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CARPENTRY. 
 
 VOL. I. 
 
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CARPENTRY: 
 
 
 BEING A COMPREHENSIVE GUIDE BOOK 
 
 CARPENTRY AND JOINERY; 
 
 ELEMENTARY RULES FOR THE DRAWING 
 OF ARCHITECTURE IN PERSPECTIVE 
 AND BY GEOMETRICAL RULE: 
 
 TREATING OF ROOFS, TRUSSED GIRDERS, FLOORS, 
 DOMES, STAIR-CASES AND HAND-RAILS, 
 SHOP-FRONTS, VERANDAHS, 
 WINDOW-FRAMES, SHUTTERS, &c. &c. ; 
 
 PUBLIC AND DOMESTIC BUILDINGS, 
 WITH PLANS, ELEVATIONS, SECTIONS, &c. &c. 
 
 VOL I. 
 
 EIGHTY-FOUR 
 ENGRAVINGS. 
 
 LONDON : 
 JOHN WEALE, 
 
 59 HIGH H 0 L B 0 R N. 
 
 WITH 
 
 ALSO, 
 
 AND 
 
 1848. 
 
LONDON : 
 
 PRINTED BY C. F. HODGSON, 1 GOUGH SQUARE, FLEET STREET. 
 
CONTENTS OF VOL. 1. 
 
 TEXT PAGES. 
 
 Preface 1 to 4 
 
 Division A. Practical Geometry, — Explanations of the Plates 1 to 30 5 to 42 
 
 Division A. Supplement^ — Practical Mathematics, Mensuration, &c 1 to 32 
 
 Division A. Practical Rules on Drawing, for the Operative Builder and Young 
 
 Student in Architecture 1 to 80 
 
 Division B. Practical — On Constructive Art ; Gerard^s College for Orphans . . 1 to 8 
 
 PLATES— PLATES. 
 
 Division A. Geometrical Plates 1 to 30 
 
 Division B. Practical Rules for Drawing, Geometrically and in Perspective . . 1 to 14 
 
 Division B. Shop Fronts 1 to 8 
 
 Division B. Gerard's College in Philadelphia ; Plans, Elevations, and Sections 1 to 5 
 
 Division B. Wooden Columns in King's College, London; Plans, Elevations, 
 
 and Details 1 to 2 
 
 Division B. Club Houses, — The Athenaeum and Arthur's ; Details, Plans, and 
 
 Elevations 1 to 4 
 
 Division B. Verandahs, Elevations, and Details 1 to 8 
 
 Division B. Hand-Rails 1 
 
 Division B. Windsor Castle : — 
 
 The Window-side of one of the Private Apartments in the 
 
 East Front ; Elevation. 
 Wainscot Window Frames and Sashes for ditto — Details. 
 Plan of one half of the Window side of ditto. 
 MuUions of Wainscot, Window Frames, and Sashes, in the 
 
 Brunswick and Octagon Towers. 
 Ditto in Private Apartments. 
 
 Plan of one half of the Oriel Windows in the State Drawing 
 
 Room. Ditto in Private Apartments. 
 The Gate at the Town Entrance to the Royal Mews 1 to 6 
 
iv 
 
 CONTENTS. 
 
 PLATES — PLATES. 
 
 Division B. Duke of Sutherland's : — 
 
 Head and Sill of the Windows in the Family Apartments, 
 Lilleshall. 
 
 Mullions of Window Frames of the Family Apartments at 
 
 ditto 1 to 2 
 
 Division B. Plans and Elevations of the Percy Arms, near Lloyd Square, 
 
 Pentonville 1 
 
 Division B. Roofs ; Exeter Hall, &c. . . . : 1 
 
 Division B. Modillions and Brackets for Shop Fronts 1 
 
 Division C. The Roof of the Nave of St. Mary, Weston-Zayland, Somersetshire 1 
 
 EIGHTY-FOUR PLATES. 
 
 The Work (Vol. IL) will be continued in Monthly Parts at 2s. 6d. each, as before, till 
 December 1848, making 2 Vols., containing near 200 Engravings, many of the highest interest in 
 Carpentry and Building ; together with the Text explanatory of all the subjects treated of and illus- 
 trated by the Plates, and Descriptive Contents of the whole Work. The present List of Contents to 
 be cancelled on the completion of the Work. 
 
 JOHN WEALE. 
 
 December 20th, 1847. 
 
PREFACE 
 
 TO THE ELEMENTARY PART OF THE WORK, 
 
 BEING 
 
 ^art 8. or Btbtsuin a. 
 — ♦ — 
 
 TO a book intended merely for the use of Practical Mechanics, much preface 
 is not necessary : — it is proper, however, to say, that whatever rules by previous 
 authors have on examination proved to be true and well explained, these have 
 been selected and adopted, with such alterations as a very close attention has 
 warranted for the more easily comprehending them, for their greater accuracy 
 or facility of application ; added to these, are many examples which are entirely 
 original, and such as will conduce very much to the accuracy of the work and 
 to the ease of the workman. 
 
 The arrangement of the subjects in this Work, is gradual and regular, such 
 as a student should pursue who wishes to attain a thorough knowledge of his 
 profession: and as it is Geometry that lays down all the first principles of 
 building, measures lines, angles, and solids, giving rules for describing the 
 various kinds of figures used in buildings ; therefore, as a necessary introduction 
 to the art treated of, such problems of Geometry as are absolutely requisite to 
 the w^ell understanding and putting in practice the necessary lines for Carpentry, 
 are now first laid down, and explained in the terms of workmen. These 
 problems duly considered, and their results well understood, the learner may 
 proceed to the practical part of the subject, in which Soffits claim particular 
 attention ; for, by a thorough knowledge of these, the student will be enabled 
 
 B 
 
2 
 
 PREFACE, 
 
 to lay down arches which shall stand exactly perpendicular over their plan, 
 whatever form the plan may be : on this depends the well executing of all groins, 
 arches, niches, &c., constructed in circular walls, or which stand upon irregular 
 bases ; wherefore the importance of rightly understanding these cannot be too 
 much insisted upon, their construction being so various and intricate, and their 
 uses so frequently required. The plates of cuneoidal or winding soffits are new, 
 and are constructed in a more simple and more accurate manner; yet this 
 method is only a nearer approximation to truth than the former one: the surface 
 of a cuneoid cannot be developed; that is, it cannot be extended on a plane: it 
 is therefore absurd to look for perfection on this subject. 
 
 The next subject which regularly presents itself is Groins ; for the construc- 
 tion of which there will be found many methods entirely new; and besides the 
 common figures, several are here shown which are difficult of execution, and not 
 to be found in any other author. A variety of methods are displayed for 
 constructing spherical niches, a form more frequently wanted than the elliptic, 
 which only has yet been explained. 
 
 Among the various methods for finding the Lines for Roofs, an entire new 
 one for finding the down and side bevels of purlines is given, so that they shall 
 exactly fit against the hip rafter ; and by the same method the jack rafter will 
 be made to fit. 
 
 Of Domes and Polygons, an entirely new method is shown, for finding 
 their covering, within the space of the board, thereby avoiding the tedious and 
 incommodious method of finding the lines on the dome itself, as has been always 
 practised heretofore ; also a method for finding the form of the boards near the 
 bottom, when a dome is to be covered horizontally. Of Dome-lights over stair- 
 cases, or in the centre of groins, a rule upon true principles is given, for finding 
 their proper curve against the wall, and the curve of the ribs : this has never 
 before been made public. 
 
 Having gone thus far in the Art of Carpentry, it is necessary to say, by 
 way of caution and guard to the ardent theorist, that there are some surfaces 
 which cannot be developed ; such as spherical or spheroidal domes, where their 
 coverings cannot be found by any other means than by supposing the curved 
 
PREFACE. 3 
 
 surface to become polygonal ; in which case such domes may be covered upon 
 true principles, as may be demonstrated. — Let it be supposed that a polygonal 
 dome is inscribed in a spherical one ; then, the greater the number of sides of the 
 polygonal dome, the nearer it will coincide with its circumscribing spherical one. 
 Again, suppose that this polygonal dome has an infinite number of sides; then, 
 its surface will exactly coincide with the spherical dome ; and therefore in any 
 thing which we shall have occasion to practise, this method will be sufficiently 
 near ; as for example, in a dome of one hundred sides, of a foot each, the ride 
 for finding such a covering will give the practice so very near, that the variation 
 from absolute truth could not be perceived. 
 
 Having gone through the constructive part of Carpentry, we next proceed 
 to examples showing the best forms of floors, partitions, trusses for roofs, truss 
 girders, domes, &c. which shall not only resist their own weight, but also any 
 adventitious load. 
 
 In that nice and elegant branch of the Building Art called Joinery, Stairs 
 and Hand-rails take the lead ; and notwithstanding the great importance of this 
 subject, it has been treated by all authors without the governing principle of 
 science. For Staircases, in general, correct methods are laid down in this 
 Work, founded on the most obvious principles ; and which, since the first 
 publication of the Carpenter's New Guide to the present time, it is satisfactory 
 to say, have been put in practice, and found to answer under every circumstance 
 and in every situation. 
 
 In this Work, a new and correct method for ascertaining the spring of 
 the plank has been introduced instead of the former, which though very nearly 
 was not exactly true, except when the plane of the plank was at right angles to 
 the chord plane. The construction of rails with butt joints is likewise shewn, a 
 method much stronger and more frequently practised than splice joints. This 
 occasioned the introduction of additional plates. 
 
 In this Work, besides the new plates, the whole have been carefully 
 examined, and every part of the explanation which did not appear sufficiently 
 clear has been better expressed, and the arrangement has been improved. 
 
 To conclude ; as I pretend not to infallibility, I hope to be judged with 
 
4 
 
 PREFACE. 
 
 candour, being always open to conviction, from a knowledge of the difficulty 
 and intricacy of science ; yet I hope that my labours may be of some use to 
 others in shortening the road, and smoothing the path through which, for many 
 years, I have been a persevering traveller for knowledge : 1 shall then be satisfied, 
 and not deem my time misspent if my labours tend to the public good. 
 
 P. NICHOLSON 
 
OF PRACTICAL GEOMETRY. 
 
 GEOMETRY is the science of extension and magnitude; it teaches 
 the construction of all right-lined and curvilineal figures, and is divided 
 into Theoretical and Practical. 
 
 The Theoretical part is founded upon self-evidence and reason ; it demon- 
 strates the construction of variously formed figures, and evinces the truth or 
 detects the falsehood on which they are made. This is the foundation of the 
 Practical part ; and, without a knowledge of the Theory, no invention to any 
 degree certain can be made in the Practice. 
 
 The uses of Geometry are not confined to Carpentry and Architecture, but, 
 in the various branches of the Mathematics, it opens and discovers to us their 
 secrets. It teaches us to contemplate truths, — to trace the chain of them, 
 subtile and almost imperceptible as it frequently is, — and to follow them to the 
 utmost extent. 
 
 Its uses are great and necessary in all the constructive sciences. The science 
 of Perspective is entirely dependent upon its principles. To enumerate its 
 many uses is beyond my power. Those who desire to become thoroughly 
 acquainted with Geometry, will do well to study attentively the Elements of 
 Euclid. 
 
 As my labours are not intended for the abstruse Mathematician, but for the 
 instruction of the Practical Carpenter, 1 shall omit all speculative demonstra- 
 tions, the sections of Cylinders and Globes excepted, (which are not to be found 
 in Euclid), and confine myself to the useful part of the science, viz. Practical 
 Geometry. 
 
 c 
 
6 
 
 PRACTICAL GEOMETRY. 
 
 PLATE I. 
 
 DEFINITIONS. 
 
 1. A POINT has position but not magnitude, 
 
 2. A line is length without breadth or thickness. 
 
 3. A superficies has length and breadth only. 
 
 4. A solid is a figure of three dimensions, having length, breadth, and thickness. 
 Hence surfaces are the extremities of solids, and lines the extremities of 
 
 surfaces, and points the extremities of lines. 
 
 5. Lines are either right, as A, curved, as C, or mixed of these two, as B. 
 
 6. A right or straight line lies all in the same direction between its extre- 
 mities, and is the shortest distance between two points, as A. 
 
 7. A curve continually changes its directions between its extreme points, as C. 
 
 8. Lines are either parallel, oblique, perpendicular, or tangental. 
 
 9. Parallel lines are always at the same distance, and will never meet, though 
 ever so far produced, as D and E. 
 
 10. Oblique right lines change their distance, and would meet if produced, 
 as F. 
 
 11. One line is perpendicular to another when it inclines no more to one side 
 than another, as G. 
 
 12. A straight line is a tangent to a curve when it is produced and touches it 
 without cutting, as H. 
 
 13. An angle is the inclination of two lines towards one another in the same 
 plane, meeting in a point, as L 
 
 14. Angles are either right, acute, or obtuse, as K, L. 
 
 15. A right angle is that which is made by one straight line perpendicular to 
 another, or when the angles on each side are equal, as G. 
 
 16. An acute angle is less than a right angle, as K. 
 
 17. An obtuse angle is greater than a right angle, as L. 
 
 18. A superficies is either plane or curved. 
 
 19. A plane, or plane surface, is that to which a right line will every way 
 coincide ; but if not, it is curved. 
 
 20. Plane figures are bounded either by right lines or curves. 
 
 21. A solid is said to be cut by a plane when it is cut through in any par- 
 ticular place by that plane, and the surface through which the plane has passed 
 is called the section of the solid. 
 
PRACTICAL GEOMETRY. 
 
 7 
 
 22. Plane figures, bounded by right lines, have names according to the number 
 of their sides, or of their angles ; for they have as many sides as angles — the least 
 number is three. 
 
 23. An equilateral triangle is that whose three sides are equal, as M. 
 
 24. An isosceles triangle has only two sides equal, as N. 
 
 25. A scalene triangle has all sides unequal, as O. 
 
 26. A right-angled triangle has one right angle, as P. 
 
 27. Other triangles are oblique angled, and are either obtuse or acute. 
 
 28. An acute-angled triangle has all its angles acute, as M or N. 
 
 29. An obtuse-angled triangle has one obtuse angle, as O. 
 
 30. A figure of four sides and angles is called a quadrangle, or quadrilateral, 
 as Q, R, S, T, U, and V. 
 
 31. A parallelogram is a quadrilateral, which has both pairs of its opposite 
 sides parallel, as Q, R, U, and V ; and takes the following particular names. 
 
 32. A rectangle is a parallelogram having all its angles right angles, as Q and R. 
 
 33. A square is an equilateral rectangle, having all its sides equal, and all its 
 angles right angles, as Q. 
 
 34. A rhombus is an equilateral parallelogram, whose angles are oblique, as U. 
 
 35. A rhomboid is an oblique-angled parallelogram, as V. 
 
 36. A trapezium is a quadrilateral which has neither pair of its sides parallel, 
 as T. 
 
 37. A trapezoid has only one pair of its opposite sides parallel, as S. 
 
 38. Plane figures, having more than four sides, are in general called polygons, 
 and receive other particular names according to the number of their sides or 
 angles. 
 
 39. A pentagon is a polygon of five sides, a hexagon has six sides, a heptagon 
 seven, an octagon eight, a nonagon nine, a decagon ten, an undecagon eleven, 
 and a dodecagon twelve sides. 
 
 40. A regular polygon has all its sides and its interior angles equal ; and if 
 they are not equal, the polygon is irregular. 
 
 41. An equilateral triangle is also a regular figure of three sides, and a square 
 is one of four ; the former being called a trigon, and the latter a tetragon. 
 
 42. A circle is a plane figure bounded by a line called the circumference, 
 which is every where equidistant from a certain point within, called its centre. 
 
 43. The radius of a circle is a right line drawn from the centre to the cir- 
 cumference, as a J at W. 
 
 c 2 
 
8 
 
 PRACTICAL GEOMETRY. 
 
 44. A diameter of a circle is a right line drawn through the centre, termi- 
 nating on both sides of the circumference, as c c? at W. 
 
 45. An arch of a circle is any part of the circumference. 
 
 46. A chord is a right line joining the extremities of an arch, as a Z> at X. 
 
 47. A segment is any part of a circle bounded by an arch and its chord, as X. 
 
 48. A semicircle is half the circle, or a segment cut off by the diameter, as Y. 
 
 49. A sector is any part of a circle bounded by an arch and two radii, drawn 
 to its extremities, as Z. 
 
 50. A quadrant, or quarter of a circle, is a sector having a quarter of the 
 circumference for its arch, and the two radii are perpendicular to each other, 
 as A 1. 
 
 51. The height or altitude of any figure, is a perpendicular let fall from an 
 angle, or its vertex, to the opposite side, called the base, as a 5 at B 2. 
 
 52. When an angle is denoted by three letters, the middle one is the place of 
 the angle, and the other two denote the sides containing that angle ; thus, let 
 n b c be the angle at C 3, 5 is the angular point, and a b and b c are the two sides 
 containing that angle. 
 
 53. The measure of any right-lined angle is an arch of any circle contained 
 between the two lines which form the angle, the angular point being in the 
 centre, as D 4. 
 
 PLATE II. 
 
 PROBLEMS. 
 
 Figure 1. — To draw a Perpendicular to a given Point in a Line. 
 
 A B \^ 2i line, and c a given point ; take a and b, two equal distances on each 
 side of c, and with the foot of the compasses in a and b make an intersection d, 
 and draw d c, which is the perpendicular. 
 
 Fig. 2. — To make a Perpendicular with a Ten Foot Rod. 
 Let a 6 be six feet, then take eight feet, and from b make an arch at c, and 
 from the point a with the distance of ten feet make a cross at c, then draw c b, 
 which is the perpendicular. 
 
 Fig. 3 — To let fall a Perpendicular from a given Point to a Line. 
 From the given point c make an arch to cross the line in a and b, and from a 
 and b make an intersection at d, and draw c d the perpendicular. 
 
PRACTICAL GEOMETRY. 
 
 9 
 
 Fig. 4. — To draw a Perpendicular upon the End of a Line. 
 
 Take any point d at pleasure above the line, and with the distance dh make 
 an arch a b c, and draw a line a d which produce to cut it at c, and draw c h 
 which is the perpendicular. 
 
 Fig. 5. — To divide a JLine in two equal Parts by a Perpendicular. 
 
 From the extreme points a and b describe two arches to intersect at c and d, 
 and draw c d, which divides the line a 6 in two equal parts. 
 
 Fig. 6. — To divide any given Angle into two equal Angles. 
 
 Take two equal distances a b and a c on each side of the angular point a, and 
 with the same opening of the compass or any other of sufficient extent, place the 
 foot in b and c, make an intersection at d, and draw d a, which will divide the 
 angle into two equal parts. 
 
 Figs. 7, 8. — An Angle being given, to make another equal to it.ftom a 
 
 o-iven Point in a rio;ht Line. 
 
 Let b a c he the angle given, and c c? a right line, c the given point ; on a make 
 an arch be with any radius, and on c with the same radius describe an arch d e, 
 take the chord of be, set it from d to e, and draw ec, then the angle ecd will be 
 equal to cab. 
 
 PLATE in. 
 
 Fig. 1. — Upon a Right Line to make an equilateral Triangle. 
 
 Take a b the given side, and from a and b make an intersection at c, and draw 
 ca and c b. 
 
 Fig. 2. — Upon a right Line to make a Square. 
 With the given side a b, and from the points a and b, describe two arches to 
 intersect at e, divide b e into two equal parts at /, make e d and e c each equal to 
 e f, draw ad, d c, and c b. 
 
 Figs. 3, 4, 5, 6. — The Side of any Polygon being given, to describe the Polygon 
 
 to any Number of Sides whatever. 
 On one extreme of the given side make a semicircle of any radius, but it 
 
10 
 
 PRACTICAL GEOMETRY. 
 
 will be most convenient to make it equal to the side of the polygon ; then divide 
 the semicircle into the same number of equal parts as you would have sides in 
 the polygon, and draw lines from the centre through the divisions in the semi- 
 circle, always omitting the two last, and run the given side round each way upon 
 these lines, join each side, and it will be completed. 
 
 Example in a Pentagon. Fig. 4. 
 
 Let ah he the given side, and continue it out to c; on a, as the centre with 
 the radius a b, describe a semicircle, divide it into five equal parts ; through 
 2,3, 4, draw a 2, ad, ae; make be equal to ab, 2 equal to 2a or join Id, 
 de, and eb. In the same manner may any other polygon be described. 
 
 N.B. This depends upon the equality of the angles upon equal arcs. See Fig. 3. 
 
 Fig. 7. — Through a given Point a to draw a Tangent to a given Circle. 
 
 Draw ao to the centre, then through a draw be perpendicular to ao, it will 
 be the tangent. 
 
 Fig. 8. A. tangent Line being given, to find the Point where it touches the Circle. 
 
 From any point a in the tangent line b a, draw a line to the centre o, and 
 divide a o into two equal parts at m, and with a radius m a, or mo, describe an 
 arch, cutting the given circle in n, which is the point required. 
 
 Fig. 9 Two right Lines being given, to find a mean Proportion. 
 
 Join ab and be in one straight line, divide a c into two equal parts at the point 
 o, with the radius oa or oc describe a semicircle, and erect the perpendicular bd, 
 then is ab, to b d, as b d, is to b c. 
 
 Fig. 10. — Through any three Points to describe the Circumference of a Circle. 
 From the middle point b draw the chords ba and be to the two other points 
 a and c, divide the chords ab and be into two equal parts by perpendiculars 
 meeting at o, which will be the centre. 
 
 Fig. 11. — To find the length of any Arc ABC of a Circle. 
 
 Draw the chord A C and produce it towards E; bisect the arc A E C in H, 
 and make AD equal to twice AE; divide C L> into three equal parts, and set 
 one out to E; then ^ ^ is the length of the arc. 
 
PRACTICAL GEOMETRY. 
 
 11 
 
 PLATE IV. 
 
 Fig. 1. — Three straight Lines being given, to form a Triangle. 
 
 Take one of the given lines a b, and make it the base of the triangle ; take the 
 other line ac, and from a describe an arch at c ; then take the third line be, and 
 from b describe another arch crossing the former at c, and join ac and be. 
 
 Note. That any two lines mnst be greater than a third. 
 
 Figs. 2, 3. — To make a Quadrangle equal to a given Quadrangle. 
 Divide the given qnadrangle.^o-. 2, in two triangles; make the triangle e/g 
 equal to a be, and eg h equal to a cd, and it is done. 
 
 Figs. 4, 5.^ — Ani/ irregular Polygon being given, to make another of the same 
 
 Dimensions. 
 
 Divide the given polygon, 4, into triangles, and in fig. 5 make triangles 
 in the same position, respectively equal to those in fig. 4 ; then will the irregular 
 polygon fg h i k be equal and similar to abode. 
 
 Fig. 6. — To make a Rectangle equal to a given Triangle. 
 Draw a perpendicular cd, divide it into two equal parts at e, through e draw 
 fg, parallel to the base ab; draw af bg, perpendicular; then will the rectangle 
 abgfhe equal to the triangle abc. 
 
 Fig. 7. — To make a Square equal to a given Rectangle. 
 Let abc d he the given rectangle ; continue one of its sides as ab out to e, 
 make be equal to the other side b c, divide a e in two equal parts at i, with the 
 radius i e ov ia make a semicircle af e, and draw bf perpendicular to ab; make 
 the square b f g h, vshich is equal to the parallelogram abed. 
 
 Fig. 8. — To make a Square equal to two given Squares. 
 
 Make the perpendicular sides ac and ab oi the right-angled triangle cab 
 equal to the sides of the given squares A and B, draw the hypothenuse c b, which 
 is the side of the square C, equal to the two squares A and B. 
 
 Fig. 9. — To make a Square equal to three given Squares. 
 Let A, B, C be the three squares ; make a b equal to the side oi B, ac equal 
 
12 
 
 PRACTICAL GEOMETRY. 
 
 to the side of A, at right angles to ab; join be, then make a d equal to be, make 
 ae equal to the side of C, join de, which will be the side of the square D equal 
 to the squares A, B, C. 
 
 PLATE V. 
 
 Fig. 1. — To draw a Segment of a Cirele to any length and height. 
 
 a ^ is the length, i h the height ; divide the length a h into two parts by a 
 perpendicular gc: divide ahhy the same method, then their meeting at g will 
 be the centre ; fix the foot of the compasses in g, extend the other leg to h, 
 make the arch a h h, which is the segment. 
 
 Fig. 2. — To draw a Segment by Rods to any Length and Height. 
 
 Make two rods c e and c f to form an angle e of, so that each may be equal 
 to a b, the opening ; place the angle c to the height, and the edges to a and b, 
 put a piece a b across them to keep them tight, then move your lath round the 
 points a, b, and the point c will describe the segment required. 
 
 Fig. 3. — To describe a Segment of a Circle at twice, upon true Principles, by a 
 
 flat Triangle. 
 
 Let the extent of the segment be a b, its height c d, from the extreme b 
 to the top d draw b d, through the point d draw e d parallel to the base a b, 
 equal in length to d b, stick a nail or pin m a, and another in d, describe one half, 
 as you see at G ; then move the nail, or pin, out of a, stick it in the point b, 
 and describe the other half 
 
 Fig. 4. — The transverse Axis a b and conjugate g c of an Ellipsis being 
 given, to draw its Representation. 
 
 Draw a d parallel and equal to 9i c, bisect it in e; draw e c and d y cutting 
 each other at m, join m c, bisect it by a perpendicular meeting c g, produced at 
 h; draw h d, cutting 6 « at A-, and make n i equal to n k; nl equal to n h; 
 through the points i, I, k, h, draw the lines, h i, k I, and i I, h k, then describe the 
 four sectors by help of the centres, i, I, k, h, and it will be the representation required. 
 
 Fig. 5. — To describe an Ellipsis by Ordinates. 
 Make a semicircle on the length a b, divide it into any number of equal parts, 
 as 16, on the end at a make a 8 perpendicular, equal to half the width, and draw 
 
PRACTICAL GEOMETRY. 
 
 13 
 
 the ordinates through all the points in the semicircle, draw the line 8 1 to the 
 centre, then « 1 8 will be a scale to set off the ordinates ; take 1 1 from the scale, 
 and set it from 1 to 1 in your oval both ways at each end ; then take 1 2 in your 
 scale, and set it to 1 2 in the oval, and find all the other points in the same 
 manner ; a curve being traced through these points will be the true ellipsis. 
 
 PLATE VI. 
 
 Fig. 1. — To make an Ellipsis with a String. 
 
 Take the half a g of the longest diameter a b, and with that distance fix the 
 foot of the compass in c, cross a b at ef, in which stick two nails or pins, then 
 lay a string round e/ c, fix a pencil at c, and move your hand round, keeping the 
 string tight, the pencil will describe the ellipsis. 
 
 Fig. 2. — To describe an Ellipsis by a Trammel. 
 
 1 2 3 is a trammel rod : at 1 is a nut with a hole to hold a pencil ; at 2 and 3 
 are two other sliding nuts; make the distance of 2 from 1, half the shortest 
 diameter of the ellipsis, and from the nut 1 to 3 equal to half the longest, the 
 points 2 and 3 being put into the grooves of the same size, move your pencil 
 round at 1, and it will describe the true curve of an ellipsis. 
 
 Fig. 3. — An Ellipsis being given, to find the Centre and two Axes. 
 
 Draw any two parallel lines a h and c at pleasure, divide each of them in 
 two equal parts at the points e and f, and through e f draw the line k I, divide k I 
 into two equal parts at the point g-, place the foot of the compass in g, with the 
 other foot make two crosses li and i, on the circumference ; draw a line h i, through 
 ff draw m n parallel to h i, also through g draw a p at right angles to m n; then 
 o p is the transverse axis, and m n the conjugate, and g the centre of an ellipsis. 
 
 Fig. 4. — To proportionate one Ellipsis ivithin another ; that is, to give it the same 
 Length in proportion to its Width, that the Length of the other has to its Width. 
 
 Let the given ellipsis he a d b c, make the parallelogram e hf <j to touch the 
 sides and ends of the ellipsis, draw the diagonals e y' and </ /* of the rectangle, let 
 r g be the width of the lesser ellipsis given, through the point q or r draw / o 
 or m n, parallel to the transverse axis at the points m and w where it cuts the- 
 diagonal, draw m Z and n o parallel to the conjugate axis, will also show its length. 
 
 D 
 
14 PRACTICAL GEOMETRY. 
 
 Fig. 5. — To describe an Ellipsis about a Parallelogram, to have the same Length in 
 Proportion to its Width, that the Length of the Parallelogram has to its Width. 
 
 Let the given parallelogram he a h c d ; let the diagonals a c and b dhe 
 drawn from the centre i ; draw the quarter of a circle, 2 1 k, to half the width 
 of the rectangle ; divide the quadrant into two equal parts at 1 ; through the 
 point 1, draw the line / 3 parallel to the transverse axis, to cut the diagonal b d 
 in the point 3 ; then draw the lines 3 2 and 3 4 : again, draw f d parallel to 2 3, 
 then if will be half the width ; and d e parallel to 3 4, and i e will be half the 
 length of the ellipsis : make i h equal to / e, and i g equal to if, which will give 
 the four points through which the ellipsis must pass ; describe the curve, and 
 the thing will be done. 
 
 Fig. 6. — To divide a Line in the same Proportion as another is divided. 
 a is a line given already divided, and </ e is a line to be divided in the same 
 proportion. Make any angle at d; join a e, draw b f and c g parallel to a e; 
 then e is divided at f and g in the ratio as a <Z at 6 and c. 
 
