keep the face of the draw- ing dry and clean and also the edges of the folders, otherwise the surface of the paper may be injured, more especially if it has much color on it. When the whole is dry, trim the edges and bind them with a narrow ribbon. When a drawing requiring heavy tints of color is intended to be backed, in order to avoid the risk of smearing in the process, it would be better to omit those tints until afterward ; they should be laid in after the paper is dry and before cutting it loose from the table, when a satisfactory result may be arrived at. TO CLEAN DRAWINGS OR ENGRAVINGS. A draughtsman occasionally desires to save a dilapidated drawing or old en- graving by backing.it, this may be done as above directed, but if it is much soiled by smoke or otherwise, it is desirable to expel the stains as much as possible before mounting. I have found the following process satisfactory : 1st. Place the engraving or drawing face downward in a pan of clear cold water, let it soak some hours, more or less, depending on the strength of the 19 146 TRANSPARENT TRACING CLOTH, ETC. paper, the longer it remains in the water without injuring the texture of the paper, the cleaner it will become. 2nd. Remove the sheet carefully from the water and lay it face downward on a clean flat table or drawing-board ; then with a soft sponge absorb the moisture, adding clean water from time to time, until all the removable stains have been extracted ; then turn the sheet over and proceed very carefully in the same way with the face ; the sponge must not be rubbed on the paper, but pressed on and wrung out until the water thus absorbed becomes clear. 3rd. Dry the drawing between sheets of thick blotting paper under pressure, when it will be ready for mounting. This process will remove most of the stains caused by water or smoke, but will not affect those made by oil or grease, these should be removed before wetting the print, by placing blotting paper over the stain and extracting the grease by pressing the paper with a hot smoothing iron. The printer's ink of an engraving is insoluble in water and not likely to take injury from this method ; the same may be said of a drawing made with India ink, but if it has been made with common ink, the drawing can only be cleaned by rubbing with soft rubber, or with crumb of stale bread. TRANSPARENT TRACING CLOTH. Is now much used because of its greater strength, in preference to tracing paper for taking copies of drawings for the workman or for preservation ; this cloth is liable to injury from dampness, and being transparent requires to be placed on a board or paper for examination. If the tracings and letterings are drawn with India ink of good quality, the whole, when the ink has become dry, may be dipped in water, which extracts the transparent preparation ; then dried and pressed with a hot iron, when you will have a permanent drawing on an opaque surface, but little liable to injury from moisture. QUALITY OF MATERIALS. The draughtsman who values his time, will always find it the more economical plan, to procure all his materials of the best quality, and also to keep his in- struments clean and in good condition ; he will thus be enabled with the same degree of effort to produce a much more satisfactory drawing. INDEX. PAG3 Abscissa, ...... .4 Absorption of light, .... .82 Accidental points, ..... . 8J Aerial perspective, .... Altitude of a triangle, .... . 1( Angles described, .... Angle of incidence, .... .8] " Visual .... .84 " How to draw angles of 45, .. Apex of a pyramid, .... .39 " of a cone, ..... .44 Application of the rule of 3, 4 and 5, 17 Apparent size of an object, .... .90 Architrave, ....... 