 Fig. 7. — To do the same by an equilateral Triangle. 
 
 a b is the given line divided : from c take two equal distances c d, c d, and by 
 drawing lines from the several points in abtoc cutting dd,dd will be divided as a b. 
 
 Fig. 8. — To make an Octagon the nearest Way from a Square. 
 
 Draw the diagonals of the square to cross at e, fix the foot of your compass 
 in c, and take the distance c e and make an arch f e g ; then set your gauge to 
 dfovbg, which will gauge off each angle. 
 
 PLATE VIL 
 
 CONIC SECTIONS BY INTERSECTING LINES. 
 
 DEFINITIONS. 
 
 1 . A cone is a solid having a circiilar base, from which the sides continually diminish in straight lines 
 to a point in which they all terminate, and this point is called the vertex. 
 
 2. Opposite cone is another cone joining the vertex of the other cone, with its sides every where in 
 the same straight line passing through the vertex as a common point. 
 
 3. A right hne joining the vertex and the centre of the basis is called the axis. 
 
PRACTICAL GEOMETRY. 
 
 15 
 
 4. If a cone be cut by a plane passing entirely through its curved surface, but not parallel to the base 
 nor to the axis, nor to a plane touching the side of the cone, the section is an ellipse, excepting in 
 one position which is a circle. 
 
 5. If a cone be cut by a plane parallel to its sides, the section is a parabola. 
 
 6. If the cone be cut by a plane passing through the opposite cone, the figure will be an hyperbola. 
 
 To describe the Ellipsis from the Cone. 
 
 Figure A. — Let B be half the circle of the base of the cone, n the vertex at 
 the top ; then n a and n d are two sides ; let the cone be cut by a plane passing 
 through g h; bisect g h at the point k, and through k draw r q, parallel to the 
 base a d; also bisect r q in m, describe the semicircle r p q, draw kp at right 
 angles to q r; and^ h is the length of the ellipsis, and hk half its width ; from 
 which the figure may be described at C, as explained in the next plate. 
 
 To describe the Parabola from the Cone. 
 Figure A. — Let i e be the axis of the parabola, parallel to the other side n d 
 of the cone, and through e draw e c at right angles to the base ; then will e c be 
 half the width of the parabola, and e i its height ; then the Jigure will be described, 
 as at D, by intersecting lines upon each ordinate, up to the crown, from the equal 
 divisions on each side. 
 
 To describe the Hyperbola from the Cone. 
 
 Figure A. — Let the axis of the hyperbola be i f cut by a plane passing- 
 through / and i, till it cut the opposite cone at draw fh at right angles to a d, 
 then is f i the height of an hyperbola, and f b half the width of the base, and i I 
 its transverse axis ; then make f i^i E equal to / i in figure A, make i I in E 
 equal to i I m. figure A, b b in E equal to twice f b in figure A ; let the base b b in 
 E be divided into ten equal parts, as at 0, 1, 2, 3, 4, 5, that is, into five equal 
 parts on each side from the centre, and draw lines to the point I through these 
 points ; likewise divide the height into five each way, and draw lines to the vertex 
 at i; this will show the points through which the curve must pass. 
 
 PLATE VIIL 
 
 To draw any Semi-ellipsis tipon the transverse or conjugate Axis, or even a 
 Semicircle itself, by a new Method of intersecting Lines. 
 
 Figures A and B. — Let the given axis be a b, and let it be divided into any 
 
16 
 
 PRACTICAL GEOMETRY. 
 
 number of parts, as 10 ; also let the height be divided into half the number of 
 parts ; make e d equal e c, that is, to the height of the arch ; then, from the point 
 r/ draw lines through the equal divisions of the axis « 6; likewise, through the 
 points 1, 2, 3, 4, 5, in the height a f, draw lines tending to the vertex at c, which 
 will intersect at the points h, i, k, I ; and lines being drawn through the divisions 
 of 6 ^ to c, at the crown, in the same manner, will give the points n, o, p, q ; a 
 curve being traced through these points, will form the true curve of an ellipsis. 
 
 The semicircle, ^<7Mre C, is drawn in the same manner, by making « y equal to 
 one half of a b. 
 
 To draw the true Segment of a Circle, hy the Method of intersecting Lines. 
 
 Figure D. — Let a b he the length of the segment, and o c its height, and draw 
 the chord b c for one half of the segment, and draw b m at right angles lobe ; and 
 from the point o divide a b each way, into five equal parts ; also from c, divide c m, 
 and c n, each into five equal parts ; and draw 1 1, 2 2, 3 3, 4 4, 5 5, on each side, 
 through the divisions 1, 2, 3, 4, 5 on a s, and 1, 2, 3, 4, 5 on b r ; draw lines to c, 
 which will intersect the other lines at the points d, e,f, g, and h, i, k, I: the curve 
 being traced, the thing is done. 
 
 To draw a flat Segment of a Circle nearly true. 
 
 Figure E. — Divide the length of the segment into equal parts each way, 
 from the centre d, as before, and draw the lines 1 1, 2 2, 3 3, 4 4, 5 5, all at right 
 angles, to the length a b ; lines being drawn to the crown at c, from the divisions 
 at each end, will show the points which the segment must pass through ; the 
 curve being traced, the thing is done. 
 
 Remark. — Although this last m.ethod is not the true segment of a circle, but 
 a parabolic curve, yet it will be found useful in practice, in tracing any segment 
 whose height is not more than one tenth part of its length : if the centre of the 
 segment is found, and drawn with a compass, the difference will hardly be visible, 
 and the flatter the segment, this difference will become the more imperceptible ; 
 but if the height exceeds one tenth of its length, the difference will be visible ; 
 for then the arch will be quicker at the vertex, and get flatter and flatter towards 
 each extreme. 
 
 In the same manner may all kinds of rampant ellipses be described, or any seg- 
 ment of them, as at Fand G, also a rampant parabola in the same manner as at H. 
 
PRACTICAL GEOMETRY. 
 
 17 
 
 PLATE IX. 
 THE SECTIONS OF A CYLINDER. 
 
 DEFINITIONS. 
 
 A cylinder is a figure generated by the revolution of a right-angled parallelogram about one of its 
 sides; consequently the ends of the cylinder are equal circles, and the line passing through the 
 centre of the cylinder, is called the axis. 
 
 The section of a cylinder, cut by any plane, is an ellipsis, or circle, or rectangle, proved by the writers 
 on Conic Sections. 
 
 To find the Section of a Semi- cylinder, by Ordinates, when it is cut at right Angles 
 to the Plane, passing through its Axis, in the Direction a b. Fig. 1. 
 
 Let the circle of the base be divided into equal parts at A, and drawn pa- 
 rallel to the axis of the cylinder, or to the line a. b, at the points 0, 1,2, 3, 4, 5, 
 &c. and from these points draw lines at right angles to a b ; then B being 
 pricked from A, as the figures direct, B will be the section of the cylinder. 
 
 DEMONSTRATION. 
 
 Conceive the semicircle A at the base to be turned at right angles to the plane, also the semi- 
 ellipsis B at right angles to the same plane ; then will the ordinates of B be parallel and perpen- 
 dicular over the ordinates of A, and every corresponding point in the circumference of B will fall 
 perpendicular to the same corresponding points in A : therefore B is the true section of the cylinder, 
 cut in this position. 
 
 T'o cut a Cylinder in the direction V, W, upo7i a Plane, passing through its Axis, 
 to make an acute Angle with that Plane. Fig. 2. 
 
 Let C, at No. 1, be the given angle, which the section at B is to make with the 
 plane of the cylinder; take a b m figure 2, that is, the radius of the base, and set 
 it from b c, at No. 1, perpendicular to i b ; draw c c parallel to i b, also from c 
 draw c e perpendicular to i b ; then take the distance c i, set it from i to f, in 
 figure 2, at B ; likewise take i e from No. 1, and set it from i to e in jigure 2, at 
 B ; draw e d parallel to m n, to cut the rake in d, and join d f ; then is d f the 
 bevel of the first ordinate of the section B. And draw the lines e c and d a 
 parallel to the axis; join a c at A; then will a c be the bevel of the first ordinate 
 of the base. Draw all the other ordinates of A parallel to a c, and at the points 
 1, 2, 3, 4, &c. in m n, draw lines parallel to the axis of the cylinder, to cut the 
 
18 
 
 PRACTICAL GEOMETRY. 
 
 raking line F TFat 1, 2, 3, 4, &c. From these points let lines be drawn parallel 
 to d f; then the ordinates of B, being pricked from the same corresponding 
 ordinates of the base at A, will give the section of the cylinder. 
 JVote. — The point / will fall beyond the sweep at the section B. 
 
 DEMONSTRATION. 
 
 Let the plane B be conceived to be turned round the line V W to make an angle at the point i, 
 with the plane n m V W equal to the angle e i c, No. 1 ; and conceive a straight hne drawn from e 
 perpendicular to the plane n m V W, the line thus supposed to be di'awn will be parallel to the plane 
 A of the base, and the triangle formed by i f, i c, and the perpendicular from e, will be equal and 
 similar to the triangle c i e, No. 1 : then because d e and the perpendicular are both parallel to the 
 base, the line that joins the points e and / will also be parallel to the base ; and because e d is equal 
 to a 6, the triangle e d f will be equal and similar to the triangle 6 « c in the plane of the base A : 
 and because 6 c and the perpendicular di'awn from e are both in a plane parallel to the axis, the plane 
 passing through d f and a c will also be parallel to the axis : but t? / is also in the plane of the section, 
 for the point d is in the intersection of V W, and the point / will be in the perpendicular drawn from 
 e, therefore if a series of planes be conceived to be di'awn through the ordinates of the base parallel to 
 the plane passing through a c and d f, the intersection of these planes on the plane of section B will 
 be parallel to the ordinates of B, and every two coiTesponding lines will be in a plane parallel to the 
 axis, and therefore as the lines formed by the intersections of the series of planes in the section B, are 
 equal to those in the base A, the extremities are in the section of the cylinder. 
 
 To cut a Segment of a Cylinder, in the Direction a b, to make an obtuse Angle 
 ivitli the Plane of the Segment. Fig. 3. 
 
 Let No. 1 be the angle given, which the section B is to make with the plane 
 of the segment ; from /in No. 1, draw f g at right angles to f c, and g e also per- 
 pendicular, to make the right-angled triangle e gf And m figure 3, at B, draw 
 y f at right angles to a h, and make y e equal to g- e at No. 1. Also, make yf at 
 B equal to y f at No. 1. Draw e d at B, parallel to 31 N, and at the point d, where 
 it intersects the line a b, join d f ; then df '\^ one of the ordinates. From e and 
 d, draw the two parallel lines e c and d 3, join c 3 ; then c 3 will also be an 
 ordinate of the base. Draw parallel lines at discretion to c 3, for the other 
 ordinates of the base ; and from their intersection upon tn n draw lines parallel 
 upon the cylinder, to cut « 6 in 1, 2, 3, 4, &c. and from these points draw parallel 
 lines to d f which are the ordinates of B; these, being pricked from the base 
 as the figures direct, will give the points through which the curve must pass, 
 which being traced, will be the true section of the segment of the cylinder. 
 
PRACTICAL GEOMETRY. 
 
 19 
 
 DEMONSTRATION. 
 
 Is the same as the preceding demonstration. 
 
 That the reader may perceive this more clearly, the best way is to di'aw those lines on pasteboard, 
 the section and the end being made to turn round, in their proper position, then the demonstration 
 will be clearly seen. 
 
 Figure 4 is to be laid down and demonstrated in the same manner as Figure 2. 
 
 Remark. — Upon these figiu-es depend the whole principles of hand-rails for stairs. The reader 
 ought to understand how to form the section of a cylinder, in any case whatever ; for the face or 
 raking mould of a hand-rail is nothing but the double section of a cylinder, as in figure 4, at B, where 
 the double circle upon the base A represents the plan of a rail, and the bevel at No. 1, figure 4, re- 
 presents the spring of the plank, and a b the pitch of the rail : therefore, it is very necessary that the 
 reader should have a knowledge of these figm-es and their demonstrations ; and not be satisfied with 
 only doing it, but read these demonstrations, and consider them with attention ; then he will be able 
 to see the reason why every line is di'a\\Ti in the manner it is. 
 
 PLATE X. 
 
 THE SECTIONS OF A GLOBE, OR ANY OTHER FIGURE STANDING UPON A 
 CIRCULAR BASE; ALSO, THE SECTION OF ANY FIGURE STANDING ON AN 
 IRREGULAR BASE. 
 
 Definition. — A globe is a figure generated by the revolution of a semicircle round its diameter, which 
 
 becomes the axis of the globe. 
 
 AXIOMS ; OR, SELF-EVIDENT TRUTHS. 
 
 1.9/. From this definition it appears, that every two sections passing through the centre, are equal to 
 each other. 
 
 2nd. Every section of a globe, cut by a plane, is a circle ; for the generating circle may be made to 
 revolve round any line, as an axis ; and therefore every point in it will generate a circle, whose 
 diameter must be twice the radius of that circle, distant from the axis of the globe. 
 
 Zrd. If a semi-globe is cut by a plane at right angles to the plane of its base, the section will be a 
 semicircle. 
 
 To find the Section of a Semi-globe cut by a Plane at right Angles to the Plane of 
 
 its Base. Fig. 1. 
 
 It appears from the last axiom, that there is no tracing required : for, let the 
 section be cut across a h, figure 1 ; divide a b m two equal parts at the point c; 
 
•20 
 
 PRACTICAL GEOMETKY. 
 
 and on c, as a centre with the radius c a or c b, describe the semicircle A, which 
 is the true section required. 
 
 The same by Ordinates. Figs. 1 and 2. 
 
 Draw any line d e through its centre, and let « & be the place of the section 
 upon the base, as before ; place the foot of your compass in the centre of the 
 globe at /, and, with a radius f c, draw an arch from c to ff, in the diameter d e ; 
 the foot of your compass remaining still in f, draw the concentric dotted circles 
 from c b to f e, and at the intersecting points 1, 2, 3, 4, 5, in f e, and likewise in 
 c b, erect perpendiculars to those lines ; then A being pricked from C, as the 
 figures direct, will give the points through which this semicircle must pass, 
 
 DEMONSTRATION. 
 
 Conceive the semicircle C to stand at right angles upon d e, also the section A to be at right 
 angles to a b ; now it is evident, if 1 is the height of the globe over the point g in the base, c 1, 
 which is equal to ff 1, must also be the height of the section, because the points c and g stand at an 
 equal distance from the centre ; and therefore the point 1 over c, is in the surface of the globe. In 
 the same manner it may be proved, that any other points carried round by the dotted lines are in the 
 same surface : but the section that stands upon a h, in A, is a semicircle ; and consequently the 
 method of tracing is also a semicircle. 
 
 Observation, — Hence appears the erroneous principle of tracing used by a late writer on this 
 subject, as you may see at figure 2, where A is the section of a globe, and the bracket at D is the 
 section across the diameter. A is truly traced from D, because the ordinates are carried round in 
 circles ; but by his method of tracing, as you see at C, upon the other side, the point of the bracket 
 C falls within the sweep of the circle, by reason of the ordinates of C being carried straight through 
 between the two bases, which I have proved to be false. And this he has apphed in bracketing up 
 the angles in the square well-hole of a staircase, to the circular curb of a sky-light, which, if truly 
 done, is nothing else but upon the same principle as the sections of a globe. 
 
 Figure 3 is done upon the same principle as Jigure 1. J is the section traced 
 from C, and wants no other demonstration than what has been given in Jigure 1. 
 
 Figure 4 is an ogee section, standing upon a circular base across the dia- 
 meter; and A is the section traced from it, upon the same principles as Jigure 1. 
 
 From these examples it is clear that this method of tracing does not depend 
 on the form of the top, but entirely upon the base. These figures are supposed 
 to be generated round an axis ; and, as every circle is carried round at an equal 
 distance from the axis, the perpendicular height of the figure, upon any circle, 
 
PRACTICAL GEOMETRY. 21 
 
 must be the same height in every point throughout the circle; which proves itself 
 to be the only method for any thing of this kind. 
 
 A Semi-globe being cut by a cylindrical Surface perpendicular to the Plane of its 
 Base, to find the Form of a Veneer that will betid round it. — Fig. 5. 
 Let d ehe drawn through the centre f ; and place the foot of your compass 
 in /, the centre; and draw the points b, 1,2, 3, 4, 5, which are equally divided from 
 the centre at b in the circular surface, draw the concentric dotted lines round to 
 the diameter d e, at 0, I, 2, 3, 4, and at these points raise the perpendiculars 
 0 0, 1 1, 2 2, 3 3, 4 4. Take the stretch-out round 6 1 2 3 4 5, which is one-half; 
 and lay it upon the base of No. 1 each way, from 0, 1, 2, 3, 4, &c., and INo. 1 
 being pricked from A, figure 5, as the figures direct, will give the points through 
 which the curve must pass for the veneer. 
 
 DEMONSTRATION. 
 
 For, since the section standing upon c? e is a semicircle, which is equal to the semicircle upon the 
 base ; and as the points 1^ 2, 3, 4, in the circular surface, stand at the same distance from the centre 
 /, as 0, 1, 2, 3, 4, in d e; now if the point o at No. 1 is made to coincide with the point b m figure 5, 
 then the height o o, standing over the point b, will be equal to the height o o at A ; but these points 
 are at an equal distance from the centre, therefore the top of each ordinate will be in the surface of 
 the globe. In the same manner every other point may be proved, when bent round and elevated, to 
 be of the same height, and at an equal distance from the centre with those of A ; and therefore No. 1 
 is the true form of the veneer. 
 
 To find the Ribs of a Gothic Niche, Fig. 6. being the Plan, and No. 1. 
 
 the Front Elevation. 
 
 Take the length of each base upon the plan, and make them the bases of 
 No. 2, No. 3, No. 4, and No. 5, respectively ; divide each base into six equal parts ; 
 also" divide the half of No. 1. into six parts, and draw the ordinates from the 
 equal divisions, perpendicular to each base; then prick each from No. 1, as the 
 figures direct, which will give the form of each rib. This wants no demonstration. 
 
 E 
 
OF 
 
 CARPENTRY 
 
 LININGS FOR SOFFITS. 
 
 nEFINITION. 
 
 The lining of a Soffit, in the Theory of Carpenti-y, signifies the covering of any concave surface of a 
 
 solid spread out on a plane. 
 Sofiit, in Architecture, is the under side of the head of a door, window, or the intrados of an arch, 
 
 and may be either plane or curved. 
 
 ♦ 
 
 PLATE XI. 
 
 To stretch out a Soffit, when a Window or Door, having a semicircular Head, 
 cuts into a straight Wall in an oblique direction. 
 
 LET C be the plan, or opening, of the window in fig. A, and let the base of 
 the semicircle B be drawn at right angles to the jambs, or sides, of the plan C ; 
 divide the semicircle into any number of equal parts, as ten, and draw the ordi- 
 nates across the plan ; extend the parts round B upon the stretch-out line, the 
 ordinates being drawn from the divisions across, and traced off from the plan C, 
 as indicated by the figures and letters ; the lining of the soffit will then be 
 completed. 
 
 If a cylinder is made to the thickness of the wall, the end of it may be traced 
 from the semicircle B. 
 
 To draw the Lining of a Soffit when the Top is a Semicircle, cutting right into 
 
 a circular Wall. 
 
 Fig. E. This and the other below are performed the same as that above, 
 with this difference, that the circular plan must be the regulating line, instead of 
 the straight plan. 
 
 E 2 
 
24 
 
 LININGS FOR SOFFITS. 
 
 Fig. /. shows the method when a circular-headed window cuts obliquely into 
 a circular wall. 
 
 Note. — In all kinds of cylindro-cylindric soffits, when the two jambs are 
 parallel, the straight line, from which the ordinates of the soffit are measured, 
 must be drawn at right angles to the jambs, as is shown in this plate : for the 
 want of this consideration such soffits are shown in books upon wrong principles. 
 
 But in the following soffits, where the jambs are not parallel, they must be 
 continued till they meet in a point, and the regulating line of the ordinates must 
 be made to form an isosceles triangle with the jambs. 
 
 PLATE XII. 
 
 To draw the Lining of a Soffit in a straight Wall, splaying equally on all sides, 
 
 ivith a circular Head. 
 
 \\ijig. A, continue the sides of the plan A, that is, a c and b d, to meet at e ; 
 then about the centre e, and from the points a and c, describe the soffit C, and 
 stretch the semicircle B along the outline of the soffit C, and it will be completed. 
 
 To draiv the Lining of a Soffit in a circular Wall, splaying equally on all sides, 
 
 ivith a circular Head. 
 
 Fig. B The stretch-out of this soffit is managed the same as in the last; draw 
 
 the ordinates of the semicircle B, from thence continue them to /, the concourse 
 of the splay ; and at the points a, b, c, d, e, where they intersect the plan, draw 
 the parallel lines a e, b f c y, &c. parallel to the base of B, and from the points 
 e,f,y, h, and i, describe arches to a, b, c, d, and e, round the centre f, which 
 will give the half of one edge of the soffit; the other half, being similar, is mea- 
 sured from it: the other edge is found in the same manner. 
 
 Note. — The ordinates of this cannot be measured from the plan as the others 
 are, as the lines round the splay are inclined, and will therefore be longer than 
 those on the plan. 
 
 DEMONSTRATION OF FIG. A. 
 
 Conceive the semicircle B to be turned at right angles to the plan A, then every point in the 
 circumference of the semicircle B will be at an equal distance from the point e, but the soffit C is 
 described with the same radius ; therefore the edge of the soffit C, that isj the arch-line a f, will 
 exactly coincide with the arch of the semicircle B, which was to be proved. 
 
LININGS FOR SOFFITS. 
 
 25 
 
 DEMONSTRATION OF FIG. B. 
 
 It is easy to conceive from the last demonstration, that if the semicircle B is turned up, and the 
 soffit at C bent round it, the points 1, 2, 3, 4, 5, at C, will coincide with equal divisions in the 
 semicircle B, and the points «, b, c, d, &c. at C, will fall perpendicularly over the points a, b, c, d, &c. 
 in the plan A ; for the arches a e,b f, c g, d h, and e i, at C, will fall over the parallel straight lines 
 e a; fb,gc, h d, i e, in the plan A, which was to be demonstrated. 
 
 The learner is advised to cut these and the following soffits out of paste- 
 board, and their demonstrations will be more clearly seen. 
 
 PLATE XIII. 
 
 To find the Linitig of an Aperture whose plan A B C D Z5 a Trapezoid, with two 
 parallel Sides A B and D C, tvhich represent the outsides and insides of the 
 Wall, and two equally inclined Sides A D and B C, which represent the Jambs, 
 
 ■ and IV hose Elevation A I B, on the inside of the Wall, is a Semi-ellipsis, and 
 that on the outside, D G C, a Semicircle, so that a straight Edge may every 
 where coincide with the lined Surface, and be parallel to the Horizon. 
 
 Produce A D and B C to E : bisect D C at F, and draw EE; produce it 
 to /, and it will cut A B 2it H : then F G and H I are equal to each other. 
 Divide the quadrant D G into any number of equal parts, (as five), and draw lines 
 through the points of division cutting the base D fat right angles ; from the points 
 of division, and in the same straight line with the point E, draw lines to cut A H, 
 and the lines so intercepted will represent the level straiglit lines on the soffit. 
 Make E J perpendicular to E D equal to F G, and mark the other divisions on 
 E J from E, at a, b, c, d, respectively equal to those in F D : then take the 
 distances of the several points in D C from E, beginning next E D, and pro- 
 ceeding to the last E F, and describe the arcs from the centres a, h, c, d, re- 
 spectively ; with the fifth part of the arc D G fix the foot of the compass in D, 
 and cross the first arc at e; place one foot of the compass in e, cross the next 
 arc at f, proceed in this manner to ^, then D, e, f g, h, i, will be the coincident 
 line of the lining or interior covering for the arc D G. Join i J and produce it 
 to K ; make the angle K J L equal to the angle K J E, and make ./ L equal to 
 J E : mark the divisions on J L, so that the distances from ./ may be equal to the 
 distances of the several divisions on J E, then the other half of the plano-cuneoidal 
 line may be found by inversion. Produce the lines a e, b f c g, d h, J, &c. to 
 
26 
 
 LININGS FOR SOFFITS AND TRACERY. 
 
 o, p, q, r, s, &c. make e o,f p, (j q, &c. in inverted order equal to the seats of the 
 lines on the soffits and the points o, p, q, r, &c. then curves being drawn through 
 the points o, p, q, r, s, &c., and through e, f, g, h, &c. will form the wall lines of 
 the covering, so that A D M N will be the whole covering or interior develop- 
 ment. 
 
 PLATE XIV. 
 
 To find the Lining of an Aperture covered ivith the same Surface as in the last 
 Plate, and terminated by a circular Wall. 
 
 Find the line D Q M, as in the last Plate, for a straight wall, then transfer 
 the distances from the straight line JJ F on the plan to the arcs, as shown by 
 e f g, h, i, representing the faces of the wall to the covering, and the edges will 
 be obtained as in the last. 
 
 PLATE XV. 
 
 7^0 draw the lining of a cylindrical Soffit, cutting right in a Wall which does not 
 stand perpendicular to the Ground, to a level Base. Fig. A. 
 
 Let a e at Z) be the level of the ground, a I the inclination of the wall, equal 
 to the radius of the cylinder ; let fall the perpendicular from / to c, in the bottom 
 line a e ; and draw / e at right angles to / a, cutting the level line at e ; make the 
 semicircle in fig. Ai, to the width of the cylinder, or the double of a I at D ; take 
 the distance a c at D, and make b a equal to it in fig. A, and describe a semi- 
 ellipsis to the length of the large semicircle d d, and to the width a b, as shown 
 by the small semicircle 0, 1,2, 3, a ; lay the equal divisions round the semicircle 
 in fig. A, at C, along the line d d, on each side of the middle point 4, then take 
 the parts e d, d c, c b, b a, from the plan B, and lay them at D respectively from 
 € towards d, c, b, a, and at the points a, b, c, d, erect perpendiculars to a e to cut 
 / e at f g, h, and ^ ; take the distances e i, i h, h g, and g f, in D, and lay them 
 on the soffit at C respectively, from 1 d, 2 c, 3 b, 4 a, each way, then will the 
 straight line d d in the soffit, when bent round, be perpendicular over the elliptic 
 line in the plan B, and the curve line d d c b a, &c. d, will fall over the points 
 d, d, c, b, a, in the plan : in the same manner the edge of the soffit may be brought 
 to answer any curve line proposed. 
 
CHURCH WORK, AND RANGING OF RIBS. 
 
 27 
 
 To draw the Arches of Groins by a neiv Method, whether right or rampant, so that 
 their Arches shall intersect or mitre truly together, from a given Arch of any form. 
 
 Let fig. E be the given arch of a Gothic form, draw the chord a c for one half 
 the arch, divide it into any number of parts, (as four), and through the divisions 
 draw lines from the centre e to terminate in the circumference at h, g, I; draw 
 lines from c through h g I to cut the perpendicular a d 2ii h, c, d ; and if No. 2 is 
 required to be wider, but the same height as fig. E, draw the two chords a c and 
 c b for each side of the arch, divide each into four equal parts, as before, and set 
 the parts a b, b c, c d, perpendicular on each end of a ^ at No. 2, and from the 
 divisions draw lines to the vertex at c, then trace the curve through the points 
 h, g, I, &c. so the arch at No. 2 will truly mitre into fig. E; in the same manner 
 the rampant curve at No. 3 will be brought to correspond with fig. E and No. 2.* 
 
 PLATE XVL 
 
 As it happens sometimes in church-work, that windows go higher than the ceiling-line, which there- 
 fore requires to be hollowed out, so that the light may be thrown down into the body of the 
 church, I shall in this place show the method of making a curb for that purpose. 
 
 To find the form of the Curb. 
 
 Let k b I he the head of the window, ^^/wre A, and let it come as high as a b, 
 above the ceiling ;f and let ab a.t No. 1 be the same height, and be the direction 
 of the light, and a c will be the length of the curb. Make a c at No. 2 equal to 
 ttc at No. 1, and divide it into six equal parts ; also divide ab, in fgure A, into 
 six equal parts, and let the ordinates be drawn as is explained in the figure ; a 
 curve being traced round the points of intersection, will give the form of the curb. 
 
 Figures B and C show the method of drawing and backing any elliptic rib 
 
 * It is hardly possible to find a more ready method in practice, because a chalk line will soon strike all the 
 radial lines, having only to move it but once from the point e up to c at the crown ; Jig. F shows the common 
 method — by dividing the basis of each into a like number of parts, and transferring the height, as the figures 
 explain, at No. 1 and No. 2; nothing is more tedious in practice than raising a number of large perpendiculars, 
 and going continually from one curve to get the height of another. 
 
 ■f The ceiling is here supposed to be level, which is seldom the case in a church ; but the method will be 
 nothing different if the ceiling-line a c fig. 1, were to incline to the horizon in any angle whatever, only observe 
 to make c ah equal to that angle. 
 
28 
 
 .CENTERINGS TO GROINS. 
 
 with a compass, which is exceedingly handy in drawing, and will be near enough 
 for the representation of an elliptic rib on paper, as no other method will be so 
 clean when done ; but for practice, a trammel, or intersecting lines, is more ready. 
 
 To draw and range the Ribs by this Method. 
 