66 Arc of a circle, ..... .1 Arcades in perspective, ..... 9 Arches Composition of .... .54 Construction of ..... 54 " Definitions of .... .54 Arch Thrust of an . . . . . . 55 " Amount of the thrust of an arch, (note) . . .55 " Straight arch or plat band, .... 55 " Rampant ....... 55 " Simple and complex arches, .... 55 " Names of arches, . . . . . .55 Arches in perspective, . . . . . . 98 to 100 Arithmetical perspective, . . . . .83 Aspect of a country house, . ;..* . . . 61 Axis of a pyramid, . . . . .39 " of a sphere, ...... 41 " of a cylinder, ...... 42 " Major and minor axes, . . . . 43, 44 " of a cone, ....... 45 " Difference between the length of the axes in the sections of the cone and cylinder, ...... 46 " of tne parabola, ...... 50 Axioms in perspective, ...... 90 Back of an arch or Extrados, ..... 55 Band, listel or fillet, ...... 68 Base of a triangle, . . ; OF THR 148 PAGE. Base of a pyramid, ....... 39 " of a cone, ...... 44 " of a Doric column, ...... 68 " line or ground line, . . . . . 86 Bead described, ....... 68 Bed of an arch, ...... 55 Bisect To bisect a right line, . . . . .14 " To bisect an angle, . . . . . 18 Bird's eye view, ... . . . .89 Cavetto, a Roman moulding, . . . . . 69 Centre of a circle, . . . . . . . .11 " of a sphere, ...... 41 " of a vanishing plane, . . , . .110 Chords defined, . . . . . 12 " Scale of chords constructed, . . . . .26 " Application of the scale of chords, ... 26 Circle described, . . . . . . .11 " To find the centre of a circle, . 22 " To draw a circle through three given points, . . .23 " To find the centre for describing a flat segment, . . 23 " To find a right line equal to a semicircle, . . .23 " " " equal to an arc of a circle, . . 23 " Workmen's method of doing the same, . . .24 " Great circle of a sphere, .... 41 " Lesser circle of a sphere, . . . . .41 " Circumferences of circles directly as the diameters, . 74 " in perspective, ...... 95 " Application of the circle in perspective, ... 101 Circular plan and elevation, . . . . .67 " objects Shading of . . . . 112 Circumference of a circle, . . . . . .11 " directly as its diameter, . . 74 Circular domes To draw the covering of . . . .52 Color How to color shadows, &c. . . . . 116 Conjugate axis or diameter, . . . . . 43, 44 of a diameter of the ellipsis, .... 44 Contents of a triangle, . . . . . .10 of a cube, ...... 37 " of the surface of a cube, . . . . .38 Complement of an arc or angle, . . . . 13 Complex and simple arches, . . . . .55 Cone, right, oblique and scalene, .... 40 To draw the covering of a cone, . . . .40 " Sections of the cone, . 45 Co-sine, ^ 13 Co-tangent, . . ... . . . 13 149 PAGE. Co-secant , . , . . . . . .13 Construct To construct a triangle, . . . . 17 " an angle equal to a given angle, . . 18 " " an equilateral triangle on a given line, . 19 " " a square on a given line, . . .20 " " a pentagon on a given line, . . 21 " a heptagon on a given line, . . 22 " " any polygon on a given line, . . 22 " " a scale of chords .... 26 " " the protractor, ... 30 Construction of arches, ...... 54 Contrary flexure Curve of . . . . 7 Arch of : 57 Cornice and piazza Effect of the ... 62 Cottage in perspective, . . . . . .106 Cohering of the cube, . . 38 " " parallelopipedon, . . . . .38 " " triangular prism, .... 38 " " square pyramid, .... 39 " " hexagonal pyramid, .... 40 " cylinder, . . . . . .40 " cone, ...... 40 " sphere, .41 " " regular polyhedrons, .... 42 " " circular domes, ... . . .52 Crown of an arch, ..... 55 " moulding, ....... 69 Cube or hexahedron, . . . . . . 37 Cubic measure, ....... 37 Cube To draw the Isometricai .... 76 " in perspective, . . . . . .104 Cycloid described, .*.... 35 Cycloidal arches, ....... 36 Cylinder, . * 40 " To draw the covering of a . . . . .40 " Sections of the . . . . .42,43 " Right and oblique ...... 42 u To find the section of a segment of a cylinder through three given points, ...... 51 " of a locomotive engine, .... 75 " in perspective, . . . . . . 102 " Shadow of a . . . . . 116 Cyme. :r cyma recta Roman, . .69 " Grecian, ..... 71 " reversa, talon or ogee Roman, . 69 c< " " Grecian, : . 71 150 PAGE. Degree defined, . . . .12 Depressed arch, ... 59 Design for a cottage, ... " What constitutes a . . . * - ' 61 Details of cottage, .... 65 Diamond defined, ..* Diagonal defined, .. Diagonal lines in perspective, ..... Diameter of a circle, ..... " of a sphere, . 41 " of an ellipsis, . . . * . 43, 44 " of the parabola, r . . . - 60 Definitions of lines, * - 7 " of angles, . . . * " of superficies, . . ... 9 " of the circle, ..... 11 u of solids, ...... 36 " of the cylinder, . . . . . 42 " of the cone, . . . . . 44 " of the parabola, . . . . . 50 " of arches, . . . . .55 " in perspective, ..... 86 Directrix of the parabola, ...... 50 Distance Points of ...... 87 Half 92 " The quantity of reflected light enables us to judge of . Ill Dodecahedron, . . . . .42 Domes Covering of hemispherical .... 52 Double shadows, . . .112 Echinus, or Grecian ovolo, . . . . 70 Effect A perspective view necessary to shew the effect of an intended improvement, . . 61 Elevation described, .... . . 61 Ellipsis False ...... 34 " the section of a cylinder, . . . . .43 " To describe an ellipsis with a string ... 43 " the section of a cone, ..... 45 " To describe an ellipsis from the cone ... 45 " To describe an ellipsis by intersections . . .46 " To describe an ellipsis with a trammel . . 47 Elliptic Arch To draw the joints of an . . . .57 Epicycloid described, ...... 36 Equilateral triangle, ...... 9 " arch, (Gothic) ..... 58 Extrados or back of an arch, . . . . .55 Fillet, band or listel, ...... 68 151 Focus Foci of an ellipsis, .. ,43 " of a parabola, . 50 Foreshortening, . . . ... 84 " The degree of foreshortening depends on the angle at which objects are viewed, .... 85, 90 Form of shadows, . . . . Ill Frustrum of a pyramid, . . . . .39 of a cone, . . . . . . 45 Globe or sphere, . . . . .41 Gothic arches described, * 58 Grades, ... 13 Grecian mouldings, ... 70 Grounds to plinth, &c., . . . . .66 Ground line or base line, . . , , 86 Habit of observation, . . . . . 1 05 Half distance, . . . . . 92 Height, rise or versed sine of an arch, . . 54 Hemisphere, ....... 41 Hexahedron or cube, . , - . .37 Hexagonal pavement in perspective, * 94 Hipped roof hipped rafter, . . .64 Horizontal or level line, . . , . 8 " covering of domes, . , . . .53 Horizon in perspective, ..,, 86 Horseshoe arch, . . . . . .57 " pointed arch, . . . . . 59 Hyperbola the section of a cone, . . . .45 " To describe the hyperbola from the cone, . . 46, 48 Hypothenuse, . . . . . . . 9, 16 " Square of the . . 16 Icosahedron, . ' . . . . . .42 Inclined lines in Isometrical drawing require a different scale, . 81 Incidence -The angle of incidence equal to the angle of reflection, . 81 Inclined lines Vanishing point of .... 89 Inclined planes Vanishing point of . . . .90 Inscribe To inscribe a circle in a triangle, . . . 19 " " an octagon in a square, . . 20 " " an equilateral triangle in a circle, . . 21 " " a square in a circle, . . . 21 " " a hexagon in a circle, . 21 c< " an octagon in a circle, . . 21 " " a dodecagon in a circle, . * - 21 Intrados or soffit of an arch, . . . . . 