 In B, let c h be the height, and c b the width ; divide the difference into three 
 equal parts, and set four such parts on each side of c, to d and d, and make an 
 intersection with the distance d d 2it e, and draw a line through e and d to i, then 
 d and e are the centres for the interior side : suppose the rib is to be backed as 
 much as a 6 upon the bottom, set a b from d to f, and from e to g, parallel to the 
 base; and draw a line through g,f, to A;; then g and f are the centres for 
 describing the ranging lines. 
 
 The rib E is traced from D, and a b being set all round on the parallel lines, 
 shows how the ranging is found for a drawing on paper : the rib C is described 
 in the same manner. 
 
 The word backing is properly applied to the upper side of any thing in 
 Carpentry or Joinery, as the back of a rafter, the back of a rail : but range applies 
 to the operation of levelling either the upper or lower edge, and explains its own 
 meaning, viz. forming the edge so as to range with the other edges, whether 
 forming a ceiling or the exterior of a roof. 
 
 PLATE XVII. 
 
 DEFINITION. 
 
 Groins are formed by the intersection of arches or vaults, and the surfaces of their meeting may be 
 considered as the sections of cyhnders, cyUndroids, &c. 
 
 BRICK GROINS.— Description. 
 P, P, P, &c. in the plan in Jig. A, are the piers which the vault is to stand 
 upon ; a b,Jig. D, is the end opening, which is a given semicircle ; and b c is the 
 opening of the side arch, which is to come to the same height as the end arch 
 ab: fix centres over the body range, A, as shown in the section at C, then 
 board them over. In Jig. A is the manner of fixing the jack ribs upon the 
 boards, which likewise shows at C. 
 
CENTERINGS TO GROINS. 
 
 29 
 
 To find the Mould for the Jack Ribs. 
 
 Take the openings of your arches in Jig. A, that is a ^ and a d, and lay them 
 down in fig. D, at ab and b c, to make a right angle. Divide one half of the 
 given semicircle into five parts, and square them across 1, 1, 1, &c., to cut d b 
 and d c, the diagonals, in 2, 2, 2, &c., and through the points 2, 2, 2, &c. draw 
 lines parallel to the base of E, both ways towards F and G ; stick in nails 
 at 1, 2, 3, 4, 5, in the arc of E, and bend a thin slip of wood round them, which 
 mark with a pencil at every nail ; this slip of wood being stretched out from 
 d,l, 2, 3, 4, 5, and squared over to G, will intersect the other lines in small rect- 
 angles : a curve being traced through the diagonals of each rectangle, will give 
 a mould to set the jack ribs. 
 
 To fix the Jack R ibs. 
 
 Bend the mould G from «, to the crown at e, in fig. A, which will give the 
 edges of your boards ; then fix a temporary piece of wood, (as a straight-edge, or 
 long level), level upon the crown, in the direction of / /, and let it come the thick- 
 ness of the boards loM^er than the crown, and it will give the height of the jack 
 ribs, which is a very sure method of placing them. 
 
 To find a Moidd to cut the Ends of the Boards. 
 
 The rib F is traced to the height of E, or got by a trammel, which will be fully 
 exemplified in the following plates. Take the parts round F, and lay them out to 
 1, 2, 3, 4, 5 ; then //will be got in the same manner as G, which will be a mould to 
 cut the ends of the board that goes upon the jack ribs against the body range. 
 
 Fig. I. Is an easy method of getting the moidds when both arches are the 
 
 same opening. 
 
 Take half the opening of the arches, whatever they are, and draw a quarter 
 circle, and divide it into six ; bend a slip round it to take its parts, then stretch 
 it out upon the base from 0 to 6, and square over the points 1, 2, 3, &c. 
 Through the points in the arch draw the lines on both sides parallel to 0, 6 ; the 
 curve being traced as before, gives both moulds of an equal and similar form. 
 
 Note. The curve F may be drawn in practice with a trammel independent 
 of the other, and the two moulds F and G may be drawn separate, without any 
 connexion of lines, as shall be shown hereafter. 
 
 F 
 
30 
 
 CRADLING TO GROINS. 
 
 PLATE XVIII. 
 
 CENTERING FOR ASCENDING OR DESCENDING GROINS. 
 
 Definition. — Groins are said to be ascending or descending when they are not built upon level 
 
 ground. 
 
 The Plan and Inclination of any Groin hei7ig given, arid one of the Body Ribs, 
 as B, also the Place of the Angles upon the Plan; to find the Form of the Side 
 Ribs, so that the Intersection of both Arches tv ill be perpendicularly over the 
 Plan. 
 
 Divide half the circumference of the given rib S into any number of equal 
 parts, and draw them to intersect the angles upon the plan ; and from thence 
 let them be returned up to the rib C, upon the side ; then C being measured 
 from the given rib at B, as indicated by the letters, will give the form of the side 
 centre. The same is shown at F, by the method of intersecting lines. 
 
 To find the two Moulds D and E for placing the Jack Ribs, to bend over the 
 Angles in the Body Range, when boarded in, so that they may be perpendicular 
 over the Angles upon the Plan. 
 
 At C, draw lines from the points a, b, c, d, e,f, g. See. where the ordinates of 
 C intersect the top of the arch, perpendicular to the rake, and draw the semi- 
 ellipsis A, to the width of the body range, and to the height a, h, of the side 
 centres taken perpendicular to the rake; and continue the ordinates of B 
 upwards to A, to intersect at 1, 2, 3, 4, 5, 6. Bend a slip round these points, 
 and mark it opposite to every point, and stretch it out along k, 1, 2, 3, 4, 5, 6, 
 between D and E, and draw lines through these points, at right angles to h 6, to 
 intersect with the perpendiculars. Begin at 6, and trace a curve both ways ; this 
 will give the edges of the two moulds for placing the jack ribs. 
 
 To cut the Jack Ribs to the Rake of the Groins. 
 Set the number of the jack ribs upon the arch B at their proper distances, 
 and take their several heights, that is, h i, k I, m n, and set them upon the arch 
 
CRADLING TO GROINS. 
 
 31 
 
 G, from a to h, and from a to c, and from a to draw lines through these points 
 parallel to the rake, which will show how the jack ribs are to be cut, so that they 
 shall range properly with the other raking centres. 
 
 Note. All the body ribs must be ranged according to the rake of tlie 
 groin ; to do this exactly, the under edges of all the ribs must be bevelled 
 according to the rake; then make a mould as B, or one of the body ribs them- 
 selves will answer instead of a mould, which being applied to each side of any 
 other rib, keeping the bottom fair with the under edge upon each side, and 
 drawing the curves by the other, it will give the ranging line. 
 
 PLATE XIX. 
 
 Given the two Side Arches of any ascending Groin, and the Inclination; to find 
 the Intersection of the Angles upon the Plan. 
 
 Divide half of the body rib B into equal parts, and draw parallel lines to 6, c, 
 d, e, and f; and from the point a, as a centre, draw the concentric dotted circles 
 round to g, h, i, k, Z; then draw parallel lines to the rake, to cut the centre C at 
 1, 2, 3, 4, 5, and a, b, c, d, e, on the other side; and from these points let lines 
 be drawn perpendicular through the plan. And on the centre g of the rib C, 
 square aline up to 5, the top of the arch C; and from 5 draw aline perpen- 
 dicular through the plan. Also through the points 1, 2, 3, 4, 5, at B, draw 
 perpendicular lines to the plan the other way ; begin at h, and trace through the 
 angles both ways, will give the place of the groin points upon the plan. 
 
 The moulds for bending over the angles are found in the same manner as 
 in the last plate, by taking the stretch-out round A, and laying it between D 
 and JE. 
 
 The reader may see such groins executed under the Adelphi Buildings in the 
 Strand, London, where the declivity is very rapid in going to the river. 
 
 The jack ribs of the groin are cut in the same manner as directed in the last 
 plate, and in practice there will be no occasion for tracing the angles, as the 
 two moulds D and B are found independent of them ; the reader will farther 
 observe that the arch B must not be used instead of the arch A, which would 
 produce a very great error in the moulds D and B, as it must be evident to 
 every one, that the section upon the square of the cylinder, or body range, must 
 be less in the height than the perpendicular or plumb section B, which in this 
 
32 
 
 CRADLING TO GROINS. 
 
 case is oblique : if these things are properly understood, there will occur nothing 
 in brick groins but what may be easily surmounted. 
 
 In all kinds of brick groins the centres or body ribs must be fixed first in 
 the same manner as if there were no side arclies cutting across them ; then the 
 centres must be boarded over; then to find the place of the angles upon the 
 boards, that is, the proper intersection of the side arches upon the plan, the 
 moulds D and E must be both bent round the boards at one time, by keeping 
 the points / and e of the moulds D and E upon the tops of the piers at o and e ; 
 then keep the top points together, and bend them round, keeping them still 
 together, then the point at 5 will fall perpendicularly over h in the plan ; round 
 the inner edges of the moulds draw a curve upon the boards, which will be the 
 proper intersection of the side arch. The jack ribs are cut in the same manner 
 as directed in the last. 
 
 Note. This differs from the last Problem inasmuch as the springings of the 
 side arches are horizontal, whereas in the last example they were all in the line 
 of Rake. 
 
 PLATE XX. 
 
 The Plan of the Diagonals of any Plaster Groin, which arestraight Lines, and one 
 of the Side Arches being given ; to find the other Side Arch and Angle Rib. 
 
 Fig. 1. Case 1. If the given rib is a semicircle or semi-ellipsis, it may be 
 described as in fig. 2, Plate VI. with a trammel, which is by far the readiest 
 method ; but if a proper trammel is not to be got, a temporary one may easily 
 be made, which will answer equally well, by fixing two pieces of wood in the 
 form of a square, that is, to make a right angle ; each leg must be as long as the 
 difference between the semi- transverse and semi-conjugate axis, and instead of 
 the sliding nuts in the rod, two brad-awls will answer the purpose, being put 
 through any straight slip of wood ; and by moving this round either the exterior 
 or the interior angles of the square, keeping the pins or brad-awls close to each 
 leg, it will describe one quarter of an ellipsis at one time. 
 
 To find the Length of the Jack Ribs. 
 
 Lay down the plan of the ribs, as at B, and draw a rib upon each opening ; 
 then draw perpendicular lines from the plan of each opening, at the extremities 
 
CRADLING TO GROINS. 
 
 33 
 
 a c e, to cut its correspondinj^- ribs at b, d,/: then the distance from b to b shows 
 the length of the first jack rib, from d to d the length of the second, and from 
 y to y the third. 
 
 To bevel the Angle Ribs, so that they shall range with each opening of the Groin. 
 
 First get the ribs out in two halves or thicknesses, as at £1 and F, then draw 
 the plan of your angle rib which is placed between E and F, and it will show the 
 true ranging upon the bottom of the rib ; then shift your hip mould parallel upon 
 the base of E and F, which will show how much wood there is to be taken 
 away; then nail the two halves together, and the rib will be completed. 
 
 METHOD I. 
 
 Fig. 2. Case 2. When the given rib is a segment of a circle, or any other 
 curve whatever, the ribs will be described as in Plate W .Jig. E, as are shown 
 at B, E, and F. 
 
 METHOD II. 
 
 When the given arch is a segment of a circle as at A, take its height b c, and 
 place it from b, the centre of the groin to c at C and D ; then take the whole 
 diameter of the arch A, that is, twice the radius a c, and place it from the crown 
 of the other arches perpendicular to their bases from c to 6 at C, and from c to d 
 at D (which represents the Angle Rib) ; then the arch may be drawn as in Plate 
 VIII, by intersecting lines; the ranging of the ribs is done in the same manner 
 as in the last groin. 
 
 Either of these two methods is much readier in practice than tracing the ribs 
 through ordinates. 
 
 PLATE XXI. 
 
 Given one of the Body Ribs, and Plan, and Angle of the Ascent of a Groin not 
 standing upon level Ground; to find the Form of the ascending Arches, and 
 the Angle Ribs. 
 
 Let b a c at B he the angle of the ascent; from the point b make b c perpen- 
 dicular to a b, and describe the curve B, as in Plate XV. (at No. 3, in ^g. E\) 
 then draw the diagonal a b at E, and make b c perpendicular to it, and equal to 
 5 e at jB ; then draw the hypothenuse a c, and describe the angle rib E, in the 
 same manner as that of B. 
 
34 
 
 CRADLING TO GROINS. 
 
 To find the LengUi of the Jack Ribs, so that they shall fit to the Rake of the Groin. 
 
 Draw lines up from the plan to the arch, as at D, in the same manner as is 
 explained in the last plate ; then the arch from a to a is the first jack rib, from 
 bio h the second, and from c to c the third, &c. 
 
 To range the Angle Ribs for such Sort of Groins. 
 
 Get the ribs out in two halves, as in the last plate, then the bottom of the ribs 
 must be bevelled agreeable to the ascent of the groin, and the plan of it must be 
 drawn upon the level, and from thence they may be drawn perpendicular from 
 the plan to the rake of the rib ; then take a mould to the form of the rib, or the 
 rib itself, and slide this agreeable to the rake, to the distance that is marked upon 
 the bottom to be backed off, which will show how much the rib is to bevel all 
 round. 
 
 PLATE XXII. 
 
 OF GROINS CUTTING UNDER PITCH. 
 
 Definition. — When the side arches of a groin are lower than the body arch, then they are called 
 
 under pitch groins. 
 
 Given one of the Body Ribs B, and the Height f g, of a Door or fVindoiv, ^c. and 
 its Width ml; to find the Side and Angle Ribs D and E, so that the Intersection 
 of the Side Arch D tvith the Body Rib B, shall be straight upon the Plan. 
 
 Draw c e perpendicular to c b, the base of B, and equal to the height of the 
 window at D, that is, equal to f g ; through e draw ea parallel to c b, cutting the 
 arch J5 in a ; let fall the perpendicular a b to c b, and continue it so as to cut the 
 line fg in £> produced to k, and draw km and kl, which is the place of the angles 
 upon the plan, or the base of the angle ribs ; then the ribs D and E may be 
 described from the given rib F, as directed in plate W.fig. E, from a centre; 
 or they may be described as at fig. F of the same plate, as you see on the other 
 side at A and C by ordinates : but the first is by far the easiest method for prac- 
 tice, for if you stick a pin or brad-awl in g, at Z), and lay a chalk line to it, you 
 may strike all the radial lines y 1, (/2, y3, y 4, &c., in much less time than the 
 parallel lines in A and C can be drawn, and with much greater accuracy : and 
 
CRADLING TO GROINS. 
 
 35 
 
 the divisions upon cn of the arch F, may be marked upon a rod, and readily 
 transferred to the arches D and E, on 7n p and : then move your brad-awl 
 out of <jr, and stick it in the crown at f, and strike lines from the divisions of mp 
 to cross the other lines, which will give the points through which the arch must 
 pass : but the reader must recollect that four or five points will not be sufficient 
 in the practice for tracing a large curve with accuracy, and therefore a greater 
 number must be found. At the other end of the groin is shown the manner in 
 which it may be fixed, sufficiently intelligible for a workman. 
 
 PLATE XXIII. 
 
 A CYLINDRO-CYLINDRIC ARCH. 
 
 DEFINITON. 
 
 A cylindi'o-cylindric arch, or Welsh groin, is an under pitch groin, whose side and body arches arc 
 both given semicircles, or they may be similar segments of circles cutting through one another, 
 whose intersections do not meet in a plane surface, that is, the place of the ribs will not be straight 
 upon the plan, but will generate a curve line. 
 
 Given the Body Rib A, and the Side Rib B, of a Cylindt o-Cylindric Arch, to 
 find a Mould for the intersecting Ribs. 
 
 Divide half the arch D into any number of equal parts, 1, 2, 3, d, or they may 
 be taken at discretion, and from these points let fall perpendiculars to a b, its 
 base; produce them at pleasure ; also from the same points 1, 2, 3, d, draw lines 
 parallel to a b, the base of J5, to intersect the perpendicular line e f ; transfer the 
 divisions from e/ to by a series of concentric circles ; then from the divisions 
 of e ^ draw lines parallel to p q, to intersect the body rib A at the points h, u,w,y ; 
 from these points draw perpendiculars to p^, its base, and continue them to 
 intersect with the perpendiculars from B, at the points k, I, m, n, between C and 
 D ; then trace a curve through these points, which will be the place of the inter- 
 secting ribs upon the plan ; then draw two other curve lines on each side of 
 k, I, m, n, &c. to make the thickness of the rib upon the plan ; on the inside of the 
 curve draw two chords for each half to their extremities, draw two other lines 
 parallel to them to touch the outside curve, then the distance between those two 
 straight lines will show what thickness of stuff it will take to make the inter- 
 secting rib ; through the points k, /, m, n, &c. draw perpendicular lines to the 
 chords ; make the heights C(i, 3 3, 2 2, 1 1, &c. at £), equal to their corresponding 
 
36 
 
 CRADLING TO GROINS. 
 
 heights at B : then D is the mould for the intersecting rib ; C is the same as D, 
 but on different sides of the rib. 
 
 To range the Ribs, so that they will stand perpendicular over the Plan. 
 At the points x, v, t, i, in the base of A, draw the parallel dotted lines to the 
 ordinates of C and D, and make their corresponding heights equal to those of 
 the arch B or A; draw the dotted curve line huwy at C, and it will show how 
 much is to be bevelled off on that side of the rib ; in like manner the other side 
 D is bevelled. 
 
 To find a Mould to bend under the intersecting Ribs, so that it shall give the 
 Place of the Angle truly upon the Plan. 
 
 Take the stretch-out round the under side of the rib D at the dots, by bending 
 a thin slip of wood round it ; mark it at each dot, and stretch it out along the 
 straight line be at draw the ordinates across, and prick them from the plan 
 that lies between D and C; then E, agreeable to the letters,, will be the mould 
 required. 
 
 Note. — The straight edge of the mould must be kept exactly to the face of 
 the rib ; when it is bent round, then draw a curve round the under side of the 
 rib, by the other edge of the mould, which will give the true place of the angle. 
 
 PLATE XXIV. 
 
 There will be no occasion for explaining the lines of this groin, as they are 
 of the same nature as those in the last plate ; but it will be proper to take notice, 
 as this is a bevel groin, the ribs must lie in the same direction as the plan of the 
 groin, which will make them longer than their corresponding given arches at the 
 top, although of the same height ; they are consequently ellipses, being the 
 sections of cylinders, therefore to make a rib over Im, across the two piers, take 
 the extent of the base Im, and the height of the given arch no, and describe an 
 ellipsis ; and to describe the side arches between any two piers, as from a to b, 
 take the extent ab, and the height of the given arch, py, at A, and describe an 
 ellipsis, it will give the proper form of the rib to stand over a b; the inter- 
 secting ribs will require two moulds, C and D, owing to the groins being bevel 
 upon the plan. 
 
CRADLING TO GROINS. 
 
 37 
 
 Note. The letters are marked the same upon D and C as they are upon E, 
 to show they are traced from it. 
 
 PLATE XXV. 
 
 To describe the intersecting or Angle Ribs of a Groin standing upon an octagon 
 Plan, the Side and Body Ribs being given both to the same Height. 
 
 Fig. 1. E is a given body rib, which may be either a semicircle or a semi- 
 ellipsis, and ^ is a side rib given of the same height; D is a rib across the angles* 
 traced from (the basis of both being divided into alike number of equal parts): 
 divide the base of the given rib A, into the same number of parts ; from these 
 points draw lines across the groin to its centre at m, and from the divisions of 
 the base of the other rib Z>, draw lines parallel to the side of the groin, then trace 
 the angle curves through the quadrilaterals, which will be the place of the inter- 
 secting ribs ; draw the chords a b and h c, then prick the moulds B and C from 
 E or D, but take care not to set them from the crooked line at the base, but from 
 the straight chords a b and b c. 
 
 To describe and range the Angle Ribs of a Groin circular upon the Plan, the Side 
 and Body Arches being given., as in the last Groin. 
 
 The ribs are described in the same manner as in the last example for the 
 octagon groin, or in the same manner as the cylindro-cylindric, Plate XXIII. 
 and the ranging is found in the same manner as is described in that Plate. 
 
 Note. E and F are the same moulds as are shown at B and D. 
 
 PLATE XXVI. 
 
 The Side Rib A, and the Angles being given straight upon the Plan, to find the 
 
 Angle Rib G, and the Body Rib C. 
 
 Let the rib A be supposed to be placed over the straight line a b, as its base, 
 which divide into any number of equal parts, as eight; from the points of division 
 draw lines to the centre of the groin to intersect the angles at a, b, c, d, e,f, g, 
 
 G 
 
38 CRADLING TO GROINS. 
 
 these points will give the perpendiculars of the ordinates of G^, which being made 
 respectively equal to those of A, will give the curve of the rib G. If from the 
 points a, b, c, &c. arcs be drawn from the centre of the groin to intersect the base 
 of C, at 1, 2, 3, 4, 3, 2, 1, and perpendiculars be drawn and made correspond- 
 ingly equal to those of A, and C be traced through these points, then C will 
 be the body rib. 
 
 To describe the Ribs of a Groin over Stairs upon a circular Plan, the Body Rib 
 
 being given. 
 
 Fig. 2. Take the tread of as many steps as you please, suppose nine, from E, 
 and the heights corresponding to them, which lay down at i^; draw the plan of 
 the angles as in the other groins, and take the stretch round the middle of the 
 steps at E, and lay it from a to 6 at i^; make d e perpendicular to e? c at B, equal 
 to rfe at draw the hypothenuse ec, draw perpendiculars from dc up to B, and 
 set off B from A, as the figures direct, then B is the mould to stand over a b ; 
 draw the chords a 4 and 4 m at the angles, make a g, A h, perpendicular to them, 
 each equal to half the height d e, at S or F, draw the hypothenuse g- 4, and h m, 
 draw the perpendicular ordinates from the chords through the intersection of the 
 other lines that meet at the angles, then trace the moulds Dand C from the given 
 rib A, which will form the moulds for the angle or intersecting ribs. 
 
 Note. The reason that the angle ribs D and Care laid contrary ways, is only 
 to avoid confusion. 
 
 It must be borne in mind that the great difference between these two species 
 of groins consists in this ; some are governed by the sweep of the groin, in which 
 case the line of intersection on the plan is curved, (except where both are level, 
 and of equal opening) ; the other, where the lines of intersection are designed as 
 straight lines upon the plan, in which case the arches are rampant, or the two 
 halves are of different curves, and the solutions are drawn from the Problems in 
 Plate XV. In one case, the pla7i is adapted to the sweep of the arch — in the other, 
 the arch is governed by the plan. 
 
CRADLING TO GROINS. 
 
 39 
 
 PLATE XXVIl. 
 
 As all the sections of a sphere are circles, and those passing through its centre are equal, and also the 
 greatest which can be formed by cutting the sphere ; it is therefore evident that if the head of a 
 niche is intended to form a spherical surface, the most eligible method is to make the plane of the 
 back ribs pass through the centre ; this may be done in an infinite variety of positions, but perhaps 
 the best and that which would be easiest understood is to dispose them in vertical planes. If the 
 head is a quarter of a sphere, the front rib, and the plate or springing, on which the back ribs 
 stand, will curve equally with the vertical ones ; but if othei-wisc, they will be portions of less circles. 
 But it is evident if the front and springing ribs are intended to be arcs less than those of semicir- 
 cles, either equal to each other or unequal, that, as they are posited at right angles to each other, 
 there can be only one sphere which can pass through them ; consequently if the places of the ver- 
 tical ribs are marked on the plan, these ribs can have only one curve : in the former case no diagram 
 is necessai-y, but in the latter it may be proper to show how the vertical ribs and their situation on 
 the front rib are found. 
 
 7b get out the Ribs for the Head. 
 
 From the centre C draw the ground plan of the ribs as at Jig. A, and set out 
 as many ribs upon the plan as you intend to have in the head of the niche, and 
 draw them all out towards the centre at C. Place the foot of the compass in the 
 centre C, and from the ends of each rib, at e and c, draw the small concentric 
 dotted circles round to the centre rib at m and n ; and draw mg and ni parallel 
 to r k, the face of the wall ; then from q round to c upon the plan, is the length 
 and sweep of the centre rib, to stand over a b ; and from / round to e, the length 
 and sweep of the rib that stands from c to d upon the plan ; and from c/ round 
 to e is the sweep of the shortest rib, that stands from e iof upon the plan. 
 
 Secondly. To bevel the Ends of the Back Ribs against the Front Rib. 
 
 The back ribs are laid down distinct by themselves at C, D, and E, from the 
 plan. Take c 1, in ^g. A, and set it from c to 1 in D, which will give the bevel 
 of the top of the rib D. And from Jig. A, take from e to 2 upon the plan, and 
 set from e to 2 in the rib £J, which will give the bevel of the top. 
 
 Thirdly. To Jlnd the Places of the Back Ribs where they are fixed upon 
 
 the Front. 
 
 From the points a, c, and e, at the ends of the ribs, in the plan. Jig. A, draw 
 
 G 2 
 
40 
 
 CRADLING TO GROINS. 
 
 the dotted lines up to the front rib, to df and w in B, which will show where 
 they are to be fixed upon the front rib. The double circle upon the front rib 
 shows the ranging. 
 
 PLATE XX VIII. 
 
 To find the Curve of the Ribs of a spherical Niche, the Plan and Elevation being 
 
 given Segments of Circles. 
 
 In fig. A is the elevation of the niche, being the segment of a circle whose 
 centre is ^ ; at £ is the plan of the same width, and may be made to any depth, 
 according to the place it is intended for, and its centre is e ; on the plan B, lay 
 out as many ribs as it will require; draw them all tending to the centre at c, they 
 will cut the plan of the front rib in g,f, e, d; through the centre c, draw the line 
 m n, parallel to a b, the plan of the front rib ; put the foot of your compass in the 
 centre at c, draw the circular lines from a, g,f, e, d, to the line mn, and make cs 
 equal to ?^ ^ ; that is, make the distance from the middle of the chord line m n to 
 s, (the centre of the arch at C,) equal to the distance from the middle of the chord 
 at the top (at fig. A,) to its centre at t ; then place the foot of your compass in s, 
 as a centre, and from the extremities m or n, describe the arch at C ; with the 
 same centre draw another line parallel to it, to such breadth as you intend your 
 ribs shall be; then C is the true curve of all the back ribs in the niche. 
 
 Note. — The points /, k, i, h, show what length of each rib will be suflficient 
 from the point ni ; from h to m is the rib that will stand over d x, from i to m is 
 the rib that will stand over e u, from Ic to m over / v, and from / to m over g iv : 
 the other half is the same. 
 
 Through the centre t, draw d e, parallel to a Z>; complete he semicircle 
 E F G D, then D E is the diameter ; through n draw n a parallel to u f/; from the 
 centre t, with the distance t a describe another semicircle, whose diameter is c b ; 
 then will the semicircle c m g a b be equal to a vertical section of the globe, 
 standing on k i, passing through its centre at c, which is the same curve as the 
 rib at C, because m a is equal to c n, and c s bisecting m n at right angles, is equal 
 to t u, bisecting e a at right angles ; therefore the hypothenuse t a, that is, the 
 radius of the circle b a m e c, is equal to s m or s n, the radius of the circle or 
 rib at C. 
 
CRADLING TO NICHES. 41 
 
 PLATE XXIX. 
 
 The Plan of a Niche in a circular Wall being given, to find the Front Rib. 
 
 B is the plan given, which is a semicircle whose diameter is a b, and a, i, k, I, m, h, 
 the front of the circular wall; suppose the semicircle B to be turned round its 
 diameter a b, so that the point v may stand perpendicular over h in the front of 
 the wall, the site of the semicircle standing in this position upon the plan will be 
 an ellipsis ; therefore divide half the arch of 5 upon the plan into any number of 
 equal parts, as five ; draw the perpendiculars 1 d, 2 e, 3/, 4 ff, 5 h-^ from the centre 
 c with the radius c h, describe the quadrant of a smaller circle, which divide into 
 the same number of equal parts as are round B ; through the points 1, 2, 3, 4, 5, 
 draw parallel lines to a b, to intersect the others at the points d, e,f,g, h, through 
 these points draw a curve, it will be an ellipsis; then take the stretch-out of the 
 rib B, round 1, 2, 3, 4, 5, and lay the divisions from the centre both ways at F, 
 stretched out ; take the same distances d i, e li,f I, g m, from the plan, and at F 
 make d i, e It,/ 1, on both sides equal to them, which will give a mould to bend 
 under the front rib, so that the edge of the front rib will be perpendicular to 
 a, i, k, I, m. 
 
 Note. — The curve of the front rib is a semicircle, the same as the ground-plan, 
 and the back ribs at C, D, and E, are likewise of the same sweep. 
 
 The reason of this is easily conceived; the niche being part of a globe, the 
 external surface curvature must be everywhere the same, and consequently the 
 ribs must fit that curvature. 
 
 Note. — The curve of the mould F will not be exactly true, as the distances 
 d i, e k,fl, &c. are rather too short for the same corresponding distances upon 
 the sofiit at F ; but in practice it will be sufficiently near for plaster work ; but 
 those who would wish to see a method more exact, may examine Plate XY.fig. 
 A, where C is the exact soffit that will bend over its plan at B. 
 
 In applying the mould when bent round the under edge of the front rib, the 
 straight side of the mould i*^must be kept close to the back edge of the front rib, 
 and the rib being drawn by the other edge of the mould, will give its place over 
 the plan. 
 