55 Isometrical drawing, ... .76 " cube, ...... 76 " circle, . . . . . .79 _ 152 PAGE. Isometrical circle To divide the .... 80 Isosceles triangle, ...... .9 Joints of an arch defined, . . . 55 " To draw the joints of arches, . * . . 56 to 59 Joists Plan of a floor of joists, ..... 65 " Trimmers and trimming . . . . .65 " Tail ....... 65 Keystone of an areh, ... . . 55 Lancet Arch To describe the . . . . . 58 Light Objects to be seen must reflect light, . . .81 " becomes weaker in a duplicate ratio, &e., . . 81 " Three degrees of ..... 82, 111 Lines Description of . . . . . 7 Line To divide a right . . . . . .25 " To find the length of a curved . * . 23 " Workmen's method of doing so^ . . .24 Line of centres (of wheels,), ..... 73 " Pitch line denned, ...... 73 " To draw the pitch line of a pinion to contain, a definite number ef teeth, ....... 74 " Ground or base line, ..... 86 " Vanishing point of a line, . .89 " of elevation in perspective, .... 95 Linear perspective defined, ...... 86 Listely band or fillet, .... .68 Locomotive cylinder,. . . .75 Lozenge defined, . . . . 10 Major and minor axes o* diameters, . . . . 43, 44 Measurements to be proved from opposite ends, ... 63 Measures Cubic . 37 " Lineal and 1 superficial .... 37 " of the surface of a cube, . .38 Middle ray or central visual ray, .... 87 Minutes, ........ 12 Mitre To fend the eut of a. . . . 18 Moresco or Saraeenie arch, . . . . . .57 Mouldings Roman ...... 68 " Grecian *..... 70 Obelisk defined, . 39 Oblique pyramid, ......... 39 " cone, . ' 40,45 " cylinder, .. 42 Oblong defined, . ^ , . 10 Octagonal plan and elevation, . * . . .67 Octahedron, ....... 42 Ogee Arch or arch of contrary flexure, . . . .57 153 PAGE. Ogee or cyma reversa Roman . 69 " " " Grecian .... 71 Optical illusion, . .... 77 Ordinate of an ellipsis, . ... 44 Ovals composed of arcs of circles, .... 33, 34 Ovolo Roman 4 ... . 69 " or Echinus Grecian ... . TO Parallelogram denned, ..... 10 Parallel lines, ....... 7 " ruler, ...... 24 " Application of the parallel ruler, ... 25 Parallelopipedon, ....... 38 Parabola To find points in the curve of the ... 32 " the section of a cone, . . . .45 " To describe a parabola from the cone, . v 46 " To describe a parabola by tangents, &c. . . .49 " To describe a parabola by continued motion, . . 50 u applied to Gothic arches, . . . .51 " Definitions of the parabola, .... 50 Parameter defined, ....... 50 Pentagon reduced to a triangle, ..... 25 " To construct a pentagon on a given line, . 21 Perimeter the boundary of polygons, ... 9 Periphery the boundary of a circle, . . : ". 11 Perpendicular lines defined, ..... 8 " To bisect a line by a perpendicular, . . .14 " To erect a perpendicular, . ... . 14, 15, 16 " To let fall a perpendicular, . . . . 15, 16 Perspective view necessary to shew the effect of a design, . 61 " Essay on perspective, . . % ,- . .81 " Linear and aerial perspective, v . . 86 " plane, or plane of the picture, . * . . 86 " " must be perpendicular to the middle visual ray, 87 " plan of a square, . . . . . 90, 91 " u of a room with pilasters, . . . .92 " " To shorten the depth of a perspective plan, . 92 " Tesselated pavements in perspective, . . .93 " Double square in perspective, . . 94 " Circle in perspective, . v . .95 u Line of elevation, . . . . . 95 " Pillars with projecting caps in perspective, . . 96 " Pyramids in perspective, ... 97 " Arches seen in front, . . . .98 " " on a vanishing plane, . . 100 " Application of the circle, .... 101 " To find the perspective plane, &c. . . . 103 _ 154 Perspective view of a cube seen accidentally, . . .104 view of a cottage seen accidentally, . . 106 view of a street, . . . .108 Piazza and Cornice Their effect on the design for a cottage, 62 Pillars in perspective, . . . . . .96 Pitch of a wheel, ...... 73 " circle of a wheel, ... 73 " line of a wheel, ...... 74 Plan A horizontal section, . . . . .60 Plane superficies, ...... 9 Planes Vanishing ...... 88 " Parallel planes vanish to a common point, . . 90 " parallel to the plane of the picture, . . .90 To find the perspective plane, . . . . 103 Platband or straight arch, . . . . .55 Platonic figures, ...... 42 Plinth Section of parlor plinth, . . . . .66 Point of intersection, . . . . . 13 " of contact, ....... 13 " Secant point, ...... 13 " of sight, ....... 87 " of view or station point, .... 87 " Vanishing points . . . . . .87 " Principal vanishing point, .... 88 " of distance, ...... 87 Pointed arches in perspective, .... 99 Poles of the sphere, ...... 41 Proportional diameter of a wheel, .... 73 circle or pitch line, . . . . .73 Polygons described, . . . . . . 9 " Table of polygons, ..... 19 " Regular and irregular polygons, . . . 19, 20 Polyhedrons, . . . . . . 37, 42 Projecting caps in perspective, .... 96 Protractor Construction of the protractor, . . . .30 Application of the protractor, ... 30 Prisms, ....... 38, 39 Pyramid, ....... 39 " in perspective, ...... 97 Quadrant of a circle, . . . . . . 12 Quadrangle defined, . . . . . .10 Quadrilateral defined, . . . . . . 10 Radius Radii, . . " . , . H Rafters Elevation of rafter, . ... . . 65 "Hip 64 Rampant arch, . . 55 155 Rays of light reflected in straight lines, . . . 83 converged in the crystalline lens, ... 83 Rectangle denned, . . . . . 10 Reduce To reduce a trapezium to a triangle, ... 25 To reduce a pentagon to a triangle, . . .25 Reflection of light, ...... 81 " The angle of reflection equal to the angle of incidence, . 81 Reflected light enables us to see objects not illuminated by direct rays, 83 Regular triangles, . . . . .9 " polyhedrons, . . . . .37 and 42 Requisites for a country residence, ..... 61 Refraction of light, . . ... 82 Retina of the eye, ...... 83 Rhomb Rhombus, . . . . . 10 Rhomboid, ....... 10 Right angled triangle, .... 9 Right line denned, ...... 7 " pyramid, ...... 39 " cylinder, ... ... 40 " cone, . . . . . 40, 45 Rise or versed sine of an arch, . . ... .54 Rise or riser of stairs, ..... 63 Roof- Hipped . . . . .64 " Section of roof, ...... 65 Roman mouldings, ...... 68 Rule of 3, 4 and 5, . . . . . 16, 17 Saracenic or Moresco arch, . . . . .57 Scale of chords, ...... 26 Scales of equal parts, ...... 27 " Simple and diagonal scales, .... 28, 29 " Proportional scale in perspective, . '* . . 96 Scalene triangle, . . . 9 Scheme or segment arch, . . . , . .56 Scotia described Roman, .... 68 " " Grecian, ...... 72 Secant Secant point, or point of intersection, . . 13 Seconds, ....... 12 Sector of a circle, . . . . . . 12 Section a vertical plan, . . .60 Sections of the cylinder, ..... 42, 43 Section " " through three given points, . . .51 " of the cone, . . . . 45 to 48 " of the eye, ...... 83 Serpentine line, ...... 7 Segment of a circle, . . . . . .12 " To find the centre for describing a segment, . , 23 156 PAGE. Segment To find a right line equal to a segment of a circle, . . 23 " To describe a segment with a triangle, . . 31 " To describe a segment by intersections, . . .33 " of a sphere, ...... 41 " or scheme arch, ...... 56 Semicircle, . . . . . . 12 Semicircular arch, ....... 56 " " in perspective, .... 99 Shade and shadow, ..... 82, 111 Shadow always darker than the object, .... 87, 90 Shadows Essay on shadows, . . . . .111 Shading of circular objects, . . . . . 