42 
 
 CRADLING TO NICHES. 
 
 PLATE XXX. 
 
 The Plan and Elevation of an Elliptic Niche being given, to find the Curve of 
 
 the Ribs. 
 
 Fig. a. Describe every rib with a trammel, by taking the extent of each base 
 from the plan whereon the ribs stand to its centre, and the height of each rib from 
 its height at the top of the niche, which will give the true sweep of each rib. 
 
 To back the Ribs of the Niche. 
 
 There will be no occasion for making any moulds for these ribs, but make 
 the ribs themselves ; then there will be two ribs of each kind ; take the small 
 distances I e, 2 d, from the plan at B, squaring each point across the rib as is 
 therein shewn, and put it to the bottom of the ribs D and E, from d to 2, and 
 e to 1 ; then the ranging may be drawn off by the other corresponding rib ; or with 
 the trammel, as for example at the rib E, by moving the centre of the trammel 
 towards e, upon the line e c, from the centre c, equal to the distance 1 e, the tram- 
 mel rod remaining the same as when the inside of the curve was struck. 
 
 Given one of the common Ribs of the bracketing of a Cove, to find the Angle Bracket 
 
 for a rectangular Room. Fig. F. 
 
 Let ^ be the common bracket, b c its base; draw b a perpendicular to be, 
 and equal to it draw the hypothenuse a c, which will be the place of the mitre ; 
 take any number of ordinates in H, perpendicular to b c, its base, and continue 
 them to meet the mitre line a c, that is, the base of the bracket at I ; draw the 
 ordinates of /at right angles to its base; then the bracket at I, being pricked 
 from H, as may be seen by the figures, will be the form of the angle rib required. 
 
 Note. — The angle rib must be ranged either externally or internally, according 
 to the angle of the room. 
 
 Having given a common Bracket K, Fig. G, for a Plaster Cornice, to find the 
 
 Mitre Bracket, L. 
 
 Proceed as in the last example, and as is shewn in the Plate, and you will have 
 the bracket required. 
 
SUPPLEMENT. 
 
 PRACTICAL MATHEMATICS, MENSURATION, ETC. 
 
 PRELIMINARY REMARKS. 
 
 OUR object in offering this Work to the Public has been to afford a good 
 book, of sound practical utility, to the most important part of the Building trade 
 — the Carpenter. And while we are sensible that great changes are now made, 
 and have been in progress for some time past, in this art, we are not forgetful of 
 the old works which have preceded these changes, and which in their original 
 constitution contain elements which can never become obsolete, and must always 
 be studied and understood before any excellence can be attained in this most 
 important branch of the arts of Construction. For this reason, as our readers 
 are aware, we have made the original work of the justly celebrated Peter 
 Nicholson the basis of the principal branch of our undertaking ; and we trust, in 
 this our supplementary matter, and in our other divisions of the subject, to add 
 such modern inventions, such new developments of fancy and taste, and such 
 productions of the bolder styles of construction, as the gigantic undertakings of 
 the present era daily call forth ; and that this Work will be acknowledged as one of 
 the most complete and useful books ever given to the Carpenter. But this con- 
 sideration forces itself upon us ; that a high and perhaps over-refined scientific 
 tone pervades the nomenclature and the details of almost all modern works, which 
 the practical man has not time nor leisure to enter into ; and yet that the progress 
 in all sorts of knowledge, and the necessity of keeping pace with that progress, 
 compels him to feel that he is excelled in his own branch by men never educated 
 therein. In short, that the vast labors of the Civil Engineers are fast causing them 
 to excel all other branches of the constructive arts. 
 
 Supp. A 
 
2 
 
 PRACTICAL MATHEMATICS, MENSURATION, &C. 
 
 For some time past it has been the case that men of science seemed to throw 
 all the difficulties in the way of students that they could : and where time is of no 
 object there is some wisdom in this course ; for as in athletic arts severe training 
 is necessary to excellence, so in the mental sciences the severer the impediment 
 which is surmounted, the more vigorous the mind and memory become ; the 
 more piercing and correct the analytic faculties ; the sounder and abler the 
 scholar is in every point. But this is only well for men of leisure. In cases 
 where we have to deal with men of active minds, and absorbing daily avocations, 
 we must pursue the contrary course. All difficulties must be lessened, all labor 
 abridged, and every pains taken to sweeten and shorten the way to the desired end. 
 
 To sum up in short, it is now necessary that every man should know many 
 things that were of little use to his predecessor; and as time now is so precious, 
 and leisure so scarce, to what it was in the days of our forefathers, it is equally 
 necessary to find the shortest and plainest methods of imparting this knowledge. 
 
 One change, and a most important one, is the rapid increase of the system of 
 calculation by Decimals. It is partly owing to the free intercourse with the con- 
 tinent. The Civil Engineers calculate their cubic and lineal measurements wholly 
 by decimals. Their tapes, rules, and height staves, are all graduated decimally ; 
 their little books of useful formulae are all so expressed ; the system is so widely 
 extending, that no person ought to be unacquainted with it. There is also great 
 probability that the same system will be introduced for our weights and measures, 
 and even into our coinage. The word " decimal " has been a sort of bugbear to 
 the student, and considered to express something very abstruse and difficult. We 
 trust to be able to afford our readers a simple and easy method by which a key 
 may be afforded to both Fractional and Decimal calculations, and that by a little 
 attention they may be proficients in both. 
 
 And these are the branches of knowledge which lead to perhaps the most viseful 
 of all, Mensuration. We hope to give in a few following pages a key to all the best 
 problems in this branch of science, so expressed that the practical man may 
 easily and speedily make himself master of them. We hope to enable the Car- 
 penter to maintain his present rank among the artisans of this industrious country, 
 making way, on the one hand, against the encroachments of machinery, and on 
 the other, against the rivalry of other branches of mechanical ingenuity. 
 
 With this view we shall commence, first, with the practice of Fractional and 
 Decimal calculations. We shall then give a series of problems in Practical 
 Geometry, as supplements to Nicholson's excellent work (which will still be 
 
PRACTICAL MATHEMATICS, 
 
 MENSURATION, &C. 
 
 3 
 
 continued with the other branches), and shall then, as may seem most suitable, 
 j^ive with them the most useful practical problems in Mensuration. 
 
 FRACTIONS. 
 
 DEFINITIONS, &C. 
 
 (1.) A fraction is a quantity that represents a part or parts of some whole 
 matter or number; as \, which is one fourth part; or f, which is three of such 
 fourth parts of some whole number or quantity. 
 
 (2.) The whole matter or number is commonly called an integer, and is sup- 
 posed to be divisible into any number of parts at will. 
 
 (3.) A simple fraction consists of two parts, divided by a straight line; as f. 
 The lower part (6) is called the denominator, and shows into how many equal 
 parts the whole matter or integer is divided ; namely, 6. The upper (5) is calletl 
 the numerator, and shews how many parts out of the six are to be taken, namely, 5. 
 The fraction is read five-sixths, 
 
 (4.) A mixed number consists of a whole number and a fraction ; as 3f . 
 
 (5.) K proper fraction is one whose numerator is less than the denominator ; as f . 
 
 (6.) An improper fraction is one whose numerator is either equal to (as f), or 
 greater than its denominator, (as ^). If equal, the fraction represents the integer, 
 or 1 ; five-fifths of course being the whole. 
 
 (7.) A fraction always represents the division of the numerator by the deno- 
 minator, as f is three integers divided by four; 18 integers divided by 5. 
 
 (8.) Any number may be expressed as a fraction by making the denominator 
 unity ; as y is 5. 
 
 (9.) Any improper fraction m^y be reduced to a mixed number by dividing 
 the numerator by the denominator, and placing (after the quotient) the remainder 
 
 A 2 
 
4 
 
 PRACTICAL MATHEMATICS, MENSURATION, &C. 
 
 over the denominator as a new numerator, and forming a mixed number ; thus, 
 5 into 18, equal 3 and 3 over, or 3f. It was supposed an improper way of 
 expression to say instead of 3f , whence its name. 
 
 (10.) A compound fraction is the fraction of a fraction, as | of f ; ^ of f of j^q. 
 
 (11.) If both the numerator and denominator be multiplied or divided by the 
 same number, the actual value is not changed : thus - multiplied by 2 is f , and 
 two fourths, or two quarters, is of course one-half; so f multiplied by 3 is i^g, 
 exactly equal in value; so \ divided by 2 is \, and divided by 3 is f. 
 
 We shall now proceed to describe the signification and use of the mathema- 
 tical signs or symbols. They are now so very extensively adopted, are so very con- 
 venient, and simplify matters so much when learned, that we propose to use them 
 frequently in this work ; and the learner must not be frightened at them, as we 
 are sure, after two or three evenings' attention, they will become familiar and 
 easy to every one, and the learner will thank us for giving him so short and clear 
 a method of describing what he means. 
 
 (12.) The sign+ oy plus, is the sign of addition, and signifies that the quantity 
 placed after it must be added to the other, as 12 + 10 is ten added to twelve. 
 
 (13.) The sign — or minus, is the sign of subtraction: thus 12— 10 is ten 
 deducted from twelve. 
 
 (14.) Where several numbers follow one another, as 10 — 2 -1- 6 — 3, the numbers 
 marked 4- must be added together, and also those marked — , and the difference 
 between the two sums taken: thus - 2 and - 3 must be added together, making 
 5 ; and 10 and +6 making 16 ; then 5 deducted from 16 leaves 11. Any number 
 standing first or alone without mark is supposed to have the sign + ; thus in our 
 example, 10 is supposed to be -f 10 ; and the whole might be written + 10 -f 6 — 2 —3. 
 
 (15.) The sign x or into, is the sign of multiplication, and signifies that the 
 quantities between which it stands must be multiplied into each other, as 2 x 3 
 signifies two into or multiplied by three, equal to 6 ; and 3 x 10 x 5 equal to 150. 
 In the higher mathematical calculations it is usual to substitute a simple point (. ) 
 
PRACTICAL MATHEMATICS, MENSURATION, &C. 
 
 5 
 
 for the sign x ; thus 3 x 10 x 5 is written 3.10.5 ; but as the point ( . ) in general 
 shews the place of the decimal, we shall always use the x as a mark of multipli- 
 cation. 
 
 (16.) When any quantity is multiplied into itself, the number of times in which 
 such operation is performed is shewn by a small figure placed above the integer 
 on the right-hand side ; thus 4' means simply 4 ; 4"^ means 4 multiplied once into 
 itself, or 4 times 4, or 16, and is usually called 4 squared. 4^ means 4 multiplied 
 into itself 3 times : thus 4 times 4 is 16, and 4 times 16 is 64, and is usually called 
 4 cubed. 4* means 4 multiplied four times, as 4 times 4 is 16, 4 times 16 is 64, 
 and 4 times 64 is 256, and is called 4 to the fourth power ; and so on to the 5th, 
 6th, &c. &c. powers. 
 
 (17.) The sign -f- divided by, signifies that the former of two quantities is to 
 be divided by the latter ; thus 12 -f- 2 signifies 12 divided by 2, or 6 ; 18 -r- 5 is 
 18 divided by 5, or (see Nos. 6 and 7) .3f . Division is also expressed fractionally 
 by placing the sum to be divided as a numerator over that by which the division 
 is to take place, as a denominator ; thus ^ is the same as 18-^5. 
 
 (18.) The sign = equal to, signifies that the quantities between which it is 
 placed are equal to one another, as 10+2 — 4 = 8, signifies the result of the 
 addition and subtraction to be equal 8 ; 4x3=12, shews the similar result of 
 the multiplication ; 4^ = 64, signifies that the triple multiplication of 4 is equal toiH. 
 
 (19.) As numbers multiplied into themselves signify the squares, cubes, &e. 
 of such numbers, so it is often very necessary in Mensuration to discover what 
 sums thus multiplied have formed the sum in question; and this number is called 
 the root of the number given ; thus 3 is the second or square root of 9 ; and 4 is 
 the third or cube root of 64 ; as has been shewn above. This is usually expressed 
 by the mark V, signifying the square root of any number, as \/16 = 4; or by 
 placing a small figure over the mark, it will show the number of times the multi- 
 plication has been made, and the degree to which it must be reduced : thus 
 '^^64 = 4, or the third or cube root of 64 is equal to 4 ; so signifies the biquad- 
 rate or fourth root; thus " ^^81 = 3, or the fourth root of 81, is equal to 3 ; and so 
 on of the 5th, 6th, &c. &c. roots. 
 
 (20.) The proportion between numbers is shewn by points ( : : : ) ; thus 
 
6 
 
 PRACTICAL MATHEMATICS, MENSURATION, &C. 
 
 6 : 3 : : 4 : 2 ; or six is to three (or bears the same ratio to three) that four does 
 to two. All rule of three sums may be thus expressed. 
 
 (21.) A quantity is said to be a measure of another when it is contained in 
 the other a certain number of times exactly, without remainder: thus 6 is a 
 measure of 12, because it is contained twice therein with no remainder; 20 is 
 a measure of 100, because it is contained 5 times exactly therein ; but 20 is no 
 measure of 98, because it cannot be divided by it exactly without a remainder. 
 
 (22.) A quantity is said to be a multiple of another when it contains it a cer- 
 tain number of times exactly when multiplied ; thus 12 is a multiple of 6, because 
 it is contained therein twice, or 6 x 2 = 12; and 100 is a multiple of 20, because 
 20 X 5 = 100 exactly and without remainder. 
 
 As the words measure and multiple are of very common use, a familiar illus- 
 tration may not be amiss. Thus, a rod 4 feet long would be a measure of 96; 
 because it may be turned over 24 times, and will end exactly at the 96th foot 
 without remainder ; but it would not be so of 98, because there would be half 
 the rod to spare after turning it over 24 times. 
 
 So again, 10, 15, 25, 60, are all multiples of 5, because 5x2=10;5x3=15; 
 5 X 5 = 25 ; and 5 X 12 = 60. 
 
 (23.) If two or more numbers are placed between ( ) or have a line drawn 
 over them, it signifies that they should be taken collectively, and are said to be 
 within a vinculum or chain. Thus 2 + 4 x 2, or (2+ 4) x 2 means 2 + 4, or 6, 
 multiplied by 2= 12 ; but 2 + 4x2 means 2 added to (4 multiplied by 2) or 8, 
 and is 2 + 8 = 10. 
 
 QUESTIONS FOR PRACTICE. 
 
 [1.] Write in ordinary words 7 + 8-6+10, and shew the result. 
 [2.] The same with 1001 + 22 - 339 - 22 - 41 + 718. 
 
 [3.] Express by signs 10 added to 7, then 3 deducted, 12 added, and 9 
 deducted, and shew the result as above. 
 
PRACTICAL MATHEMATICS, MENSURATION, &C. 
 
 7 
 
 [4.] The same, .3002 added to 19, and then to 331 ; deduct 478, and also 92, 
 and add 573, and deduct 45 and 39. 
 
 [5.] How would you express the result of 491 — 763 + 42 + 19 — 374 ? 
 
 [6.] The same of - 7854 + 31092 - 26066 + 312 + 999. 
 
 [7.] Write in ordinary words 6 x 5 x 10 x 22= ; and shew the result of 
 this multiplication after the sign of equality. 
 
 [8.] The same with 33 x 21 x 718 x 9 = 
 
 [9.] Express by signs 29 multiplied by 36, then by 4, then by 523, and shew 
 to what sum it is equal. 
 
 [10.] The number 12 is to be multiplied into itself 3 times ; express this by 
 the proper sign, and state the mathematical phrase for this operation. 
 
 [11.] The number 9 is multiplied into itself, the product is again multiplied 
 by 9, then again, then again ; express these operations by the proper signs and 
 proper phrases. 
 
 [12.] Write in ordinary words 3112-7-51 ; 793^22; 4192-^17. 
 [13.] Why can 7 -r 33 be written 3^3 ? 
 
 [14.] Express by signs 10 divided by 5 ; 11 divided by 33; 4786 divided by 
 222 ; and also write the same fractionally. 
 
 [15.] The number 14 multiplied once into itself is equal to 196; and this 
 result also by 14 = 2744. What is the square root of 196 ; and what is the cube 
 root of 2744 ; and express these results by the proper signs ? 
 
 [16.] Write in the ordinary manner 899^ = 808201; 151^ = 22801; ^^50176 = 224; 
 V531441=81. 
 
B 
 
 rHACTICAL MATHEMATICS, MENSURATION, &C. 
 
 [17.] The same 155^ = 24025; 999^ = 998001; v961 =31 ; V729=9. 
 
 [18.] Express by signs 6 squared; 9 cubed ; 713 to the fourth power; the 
 square root of 1714 ; the cube root of 999 ; the fourth or biquadrate root of 10033. 
 
 [19.] 8x8 = 64; 64x8=512; 51 2x 8 = 4096. Express these by the signs 
 of the different powers. 
 
 [20.] Write in ordinary words 9 : 81 :: 27 : 243 ; 14 : 7 : : 1000 : 500. 
 
 [21.] Express by signs, as 30 is to 120, so is 120 to 480. As 729 is to 9, so is 
 81 to 1. As 6 is to 36, so is 36 to 216. 
 
 [22.] Shew the difference between 13 x 2 + 21, and 13 x 2 + 21 ; also 
 (6 + 3 + 4 + 2) X 5 and (6 + 3 + 4) + (2 x 5) ; also 7 x 10 + 7, and 7 x 10 + 7 ; and 
 shew all the results after a sign = or ecpiality. 
 
 [23.] Shew by signs the difference between 99 added to 9 and multiplied by 
 81, and 99 multiplied by 9 added to 81, and express the different results after a 
 sign = as in the former cases. 
 
 The first thing necessary to be learned in Fractional Arithmetic is what is 
 usually called the Reduction of Fractions; but as it frequently happens that in 
 the course of this operation some fractions have to be increased in apparent 
 amount for the purposes of simplifying their ultimate form, we think it will be 
 better to call this branch of our subject 
 
 THE CONVERSION OF FRACTIONS. 
 
 (24.) To convert a whole number or integer into a fraction with a given 
 denominator : 
 
 Multiply the proposed number by the given denominator, make this the 
 numerator, and place the given denominator below it. 
 
 Thus, convert the number 9 into a fraction whose denominator shall be 5. 
 Now 9 X 5 = 45. Therefore the fraction is ^. See (11.) 
 
PRACTICAL MATHEMATICS, 
 
 MENSURATION, &C. 
 
 (25.) To convert a mixed number to an improper fraction : — or, in other 
 words, to merge the integral part into the fraction : 
 
 Multiply the integral part, that is, (as has before been explained) the whole 
 number, or integers, which stand at the left hand of the fractional part, by 
 the denominator of such fractional part : add the numerator thereof to the 
 product, make this last sum the new numerator, and keep the original deno- 
 minator. 
 
 Thus, convert 3f into an improper fraction; then (3x5) + 3 = ^. See (9.) 
 Also convert 19 into an improper fraction; then (19 x 10) + 7 = ^i^. 
 in ordinary figures : 
 
 19 
 10 
 
 190 
 7 
 
 197. Place this above the denominator for 
 
 a new numerator ; ^y^. 
 
 (26.) To convert an improper fraction to a mixed number; this is the 
 reverse of the last operation, and in fact is in other words to separate the integei 
 from the fractional part : 
 
 Divide the numerator by the denominator, put down the quotient for the 
 integer, and place the remainder over the denominator for a new numerator. 
 
 Thus, convert to a mixed number; 197 divided by 10 is 19 and 7 over; 
 then the fraction is 19i-o. 
 
 Convert ^ t^i'^ to a mixed number ; then 
 
 21) 14534 (692 
 126 
 
 193 
 189 
 
 44 
 
 42 
 
 Supp. 
 
 2 Rem. 
 
 B 
 
I 
 
 10 PRACTICAL MATHEMATICS, MENSURATION, &C. 
 
 Or in mathematical signs 14534 21 =692+ 2 -=- 21, and therefore the improper 
 fraction is 6922^1- 
 
 (27.) To convert a compound fraction to a simple fraction : 
 
 Multiply all the numerators together for a new numerator, and all the deno- 
 minators for a new denominator. 
 
 Thus 4 of f = f ; f of I of f = yf. To make this rule clear, it is necessary to 
 anticipate a part of one branch of our subject, and explain the method of mul- 
 tiplying- and dividing a fraction by a number. The rules are : 
 
 To multiply any fraction by a whole number \ multiply the numerator by that 
 number, and retain the denominator. 
 
 To divide a fraction by a whole number; multiply the denominator by 
 that number, and retain the numerator. Or in other words, if the whole 
 number be treated as a fraction, invert this fraction, and proceed as in mul- 
 tiplication. 
 
 We have already shewn (8), that any whole number may be expressed as a 
 fraction, by making the denominator unity ; as y is 5. Now to multiply the 
 fraction by 5, x ^ = The unit in both the two fractions and ^ is 
 divided into 20 parts ; and as of course 3 of such parts are taken in the first 
 instance, and 15 in the second, and as 3 times 5 are 15, it is clear that 5 times as 
 many parts are taken in the second instance as the first, and therefore the frac- 
 tion is multiplied by 5. 
 
 The converse of this is shewn in the division of fractions : To divide a fraction 
 by any whole number ; multipli/ the denominator by that number, and retain the 
 same numerator. Thus divided by 5 is -f- f = i^^o i oi" io X i = 'roo- 
 
 The unit in the first fraction is divided into 20 equal parts, but in the second 
 it is divided into 100 equal parts, each being of course | of the former, because 
 5 X 20= 100; now as the same number, 15, of such equal parts is taken in both 
 cases, and as each of the parts is respectively only \ of the other parts, it is 
 clear the fraction must be divided by 5. 
 
 The learner is requested to peruse this part very attentively, till he un- 
 derstands it thoroughly; as the wliole theory of fractions depends upon a 
 clear notion of the calculations of equal parts or divisions of one whole thing or 
 integer. 
 
 To give a familiar illustration: multiply of a pound by 5; io of £l. is of 
 
PRACTICAL MATHEMATICS, MENSURATION, &C, 
 
 11 
 
 course a shilling, because 20 shillings make one pound; here ill. is the unit or 
 integer, see (2); each fractional part is 2-^, or a shilling ; 3 of these multiplied by 
 
 5 make 15 shillings, or ^ of £\. If we spoke in terms of shillings nothing could 
 be simpler ; 3 x 5 = 15, or, see (8), f x = 15 ; but as we speak in terms of 
 
 4?1 Sv*^— 3w5._J__5 
 ''■'^•f 20 ^ ^~20 I — '2 0* 
 
 The same course of reasoning will prove the method of division, and the reader 
 is requested to work it out himself, as a right understanding of fractional arith- 
 metic depends on this. 
 
 The expression "in the terms of" may also be familiarly illustrated thus : 
 
 6 pence is ^ in terms of £l., and | in terms of a shilling ; 3 pence is | in terms 
 of a shilling, | in terms of (i pence, and ^ in terms of a pound. 
 
 It follows, from what has been said, that division may be performed by 
 multiplication, by inverting the fraction; thus, To-^5 = ^-^y= as has 
 
 been shewn: but the same result is found x ^ = The idea being, 
 
 that a fraction always represents the division of the numerator by the denomi- 
 nator. See (7.) 
 
 From what has been said, it will also be easy to understand the conversion of 
 a compound fraction to a simple one. Thus, by the rule, fof|^ = f ^ to = loo- 
 It was shewn above, that ^ of was j\)-o ; and it is clear that f, or twice that 
 quantity, is ^^q. 
 
 Thus I of 5 is I of -f- = = 4 |. See (26.) 
 
 And 1 nf \^ 1 ^ 3 — 2.1. 
 
 And ^ 1 nf ^ — ^ 
 /inu g oi 8 or n) — 480* 
 
 As whole numbers must be converted into fractions, so mixed numbers must 
 be converted into improper fractions before the rule can be applied ; thus, 
 I of f of 9 i is equal to ,\ offi = ff«. 
 
 (28.) To convert a fraction into another of greater terms : 
 Multiply both the numerator and denominator by the term given. 
 Thus, ^ in terms of double amount, is f , and of treble amount, ; in terms 
 6 times greater, is f^; 8 times greater, See (11.) 
 
 (29.) To convert a fraction into another of lower terms : 
 
 If the numerator and denominator can each be divided by any number w ithout 
 a remainder, such number is called a "common measure." A "measure" has 
 
 B 2 
 
12 
 
 PRACTICAL 
 
 MATHEMATICS, 
 
 MENSURATION, &C. 
 
 already been defined (22). A " common measure " is that which may be applied 
 to two or more numbers. 
 
 Find any common measure, divide both numerator and denominator by it, and 
 the fraction is then reduced to lower terms. 
 
 Thus, in the fraction ff , the numerator and denominator may both be divided by 
 8, which is therefore a common measure of them, and (ff ) -f- 8 = f . This, again, 
 may be divided by 2, and (f ) -f- 2 = f . 
 
 (30.) The greatest common measure of two numbers, is the greatest number 
 that will divide them both without a remainder, and is found thus : 
 
 Divide the greater by the less ; then divide the first divisor by the remainder, 
 which forms a fresh divisor ; then that fresh divisor by the last remainder, and 
 so on, till nothing is left ; the last divisor is the greatest common measure. 
 
 If only one be left as a remainder, the two numbers have no common measure, 
 
 and are said to be prime the one to the other. 
 
 Thus, what is the greatest common measure of 189 and 1188 ? 
 
 189) 1188 (6 
 1134 
 
 54) 189 (3 
 162 
 
 27) 54 (2 . 
 54 
 
 Then is 27 the greatest common measure of these two sums. 
 
 Of course 9 or 3 would be a common measure of these sums, but 27 is the 
 greatest common measure. 
 
 This rule is all that is usually necessary for fractional arithmetic, but it may not 
 be amiss to give an example where three numbers are in question. 
 
 The rule is, find the greatest common measure between two of such numbers, 
 and then the greatest common measure between the remainders. 
 
 Thus, what is the greatest common measure of 56, 512, and 768 ? 
 
 512) 768 (1 
 512 
 
 256) 512 (2 
 
 Therefore 256 is the greatest common measure between 512 and 768. 
 
• 
 
 PRACTICAL MATHEMATICS, MENSURATION, &C. 13 
 
 Then, what is the greatest common measure between 56 and 512 ? 
 
 56) 512 (9 
 504 
 
 8) 56 (7 
 56 
 
 Therefore 8 is the greatest common measure between 56, 512, and 768. 
 On this operation depends the very important rule in fractions, viz. : 
 
 (31.) To convert a fraction into another of equal value, but expressed in 
 the lowest terms. Here the word "reduction" may be properly applied, 
 and the rule may stand as in ordinary books, " to reduce a fraction to its lowest 
 terms. ' 
 
 Rule: Find the greatest common measure between the numerator and 
 denominator (as if they were two independent sums); divide them both by such 
 common measure, and the two quotients will form respectively a new numerator 
 and new denominator, and you will have a fraction converted into a new one of 
 equivalent value with the former one, but in the lowest terms. 
 
 To make this branch of our subject more clear, the student will easily see 
 that the fraction |4 equal in actual value to the fraction yf, and also 
 to the fraction f , because each fraction sheWs a similar proportional quantity, 
 taken out of a whole number. Now, f is equal also to f ; but it is impossible to 
 express f in any lower terms, because the two essential parts of the fraction, the 
 numerator and denominator, cannot be divided by the same sum without 
 remainder. Now we have before shewn (11), that to form one fraction equiva- 
 lent in value to another, both the numerator and denominator must be divided by 
 the same sum ; but as the numbers 3 and 4 cannot be divided by any sum (except 
 unity) without a remainder, it is clear this fraction, and indeed every fraction 
 that cannot be divided by any greater number than unity (or 1), is already in its 
 lowest terms. 
 
 Ex. : To reduce to its lowest terms. 
 
 Now we have already shewn (30), that 27 is the greatest common measure of 
 
14 PRACTICAL MATHEMATICS, MENSURATION, &C. 
 
 these two sums ; both the numerator and denominator are to be divided by 
 it. Thus, 
 
 27) 189 (7 and 27) 1188 (44 
 
 189 108 
 
 108 
 108 
 
 The fraction in its lowest terms, therefore, is The student will see that, 
 though 44 may be divided by several numbers, 11, 4, and 2; yet the number 7 
 cannot be divided by either of these. 
 
 Again, to reduce to its lowest terms. We have shewn above (30), that the 
 
 greatest common measure of these two sums is 256 ; and 
 
 256) 512 (2 , 256) 768 (3 
 
 512 768 
 
 Therefore this formidable looking fraction is, after all, nothing but f . 
 
 Now the worth of this rule of conversion becomes developed by degrees. 
 
 (32.) To convert fractions into others having one common denominator. 
 
 First. If there be any compound fractions in the question, convert them into 
 simple fractions, see (27) ; and if there be any mixed numbers, convert them into 
 improper fractions, see (25) ; then proceed thus : 
 
 Multiply each numerator by all the denominators in the question successively, 
 except its own, and put down their product as a new numerator ; having done this 
 by all the numerators, and having thus found as many new numerators as there 
 were old ones, multiply all the denominators together for a common denominator, 
 that is, a denominator which is to be affixed to all. 
 
 Thus : Convert \, f, f , to a common denominator. 
 
 Now the common denominator will be, as last shewn, 2 x 4 x 6 = 48. 
 
 The first new numerator will be, 1 x 4 x 6 = 24. 
 
 The second, 3 x 2 x 6 = 36. 
 
 The third, 5 x 2 x 4 = 40. 
 