112 Shadow Lightest and darkest parts of a . . . .116 Sight Method of sight, . . . .81,83 " Point of sight, ...... 87 Sills of window, . . . . . 66 Simple and complex arches, . . . . .55 Sine, ........ 13 Skew-back of an arch, ...... 56 Soffit or intrados of an arch, . . . . . 55 Span of an arch, . . . . . . .54 Sphere Definitions of the sphere, .... 41 " To draw the covering of a sphere, . . . .41 Springing line of an arch, ..... 54 Square, . . . . . . . .10 Square corner in a semicircle, . . . . . 15 " " by scale of equal parts, . . . .16 " of a number, ...... 16 " of the hypothenuse, . . . . . .16 Stairs To proportion the number of steps of stairs > . . 63 Station point or point of view, . . . . .87 Stiles of sash, &c. ...... 66 Step or tread, ....... 63 Straight or right line, ...... 7 " arch or plat band, ...... 55 Street in perspective, . . . . . . 108 Subtense or chord, . . . . . . .12 Summit of an angle, ..... ^. 8 " of a pyramid, ...... 39 " of a cone, ..... 40, 44 Superficies or surface, ..... 9 Supplement of an angle or arc, . . . . . 13 Table of the names of polygons, ..... 19 " " the angles of polygons, .... Plate 10 Tail joist, ....... 65 Talon or Ogee Roman, ..... 69 157 PAGE. Talon or Ogee Grecian, . . . . .71 Tangent defined, . . . . 13 Teeth of wheels To draw the teeth of wheels, . . .73 " " Pitch of the " ... 73 " " Depth of the " .... 73 Tesselated pavements in perspective, .... 93 Tetragon defined, . . . . .10 Tetrahedron one of the regular solids, .... 41 Torus described, . ...... 68 Trapezium defined, ...... 10 " reduced to a triangle, . . . . .25 Trapezoid defined, ...... 10 Transverse axis or diameter, . . . 42, 43 Tread or step, ...... 63 Trigons or triangles, ...... Trisect To trisect a right angle, 18 Trimmers and trimming joists, . ... 65 Truncated pyramid, ...... " cone, . . . .45 Tudor or four centred arch, ..... 59 Vanishing points, .. " Principal vanishing point, " planes, . Versed sine of an arc, . . " " or rise of an arch, ... .54 Vertex of a triangle, . . " of a pyramid, ... . " of a cone, .. 44 " of a diameter of the ellipsis, " Principal vertex of a parabola, . . . 50 " of a diameter of the parabola, .... 50 Vertical or plumb line, .. " coverings of domes, . . ... Visual angle, . " rays, ....- Voussoirs of an arch, . ... Wheel and pinion Drawing of a " To proportion the teeth of a . ?4 Wheel viewed in perspective, Windows Details of windows . . . 65, 66 159 INDEX TO THE ESSAY ON OOLOE. PAGE. Absorption of light by all bodies, . 127 Advancing colors, ... 125, 134 Application of colors to Geometrical Drawings, . . , . 139 Artificial light, its effect on colors, .... 126, 135 Ash-color, slate, lead-color, etc., modifications of gray, . . 133 Auburn, hazel and dun, modifications of brown, . . . 133 Black, the total absence of light, . . . . 123 " How the term is usually applied, ... . 123 " Neutral Compounded of the three primaries, . . 132 Blue, its properties and effects on other colors, . . 130 " Light tints of known as pearl white, silver gray, &c., 133 Bodies, have no inherent color, ..... . 127 Brass or Bronze, the colors for indicating them, . . . 142 Brick, the colors used to represent it, . . . . . 141 Broken colors, the general acceptation of the term, . .132, 134 Browns, Composition and qualities of . . . . . 133 Citron or yellow hue, ...... 124, 131 Color, the term defined, . . . ... .118 Colors, Broken colors, ...... 132 " Contrasting colors, ..... . 120 " Complementary colors, . . . . . . 120 " Cool colors, retreating from red toward black, . . 125, 134 " Conventional colors, to denote the construction, . . 