 The fractions then will be ff , ff , and ff ; and the student will readily per- 
 ceive, by reference to what has before been explained, that the fractions are 
 respectively of the same value. 
 
PRACTICAL MATHEMATICS, 
 
 MENSURATION, &C. 
 
 15 
 
 Thus ft = 24 being the greatest common measure, 
 ff = f ; 12 being the greatest common measure. 
 11^ = A ; 8 being the greatest common measure. 
 Again, to reduce |, | of f, and 3^, to a common denominator: now i of | = f, 
 see (27), and 3^ = see (25). The fractions will therefore stand |, |, ^. The 
 common denominator as shewn above will be 4 X 6 X 3 = 72, and 
 The first numerator 3 X 6 X 3 = 54 ; 
 The second 2 X 4 X 3 = 24 ; 
 
 The third 10 X 4 X (J = 240; and the three fractions will be ^, ^, 
 
 and 
 
 (33.) From these rules it is clear that a simple rule of proportion may be 
 found between two fractions, by reducing them to common denominators. 
 
 Thus If is to If as 24 is to 36; for by simple proportion H : ff : 
 The integer being divided by the same number, and the parts being all equal, 
 they must bear an equal ratio to each other. 
 
 (34.) To convert a fraction of a given denomination into integers equal in 
 value, but in terms of a lesser denomination. 
 
 By this rule fractions of pounds may be converted into shillings, and (subse- 
 quently indeed) into pence; fractions of feet into inches; fractions of days into 
 hours, minutes, seconds, and so on. 
 
 Rule. Multiply the fraction by the number of integers of the lesser denomi- 
 nation contained in owe integer of the higher. The product is the value required. 
 
 Thus, what is the value of f of a pound? INow £l. is equal to 20 shillings, 
 and £1. is the greater integer, and one shilling the less. 
 
 ]Now it is clear that f of £l. is of the same value as f of 20 shillings, or f of ^ 
 shillings, see (8), and f X -y- = ^"ip ; or, see (26), 1 1^ in terms of shillings. 
 
 Next, what is the value of ^ of a shilling ? The lesser denomination to a 
 shilling is a penny, and 12 pence make a shilling. INow, by the rule, multiply 
 ^ by 12, or ^ X nr = ^ ^ or, see (26), If, or one penny and f . 
 
 JNow reduce this fraction to terms of the next lowest denomination ; this is 
 a farthing, and as there are four farthings in a penny we proceed as above, and 
 I X T = or, see (26), If ; and therefore the fraction f of £l. may be converted 
 into 11 shillings, 1 penny, 1 farthing, and f of a farthing. 
 
 Convert f of a chain to terms of feet and inches : a chain is 66 feet ; then 
 
16 
 
 PRACTICAL MATHEMATICS, MENSURATION, &C. 
 
 I X ^ = ^ ; or, (26), 52|. There are 12 inches in a foot, and f of a foot is the 
 same as | of 12 inches ; or | X ^ = ^ = 9|. To reduce this to eighths of an 
 incli, f X f = ^ = 4|. Therefore | of a chain is 52 feet, 9 inches, 4 eighths, 
 and f of an eighth. 
 
 Or at length, what is the vahie of of a cwt. ? 
 
 9 
 
 4 quarters in cwt. 
 
 15) 36 (2 qrs. 
 30 
 
 6 
 
 28 lbs. in a quarter. 
 
 15) 168 (11 lbs. 
 15 
 
 18 
 15 
 
 3 
 
 16 oz. in a lb. 
 
 15) 48 (3 oz. 
 45 
 
 Rem. 3-15 parts of oz. 
 Then of a cwt. = 2 qrs. 1 1 lbs. 3 oz. 
 
 It is not necessary to go through all the intermediate values, but we may find 
 the value of a fraction in terms of any lower denomination by multiplying by the 
 total number of integers contained in the one of higher amount : thus, what is 
 the value of f of £l. in terms of a farthing? Now 20 X 12 X 4 = 960, the number 
 of farthings in £l., then 
 
 6 
 
 960 
 
 7) 5760 
 
 822. 6. Therefore f of £l. = 822f farthings. 
 
PRACTICAL MATHEMATICS, MENSURATION, &C. 
 
 17 
 
 (35.) To convert any quantity into a fraction of any denomination : 
 
 [Case 1.] If the given quantity be a simple sum, make it the numerator, and 
 take the number of integers of its denomination which make up one integer of 
 the new or proposed denomination ; make this the new denominator, and the 
 fraction is obtained. 
 
 Thus, what fraction of a pound is 11 shillings ? 
 First, the new numerator will be the number of integers, or 11. 
 The new denominator will be the number of times the integer named, or 
 shillings, is contained in a pound, or 20. 
 Therefore the new fraction will be 
 
 What fraction of a load of timber is 10 cubic feet? 
 
 The numerator as above is 10 ; 50 cubic feet make a load ; therefore the new 
 denominator will be 50, and the fraction will be or \. 
 
 [Case 2.] If more whole numbers or integers than one be given. 
 
 Reduce the whole to the lowest number named, and make this the nume- 
 rator, and find the number of parts the lowest integer contains in the denomi- 
 nation required, and make this the denominator. 
 
 Thus, what fraction of a pound is 9s. 7^d. ? Bring the whole sum into far- 
 things, and find how many farthings, or lowest integers, are contained in £1., or 
 the highest integer : thus. 
 
 Therefore the required fraction is ^|-^. 
 
 What fraction of 1 cwt, of lead is 1 qr. 14 lbs. ? 
 
 JNow 1 cwt. = 112 lbs. 
 
 And 1 qr. = 28 lbs. Then 28 lbs. -f 14 = 42lbs. 
 And the fraction is -pp^"- 
 [Case. 3.] When a common denominator can be found between the two 
 quantities, they may be converted, (32), and the fraction is obtained. 
 Supp. c 
 
 9s. 7id. 
 12 
 
 20s. in £1. 
 12 
 
 115 
 4 
 
 240 pence in £l 
 4 
 
 461 
 
 960 farthings in £l. 
 
18 
 
 PRACTICAL MATHEMATICS, MENSURATION, &C. 
 
 Thus, as above, 1 qr. is ^ of 1 cwt., and 14 lbs. | of |. 
 
 Theniofi(27)=i; 
 
 And i (28) = f. 
 Then the fraction will be f of 1 cwt. 
 
 Any common denominator would reduce the fraction, but if the greatest be 
 taken, the fraction is then in its lowest terms. 
 
 The above case may be proved thus : the greatest common measure between 
 
 42 
 
 112 
 
 IS 
 
 42) 1 12 (2 
 84 
 
 28) 42 (1 
 28 
 
 14) 28 (2 
 
 And therefore 14,(30), is the greatest common measure. Now 42 -^14= 3, 
 and 112 -T- 14 = 8; and therefore the fraction, as above stated, is f. 
 
 What fraction of a guinea is 3s. (id. ? 
 
 The greatest common denominator between them is sixpence. (They both 
 contain pence and farthings, which are common denominators, but the greatest 
 common denominator is sixpence.) 
 
 Then as 3s. 6d. = 7 sixpences ; 
 And as a guinea = 42 sixpences ; 
 The fraction is and the common measure is 7, and the fraction, 
 
 reduced to its lowest terms, (31), is ^. 
 
 (36.) To convert a fraction into another fraction of any denomination. 
 
 Take an integer of the proposed denomination, and find what fraction that is 
 of an integer of the given fraction, then find its result as a compound fraction, 
 as directed in (27). 
 
 Thus, what fraction of a pound is f of a shilling? Now 20s. = £1., and 
 Is. = 2^ of £1. Then the fraction '\s ^ oi oi or, (27), y^oi£\. 
 
 What fraction of a chain is three quarters of a yard? 
 
 Now 1 yard is -^^ of a chain, and the fraction will be f of or ^ of a chain. 
 
 What fraction of a day's work is f of an hour? 
 
 The working day is 10 hours ; then the fraction is f of or 
 
PRACTICAL MATHEMATICS, MENSURATION, &C. 
 
 19 
 
 ADDITION OF FRACTIONS. 
 
 (37.) \_Case 1.] If fractions have the same or a common denominator, add 
 the numerators together, and retain the old denominator. 
 
 Thus -3_ I _4_ _ _7_ 
 
 This is clear, because each represents a certain number of integers of the 
 same value; thus i + | of a penny, is f ; just as 1 farthing + 2 farthings = 3 
 farthings. 
 
 Again, i + | + | = f , (as may be seen by inspecting any common 2 foot rule,) 
 [Case 2.] If the fractions have not a common denominator, reduce them to 
 a common denominator, and then proceed as in Case 1. Thus, what is the sum 
 off) ¥ • Then (32) the fractions converted into a common denominator are, 
 tH' iff' tM' and therefore the sum of these is or (26) 2^^^, or (31 ) 2|t = m. 
 
 [Case 3.] If mixed numbers or compound fractions are to be added together, 
 reduce them to improper fractions, and then proceed as before. 
 Thus, add together 2^, 3;^, 5^. 
 Then (25) the fractions are f , 
 Reduce these to a common denominator (32). 
 
 The common denominator is 3 X 4 X 6 = 72, 
 The first numerator 7 X 4 X 6 = 168, 
 
 The second 13 X 3 x 6 = 234. 
 
 The third 31 X 3 X 4 = 372. 
 
 The fractions will be -j- + = ; and reducing this to the lowest 
 terms, find the greatest common measure (30), = 18, and then dividing by it, 
 (31), we obtain the fraction or (26) lOf. 
 
 SUBTRACTION OF FRACTIONS. 
 
 (38,) To subtract one fraction from another, or, more properly, to find the 
 difference between two fractions, proceed as follows : 
 
 [Case 1,] If both fractions have a common denominator, take the difference 
 between the numerators, and retain the denominator. 
 
 Thus, what is the difference between f and ^ of an inch ? 
 
 Now as the unit is divided into 8 equal parts, and as we suppose three 
 of those parts or eighths are to be taken from seven of such parts, it is clear the 
 remainder is four, or f , 
 
 c 2 
 
20 
 
 PRACTICAL MATHEMATICS, MENSURATION, &C. 
 
 [Case 2.] If they have no common denominator, convert them into fractions 
 having a common denominator, and deduct as above. 
 
 Thus, take f from ^. 
 Now it must be borne in mind that this system of subtraction is a system of 
 differences ; f being greater than the former cannot be taken from the latter in 
 the ordinary sense of the words ; but if we convert the fractions to others of a 
 common denominator, (32), we can find their difference : 
 7 X 5 = 35 for the new denominator ; 
 And 6 X 5 = 30") ^j^^ numerators. 
 
 And 1 X 7 
 
 The fractions are therefore ff and 3-5, and the difference ff — st = ff • 
 [Case 3.] Where one or both fractions are mixed numbers, or compound 
 
 fractions, convert them into improper fractions, and proceed as before. 
 
 This is the converse of the rule for addition, and has been so amply explained 
 
 that we think no example necessary. 
 
 MULTIPLICATION. 
 
 The reader is requested to reperuse (27), where this branch of our subject has 
 been in part anticipated, and to give his earnest attention to the following para- 
 graphs, as they are of the greatest importance to understanding this branch of 
 mathematics. 
 
 (39.) To multiply one fraction by another, multiply the numerators for a new 
 numerator, and the denominators for a new denominator. 
 
 Thus f X I = M . 
 
 Anrl io V — 80 
 
 Jr%.llU 32 A 9 — 288* 
 
 To avoid the confusion sometimes caused in the mind of the young student, 
 who expects that multiplication of two quantities must necessarily give an 
 increased product, and who finds the results invariably less ; as ^ X v = i ; it 
 must be borne in mind that in reality to multiply one fraction by another is to 
 take such parts of one fraction as the other expresses j thus ^ X ^ = ^ is (in 
 common words) the half of a half, which is of course a quarter. It is in reality 
 not a quantity multiplied by another, as half a shilling is sixpence, and 6 times 6 
 pence = 36 pence ; but having a half, as 6 pence, we take the half of that again, 
 or 3d., which of course is :j of a shilling. 
 
PRACTICAL MATHEMATICS, MENSURATION, &C. 
 
 21 
 
 Mixed numbers must be converted into improper fractions, and compound 
 fractions into simple fractions. 
 
 Thus, multiply 3^ by 9f . 
 Then (25) we have ^ X ^ = 
 
 Multiply f of I by of |. 
 
 Then (27) | of | = if, and of | = f^, and if X f§ = 3^4^, the greatest 
 common measure of which is 90, and the fraction -j^. 
 
 If it is required to multiply a fraction by a whole number, express such 
 number fractionally, as has been shewn, (8), and proceed as above. 
 
 Multiply I by 20. 
 This is I X ^ = H^. 
 Multiply ^ by 9. 
 
 3 V 9. 2_ 7. 
 
 6 3-^ 1 — 6 3" 
 
 It will be seen that 9 is a common measure of this fraction, and that it may 
 be expressed as f. As 63 9 = 7, it may be seen that a fraction may be multi- 
 plied by another by dividing the denominator by that sum, and retaining the 
 numerator, if such division can be made without remainder. 
 
 In all cases of fractional arithmetic where whole numbers are used, we should 
 advise the student immediately to express them as fractions, (8) ; it will save 
 much trouble, and avoid errors. 
 
 DIVISION. 
 
 (40.) To divide one fraction by another, invert the numerator and denomi- 
 nator of the division, and multiply them together, as (39.) 
 
 Thus, divide | by 
 
 "8^3 — 24' ■^24* 
 
 Divide i by f. Then fxf=t=i=l. 
 
 It was shewn (39) that multiplication is in reality the taking such part of one 
 fraction as the other expresses ; to multiply f by f is to take three quarters of five 
 sixths. Division is the reverse of this. To divide one fraction by another is to 
 find how often one is contained in the other ; thus it is clear that ^ is contained 
 once in |. As above, convert f and ^ into fractions with a common denomi- 
 nator (32), they will be |f and or (33) \^ and ^, or (8) 70 - 24 or (7) |f . 
 
 Mixed numbers must in all cases be converted into improper fractions, and 
 compound fractions into simple ones. 
 
22 PRACTICAL MATHEMATICS, MENSURATION, &C. 
 
 To divide a fraction by a whole number, express the number fractionally, and 
 proceed as before. 
 
 Divide by 5. 
 
 5 is |. Then ^ X i = rb- 
 
 QUESTIONS FOR PRACTICE. 
 [24.] Convert 12 into a fraction whose denominator is 144. 
 
 [25.] Convert 397 into a fraction whose denominator is 41, and 172 into a 
 fraction whose denominator is 14001. 
 
 [2(1.] Convert 2ff, 9-^^% 37 43^^ into improper fractions. 
 
 [27.] A carpenter has worked 4 single days at a job, and 3 single quarters ; 
 express as an improper fraction how many quarters he has worked. 
 
 [28.] Convert ^5^, -2^1^' ^3!^, f|-|, fff-, each respectively to mixed numbers. 
 
 [29.] Convert 3^ of yy to a simple fraction. 
 
 [30.] The same with l,^^ of H; of ; M of ^. 
 
 [31.] Convert ff into fractions of 3, 4, 5, 6, 7, and 8 times its present terms. 
 
 [32.] I hold five sixths of the chances at a raffle ; the amount is doubled : 
 express the fractional part of the interest 1 have in double the terms. 
 
 [33 ] Convert tS^, ff , ^^H^, into fractions of half their terms. 
 
 [34.] Convert -y^, ^3-, into fractions of one third lower terms. 
 
 [35.] 96 and 144 may both be divided by 12. What is the fraction con- 
 verted into another of 12th lower terms ? 
 
 [36.] What is i of i of 1 of i of i of i? 
 
PRACTICAL MATHEMATICS, MENSURATION, &C. 23 
 
 [37.] What is i of I of I of I of I of f of I ? 
 
 [38.] What is the greatest common measure of 332 and 2656? of 7956 and 
 9168? of 99 and 1881 ? 
 
 [39.] What is the greatest common measure of 51, 136, and 170? of 764, 
 1146, and 2292? 
 
 [40.] ¥ hat is the greatest common measure of 1332, 1665, 1998, and 2331 ? 
 of 88, 528, 6336, and 76032 ? 
 
 [41.] Convert to9^^ iiiii' their lowest terms, 
 
 [42.] Two places are 15 degrees, or of the earth's circumference, apart. 
 State this distance in its lowest fractional terms. 
 
 [43.] There are 120 shares in a mining company. A. has bought 25. What 
 fractional part of the concern does he possess, expressed at its lowest terras ? To 
 possess the fourth part of the shares qualifies to be chairman : how many more 
 must he buy for this purpose ? 
 
 [44.] Convert ^, to a common denominator. 
 
 [45.] Convert f, 7^, ^§ to a common denominator. 
 
 [46.] Shew how ^ and fff may be stated, so as to shew a simple method of 
 proportion between them. 
 
 [47.] What is the value of of a pound? 
 
 [48.] What is the value of of a chain? 
 
 [49.] What fraction will express the third of a rod of brickwork in terms of 
 cubic feet ? 
 
 [50.] At the close of a job a builder gives a lot of old stuff to be divided 
 
24 
 
 PRACTICAL MATHEMATICS, MENSURATION, &C. 
 
 among 15 carpenters; it measures a load and a quarter cube; two of them buy 
 up the shares of the rest ; thus, one buys 7 shares, the other the remainder, except 
 of one person, who will not sell his share. What fractional parts has each got, 
 and how may they be expressed in terms of a foot cube ? 
 
 [51.] What fraction of £1. is 2^d. ? 
 
 [52.] I give 35 carpenters a pot of beer each, to 20 labourers a pint each, 
 and to six boys half a pint each. What fractional part is this of a butt of beer ? 
 
 [53.] What fractional part is a three-out glass of a gallon of rum 1 
 
 [54.] I agree to prepare and fix some rough fence to a railway at per mile, 
 and find during the first week that 532 ft. 9 in. has been done. What fraction is 
 that of the mile? 
 
 [55.] What fraction of a ton is one hundredweight, one quarter, and one 
 stone horseman's weight ? 
 
 [56.] What fraction of a mile is a chain and a fifth ? 
 
 [57.] Add together i |, |, |, and |. 
 
 [58.] Add together |, |, f, i 
 
 [59.] Add together l^ 2|, 3|, 4|. 
 
 [60.] Add together iff, 3^%, \\K 
 
 [61.] Subtract ,^^¥4 from eWa^,. 
 
 [62.] What is the difference between ten and a ninth, and nine and a tenth 1 
 [63.] What is the difference between ^ and yo ' 
 
 [64.] The Excise allow one tenth for waste off the duty on bricks, and 
 
PRACTICAL MATHEMATICS, MENSURATION, &C. 25 
 
 charge on (he less sum. 1 am charged for a million and acjuarter of bricks, how 
 many have actually been made ? 
 
 [65.] What is the difference between a third of a third, and a third and 
 a third ? 
 
 [60.] Multiply half an inch by one quarter of a yard, all expressed in terms 
 of a foot. 
 
 [67.] Multiply yo of a chain by of a chain, and state the product in terms 
 of a foot. 
 
 [68.] Multiply a half, by a half of a half. 
 [69.] Divide i by 3J3. 
 
 [70.] Divide a fifth, by a fifth of a fifth. 
 
 [71.] Divide |f by in the shortest manner, and express the same in the 
 lowest terms. 
 
 [72.] What is the third and half the third of five-pence halfpenny? 
 
 [73.] What are the two-tenths and half a tenth of a shilling? 
 
 [74.] At the Saw Mills they cut 16 veneers out of an inch; the saw curf is 
 \ less than the veneer; how thick is the veneer in terms of an inch, supposing 
 the flitches of mahogany to be wrought 4 inches thick ? 
 
 [75.] I send a piece of rosewood 8 inches thick to be cut as above ; what 
 proportion is each veneer of the whole log, supposing the outside flitches to have 
 been cut off, and the log left parallel and square ? 
 
 DECIMAL FRACTIONS. 
 
 It has been seen that however valuable fractional calculations may be, 
 there is great labour and trouble in all cases of addition and subtraction where it 
 is necessary to bring the fractions to a common denominator ; particularly where 
 many fractions are in the operation. To facilitate these calculations, the system 
 of Decimals was invented; a system of immense value, affording facilities in 
 calculation few would believe who do not understand them ; and of such daily 
 Supp. , D 
 
26 
 
 PRACTICAL 
 
 MATHEMATICS, MENSURATION, &C. 
 
 increasing use — a use probably to extend to our coinage, and weights, and mea- 
 sures — that we trust our readers will give their most earnest attention to this 
 branch of practical mathematics. 
 
 (41) Decimal fractions are those whose denominators are always 10, or some 
 product of 10 multiplied into itself, or (16) power of 10, as 10, 100, 1000, 
 10,000, &c. &c. 
 
 (42.) The place where the hgure 1 in the denominator would be, if fractionally 
 expressed, is denoted by a point ( . ), usually called the decimal point ; thus the 
 following fractions are expressed decimally as under, 
 
 2 9 9 4 7 2 3 5 7 3 
 
 10' 100' 1000' 10000' 
 
 or .2, .99, . 472, . 3573. 
 This saves the trouble of drawing a line, and writing the fraction in full. 
 So also U-^ = -W = 14.4. 137T4fo- = HU^ = 137.45. 
 
 (43.) Should there be more cyphers in the denominator than there are figures 
 in the numerator, then the decimal is expressed by adding cyphers to the left 
 of the numerators to make up the surplus, the point ( . ) being always supposed 
 to represent the figure 1. Thus, 
 
 Too — •^^' 1 000 0 — -0053, 100000 — • 01007. 
 
 (44.) The figures to the left of the decimal point (.) are integers, and are 
 always treated as such, without regard to those which follow; those to the 
 right are decimal fractions of such integers in proportions as described above. 
 Thus 19.7 = 19^, 21-1^0% = 21.33, 44^^ = 44.07, 3^0^^ = 3.00017. 
 
 (45.) The value of every figure in a decimal expression becomes less by one- 
 tenth as it is placed to the right ; every figure in a decimal expression is ten 
 times greater than the one following, or to the right of it ; and ten times less 
 than the preceding one to the left of it. 
 
 Thus .5, .05, .005, .0005 are respectively 
 
 5 5 5 nnrl 5 
 
 10' 100' 1000' 10000* 
 
 The decimal .9763 is + + -f twoo- 
 
 All these are divisions of the integer or unit, or the first figure to the left of 
 the decimal point (.) 
 
 (46.) If we add cyphers to the right of a decimal, it does not affect its value ; 
 
PRACTICAL MATHEMATICS, MENSURATION, &C. 
 
 27 
 
 thus .7, .70, .700, are all of equal value, because they represent the several frac- 
 tions i^nnj, all of which fractions are equal in value (11). 
 
 So if it be necessary to express decimals as fractions, it is clear they may 
 be converted into those of a common denominator, by adding cyphers (to the 
 right of course), both to the numerator and denominator. 
 
 ADDITION OF DECIMALS. 
 
 (47.) To add decimals together, place the figures so that those of the same 
 denomination may stand under each other, and the points of course will be 
 always in the same line ; add them together as in ordinary arithmetic, keep the 
 decimal point always in the same relative place, and if any numbers are carried 
 beyond the decimal point to the left in the course of addition, they then become 
 integers, and are treated as in common addition. 
 
 Add together .9, .17, .023, .7302. 
 
 Now these, as above stated, if considered as fractions reduced to a common 
 denominator, would be -^So + roinyo + ToTO + "nnnfo' and the addition 
 would be as they are all represented by one common denominator. 
 
 .9000 + .1700 + .0230 + .7302 = 1.8232. 
 
 Or without the cyphers to the right hand, which, as has been shewn above, 
 are superfluous, 
 
 .9 + .17 + .023 + .7302 = 1.8232. 
 
 Perhaps it may be well here to remark, that a species of numeration analogous 
 to that of integers is clearly to be derived from what has been said : thus, the 
 first figure in any sum is the unit; this place decimally is taken by the point ( • ); 
 the first place to the left of the unit represents the tens, the first to the right of the 
 decimal })oint the tenths ; the second to the left hundreds, the second to the 
 right hundredths, &c. &c. ; the value of this system is that the whole number 
 and the decimals may be so readily added or subtracted, without conversion of 
 the integral part to improper fractions, or without assuming the fractional form, 
 id est, the form of numerator and denominator. 
 
 SUBTRACTION. 
 
 (48.) Subtraction is performed in exactly an analogous manner : thus, place 
 
 D 2 
 
28 
 
 PRACTICAL MATHEMATICS, MENSURATION, &C. 
 
 the figures under each other, being very careful always to keep the decimal 
 
 points under each other, and proceed as in common arithmetic. 
 
 Thus, from 7.850' take 3.947. 7. 856 
 
 3 947 
 
 3.909 
 
 If there be more figures in the first than in the second sum, supply the defi- 
 ciency either by adding cyphers to the right, or suppose them to be added ; they 
 then, as above stated, (45) and (46), become converted to a common denominator. 
 Thus from 71.5 subtract 5.0732. Fractionally expressed these sums are 
 and TFofI, and converted to a common denominator become ^^^nnP and 
 HiM' and of course 71.5000 
 
 5.0732 
 66.4268 
 
 Or without the cyphers 71.5 — 5.0732 = 66.4268. 
 
 Great care must be taken, as in Addition, to keep the decimal points under 
 
 each other. Thus from 3402.005 take .006679. 
 
 Then 3402.005 
 
 .006679 
 
 3401.998321 
 
 MULTIPLICATION. 
 
 (49.) Multiplication of Decimals is performed exactly as that of integers; 
 afterwards count how many decimal places there are in the multiplier and mul- 
 tiplicand both together, and cut oft the same number of figures from the end of 
 the product by a point ; those to the left of the point will be the integers, and 
 those to the right decimals. 
 
 Thus, multiply 71.5 by 6.3. 715 
 
 63 
 
 2145 
 4290 
 
 45045 
 
 Now as there are two decimals in the multiplier and multiplicand, one in 
 each, we must point off* two from the end of the product, which will be 450.45. 
 The reason of this is, that if converted into fractions they would be, (42), 
 X Yo, and of course = ; this converted into a mixed number, (26), is 
 
 450 and -^^^ (42) = .45. Therefore the product is 450.45. 
 
PRACTICAL MATHEMATICS, MENSURATION, &C. 
 
 29 
 
 (50.) If there be not enough figures in the product to make up the number 
 of decimals in the multiplier and multijilicand taken together, there must be as 
 many cyphers added to the left of the product, so that the whole number of 
 decimal places may be made up. See (45) and (46). 
 
 Thus multiply .256 by .107. Then: 256 X 107 = 27392. 
 Now here are only five figures altogether, whereas there are six in the multiplier 
 and multiplicand ; I must therefore add one cypher to the left, and place the 
 decimal point before it, and the product will be .027392. 
 
 DIVISION. 
 
 (51.) Division of Decimals is performed exactly as in integers, and the 
 
 decimal places in the quotient are found by pointing ofi as many figures from the 
 
 end of the quotient, as the number of decimals in the dividend exceeds the 
 
 number of decimals in the divisor. 
 
 Thus, divide 450.45 by 71.5. 715) 45045 (63 
 
 4290 
 
 2145 
 2145 
 
 Now as there are two places of decimals in the dividend, and only one in the 
 divisor, I point off the difference, namely, one figure, and the (quotient is 6.3. 
 The reason of this will be seen by referring to (49). If 71.5 X 6.3 = 450.45, it 
 is clear 450.45 71.5 = 6.3. Now we have seen that the number of decimals 
 in the product was found by adding the number of those in the multiplier and 
 multiplicand together, namely 2 ; and therefore it is clear, that in the second ope- 
 ration (that of division) we must take the diflference, namely 1. 
 
 (52.) In the same way, if there are not figures enough to make the dift'erence, 
 
 cyphers must be added to the left of the quotient till the number is complete. 
 
 Thus, divide .864 by 14.4. 144) 864 (6 
 
 864 
 
 Now as there are three decimals in the dividend and only one in the divisor, 
 1 must have two decimals in the quotient, which will be .06. The reason is, that 
 if expressed fractionally (42) they will be -^^^^ and Yo^; and (40), ^^o^o 'nr 
 
30 
 
 PRACTICAL MATHEMATICS, MENSURATION, &C. 
 
 = UHH) X = 144000 ? and this reduced to its lowest terms (31) = which 
 is expressed decimally (42) .06. 
 
 (53.) The dividend must contain at least as many decimal places as the di- 
 visor ; if it does not, cyphers must be added to the right till there are enough. 
 Thus, divide 108 by .018. 
 
 Now 108 = 108.000, (46), and 108.000 -r .018 = 6000. 
 
 This may be proved by multiplication — thus, .018 X 6000= 108.000, or 108. 
 
 (54.) The same system must be pursued when there are remainders in the 
 division. 
 
 Divide 4 by 62.5. 625) 4000 (64 
 
 3750 
 
 2500 
 2500 
 
 Here are four decimals in the dividend, and only one in the divisor ; the quo- 
 tient must consist of three decimals, and is of course .064. 
 
 CONVERSION OF DECIMALS. 
 
 (55.) To convert a vulgar fraction to a decimal, divide the numerator by the 
 denominator, adding cyphers as before if necessary, and observing the rules of 
 the three preceding articles. 
 
 Convert ^ih to a decimal. 4) 1.00 
 
 ~25 or .25. 
 
 Convert fths to a decimal. 8) 3.000 
 
 375 or .375. 
 