141 " How to use them, ..... 142, 143 11 Homogeneous yellow, red and blue, . < : . 122 11 List of for architectural and mechanical drawings, . . 139 11 Neutralizing powers of the primary . . . 124 " Nomenclature of colors uncertain, . . . 133, 134 11 Neutral and semi-neutral .... 132, 133, 134 " Primary 123, 134 11 Secondary and tertiary . . . . 124, 134 " Of the solar spectrum, . . . . . 121 " Scales of 123, 124, 125 " Transparent and opaque .... 139 Copper, the color used for indicating it, Crimson, pink, peach-blossom, rose-color, lilac, violet, &c., combinations of red and blue, ...... 133 Decorations, that are to be viewed by artificial light, require peculiar treatment, ...... 126 Definitions' of terms, recapitulated, .... 134 160 INDEX TO ESSAY ON COLOR. PAGE. Detail Drawings, how to make and tint them, . . . 136 Drawing Boards, the best way to make them, . . . 136 " paneled for stretching the paper on, . . 137 Drawing Instruments and requisites for geometrical drawing, . 138 " methods of using them, ... . 138 Drawing Paper, Regular sizes of . . . . 137 " which side of the sheet to draw on, . . .137 Method of stretching .... 137 " to back it with muslin, ... . 145 " to correct a greasy or glazed surface, . . 143 Drawing Pins, for holding the paper on the board, . . . 136 Engravings, to clean and mount them, .... 145 Experiments on light by Sir Isaac Newton, ... . 121 " " by Sir David Brewster, . . . 122 " with red and green wafers, . . . .120 " with a circular disk, (note,) .... 124 Flowers which indicate the primary colors, . . . .117 " Blue natural flowers extremely rare, . . . 117 Geranium, the red geranium to represent the primary red, . .117 Geometrical Drawings described, ..... 136 Glazing, the laying a transparent color over another, . . 139 Gold-color, giraffe, scarlet, &o., combinations of yellow and red, . 133 Gray, Neutral ....... 123, 134 Green, its properties and effects, . . . . . 131 Grass-green, Emerald-green, etc., .... . 133 Harmony of colors, definition of the term, . . .119, 120 Hue, definition of the term, . . - . . . . .119 Illustration by drops of color, . . . . . 118 Ink, the common writing ink should not be used for drawing purposes, . 138 11 India or China ink, ...... 138 " Carmine aids it to flow more freely, . . . .138 Iron, wrought and cast, the colors for indicating, . . . 141, 142 Jasmine, the yellow to represent the primary yellow, . .117 Keeping, definition of the term, . . . . . 119 Key or Tone, the prevailing hue of a landscape, etc., . . .119 Lakes, Lilac, see crimson. ...... Lead and Tin, the colors for indicating them, . ." - . 142 Leather, " " . ... 142 Light, shade and shadow, . . . . - . .120 " and shade, represented by white and black, . . 123 " Artificial its effects on colored surfaces, . . . 126 " " always inclines to yellow, . . . 126 " Absorption and reflection of .... . 127 Lights, in water color drawings, should be represented by the surface of the paper, ...... . 140 Manipulation of the colors, ..... 148 Materials should be of the best quality, ... . 146 Melody, the term defined, ...... 120 INDEX TO ESSAY ON COLOR. 161 PAGE. Melody and Harmony, applied to colors as in music, . . . 120 Melodizing colors, those which follow a certain natural scale, . 120 Mounting drawings, .... 145 Murrey, puce, etc., modifications of maroon, . , . 1^3 Nature, All the various tints and hues of produced by three colors, . 117 Natural scale of colors, ...... 125 Neutral blacks and grays, semi-neutral, etc., . . 132, 133, 134 Neutral grays, the gradations between white and black, . .123 Neutral tint, a mixed pigment for shading, etc., . . . 