 Convert xlr to a decimal. 4.000 ^ 125 = 32. 
 
 Now as there are three decimals in the dividend and none in the divisor, this 
 will be .032. 
 
 The reason of this is apparent from the rules given for the conversion of 
 fractions, (24, &c.), the decimal being a fraction whose denominator is 10, 100, 
 iOOO, &c. &c. 
 
PRACTICAL MATHEMATICS, MENSURATION, &C. 
 
 31 
 
 (56.) It often happens that fractions are not formed of numbers which are 
 wholly composed of tenths, hundredths, &c. ; in this case, on proceeding to di- 
 vision, we constantly find a remainder, and the division would go on for ever. 
 Thus, f rds cannot be made a perfect decimal. 
 
 3) 2.0000 
 
 .6666 &c. for ever. 
 
 Thus 1= 6) 5.0000 
 
 .8333 &c. 
 Thus ^ = ^ff^ = .121212, &c. 
 Thus Yh = &c. = .0148148148, &c. 
 
 These are called circulating or recurring decimals. 
 
 [57.] To convert a decimal into another of lower value, or of lower deno- 
 mination. 
 
 Consider the decimal as a fraction whose denominator is 10, or 100, or 1000, 
 &c. &c. &c., proceed as in (34, &c.) 
 
 Or, at length; multiply the quantity by the number of integers of its denomi- 
 nation contained in one of the inferior denominations, and (repeating this as often 
 as necessary) the products are the successive decimals required. 
 
 Thus, what is the value of .1925 of £1 ? 
 
 Then .1925 
 
 20 shillings in £1 . 
 
 3.8500 
 
 12 pence in a shilling. 
 
 10.2000 
 4 
 
 .8000 or 3s. lOd. and -j^ ofa farthing. 
 
 What is the value of .725 of a chain in terms of a foot ? 
 
 .725 
 
 66 feet in a chain. 
 
 4350 
 4350 
 
 47.850 
 
 12 inches in a foot. 
 
 10.200. Or 47 feet 10 inches i^ths of an inch. 
 
32 
 
 PRACTICAL MATHEMATICS, MENSURATION, &C. 
 
 [58.] To convert a quantity into a decimal of a greater denomination. 
 
 Proceed as in the reverse of the last article, and on the principles laid down 
 before in the conversion of fractions (24, &c.) 
 
 Or, more shortly. Divide the quantity by the number of integers of its own 
 denomination contained in one of the greater denomination, the quotient is the 
 decimal required. If there be any intermediate denomination, proceed in the 
 same way as shewn in fractions, till you arrive at the proper denomination. 
 
 What decimal of a foot is 3 inches? 
 
 Now 12 inches make a foot. 
 
 12) 3.00 
 
 .25. Suppose we consider the denomi 
 nation as feet, then its value as feet must be -jfath its value as inches. 
 
 What decimal of a pound is 16s. 7^d ? 
 
 Then, 4) 1.00 
 12) 7. 25~ 
 20) 10.6041(36 
 
 83020833 
 
 In effect, we find first what decimal of a penny a farthing is, then what deci- 
 mal of a shilling seven pence is with the previous decimal attached to it, and so 
 on till we arrive at the decimal of a pound. 
 
 We have now reached what may be considered the conclusion of the first branch of Practical 
 Mathematics, that of Fractional and Decimal Arithmetic. There is, however, one branch closely 
 connected with Decimal Arithmetic, — the subject of Involution and Evolution, — on which we 
 wish to treat before giving any questions for practice. We again earnestly entreat the attention 
 of our readers to this branch of our subject. It is easy to master with patience; and whoever 
 wishes to march with the times, must make up his mind to the exertion, or he may be sure that that 
 branch of the building business, the Carpenter's, will sink in the scale, instead of maintaining the 
 respectable position it always has supported. 
 
PRACTICAL RULES ON DRAWING, 
 
 FOR THE 
 
 OPERATIVE BUILDER AND YOUNG STUDENT IN ARCHITECTURE. 
 
 CHAPTER I. 
 
 INTRODUCTORY. 
 
 Much of the art of Drawing and Painting — one of the most useful, as well 
 as one of the most elegant, of acquirements — is to be attained by patient and 
 well directed industry, without the aid of a master. In evidence of this may 
 be adduced the fact that many ancient and modern painters, almost entirely 
 self-taught, have arrived at the greatest eminence in their profession. Within 
 the writer's recollection, many personal friends and acquaintance, men literally 
 sprung from nothing, by an assiduous application of the little leisure afforded 
 them from those pursuits on which depended their daily bread, have gradually 
 raised themselves to the highest rank among their professional brethren. Five 
 individuals, whose occupations were those of a groom, a bottle cleaner, a 
 waiter, a tailor, and a plough boy — three of whom are. now living, and all 
 of whom were personally known to the writer — by dint of their talent and 
 unflinching perseverance, placed themselves in the highest rank of artists, 
 and became the courted guests of some of the first nobility and gentry of 
 the country. The world too generally attributes the rise of such men to their 
 genius alone, rather than to that persevering industry and courageous battling 
 with difficulties, which in the long run must overcome every thing that opposes 
 them ; and though to assert that the talent or capacity of all men is equal 
 would be absurd, yet it is generally admitted, that a moderate capacity, 
 
 (Ov I) HAWING.) A 
 
2 
 
 PRACTICAL RULES ON DRAWING. 
 
 backed by steady industry, is more successful than brilliant talents without 
 perseverance. In the discourses on painting by Sir Joshua Reynolds, a work 
 the perusal of which cannot be too strongly recommended to every stu- 
 dent, occurs the following passage: "There is one precept in which I shall 
 only be opposed by the vain, the ignorant, and the idle. I am not afraid that 1 
 shall repeat it too often; — you must not depend on your own genius. If you 
 have great talentse industry will improve them ; if you have but moderate abili- 
 ties, industry will supply their deficiency. Nothing is denied to well-directed 
 labour, nothing is to be obtained without it. Not to enter into metaphysical 
 discussions on the nature or essence of genius, 1 will venture to assert, that assi- 
 duity unabated by difficulty, and a disposition eagerly directed to the object of its 
 pursuit, will produce effects similar to those which some call the result of natural 
 powers." Experience has proved the truth and value of these observations ; 
 the young student must not, therefore, suffer himself to be discouraged by any 
 want of success that may attend his first endeavours, neither let him be too much 
 elated by injudicious praise. The acquirement of Drawing is too much looked 
 upon as a mere accomplishment, and its usefulness undervalued ; the knowledge 
 of Painting is absolutely the acquisition of a new sense. In the writer's experience 
 as a teacher, it has been lamentable to witness how many highly gifted young people 
 have been debarred from learning this eminently useful art, by the folly and igno- 
 rance of their parents, whose constant cry has been : " It is of no use paying 
 money for lessons for my son or daughter, for they have no taste." It would be as 
 absurd to prevent a youth from studying the mathematics, from the fear he would 
 never make a Newton, as it is to withhold him from the pursuit of drawing, 
 because he may not, forsooth, be capable of becoming a Lawrence or a Turner. 
 In the present day, such facilities are to be met with for the acquirement of every 
 branch of knowledge, that ignorance can only be the consequence of idleness. 
 Few individuals are so constituted but that, by proper direction and diligent 
 application, they may, by their own exertion, obtain such a fund of information, 
 as will fit them to pass through the world as useful and well-informed members 
 of society. It is our task to furnish the means by which those totally unac- 
 quainted with drawing may, by careful application, teach themselves this art. 
 
 The art of Drawing is entirely an imitative one; for the finest works unques- 
 tionably are those the truest to their prototype — Nature. It is necessary, in the 
 study of this, — as in the study of all other things to be well understood, — to 
 begin at the beginning. A picture consists of various individual parts or objects, 
 
PRACTICAL RULES ON DRAWING. 
 
 3 
 
 so combined as to form an agreeable whole, whether portrait, landscape, or 
 architecture ; and it is indispensable that the student understand and draw these 
 separate parts before he can combine them to form a picture. The first requisite 
 is correct outline — this necessarily requires a knowledge of perspective; next is 
 an acquaintance with the principles of light and shade, artistically termed chiaro 
 oscuro ; then a knowledge of the combination of colours ; and, lastly, the prin- 
 ciples of composition. We shall, therefore, divide our subject into four parts : — 
 1st. Outline, which will embrace elevation and perspective drawing; 2nd. Light 
 and Shadow ; 3rd. Colour ; and 4th. Composition. Though the main object 
 of this treatise is to convey information for executing drawings of buildings, it 
 is proposed to touch cursorily on landscape in combination with them. The 
 limits of the work will not admit of entering into all the minutiae of detail ne- 
 cessary to fully carry out the writer's ideas ; he proposes, therefore, to furnish 
 information sufficient to enable the student to comprehend the leading principles 
 of each division of his subject; referring him from time to time to such works and 
 examples as have more deeply entered into their consideration. 
 
 All objects are represented in outline by lines, either straight or curved, or 
 by a combination of straight and curved lines together; and the first step towards 
 the acquirement of drawing, and more particularly of that branch of which it is 
 our province to treat, is to attain a knowledge of certain established forms, both 
 regular and irregular, to learn their definitions, and to become acquainted with the 
 rules for drawing them. This first necessary preliminary to the study of drawing, 
 is called Practical Geometry ; but as the subject is already treated of in the 
 preceding Numbers of this Publication, it would be superfluous to go again over 
 the same ground; the student must therefore make himself thoroughly acquainted 
 with that portion of this Work devoted to the study of Practical Geometry. 
 
 There are two different ways of representing Architecture by drawing ; — that 
 of elevation drawing, representing the objects geometrically; and perspective 
 drawing, representing the objects as they appear to the spectator from different 
 points of view. Perspective representations convey a far better notion of objects 
 than elevations, and are consequently used by architects, as examples, to shew 
 the appearance of their designs in execution ; but they are not of much practical 
 utility. In perspective drawing, things are represented as they really appear; 
 but strange as it may seem, no form viewed at an angle, save that of a sphere, is 
 ever seen in its positive shape ; and as every line of an object drawn perspec- 
 
4 
 
 PRACTICAL RULES ON DRAWING. 
 
 tively changes its size according to its distance, the proportions of the various 
 parts of a building represented by a perspective drawing would not be available 
 for practical purposes ; whereas in elevation or geometrical drawing, every thing 
 is represented in just proportion, either of its real size, or according to a scale. 
 The student must be aware, that if an elevation or geometrical drawing of any 
 very large object were to be made of the real size, it would be difficult, if not 
 impossible, to procure paper and instruments of sufficient magnitude ; it is usual 
 therefore, before commencing any drawing, to arrange a scale to draw by, so that 
 all the parts may have their due relative proportion one with another, be the size 
 of their representation what it may ; the proportion or size of the scale depending 
 on the judgment of the draughtsman, or fixed on by his employer. To represent 
 a building of the real size of 100 feet, on a scale of one inch to a foot, would 
 require all the necessary tools of a very large size; consequently it would be better 
 to choose a smaller scale, say of ^ or even ^ of an inch to a foot. A scale is 
 commonly given at the bottom or side of any mechanical drawing, to the 
 extent of from 10 to 50 or 100 feet, according to circumstances, for the con- 
 venience of measurement in working, and for the advantage of ascertaining any 
 proportions when the drawing is completed. By referring to the plate of the 
 elevation of the principal front and plan of the Athenaeum Club House, in 
 Part I. of this Work, the student will find the scale from which it has been 
 drawn, and by which he may ascertain the height or width of any portion of the 
 representation. 
 
 Before a perspective drawing of any object can be made, it is necessary, 1st, 
 to have a ground-plan of the building ; 2ndly, elevations of those fronts to be 
 represented ; and, 3rdly, sections of all the various projections, such as cornices, 
 architraves, mouldings, &c., with their exact measurements. Much less of plan- 
 drawing is required for the purpose of drawing the exterior of a building already 
 constructed, than is necessary for the knowledge of how to construct it ; in fact, 
 all that is required is the outline of the outer form of the building, carefully 
 marking all the projections, recesses, spaces for doors and windows, &c. In plan- 
 drawing for construction, the whole of the interior must be carefully laid down, 
 the thickness of walls, doors of communication from one room to another, fire- 
 places, &c. &c. &c., and this on every floor, from the basement to the roof. It 
 must be borne in mind that all measurements of height, both in geometrical and 
 perspective representations, must be made perpendicularly ; or, as it is more 
 
ON OUTLINE. 
 
 5 
 
 familiarly understood, by plum lines ; and the measurement of the width of all 
 objects, horizontally, or at right angles to the sides of the building, or to lines 
 parallel with them, as the doors, windows, &c. &c., which are perpendicular. 
 Supposing the student to require to measure the distance between two perpen- 
 dicular lines ; as for instance, to get the width of a door or window. In Fig. 1. 
 are two perpendicular lines, A, B ; C is a horizontal line between them. D is a 
 line drawn obliquely between them, and consequently does not give the correct 
 width. The greater the angle made by the oblique line with the horizontal line, 
 the greater would be the inaccuracy ; for, as a horizontal line is the shortest 
 distance between two perpendiculars, the more any line between them diverges 
 from the horizontal direction, the greater must be its length. In mechanical or 
 architectural drawing, the eye must not be trusted ; and nothing can be depended 
 on but actual measurement. Sections, also, of any parts of a building must 
 always be made vertically or by plum lines; as the section of a moulding of any 
 kind, made in an oblique direction, would materially alter its form. 
 
 CHAPTER 11.— DIVISION 1. 
 
 ON OUTLINE. 
 
 Plans and Elevations. 
 
 Before commencing any mechanical drawing, the learner must provide himselt 
 with a proper drawing-board on which to strain his paper ; taking care that the 
 bottom edge of his board be perfectly straight, and one of the sides exactly 
 perpendicular to it, or at a right angle.* He must also provide himself with a T 
 
 * The best material for making drawing-boards is deal, from that wood taking the paste better than any 
 other. The surface on which the paper is to be laid should be perfectly level, as the slightest warping of the 
 board will interfere with the straightness of the lines. Should the learner be unacquainted with the manner 
 of stretching or mounting his paper, — a most essential operation, — the following directions will be found 
 useful ; and it is better to mount the paper double, as in laying on large washes of colour it obviates, in a 
 
6 
 
 PRACTICAL RULES ON DRAWING. 
 
 square and a pair of compasses or dividers. The T square is a straight flat rule, 
 having at one end a cross piece attached to it, exactly at a right angle, the cross 
 piece being made thicker than the rule inserted into it, in order that it may slide 
 along the edge of the board, the rule always preserving the same direction. 
 (Fig. 2.) The straight rule A, the two sides of which, 2 2, are parallel, is so 
 placed in the cross piece B, as to be perpendicular to the plane 1. (Fig. 3.) A, B, 
 C, D, represent the four sides of a drawing board, of which the side B is perpen- 
 dicular to the bottom A. If the flat part of the T square, 1, be placed against 
 the edge A of the drawing board, the sides of the rule 2 2, will be perpendicular 
 to the edge A; and if the rule be slid along the edge of the board, and lines ruled 
 from either edge 2 2, as at a a a, these lines will be all parallel to each other, and 
 perpendicular to the line A of the board. In like manner, if the edge 1 of the T 
 square be placed against the side B of the drawing board, slid along it, and lines 
 ruled from either edge 2 2, as at Z» 6 b, these lines will be perpendicular to the side 
 B of the board, parallel to each other, and to the bottom line A. It is 
 seen, then, that the lines a a a are perpendicular to the line A ; and the lines 
 b b b being parallel to the line A, the lines a aa are also perpendicular to them. 
 The edge of the T square 1, therefore, being placed against the edge A of the 
 drawing board, and slid along, perpendicular lines may be drawn on any part of 
 the board from the edge 2 of the T square ; and the edge of the T square being 
 placed against the edge B of the board, will enable you to draw horizontal lines 
 wherever they may be required. The T square is the readiest and most conve- 
 nient instrument for drawing all horizontal and perpendicular lines. 
 
 As a first essay, let us suppose we have to draw the plan, from which to make 
 an elevation of a cottage standing on a parallelogram of 20 feet by 10, to be 
 
 great measure, the paper rising in ridges, or, as it is familiarly called, cockling. Lay on a flat board, or table, 
 a sheet of paper the size of the drawing required, and a second piece about one inch larger every way ; 
 saturate both pieces with water by gently sponging them, and then lay the smaller sheet on the larger one, as 
 B on A, (fig. 4), wiping the wet off the surface of the paper with a dry cloth ; then, with a large hog's-hair brush, 
 cover the whole exposed surface of the paper with paste made of flour and water only. When thoroughly 
 pasted, remove the upper sheet B, and place the sheet A, the pasted side downward, on your drawing-board, 
 wiping it carefully with a cloth to prevent any bubbles of air remaining underneath ; this done, place the sheet 
 B, the pasted side also downward, in the middle of the sheet A, wiping it gently with a cloth, so that every part 
 of the upper piece of paper is in close contact with the under piece ; lay it flat till quite dry. It will be obvious 
 to any one, on a little consideration, that when the drawing is completed, it can be cut from the board with 
 a. penknife, the edge only of the under sheet, which projects beyond the drawing, adhering to the board. 
 
ON OUTLINE. 
 
 7' 
 
 drawn on a scale of ^ of an inch to a foot.* First draw a horizontal line at the 
 bottom of your paper, on which mark off 10 spaces of ^ of an inch each, (fig. 5). 
 marking them 1, 2, 3, 4, 5, 6, 7, 8, 9, 10; this will answer as a scale for the first 
 10 feet — and continue it 5 at a time on to 20 or 30, marking them 15, 20, 25, &c. ; 
 this done, draw a long horizontal line A B at the lower part of the paper, and 
 measure off" on it 20 feet from the scale, as at a h, from each of which points draw 
 a perpendicular line, and measure off on each 10 feet of the scale, as at c and d; 
 join c and d, and you will have the outer boundary of the parallelogram on which 
 the cottage stands. The walls of the cottage are one foot thick — a distance of 
 one foot of the scale must be measured on each of the lines forming the paralle- 
 logram, as at g, h ; from each of the points y'and h perpendicular lines must 
 be drawn, and from the points e and g horizontal lines must also be drawn, 
 forming, by their intersection with the lines drawn from f and h, a second paral- 
 lelogram within that first drawn ; the space between the two parallelograms 
 representing the thickness of the wall. In the middle of the front of the cottage 
 is a door four feet wide ; and midway between each side of the door and the sides 
 of the cottage, a window three feet wide. If eight feet of the scale be measured 
 off from each side of the plan a and b, as at i i, it will leave a space between 
 equal to four feet, to represent the position of the door ; from each of the points 
 i i a perpendicular line must be drawn, to meet the line of the inner parallelogram ; 
 from each side of the door and the sides of the cottage, a distance of two feet and 
 a half must be measured off", as at k kkk; this will leave two spaces of three feet 
 wide each, which will represent on the plan the positions of the windows ; from 
 each of the four points marked k, a perpendicular line must be drawn to meet the 
 inner line, denoting the thickness of the wall. This completes, according to the 
 directions given, the ground-plan of a cottage containing one door and two 
 windows. 
 
 We have here drawn more than is absolutely necessary for the mere elevation 
 of the front and side of the cottage we are about to give directions for; but it has 
 been considered desirable to proceed thus far in the directions for the foregoing 
 plan, as they embrace all that is essential for drawing any plan the parts of whicli 
 
 * The examples are for the most part made on a very small scale, for the convenience of printing ; so 
 long as they are sufficiently large to avoid any confusion in the references by letters and figures to the points 
 and lines, it is of little moment of what size they are given ; but as drawing with accuracy on a very small 
 scale requires considerable practice, the student is advised to make his own drawings on a scale three or four 
 times larger than the figures are here represented. 
 
8 PRACTICAL RULES ON DRAWING. 
 
 can be represented by right lines; for, in the same manner that the length, width, 
 and position of what is here laid down have been accomplished, the length, width, 
 and position of any other parts may be represented. The width and position of 
 any partition walls, in their relative position with the outer wall, being known, 
 they might easily be drawn ; as also any back door or doors of communication 
 from room to room ; their width and distance from any part already drawn being 
 described, may be represented in the same manner as were the positions for the 
 door and windows in the example given. It is usual, after the outline of a plan 
 is completed, to shade that portion of it which represents the wall, leaving the 
 spaces for the door and windows white. 
 
 From the plan and description of the dimensions of the different portions of 
 the cottage to be represented, we will now proceed to draw the elevation (Fig. 6.) 
 First draw the horizontal line A, B, on which mark off the points indicating the 
 positions of the sides of the cottage, and the door and windows on the plan 
 marked 1, 2, 3, 4, 5, 6, 7, 8, from each of which points raise perpendicular lines. 
 On each of the two extreme lines representing the outer lines of the wall of the 
 cottage must be measured off the height of the wall, twelve feet, at 9 and 10, 
 and a horizontal line must be drawn between them ; this represents the upper 
 line of the wall from which the roof springs. The two middle perpendicular 
 lines drawn from 4 and 5, represent the sides of the door, which must be made 
 seven feet high; measure off on either of the lines seven feet, as at 1 1 or 12, and 
 from either point draw a horizontal line between the two perpendiculars ; this 
 completes the elevation of the doorway. The lower parts of the windows are 
 • each of them three feet from the ground, and the windows five feet in height. 
 
 On either of the lines 2, 3, 6, or 7, measure off three feet for the distance of the 
 windows from the ground, as at 13, 14, 15, or 16, through either of which points 
 between the lines 2 and 7, draw a horizontal line. On any one of the same lines 
 2, 3, 6, or 7, from either of the points 13, 14, 15, or 16, measure ofi'a distance of 
 five feet, for the height of the windows, as at 17, 18, 19, or 20, and through either 
 of them draw another horizontal line between the lines 2 and 7. The horizontal 
 lines between 13 and 14, and 15 and 16, form the bottom lines of the windows; 
 those between 17 and 18, and 19 and 20, marking the upper lines of the windows. 
 Having accomplished the drawing of an elevation of the wall of the front of a 
 cottage with a door and two windows, let us now proceed to the roof. The 
 roof, as will be seen by the side elevation, is a gable roof; now as all measure- 
 ments are taken perpendicularly, the height must be measured in a perpendicular 
 
ON OUTLINE. 
 
 9 
 
 direction from the top of the roof to the ground, the oblique lines or slopes of 
 the roof being left to be shewn in the side elevation. Measure off on the 
 perpendicular line from the point 8, eighteen feet, as at 21, through which draw 
 a horizontal line, carrying it on each side a little beyond the outer lines of the 
 wall of the cottage. The roof of the cottage projects on either side one foot 
 beyond the wall of the cottage; a space of one foot must therefore be measured 
 oft" on either side of the lines 1, 9, and 8, 10, as at 22, 23, through which draw 
 two perpendicular lines from the top of the roof, drawing each of them below 
 the upper line of the wall. The slope of the roof projects beyond the front of 
 the cottage, one foot perpendicularly below the top of the wall ; a distance of 
 one foot must therefore be measured off below the top of the wall on the line 
 9, 10, as at 24, and a horizontal line drawn through it betw een the perpendicular 
 lines drawn from the roof through the points 23, 24. Having thus completed 
 the front elevation of the roof, and with what has been before done, the complete 
 outline of the elevation of the front of a cottage, let us now proceed to draw the 
 elevation of one of the sides. 
 
 As in the elevation of the front, first draw a horizontal line A B (fig. 7), 
 on which measure off the depth of the cottage from the plan (10 feet), at the 
 extremities of which, 1, 2, raise two perpendicular lines twelve feet high, as at 
 3, 4, and draw a horizontal line between the two points.* This side, which 
 forms a gable of which the apex is eighteen feet from the ground, the point of 
 the gable situated exactly above the centre of the parallelogram just drawn, must 
 be found, from which the oblique lines to the points 3 and 4 must be drawn. 
 Draw a perpendicular line dividing the parallelogram into two equal parts ; (a 
 certain and quick mode of accomplishing this is, by drawing two diagonal lines, 
 from the points 1 to 4, and 2 to 3; the point of intersection of these diagonals is 
 the exact centre of the parallelogram, through which draw the perpendicular line,) 
 and on it measure off the height of eighteen feet ; this will be the point marking 
 the apex of the gable, from which draw the sloping lines of the roof, 5 4, 5 3. 
 The roof projecting one foot perpendicularly below the top of the wall, measuie 
 off on one of the sides, 3 1 or 4 2, one foot, as at 6, through which draw a 
 horizontal line a little beyond either side of the cottage, and continue the sloping 
 lines of the roof 5 3 and 5 4, till they meet this line at the points 7 and 8, which 
 will complete the elevation of the side front of the cottage. 
 
 * This is done in precisely the same manner as the elevation of the front wall of the cottage, (fig. 6.) 
 
10 PRACTICAL RULES ON DRAWING. 
 
 The foregoing examples are given with a view only of shewing the rules by 
 which a plan and elevation of a building may be drawn from description, or 
 what is nearly the same thing, from your own design ; and the subject chosen is 
 of the most simple kind, intended solely to illustrate the mode of drawing 
 rectangular forms by description ; the fewest possible number of lines have been 
 employed, to render the rules for drawing them clearly intelligible ; but simple as 
 are the forms and descriptions here given, they contain the essential information 
 necessary for drawing plans and elevations consisting of right lines. According 
 to the number of parts in the subject, many or few points from which to draw 
 the required forms may be necessary ; but be they ever so numerous, or ever so 
 few, the rules here given will enable the student to hnd them. Every line except 
 the mere outline of the different parts has been carefully avoided, to prevent 
 confusion, and for the same reason the chimney has been omitted ; but having 
 proceeded thus far, it is easy to add to what is already done, any parts that may 
 be required. Let us take a portion only of fig. 6, (which is all that is required 
 for our purpose), viz. the door, which we will say contains tw^o panels, the upper 
 one, two feet high by three feet wide ; the lower one, three feet six inches high 
 by three feet wide ; with a frame work round each of the panels six inches wide. 
 Let A, B, C, D, (fig. 8) represent the sides and top and bottom of the door, drawn 
 on double the scale of the preceding figures. On either of the lines D or B, from 
 the ground line, first measure off six inches for the frame work at the bottom of 
 the lower panel at 1, then a space of three feet six inches at 2, for the height of 
 the lower part, above that another space of six inches for the frame work 
 between the two panels at 3, and lastly a space of two feet for the height of 
 the upper panel at 4 ; this will leave a space of six inches between the point 
 
 4, and the top of the door, for the frame work above the upper panel ; from 
 each of these points 1, 2, 3, 4, draw a horizontal line to the opposite side of the 
 door ; then, on either of the lines A or C, measure off a distance of six inches, at 
 
 5, for the width of the frame work ; then three feet for the width of the panels at 
 
 6, which will leave a space of six inches between that point and the side of the 
 door, for the width of the frame work on this side of the panels ; from each of 
 these points 5 and (3, draw a perpendicular line from the top to the bottom of 
 the door, intersecting the horizontal lines at the points 7, 8, 9, 10, 11, 12, 13, 14; 
 by strengthening the horizontal and perpendicular lines between these points, 
 the drawing of the panels and frame work required is completed. In like 
 manner, suppose each window to contain nine panes of glass, if one of the sides 
 
ON OUTLINE. 
 
 11 
 
 be divided into three equal parts, and the top or bottom line also divided into 
 three equal parts, as in fig. 9, at 1 2 and 3 4, and horizontal and perpendicular 
 lines drawn from each of the points of division, the lines denoting the frame 
 work and the nine equal divisions will be found. In careful drawing, however, 
 this mere division into nine equal parts would not be sufficient, the wood-work 
 containing the panes of glass must also be drawn ; this is not attended with any 
 additional difficulty, it requires only care and attention. Let A, B, C, D, 
 represent the outline of the space for the window, and suppose the nine panes 
 of glass to be contained in a frame three inches wide, the bars containing the 
 separate panes being half an inch wide. The following figure (10,) is drawn on 
 a scale four times that of fig. 6. Measure off a space of three inches on 
 each line A, B, C, D, to represent the width of the outer frame, and draw the 
 lines to represent it precisely in the same way as was done in drawing the 
 thickness of the wall in the plan (fig. 5 ;) divide the inner parallelogram into 
 nine equal parts, as in fig. 9; on each side of the lines forming the divisions of 
 the nine spaces, measure off one quarter of an inch, and draw at that distance, on 
 either side of these lines, parallel lines from top to bottom and from side to side, 
 the intersections of these lines 1, 2, 3, 4, &c. giving the outline of the forms 
 required, half an inch wide^ 
 
 To the mere skeleton of the plan and elevation, (figs. 4, 5, and 6,) here 
 given, the student will perceive how easily any additions might be made, — with 
 what facility a chimney, portico, or any other object, might be added to the plan, 
 and from the plan to the elevation, — how any window or recess might be added 
 to the side elevation, — the divisions of stones on the walls, or those of slates or 
 tiles on the roof, &c. Sufficient is here considered to be done however for a 
 first attempt, and the representation of cornices, mouldings, and other ornamental 
 work, must be left for future consideration. 
 
 CHAPTER III. 
 
 Plans and Elevations — (continued.) 
 