142 Obstacles to be removed from the path of the learner at once, . 145 Olive or blue hue, ....... 124, 132 Opacity and transparency, not absolute terms, . . .127 Orange, its properties and effects, . . . . . 129 Perspective views if fancifully colored, often convey a wrong impression of the building to be erected, . . . 141 Pencils, The qualities required in good drawing . . . 138 should always be cut with a keen knife, . . 138 Pigments often act chemically on each other, . , . 139 by different makers seldom correspond in hue, . . . 134 the fewer used to make a tint the better, . . . 140 Pink, composed of red and blue, ..... 133 Plans and sections should be colored to show the materials designed for the construction, . . . . 140 Positive colors, ....... 134 Primary colors, homogeneous, etc., .... 122, 124, 134 Prism of glass, for the refraction of light, . . . 121 Purple, its properties described, .... . 130 Quality of materials, . . . . . . 146 Refraction of light in the solar spectrum, . . . .121 Eefrangibility of the primary colors unequal, . . . 121 Reflections Effects of from colored surfaces, . . . . 12'6 Red, its properties and effects, . 129 Red Light tints of known as rose white, carnation, coquelicot, etc., . 133 Retreating colors, . . . . . .125, 134 Rose color, see crimson Rubble stone walls, how to color them in sections, . . . 141 Russet or red hue, * . ' . . . . 124, 132 Sage The blue to represent the primary blue, . . .117 Scarlet, 129, 133 Scales of the colors, ... ... 123, 124, 125 Secondary colors, ....'... 124, 134 Sections and plans, colored to show the construction, . . . 140 Shade, definition of the term, .... 119 Shading detail drawings by ruling parallel lines, . .136 Shadows on plans and sections to be at an angle of 45, . 140 Solar spectrum, . . . . . . . .121 Steel or iron, the colors for indicating them, . . . 142 Stone, " " " ... . 141 21 162 INDEX TO ESSAY ON COLOK. Tertiary colors, ....... Theory of color, .. Tint, the term defined, . . . . . Tone, " " Tracing cloth, for copying drawings, . Translucent substances transmit light but not vision, . , Transparent media, permit light and vision to pass freely, Transparency and opacity, only relative terms, . . Violet in the solar spectrum, . . . " composed of red and blue, . . . Vulcanized Rubber, colors to represent it, . Wafers, experiment with red and green wafers, . . Warm colors, red, orange, yellow, ..... Wash, to correct the greasiness of paper to be colored, . White the representative of light, its properties and effects, . White light may be resolved into the three primary colors, Woods, how to color them in sectional drawings, . . . Yellow, its properties and the effect it produces, . " its character entirely changed on coarse fabrics, . . " should not be used in large masses, . . . " Light tints of called primrose, straw color, cream color, etc., PAGE. 124, 134 . 117 119 . 119 146 . 127 127 . 127 121 . 133 142 . 120 125, 134 . 143 126 . 123 142 . 128 128 . 128 133 DIRECTIONS TO THE BINDER. Plate 54 to face the title page. " 1,2,3 to follow page tc " 4,5 " 6, 7 " 8,9 11 10, 11 " : 1-2, 13 " 14, 15 16, 17, 18, 19 " 20, 21 " 22, 23 " 24, 25 " 26, 27, 28, 29 " 11 30, 31, 32, 33 " 8 16 20 24 32 36 40 44 48 52 56 64 Plate 34, 35, 36, 37 to follow page " 38, 39 " 40, 41 " 42, 43 " 44, 45 " 46,47 48, 49 50 to face " 51 " 52 " 53 " 55 " 56 7,2 76 80 84 92 96 100 102 103 105 106 113 116 THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO SO CENTS ON THE FOURTH DAY AND TO $1.OO ON THE SEVENTH DAY OVERDUE. EPT. 44961 LD 21-100m-7,'40 (6936s) 53684 UNIVERSITY OF CALIFORNIA LIBRARY