 The figures given in Plate I., illustrative of the foregoing matter, contain no 
 drawing of any ornamental projections; and in the simple outline there given of 
 
 B 2 
 
12 
 
 PRACTICAL RULES ON DRAWING. 
 
 the windows and door, there is no more than the mere indication of the form and 
 position they would occupy traced on the wall by lines. To shew that the 
 windows or door are not on the same plane as the wall of the cottage, or in other 
 words that they form recesses, would be easily done in a shaded representation, 
 as the shadows, which in architectural drawings are laid in according to regular 
 rule, determine by their depth and width the thickness of the projections that 
 cast them. This is only mentioned incidentally, as we shall treat of hght and 
 shade separately. 
 
 In referring to any elevation-drawing in simple outline, of which several are 
 to be found in this Work, certain mouldings can only be represented by a series 
 of parallel lines at unequal distances, shewing the perpendicular widths of the 
 different parts of the moulding ; but it is impossible from these lines only, to 
 ascertain either the forms of the various parts, whether rectilinear or curvilinear, 
 or the extent of their projection ; it is therefore necessary, in order to become 
 acquainted with their forms and projections, to draw sections of them. If lines 
 be merely drawn round any part of a building, such as a door, window, recess, 
 &c., as in tig. 1, Plate II., we know that they represent the lines of a moulding, 
 with the width of its several parts ; but in order to shew their form and projec- 
 tion, a section must be drawn from description. Suppose the lines a & to repre- 
 sent a rectangular projection of two inches by a perpendicular height of one 
 inch, those of 6 c to represent a convex projection of If inch by if perpendicular, 
 and that of c </ a semicircular band of f of an inch diameter. First draw a per- 
 pendicular line A B, (tig. 2),* to represent the plane from which the moulding 
 projects, and at right angles with it draw the line 1 2, two inches long; draw the 
 parallelogram 1, 2, 3, 4, one inch perpendicular height, which will represent a 
 section of the upper projection. With a pair of compasses from the point 4, 
 with a radius of one inch and three quarters, describe the arc of a circle from 
 5 to 6, this gives the section of the convex moulding ; then with half the diameter 
 of the lowest projection, three eighths of an inch, from the point 7 describe the 
 semicircle 6, 8, 9, which gives the section of the bead, the whole forming a 
 section of the moulding, (tig. 1). It is of no moment whether a moulding consist 
 of many or few parts ; if the elevation be correctly drawn as to the width of its 
 several parts, and correct descriptions of the form and projection of the different 
 portions be given, any moulding can be drawn from them ; always supposing the 
 
 * The references to the figures throughout this Chapter must be understood to relate to Plate II. 
 
ON OUTLINE. 
 
 13 
 
 student to be acquainted with the names of the various forms of the projections, 
 and the rules for drawing them. It is therefore necessary, before proceeding 
 further, that the student make himself acquainted with the names of the different 
 forms, and the mode of representing them. 
 
 Mouldings are projections consisting of figures composed of curved and 
 straight lines; the curves vary considerably in their form, particularly those used 
 in the mouldings of Grecian architecture. The curves in Roman architecture 
 are all portions of a circle ; in Greek architecture they are portions of the ellipse, 
 parabola, or hyperbola. It has been already stated that all sections for drawings 
 should be made, or supposed to be made, perpendicularly in the case of 
 mouldings, at right angles with the perpendicular or horizontal lines representing 
 them in elevations. In illustration of the necessity for this, we will take certain 
 solids, and shew what a wide difference the form of the section assumes 
 according to the direction in which it is cut. In making a section of a cube, if it 
 be cut parallel to the plane of either of its faces, the form of the section will be 
 a square, (fig. 3); if the slightest deviation be made, so as to take an oblique 
 direction, it will take the form of an oblong, (fig. 4). If a cylinder be cut 
 longitudinally, or at right angles with either end, the section is a rectangular 
 parallelogram, (fig. 5). If it be cut transversely, or parallel with its circular 
 ends, the section will be a circle, (fig. 6) ; and if cut in an oblique direction, the 
 section will take the form of an ellipse, (fig. 7.) The sections of a cone shew 
 the greatest diversity of form, according to the directions in which they are made. 
 If a cone be cut, passing through the vertex to any part of the base, the section 
 is of the form of a triangle, (fig. 8). If cut transversely, parallel to the plane of the 
 base, at any point between the base and the apex, the section is a circle, (fig. 9). 
 If the section be made obliquely from side to side, the form is that of an ellipse, 
 (fig. 10). When a cone is cut through in a direction parallel to one of its sides, 
 the section is called a parabola, (fig. 11). When the cone is cut from its base, 
 in a direction so that a continuation of the cutting line would meet a continued 
 line of the opposite side of the cone, somewhere above the vertex, the form of the 
 section is an hyperbola, (fig. 12.) 
 
 All the forms made by these sections are met with in Architecture, and re- 
 gular rules are laid down for their construction. Rectilinear figures can be 
 drawn with a ruler ; all circles, or portions of circles, may be described with a 
 pair of compasses. There are various modes given in works on practical 
 geometry for drawing ellipses, — by means of the trammel, by fixing nails or pins in 
 
14 
 
 PRACTICAL RULES ON DRAWING. 
 
 the foci and drawing the curve with a string, by ordinates, &c. &c. For our pur- 
 pose, the finding certain points through which to draw the curve, by means of 
 the intersections of straight lines, will be found the most desirable, the more 
 especially when we come to treat of perspective drawing ; — certain rules for 
 describing them, by points of intersection, will be given in the directions for 
 drawing the curves of mouldings.* 
 
 1 . The rectilinear portion of a moulding, either above or below, is usually 
 termed a fillet ; the way of drawing these has been already explained, a b, figs. 
 1 and 2. 
 
 2. Any narrow semicircular portion of a moulding is termed ahead, cd, fig. 1. 
 
 3. A moulding similar to a bead of a large size, with a fillet above or below 
 it, is termed a torus. 
 
 4. When the contour of a moulding is convex, and the curve the segment of 
 a circle equal to or less than a cjuadrant, it is termed an ovolo ; if the contour 
 be convex, and the curve not a portion of a circle, but of some other conic sec- 
 tion, it is called Grecian ovolo. 
 
 To draw the ovolo : If the contour of the moulding be an exact quadrant, 
 the point of contact with the lines denoting the plane and the projection, will be 
 the centre from which to describe the arc of the circle, as at 4, fig. 2 ; but if the 
 curve be less than a quadrant, the point will not fall on either line. The two 
 
 * In drawing arches, the contour of mouldings, <S:c., the various curves produced by the different sections 
 of the cone occur so constantly, that the student would do well to make himself acquainted with that portion 
 of practical geometry. The study of practical geometry, in fact, is of such great utility, that he is strongly 
 recommended to make himself thoroughly master of it in all its branches. Many excellent works have been 
 written on the subject, but for those who have not thte advantage of a master, the " Complete Course of Practical 
 Geometry," by Major Gen. Sir C. Pasley, will be found most serviceable; it is barely possible to imagine any 
 one so dull of comprehension as not to be able, with moderate attention, to understand the problems, from the 
 simple and clear manner in which they are there described. In writing directions for describing curves, it is almost 
 impossible to avoid certain technical terms ; and, as the want of knowledge of these terms may be attended, in 
 our progress, with much inconvenience, a diagram of an ellipse, with certain lines and points, and their defi- 
 nitions, is here given at Fig. 13. A is the transverse axis, or longest diameter; B, a line perpendicular to it, 
 the conjugate axis, or shortest diameter ; C, the point of intersection of ^^the axes, the centre of the ellipse. 
 D D, the vertices, the two most distant opposite points of an ellipse, between which the transverse axis always 
 lies ; E E, points on the transverse axis called the foci, found by intersections of the arcs of circles, described 
 from the extremity of the conjugate axis, with the radius of the semi-transverse axis; any straight line passing 
 through the centre, and touching the curve on either side, is a diameter, as F, (the transverse and conjugate 
 axes are both diameters). Any line drawn from a diameter to the curve, parallel to a tangent, as G G, touching 
 the extremity of the diameter, is called an ordinate, as H H ; if an ordinate be continued through, to meet 
 the curve on the opposite side, it is called a double ordinate, as J. 
 
ON OUTLINE. 
 
 15 
 
 points being given, A and B, (fig. 14), between which the curve is to be drawn, 
 with the length of a straight line between these points as a radius, draw from 
 either point the arc of a circle, intersecting each other at C, from which point, 
 as a centre, with the same radius, describe the curve A 1) B, which is the required 
 form. 
 
 If the convex moulding be the quarter of an ellipse, the length of the pro- 
 jection will be the semi-transverse axis, the perpendicular height of the moulding 
 the semi-conjugate. Let A B, (fig. 15), be the length of the projection, and 
 B C the width of the moulding. Construct the parallelogram A B C D, and draw 
 the whole length of the transverse axis to E. Divide the semi-conjugate axis 
 into any number of equal parts, numbering the points of division from the bottom 
 upwards, as at 1, 2, 3, 4, 5, 6, 7. Divide the line D C (which is of the same 
 length as the semi-transverse axis) into the same number of equal parts as the 
 semi- conjugate B C, mnnbering the points of division from right to left, as 
 1, 2, 3, 4, 5, 6, 7. From each of the points on D C draw a line to the 
 vertex A of the ellipse, and from the opposite vertex E, through each of the 
 points of division on the semi-conjugate axis, draw a line to intersect those 
 already drawn at the points 9, 10, 11, 12, 13, 14, 15, through which points draw 
 a curve, which will be the contour required. It must be evident to the learner 
 that the greater the number of points, the more correct will be the curve. The 
 projection here given is much longer than the depth of the moulding ; but if 
 this were reversed the mode of drawing would be similar, the contour taking the 
 appearance of an upright ellipse, instead of a horizontal one. (See fig. 16.) 
 
 5. If the contour of a moulding be the exact reverse of an ovolo, that is 
 concave, instead of convex, and equal to or less than a quadrant, it is called a 
 cavetto, (fig. 17.) This is drawn in the same manner as the ovolo, (fig. 2), 
 with this difference only, that the central point 4, (fig. 2,) from which to draw the 
 curve, is found on the opposite side, as at a, (fig. 17). 
 
 6. When the contour of a concave moulding is a semi-ellipse, it is called a 
 scotia, and is drawn by the same rule as fig. 15, the divisions being made on 
 the opposite lines. By looking at fig. 15 upside down, the mode of drawing the 
 curve this way will be seen. 
 
 7. If the contour of a moulding be partly convex and partly concave, it is 
 called a cimatium. When the convex part of the contour projects beyond the 
 concave, it is called a cima reversa or ogee curve ; if the reverse be the case, 
 that the concave projects beyond the convex, it is called a cima recta. Let 
 
16 
 
 PRACTICAL KULES ON DRAWING. 
 
 A B, (fig. 8), represent the projection of a cima reversa, and B C the per- 
 pendicular height of the moulding. From A to C draw a right line, and divide 
 it into two equal parts at D ; from each of the points A and D, with the radius 
 A D, describe an arc of a circle intersecting each other at E, from which point 
 describe the arc A F D ; from the points D and E, with the same radius, 
 describe two arcs on the opposite side, intersecting each other at G, from which 
 point describe the curve D H C, which will complete the contour of the ogee or 
 cima reversa. The cima recta is drawn in a manner precisely similar, describ- 
 ing the concave portion at the top and the convex at the bottom. 
 
 If the curves of the contour be not parts of a circle, but portions of an ellipse, 
 they may be drawn by the same rule as that described in fig. 15. Let A B, 
 (fig. 19.) be the projection, and B C, the depth of a cima recta moulding, the 
 curve of which is formed by two quadrants of an ellipse. Construct the 
 parallelogram A, B, C, D, which divide into four equal parts. In the upper 
 division, to the left, describe the concave quadrant of an ellipse by the rule given 
 in fig. 15. In the lower division to the right, describe the convex quadrant of an 
 ellipse. The two quadrants will give the contour of the moulding required. By 
 reversing the order of drawing the two quadrants, putting the convex in place of 
 the concave and vice versa, the curve will be that of a cima reversa. 
 
 The foregoing are general rules for drawing the contour of mouldings, but the 
 student will find on examination of different buildings or representations of 
 them, that the form of the mouldings presents an almost endless variety ; our 
 space does not admit of giving more than the ordinary examples; a thorough 
 acquaintance with practical geometry will however enable the student to draw 
 any form of moulding. In addition to the directions already given for drawing 
 elliptic curves, the following, as a general rule for describing the curves of conic 
 sections in mouldings, when certain points are given, will be found useful. 
 Let A (fig. 20.) be the extreme projection of the curve, and B the point of 
 contact of a tangent B C, at the bottom of the moulding. Draw the line B D, a 
 continuation of the upper line of the lower fillet, to the length of the projection 
 at D, from which point draw a perpendicular line to A, cutting the tangent B C, 
 at the point C, which will complete the parallelogram A E B D. Continue the 
 line B E, to F, and make E F equal to E B. On the same line from the point 
 E, set up the distance from D to C, at G, making E G equal to D C ; draw 
 a line from A to G. Divide each of the lines A G and A C, into a like number 
 of equal parts, numbering the points of division on each line from the point A. 
 
ON OUTLINE. 
 
 17 
 
 draw lines intersecting those drawn from A C to B, at the points G, 7, 8, 9, 10, 
 which are the points through which to draw the curve. Fig. 21. represents a 
 curve drawn in the same manner, but of a different form ; the flatness or roundness 
 of the contour depending on the point at which the tangent B C intersects the 
 line A D. If the distance from A to C be less than one half of the line A D, 
 the curve is elliptic ; if the distance be exactly one half, the curve is parabolic ; 
 if more than half, it is hyperbolic. 
 
 In architraves, cornices, capitals of columns, &c., when the mouldings project 
 from the side the same as from the front, the projection and form of the mould- 
 ings are shown in elevation drawings, by representing a section of them on the 
 side. In drawing the elevation of a column, the form of the moulding of the 
 capital which projects equally all round, is represented on either side of the 
 lines denoting the diameter of the shaft ; so likewise in architraves and cornices, 
 the horizontal lines representing the perpendicular height of the mouldings, are 
 carried beyond the line of the wall, the extent of the projections is marked on 
 the several lines, and the contour of the mouldings is drawn according to 
 description from the projecting points. 
 
 From the information contained in the preceding pages, let us proceed to 
 draw a plan with the front and side elevation of a portico and steps, 
 
 Letaa(fig. 1, Plate III.), represent the front wall of the building (or fascia) with 
 a space b for a door-way.* 
 
 c c. The plan of the outer walls from which the side walls of the portico 
 spring. 
 
 d d. The side walls of the portico, with spaces for windows in each. 
 e e. Plan of the pilasters and plinth. 
 
 / f. Plan of the pilasters and plinth at the back, half the depth of those in 
 front. 
 
 g. Plan of the stairing. 
 
 h h. Plan of the coping at each side of the steps. 
 
 From the above plan and the following description, proceed to draw the front 
 elevation, (fig 2). First draw a horizontal line A B, and an upright line C D 
 perpendicular to it, as lines of projection on which to make all the required 
 
 * The dimensions of the various parts are not specified, as they may be found by measurement from the 
 scale at the bottom of the plate, a quarter of an inch to a foot. 
 
 On Drawing. ^ 
 
18 
 
 PRACTICAL RULES ON DRAWING. 
 
 measurements.* On the line A B mark the distance between the outer lines of 
 the wall cc of the plan, at 1, 1, and from each of the points draw a perpendicular 
 line through the base line of the building E, to represent the outer lines of the sides 
 of the wall ; on the line C D, from the base line E, put up the distance at 2 ; from this 
 point, which determines the height of the lines 1,1, draw a horizontal line shewing 
 between these lines the top line of the upper flag-stone of the steps, and the upper 
 line of the outer walls. On either side of the points 1, 1, on A B measure off the 
 width of the plinth and pilaster at the several points marked 3, 4, through each 
 of which points draw perpendicular lines. On the line C D measure off the 
 height of the plinth at 5, that of the cavetto moulding at 6, and that of the fillet 
 at 7 ; from each of which points draw horizontal lines : that from 5, between the 
 lines 3, 3, gives the drawing of the plinth of the pilaster; that from 6, the perpen- 
 dicular height of the cavetto moulding; and that from 7, the perpendicular 
 height of the bead : these parts to be drawn as before described in treating of 
 mouldings. From the point 8 on the line C D set up the height of the shafts of 
 the pilasters, from which draw a horizontal line, which will complete the 
 drawing of the shafts where it passes between the perpendicular lines drawn 
 from the points marked 4 on A B. Above the point 8, measure off a space for 
 the height of the capital of the pilaster at 9, from which draw a horizontal line. 
 Between the points 8 and 9 on the line D C, mark off the perpendicular 
 heights of the various parts of the capitalf at 10, 11, 12, 13, 14, 15, and from 
 each of these points draw a horizontal line entirely across the elevation, to give 
 the height of the various parts on each pilaster ;1: their projections must be 
 measured off on either side, from the continuation of the lines representing the 
 sides of the shaft as at a, b, c, &c., and the contour of the mouldings drawn from 
 point to point. From the point 9 set up the whole height of the superstructure 
 at 16, and between these points 9 and 16 mark off the perpendicular heights of 
 all the intermediate parts, carrying a horizontal line from each across the 
 elevation. The sides of this superstructure are perpendicularly over the outer 
 
 * This mode of using lines of projection for the width and height of the various parts to be drawn will be 
 found generally useful, as it obviates the necessity of pricking holes with the dividers, or making a multitude 
 of marks on the drawing itself. 
 
 f In the following chapter, which will contain directions for drawing the different orders of architecture, 
 the various parts of a column, capital, &c., will be described : it must suffice here to copy the forms given. 
 
 X The lines in the example are not carried right across, the right hand side has all the lines left that 
 were employed to make the elevation ; on the left hand side they are erased. 
 
ON OUTLINE. 
 
 19 
 
 lines of the pilasters, these lines must therefore be continued through to the top, 
 and the length of the projection of the several parts set off on either side at the 
 points d, e, f\ &c., and the forms of the different mouldings drawn from them. 
 On the line A B, from the plan mark the position and width of the doorway at 
 17, 17, from each of which points raise two perpendicular lines;* put up the 
 height of the doorway on C D, at 18, either its height from the base line E, or 
 from the top of the upper flag of the steps at 2, and draw a horizontal line 
 from the point 18 between the upright lines of the sides ; this completes the front 
 elevation above the steps. On the line of projection A B, mark from the plan 
 the several points 19, 20, for the square post and base, forming the front of the 
 side walls of the steps, carryingup a perpendicular line from each, and on the line 
 of projection C D, measure off the heights of the different parts, between the line 
 E and the point 21 ; draw horizontal lines from each, intersecting the perpendiculars 
 from 19, 20, measuring the projecting parts from the side lines of the post drawn 
 from 20, 20. Mark the height 22 on C D, where the sloping part of the coping meets 
 the pilaster, and draw a horizontal line across ; find the width of the wall, from 
 the points 23 on A B ; the lines of the coping on either side, which in the elevation 
 drawing come perpendicularly over the sides of the part drawn from 20, 20 ; above 
 the point 21, and below 22, measure off a space the width of the projection of the 
 coping 24, 24, on C D, and from them draw two horizontal lines ; where the upper 
 one intersects the outer lines of the coping, and the lower one the inner lines, will 
 be the points for drawing the oblique lines of the coping. Divide the space between 
 the base line E and the point 2 on C D, into six equal parts, from the points of 
 division draw horizontal lines to represent the height of the steps, shewing by the 
 perpendicular lines drawn from 25, 25, on A B, that the first step has the greatest 
 projection, as shewn in the plan at D. Set up the height of one step above the 
 point 2, on C D, and from it draw a horizontal line between the sides of the 
 doorway, to represent one step within, and the front elevation is completed. 
 
 In drawing the side elevation (fig. 3.) proceed precisely as in the front eleva- 
 tion. The line of projection C D is placed between the two elevations in order 
 that the perpendicular heights of the same parts represented in fig. 2. may serve 
 
 * The lines are represented in the plate, drawn from the points on A B, right through, to shew clearly the 
 way in which the distances are got in the elevation ; but it is not necessary that the lines should be drawn 
 beyond their absolute length. The points being fixed on the line of projection, the T square may be placed 
 against them, and the lines drawn only where they are intended to represent the object, as is shewn on the 
 left side of the door, and in fact on the whole of the left side of the elevation. 
 
 c 2 
 
20 PRACTICAL RULES ON DRAWING. 
 
 for fig. 3. Thus in executing a drawing of arches, columns, &c. of the same 
 form and size, one line of projection will serve for all. In order to avoid confusion 
 between the two elevations, the references to all those parts that relate to the side 
 elevation only, are put, both in figs. 1 and 3, in capital letters. It is unnecessary 
 to go over again the pilasters, superstructure, &c. ; we will proceed at once to those 
 parts distinct from what has been described, fig. 2. Set off on the line of projection, 
 A the face of the wall of the building, B the projection of the wall on which 
 the portico stands, C the projection of the base of the post of the side wall of 
 the steps, D the projection of the forwardest step ; through each of these points 
 draw perpendicular lines to their respective heights determined by the lines from 
 the points denoting the heights of the different parts on D C, carried through 
 from the front elevation. Mark on the line of projection the points E and F, 
 the position and width of the window, through each of which points draw two 
 perpendicular lines. Mark on the line C D, at G, the height of the bottom of 
 the window, and at H the height of the perpendicular lines of the sides, and draw 
 horizontal lines from each of the points ; from the centre O of that part of the 
 upper line, passing between the lines drawn from E and F, with a radius of half 
 the width of the window, describe the semicircular top ; on either side of the points 
 E and F measure off the width of the moulding round the window J K, from which 
 draw up the perpendiculars ; below the point G at L place the width of the mould- 
 ing at the bottom of the window, from which draw a horizontal line. From the 
 centre O describe the outer semicircle of the top of the window. Measure off 
 on the line C D, from the point 9, the depth of the key-stone at M, from which 
 draw a horizontal line, mark the width at the top and bottom, and draw the inclined 
 lines.* The points for all parts of the side elevation similar to those drawn in the 
 front elevation, fig. 1, are marked on the horizontal line of projection, and figured 
 the same as those in fig. 2 ; but the lines have been erased. The perpendicular 
 heights are already marked on the line D C, and the horizontal lines determining 
 the height of the various parts in fig. 2, are carried through to fig. 3, so that the 
 side elevation may be completed without additional directions. 
 
 The space allotted to this portion of the work, (Drawing), is unavoidably so 
 limited, that plans and elevations of all the different parts of the interior of a 
 building would occupy more than could be spared from other necessary subjects; 
 
 * There are rules for drawing tlie inclination of the lines of stones surrounding arches, which are treated 
 of in Practical Geometry. 
 
ON THE GRECIAN AND ROMAN ORDERS. 
 
 21 
 
 neither is it exactly the province of the writer to enter into all such minutiae ; it 
 belongs rather to the teacher of building ; the information already given being 
 sufficient for drawing the plan of any part of a building, the form and position of 
 which is understood. To those however who may be desirous of exercising 
 themselves on the preceding instructions, ample opportunity is afforded in 
 other portions of this work ; the plans, elevations, and sections of a door and 
 window of the Arthur Club-house, Part II. ; the plan, section, and elevation of a 
 shop-front in Mount Street, also in Part II. ; and interspersed throughout the work, 
 the plans, sections, and elevations of various buildings and their separate parts 
 in detail, will give ample employment and afford excellent information to the 
 student at this stage. We will proceed to the drawing of the several parts of 
 the different orders of architecture, endeavouring to point out the distinguishing 
 features in each. 
 
 CHAPTER IV. 
 
 ON THE GRECIAN AND ROMAN ORDERS. 
 
 In architecture, so various are the styles, and so much variety is to be found 
 in each, according to the period in which it has been used, and the purpose to 
 which the structure has been applied, that to become thoroughly acquainted Avith 
 the subject, requires a life of study ; there are the Egyptian, the Hindoo, the 
 Grecian, the Roman, the Arabesque, the Gothic or pointed style, &c., all distinct 
 from each other, and each possessing its peculiar attributes. We are much 
 indebted to our predecessors for their patient researches, and to our contem- 
 poraries who have so far reduced these researches into a system, as to enable us 
 to understand and appreciate with comparative ease the beauties of the construc- 
 tions of former ages. Simplicity and harmony constitute the great excellencies 
 of architecture ; and for these qualities the Grecian styles perhaps stand un- 
 rivalled. 
 
 Certain arrangements of certain solids, from which stand out various pro- 
 jections, supported on columns, the form and magnitude of the parts bearing 
 a relative proportion one with another, according to rules deduced from 
 
22 
 
 PRACTICAL RULES ON DRAWING. 
 
 ancient models, the leading features of which vary so as to admit of their being 
 classed under different heads, are denominated orders of architecture. In the 
 Grecian style these have been arranged under three heads, viz., 1. The Doric, 
 so called from the earliest specimens having been found in the Dorian States ; 
 2. The Ionic, taking its name from the earlier prototypes being found in 
 Ionia ; and 3. The Corinthian, from this style having been first invented and 
 used in the neighbourhood of Corinth. The Romans however, not satisfied 
 with the beauty and simplicity of the Greek models, have made use of the three 
 Grecian orders, materially altering their proportions, and have added to them 
 two other orders : the Tuscan, which may be considered a mere modification of 
 the Doric, and the Composite, a mixture of the Ionic and the Corinthian. 
 
 Each order of architecture has necessarily some of its parts distinct from 
 those of the others, but there are certain definitions that belong to all, with the 
 terms of which it is necessary to be acquainted before the descriptions of the 
 several orders can be understood ; such as, 
 
 1. The Column. The whole of the pillar supporting the superstructure; 
 divided into — 
 
 The Base ; an assemblage of mouldings surrounding the lower part of 
 the column. 
 
 The Capital ; an assemblage of mouldings surrounding the upper part of 
 the column. 
 
 The Shaft ; the plain part of the column between the base and the capital. 
 
 2. The Entablature. The horizontal portion of a structure resting on the 
 columns, divided into — 
 
 The Architrave ; the lowest portion ; that which comes in contact with the 
 uppermost portion of the column ; also called epistylium, or epistyle. 
 
 The Frieze ; the middle portion of the entablature, on which any orna- 
 mental sculpture is displayed. It goes also by the name of Zoophorus. 
 
 The Cornice ; the highest portion of the entablature, resting on the frieze. 
 
 3. Diameter. The scale from which the measurements of the various parts 
 of an order are taken ; divided into — 
 
 Module ; half a diameter. 
 
 Minute ; the sixtieth part of a diameter. 
 Note. — The diameter of the base of the shaft, with its subdivisions, is the 
 scale commonly made use of for the several parts of a columnar structure ; thus 
 in speaking of an entablature or any other part, its height is said to be so many 
 
ON THE GRECIAN AND ROMAN ORDERS. 
 
 23 
 
 diameters or modules, and so many minutes, instead of so many feet and inches. 
 The diameter of the top of the shaft, from its position, is called the superior 
 diameter ; and that of the base, the inferior diameter. The latter is, however, the 
 longest, as the shaft invariably tapers upwards. In speaking of a diameter as a 
 scale of measurement, the inferior or lower diameter is always intended. 
 
 4. Abacus. The square tablet at the top of a Doric column, on which the 
 architrave is placed. The uppermost portions of the other columns on which 
 the entablature rests are also so termed, but strictly speaking the abacus applies 
 to the Doric. 
 
 5. Intercolumniation. The shortest distance between the base of one 
 column to the base of the next. 
 
 The first, which is the most simple and the most symmetrical of the Grecian 
 orders, is the Doric, of which the following is a general description. The whole 
 structure from the base to the top of the entablature, is divided into three parts. 
 The individual parts here mentioned may be understood by referring to Plate 
 IV., figs. 1 and 3. First, The steps, on the upper slab of which the columns are 
 placed, termed the stylobate, being in general proportion about half a diameter. 
 Second, The column, including the capital and shaft, varying in different examples 
 from four to six diameters in height, of which the capital takes about half a 
 diameter, the shaft gradually diminishing from the base to the top ; the upper 
 diameter being from about two-thirds to four-fifths of the lower diameter. 
 The capital is also divided into several parts ; the necking or hypotrachelion, 
 that portion of the column from the upper groove to the lower ring, occupying 
 the space of about one-fifth the height of the capital ; the echinus or ovolo, 
 with the rings, called annulets, in perpendicular height about two-fifths of the 
 capital; and the abacus, also taking about two-fifths. Third, The entablature, 
 varying in its proportions to the column of from one and three quarters to a 
 little beyond two diameters. It has three principal divisions: 1. The architrave, 
 about two-fifths the height of the entablature, its face being perpendicularly over 
 the outer extremity of the inferior diameter, and having at its upper part a pro- 
 jecting fillet called the taenia, about one-tenth of the architrave, under which at 
 regular intervals are projections called regulae, from each of which depend six 
 small drops or guttae, the height of the regulae with the guttae being the same as the 
 taenia. 2. The frieze, about two-fifths of the entablature, is bounded on the top by 
 a slightly projecting fascia about a seventh or eighth part of the height of the 
 frieze, the remaining portion of the frieze being divided horizontally into spaces 
 
24 
 
 PRACTICAL RULES ON DRAWING. 
 
 termed alternately triglypli and metope. The triglyph, the plan of which is given 
 tigs. 3 and 4, PI. 111., projects a little in front of the metopes, which latter are 
 frequently ornamented with sculptures.* 3. The cornice, about one-fifth of the 
 entablature, consists of three parts ; the rautules, similar to the regulae of the 
 architrave, each also having six guttae depending from it, with a slightly pro- 
 jecting fillet above them; the mutules, guttae, and fillet, taking about one fourth 
 of the cornice. The mutules project from the face of the entablature about two- 
 thirds of their length, not at right angles, but in a sloping direction, as shewn in 
 fig. 3, PI. 111., and having two other rows of six guttae each, at equal distances from 
 the front row shown in the elevation. The corona, the second division, taking 
 two fourths of the cornice, and the remaining fourth to a combination of 
 mouldings, forming the third division, varying in the different examples, but 
 finished by a fillet at the top. The length of the regulae and mutules is decided 
 by the trigiyphs, one of the former being placed under, and one of the latter 
 above, each triglyph ; the number of the mutules, with their guttae, is more than 
 that of the regulae, one being placed immediately over the centre of each of the 
 metopes. The trigiyphs are so arranged that the middle of the triglyph must 
 come on the centre line of each column over which it is placed, excepting at the 
 two extremities of the frieze, when one side of the triglyph must come over the 
 centre line of the column, and the other perpendicularly over the angle of the 
 architrave. 
 
 The descriptions here given are an average of proportions, derived from a 
 comparison of the most approved ancient relics. The proportions of the several 
 parts in the plate are such as to render the drawing of it as simple as pos- 
 sible ; at the conclusion of the subject, the student will be referred to autho- 
 rities from which, by actual measurement, he will be enabled to draw any indi- 
 vidual example of Grecian architecture. 
 
 Fig. 1, Plate IV., is the elevation of a Grecian Doric column, standing on the 
 stylobate and supporting the entablature. First draw a horizontal line A B, at 
 discretion, to represent the inferior diameter of the shaft of the column, from the 
 centre of which dravt^ up a long perpendicular line on which to measure the 
 perpendicular distances. Make a right line at the foot of your drawing, fig. 2, 
 which divide into two or three equal parts, each of the length of the line A B, 
 and subdivide one of the divisions into sixty parts, to serve for a scale. From 
 
 * A part of the celebrated sculptures, known to every visitor of the British Museum by the name of the 
 Elgin marbles, formed a portion of the frieze of the Parthenon at Athens. 
 
ON THE GRECIAN AND ROMAN ORDERS. 
 
 25 
 
 the point C on the line C D, measure off six diameters at E for the height of the 
 column ; through which draw a horizontal line. From the point E measure 
 down half a diameter at F for the height of the capital, and through this point 
 draw another horizontal line. The diameter of the shaft at the top F is one 
 quarter less than that of the base ; measure off then, on either side of the point F, 
 a space of three eighths of a diameter, or twenty-two and a half minutes, at 
 G and H ; draw the lines G A and H B, which will give the outer form of the 
 shaft. Mark off from the point E two diameters at D, for the height of the 
 entablature ; giving three quarters of a diameter, or 45 minutes, to the architrave 
 from E to J, an equal distance to the frieze from J to K, and half a diameter to 
 the cornice from K to D; through each of which points draw horizontal lines, and 
 draw the outer line of the architrave perpendicularly over the outer point of the 
 lower diameter. In order more clearly to understand the different parts, the 
 capital of the column and the entablature are drawn at fig. 3. on a scale three 
 times the size of that of fig. 1 ; lettering the parts the same up to the letter K. 
 Divide the space allotted to the capital from E to F, into five equal parts as 
 between a b; on the line L (fig. 3.) take one fifth of these divisions for the neck 
 (hypotrachelion) of the capital, two fifths for the annulets and the echinus or 
 ovolo, and the remaining two-fifths for the height of the abacus. Divide that 
 portion allotted to the echinus and annulets into four equal parts, as at c d, 
 giving one fourth of the distance for the annulets, and the remaining three-fourths 
 to the echinus ; divide the space allotted to the annulets, of which there are three, 
 into three equal parts at e f, and subdivide these again for the convenience of 
 drawing the annulets ; through all these points above described draw horizontal 
 lines across the space for the capital. Make the width of the abacus one 
 diameter twelve minutes, by measuring off on either side of the point E at N and O, 
 one module six minutes (or thirty-six minutes), and draw from them two perpen- 
 diculars to the line determining its height. From the points G H, from whence 
 the neck of the capital springs, continue the sides from the top of the shaft per- 
 pendicularly for about three-fourths the height of the neck, when curve the lines 
 outwards to meet the points P and Q on the upper line of the top annulet from 
 where the echinus commences. Find the greatest point of projection of the 
 echinus on either side, and from that point and the point P or Q describe the 
 contour as directed, figs. 20 and 21, Plate II. It will be seen that though the 
 echinus springs from the point Q, from its commencement at the lower line 
 of the abacus, if the line were continued through to the point H, it would form 
 
 (Os Drawing.) D 
 
•26 
 
 PRACTICAL RULES ON DRAWING. 
 
 a cimatium ; it is in fact a ciraa reversa encompassed by three rings ; from this 
 curved line on the lines already drawn denoting the width of the rings draw their 
 contour ; a section of them is given fig. 4, that the form may be more clearly 
 understood. Though the proportions may vary, any Grecian Doric capital may 
 be drawn by the foregoing directions. Divide the height of the architrave E to J 
 into ten equal parts as from h to g, of which give the uppermost tenth for the fillet 
 called the taenia, and the next tenth for the regulae with the guttae, drawing the 
 taenia to project beyond the architrave two minutes and a half. The width of the 
 regulae depending on the horizontal divisions of the frieze, we must return to them 
 after drawing the triglyphs. Divide the height of the frieze from J to K into eight 
 equal parts, as from g to h, of which take the upper eighth for a fillet, having a 
 slight projection called the capital of the triglyphs, the remaining seven eighths 
 being divided into alternate spaces called triglyph and metope. The inner line of 
 the first triglyph comes immediately over the vertical line of the column, the outer 
 line over the outer point of the diameter; draw the outer line perpendicularly over 
 the outer line of the architrave, and the inner perpendicularly over the middle of 
 the diameter of the shaft. In order better to understand the form of the triglyph, 
 a horizontal section is given at fig. 4, Plate III., where the divisions in an elevation 
 would be equal. Fig. 5, Plate III., is a section, in the elevation of which the 
 flat faces would be a little broader than the angular recesses. The two angular 
 channels a a, (figs. 4 and 5), are called glyphs ; the two half channels on either 
 side h b, hemi-glyphs, making three in each figure ; hence its name, tri-glyph. 
 In the example before us, fig. 4 is chosen; the space between the outer and inner 
 lines must be divided into nine equal parts, and through each point of division a 
 perpendicular line drawn, which is all that can be shewn of a triglyph in an 
 outline elevation. The top of each glyph is sometimes terminated by straight 
 lines, sometimes by curves, as is shewn on that partly drawn to the right. From 
 the inner line of the triglyph measure off" the width of the metope ; the metopes 
 are generally square, or nearly so ; set off therefore, from the inner boundary of 
 the triglyph, the height of the frieze for the width of the metope, and at that dis- 
 tance draw a perpendicular line for the commencement of another triglyph. The 
 length of the regulae depending on that of the triglyphs, may now be put in ; 
 give two-thirds of the distance allotted for the regulae and guttae for the depth of 
 the regulae, and draw perpendicularly under the outer lines of the triglyph its 
 boundary lines; divide the space left for the guttae, which is equal to the regula, 
 into eleven parts, six equal parts for the guttae, and five for the intervals between 
 
ON THE GRECIAN AND ROMAN ORDERS. 
 
 27 
 
 them, making the latter a little wider than the former; mark the points for the 
 guttiE and their intervals alternately, and draw their perpendicular sides. The 
 cornice from K to D is divided into so many unequal parts, that we must take 
 each part separately ; first, immediately over the frieze draw a bead one minute 
 thick, and above this mark off the space for the mutules, with their guttaj, four 
 minutes and a half (the same height as the regulae and guttse), place the mutule 
 immediately over the triglyph, as the regula is placed under, and draw it every 
 way similar ; the face of it, being one and the same, the situation only excepted, 
 being placed above instead of below. Over the centre of the metope, another 
 mutule, with its depending guttai, similar to that over the triglyph, must be drawn, 
 and so on over every triglyph and metope in an elevation. The projection of 
 the mutule is two-thirds of its length beyond the face of the architrave, not at 
 right angles, as seen by the projection of a side mutule, but at an inclination, 
 which shews also the first of each of the three rows of guttse. Above the mutules 
 draw a fillet one minute and a half wide, projecting beyond the mutules half a 
 minute; above this draw the corona, ten minutes in height, projecting beyond the 
 fillet below one minute; above the corona draw a narrow fillet half a minute high, 
 and again above this a small echinus two minutes and a half high, having its 
 greatest projection beyond the corona two minutes and a half; over the echinus 
 a fillet two and a half minutes high, projecting half a minute beyond the echinus; 
 above the last fillet draw a broader echinus six minutes high, with a projection 
 of six minutes; and above all a fillet two minutes high, projecting one minute 
 cind a half beyond the echinus. 
 
 The circumference of the shaft is divided into twenty equal parts, each divi- 
 sion being slightly hollowed; these are called flutes. Fig. 6, Plate III., repre- 
 sents the horizontal section of a column with twenty flutes. To draw the flutes 
 of a column, describe a circle, the diameter of which shall be the same length as 
 that of the shaft ; divide the circumference into as many equal parts as there are 
 flutes in the column, in this instance twenty. 
 
 In making the divisions for the lines, to represent the edges of the flutes on 
 the shaft, there will be nine whole flutings to represent, and a half-fluting on 
 each side. The diameter A B, (fig. 6, Plate III.), must be drawn parallel to the 
 horizontal lines of the elevation, and the diameter C D at right angles with it ; 
 the divisions for the flutes on the circumference must be so arranged that each 
 of the extremities of these diameters must come in the middle of the space for a 
 flute. Let E F represent the diameter of the shaft on which the flutes are to be 
 
 D 2 
 
28 
 
 PRACTICAL RULES ON DRAWING. 
 
 drawn, the points E and F being placed perpendicularly over the points A B. 
 From each of the points of division for the flutes 1, 2, 3, 4, &c., on the semi- 
 circumference A D B, draw a perpendicular line to the diameter E F ; lines 
 drawn from these points, the length of the shaft, will give the correct diminution 
 in the apparent width of the flutes in the elevation of a column; bearing in mind 
 that, as the shaft is narrower at the top than at the bottom, both the superior 
 and inferior diameters must be divided into a like number of parts, and lines 
 ruled the whole length of the shaft from the corresponding points. The number 
 of flutes varies in different examples ; being in some more, in others less than 
 twenty. The points a a a shew the centres for drawing the curves of the flutes. 
 The rule given in the problem, fig. 10, Plate III,, in Practical Geometry, will enable 
 the student to find these points, whether the concavity is great or little. 
 
 The stylobate, the perpendicular height of which is half a diameter, must be 
 divided into three equal parts for the width of the steps; horizontal lines must 
 be drawn through each point of division ; the projection of the upper step in 
 front of the columns, and the projection of the two steps below must be marked 
 ofl, and perpendicular lines drawn from each point, which will complete the eleva- 
 tion of an example of Grecian Doric. It has been the Author's endeavour to make 
 the necessary directions for drawing this elevation as intelligible as possible, at the 
 risk of being, by many, considered too prolix; but these directions will be ample to 
 enable the student to draw any example of Doric, and the greater portion of either 
 of the other orders of architecture. An example will be given of each of the other 
 orders, with the measurement of their several parts, and the directions for drawing 
 them will be necessary for those parts only which form the distinguishing features 
 of the order. It will be seen by comparison of the elevations. Plates lY. and VI., 
 though only one example of each order is given, that the form of the capital of 
 the column constitutes the most essential distinction of an order, both in the 
 Grecian and Roman styles. The rectangular abacus, with the echinus and rings 
 denoting the Doric ; spirals at either end of the capital, immediately under the 
 abacus, forming what are termed volutes, denoting the Ionic capital, fig. 1, 
 Plate VI, Foliage composed of rows of leaves surrounding the capital with 
 tendrils terminating in volutes, form the distinguishing features of the Corinthian, 
 (fig. 2, Plate VI.) The terms made use of for the several mouldings are the same 
 in all the orders, as also the names of the several parts, as column, entablature, 
 &c., with their subdivisions. In the Ionic and Corinthian orders, the lower part 
 of the shaft of the column where it joins the base forms a concave sweep, 
 
ON THE GRECIAN AND ROMAN ORDERS. 
 
 29 
 
 similar to the cavetto, from the top of which the shaft springs ; it is termed 
 the apophyge. In the fluting of the columns of these orders, it will be observed 
 that instead of the divisions between the flutes coming to an edge, the intervals 
 form a hllet, the fillets being usually about one fourth, a little more or less, the 
 width of the flutes. The mode of drawing these is similar to that given for 
 drawing the flutes of the Doric column, (fig. 6. Plate III.) ; with this addition, that 
 whatever may be the number of flutes, there must be an equal number of fillets; 
 the circumference must therefore be divided into so many parts for flutes and a 
 like number for fillets, (fig. 1, Plate V.,)* placing them alternately, taking care 
 always that the centre of the flutes comes on the lines of the diameters as 
 described fig. G, Plate III. The curves of the flutes when elliptical must be 
 drawn by joining the extremes of the curve by a straight line, to serve as 
 a transverse diameter, from the centre of which a perpendicular line must be 
 drawn inwards, to the full depth of the concavity, to serve as a semi conjugate 
 diameter. 
 
 To draw an Ionic elevation, proceed in precisely the same manner as in that 
 of the Doric. Divide the three principal parts, stylobate, column, and entabla- 
 ture, into their several component parts, according to the measurements given in 
 the plate. The base of the column, which is not found in the Doric, consists of 
 a series of mouldings already described; the entablature presents a fresh feature 
 in the rectangular parts called dentils, which may also be drawn without further 
 directions; — any ornamental work on the mouldings must be for after consideration, 
 when we treat of drawing foliage, &c. The main point in the representation 
 of the Ionic order is to describe the spiral lines forming the volutes. A variety 
 of modes are given by different authors for the construction of spirals ; some 
 recommending the drawing the curve through certain points ; others, the finding 
 certain points as centres from which to describe a series of quadrants of circles. 
 We give one example of each mode of describing a spiral. 
 
 Let A (fig. 2, Plate V.) represent the centre of the spiral, and B its greatest 
 distance from the centre ; from the point A as a centre, with the radius A B, 
 describe a circle. Divide the circumference into any number of equal parts, (in 
 the example ten is chosen), and from the centre A to each of the points of di- 
 
 * In the plan of the half shaft here given, the column is represented as having but twelve flutings. From 
 the unavoidably small size of our figures this number will enable the student to understand more clearly the 
 figure than if it had been represented with twenty-four. 
 
30 
 
 PRACTICAL RULES ON DRAWJNG. 
 
 vision draw a straight line, as A a, A b, A c, &c. Divide the radius A B into 
 the same number of parts as there are to be revohitions of the curve, (in this case 
 three), and divide each third into as many equal parts as the circumference has 
 been divided into, (ten), which will give thirty divisions on the line A B; measure 
 from A towards a twenty-nine of these divisions, from A towards b twenty-eight, 
 from A towards c twenty-seven, and so on round and round, making each radius 
 less by one thirtieth till you arrive at the centre, figuring each with the number 
 of divisions, as at twenty-nine A a, twenty-eight A b, &c. &c. Through each 
 of these points, commencing at B, (the greatest height of the volute), draw by 
 hand a continuous curve, till you arrive at the centre-point A, which will de- 
 scribe the spiral. The number of divisions depends entirely on the pleasure of 
 the draughtsman ; in small spirals, the division of the circumference into ten 
 parts will be found quite sufficient; where the spiral is large, and great accuracy 
 is required, the number of divisions must be increased. 
 
 To draw a spiral, of a given height, to meet the circle in the centre, (called the 
 eye), to any number of revolutions. Let A B (fig. 3, Plate V.) represent the whole 
 height of the proposed spiral; cut off from the end of it the length of the diameter 
 of the eye, at C; divide the remainder, C to A, into as many equal parts as there 
 are to be revolutions in the spiral, and subdivide each of these parts into four 
 equal parts, so that the line from A to C is divided into four times as many equal 
 parts as there are revolutions in the spiral. Three revolutions are here proposed. 
 With a radius of half B C describe a circle to represent the eye of the volute, 
 through the centre O of which draw a horizontal line and a long perpendicular 
 line; set up on this perpendicular line, at D, half the length of the line from A to C 
 with one division added to it, (seven twelfths of A C), and also one half the 
 diameter of the eye; this will give the extreme distance at which the revolutions 
 terminate from the centre. Take a distance of half of one of the divisions of 
 A C, (one twenty-fourth), and set it up on this perpendicular at a,* through 
 which point draw a horizontal line, and measure off on either side a space equal 
 to that from the centre to a, at b and c, from each of which points draw diagonal 
 lines to the centre. From the points b and c draw down perpendicular lines, 
 cutting the semi-diameters of the eye into two parts at d and e ; divide 
 each of the spaces from O to e and O to d into as many equal parts as there 
 are to be revolutions, and according to which side the volute is to be represented, 
 
 * This part being extremely minute, the figure for finding the centres is given on a larger scale, (see fig. 4.) 
 
ON THE GRECIAN AND ROMAN ORDERS. 
 
 31 
 
 divide on that side the nearest of these divisions into two equal parts, as at f; 
 from this point, parallel to Z> O and c O, draw the diagonal lines / h and / g ; 
 through each of the points of division on d O and e O, perpendicular lines must 
 be drawn, cutting the diagonal lines at the points iklmnopq; join the oppo- 
 site points on each diagonal, and you have the centres required for describing 
 the spiral with the compasses, equal in number to the divisions on the line from 
 A to C, — a quadrant from each centre ; — twelve quadrants necessarily producing 
 three revolutions, the number proposed. From the point c as a centre, with the 
 radius a D, draw the arc of a circle till it meets the continued line c a ft at G, 
 which forms the first quadrant F c G ; from the point 6 as a centre, with the 
 radius h G, describe from the point G the arc of a circle till it meets the conti- 
 nuation of the line 6 c?^ at H, which will form the second quadrant G 6 H ; 
 from the point g, with the radius g H, describe the third quadrant H </ J, and so 
 on, taking the points for centres in the order they are lettered, when the last quad- 
 rant from the point q will touch the inner circle (the eye) on the perpendicular line 
 above a at r. It is to be hoped the student clearly understands the mode here 
 given for drawing spirals, and that, whatever may be the number of revolutions, 
 he would be enabled to describe them; were there twenty revolutions the process 
 would be precisely the same, the number of centres only being increased. To 
 draw a spiral with twenty revolutions would require eighty centres, from which 
 to describe the quadrants ; more lines would of necessity be required ; but if the 
 spaces from dio O and e to O were each divided into twenty parts, and perpen- 
 dicular lines drawn through each to meet the diagonals, the twenty centres would 
 be as easily found as in the foregoing example. Fig. 5 is a diagram, shewing 
 the centres for describing a spiral of five revolutions. The diameter of the eye 
 being given with the height of the spiral, the length of the line a O is assumed. 
 All proceeds the same as in the former example, and the points from which to 
 describe the quadrants are lettered consecutively from the first centre to the last. 
 
 The volute is composed of many spirals one within another, the lines of the 
 curves gradually approaching each other till they meet at the same point on the 
 eye. In the example before us the line from the centre O to D is the greatest 
 distance of the spiral ; if the half diameter of the eye be taken from it there remains 
 a space r D, consisting of seven equal parts taken from A B. Suppose the 
 greatest distance of the second spiral line of the volute from the outer spiral that 
 from D to L, the distance from L to r maybe divided into seven equal parts as a 
 scale similar to A C. Set up, on the perpendicular line O D, half of one of these 
 
32 
 
 PRACTICAL RULES ON DRAWING. 
 
 sevenths, and you will have a point for finding the centres of the second spiral, 
 corresponding with the point a of the first ; from which proceed according to the 
 directions laid down for the first. Spirals may be drawn from the points MNP 
 in the same manner, dividing the different spaces M r, N r, P r, each into seven 
 equal parts as a scale, and so on to any number of revolutions required. The lines 
 connecting the opposite spirals of the volutes, are some straight, others curved ; 
 the straight lines of course can be drawn with a ruler; the curves must be drawn 
 by hand : measure on a perpendicular line drawn through the centre of the 
 column, the greatest distance one curve is from another ; and from the termina- 
 tions of the corresponding spirals, draw the curve lines through these points. 
 Where great accuracy is required, perpendicular lines may be drawn at equal 
 distances from either side of the middle perpendicular line, and the distance of 
 one curve from the other at corresponding distances marked on them, through 
 which the curves may be drawn, — as is shewn fig. 3. 
 
 Before commencing to draw an elevation of the Corinthian order, the dis- 
 tinguishing feature of which is the foliated capital, it is necessary to turn our 
 attention a little to the drawing of foliage. Hitherto our progress has been 
 easy, from all the lines and curves being determined by rule; much of orna- 
 mental drawing must however depend on the correctness of the eye and 
 elegance of handling. Practice is the only way by which the student can 
 expect to accomplish with beauty and facility this branch of drawing. He is 
 recommended in the first instance to practise drawing curves of various kinds to 
 be found in mouldings, volutes, &c. by means of points, commencing, in order 
 to ensure correctness, with numerous points through which to draw the curve 
 lines, and gradually diminishing their number, till his eye and hand become accus- 
 tomed to their various forms. 
 
 The prototypes for drawing leaves, being found in nature, the student, after 
 he has accustomed his eye and hand to execute a variety of geometrical curves, 
 would do well to draw from nature the leaves and tendrils of various plants. 
 The leaves surrounding a Corinthian capital, though similar in form, present a 
 great variety to the eye, from the different angles at which they present them- 
 selves ; those immediately in front of the column in an outline not shewing any 
 projection, and those at the sides shewing the mere profile of the leaf and its 
 greatest projection. In drawing an elevation where a number of columns are 
 to be represented, it is necessary that they should be symmetrical, and the leaves 
 of the same size and form ; to perform this it is best to describe a right lined 
 
ON THE GRECIAN AND ROMAN ORDERS. 
 
 33 
 
 figure about the leaf, intersecting various parts of the curves within it by straight 
 Hnes, then constructing a similar rectilinear figure, find points corresponding with 
 the intersections of the example, and draw the curves through these points, as 
 the curves of ellipses, spirals, &c., have been described ; this will insure uniformity. 
 Or the rectangular figure may be divided into compartments, which is perhaps 
 preferable, as by this mode the form may be drawn to any proportion. Fig. 0, 
 Plate v., is the representation of the profile of a leaf from a Corinthian capital, 
 enclosed in a rectangular figure, giving its greatest height and projection, divided 
 into sixteen compartments, figured from 1 to 16. The greater the number of com- 
 partments, the easier it will be found to draw the representation. 
 
 Let the student draw a parallelogram double the height and double the width 
 of that in fig. 6, Plate V., and mark the different compartments from 1 to 16, as 
 in the example ; then mark the points on the several lines through which the 
 outer curve of the leaf passes, corresponding with the points a, a, a, &c., fig, 6, and 
 draw the line through them, endeavouring to keep one even, continuous sweep. 
 It is best to hold the pencil at some length, and sketch the lines very lightly at 
 first, going over and over the whole length of the line, till the contour is satis- 
 factory ; then draw over the sketching lines firmly, one continued line, and rub out 
 the sketching lines with bread. Take points from where to commence the principal 
 parts of the leaf as at b, b, b, figs. 6 and 7, and other points c, c, c, to where the lines 
 are to terminate and others again to commence, and draw the curves from point to 
 point. Where certain portions may be more intricate than others, and great accu- 
 racy is required, the division containing them may be subdivided into any number 
 required ; practice enables the draughtsman to execute ornamental drawings with 
 very few points ; his eye gradually getting accustomed to the work, enables him 
 to take distances with great accuracy without measurement. Only a small part of 
 the leaf is represented in fig. 7, sufficient, however, to show the young draughtsman 
 how to proceed with the remainder ; to execute the long curves, let him mark the 
 highest point first, with its greatest distance from that first drawn ; and then its 
 lowest point and closest distance to the first drawn ; then draw the line, noting 
 carefully its gradual diminution in distance from the other. Fig. 8 is the same leaf 
 reversed ; supposing fig. 6 to be the outer leaf on the right of a capital, fig. 8, 
 when finished, would represent a similar leaf on the left. By reversing the 
 figuring of the compartments 1 to 16, as shewn in the figure, the drawing is 
 accomplished as easily as that of fig. 7. The points a, a, a, in fig. 8, are marked 
 
 ^On Drawing.^ E 
 
34 
 
 PRACTICAL RULES ON DRAWING. 
 
 in the corresponding- compartments to those of fig. 6 ; the highest point of one 
 of the long curves, with its greatest distance from the outer curve, is marked at 
 d, and its nearest distance at the point e and tlie line drawn through ; several 
 other points are also marked as directions for drawing the separate parts 
 of the leaf. 
 
 Fig. 1, Plate VIL, is a front representation of the same leaf of which we have 
 given the profile, fig. 6, PI. V. : one side is represented completed ; the other 
 divided into compartments, with portions only of the leaf drawn to enable the 
 student better to understand the mode of proceeding. He is recommended to 
 copy in this manner any good examples he can procure ; and after completing 
 one copy, to make another from it, as perfect a fac-simile of the first as possible ; 
 uniformity being one of the most essential parts of ornamental drawing. 
 
 Plate VI. contains an example of each of the four remaining orders of archi- 
 tecture ; the Ionic and Corinthian both from the Greek models. After the copious 
 remarks given in the preceding pages for drawing the example given PI. IV., 
 little remains to be said to enable the student to draw the examples before him ; 
 the directions given for drawing spirals, it is to be hoped, will enable him to draw 
 the volutes ; and those for drawing leaves, suflicient to insure uniformity in the 
 representation of any number of them, an example of one leaf being given. Each 
 specimen of an order is divided into its several divisions and subdivisions, and 
 the dimensions marked against each. The mouldings vary in their arrangement ; 
 but as the name of each moulding, according to its form and the mode of de- 
 scribing the contour, has been given in the third Chapter, and illustrated in 
 Plate II., they may be drawn without farther direction. The modillion, an 
 ornamental bracket, is a feature in the cornice of the Corinthian order, not yet 
 described ; to give a representation of this requires no further directions, either 
 for the front view or for the profile ; an example of a modillion in profile is given 
 at fig. 2, PI. VII. The representation of modillions in an elevation is generally 
 so small, that their size and position may be drawn by right lines, like the regulae 
 ormutules, and the form of the modillion drawn within the lines by eye; if of 
 a large size and much enriched, the right lined figure containing the modillion 
 may be divided into compartments similar to the profile of the leaf, fig. 6. PI. V., 
 marking the points through which the various lines pass. 
 
 For the advantage of representing the bases and capitals of the columns, as 
 well as the several parts of the entablature, of as large a size as the limits of 
 
ON THE GRECIAN AND ROMAN ORDERS. 
 
 35 
 
 our plates will permit, the whole length of the shaft is not drawn ; the superior 
 and inferior diameters, however, are both given, and the length of the shaft 
 marked between them ; the student can therefore find no difficulty in representing 
 them in his drawing of their entire length. 
 
 The Doric, as we have before mentioned, is for the most part almost bare of 
 ornament; the other orders, and more particularly those of the Roman style, are 
 on the other hand occasionally highly enriched ; to so great a pitch has the en- 
 richment of some of the Roman orders been carried, that there are examples 
 where scarcely a portion is to be found that is not covered with some kind of 
 ornament. The friezes frequently contain series of figures, representing proces- 
 sions, &c., sometimes combinations of foliage and animals ; the representation of 
 these depends less on rule than on the correctness of the eye and hand. Much 
 of the decorative part of both Grecian and Roman architecture consists of foliage ; 
 and though the representation of this depends mainly on skilful handling, still 
 the division of the examples into small rectangular compartments will enable the 
 draughtsman to represent the curves with tolerable accuracy, and preserve a 
 uniformity in form and size. A common mode of ornamenting mouldings is by 
 carving on them an ornament called the egg and arrow, or egg and tongue. Fig. 
 3, Plate VII., is an example of the egg and tongue ornament on a Grecian ovolo ; 
 upright lines are drawn at regular intervals to shew how the uniformity of the 
 different parts may be preserved in a long series of these ornaments. Fig. 4, is 
 an example of a Grecian cima recta, enriched with carvings of foliage. Fig. 5, an 
 enriched cima recta of a Roman cornice. Fig. 6, an elegant ornament taken from 
 the neck of a Grecian capital. These figures are intersected by lines, and 
 similar lines are drawn at the sides to shew how they may be copied, and por- 
 tions of each are commenced at various parts, always commencing with the prin- 
 cipal curves. All these figures may be drawn as described for fig. 6, Plate V., 
 and any other examples may be copied in the same manner. 
 
 There is a part not yet mentioned, belonging to an order placed upon the 
 entablature, at the front or end of a temple, or over a portico, called the pediment. 
 (Fig. 7, Plate VII.) The simple outline of a pediment, as is seen by the figure, 
 is similar to the common gable end of a house, the angle formed by the junction 
 of the oblique lines being generally more obtuse. The sides of a pediment are 
 formed by a series of mouldings, sometimes similar to the cornice ; the space 
 contained in the triangle formed by the top of the entablature, and the sides of 
 the pediment, is called the tympanum ; this is commonly filled up with figures ; 
 
 E 2 
 
36 
 
 PRACTICAL RULES ON DRAWING. 
 
 it is also very common to find ornaments or figures placed on the apex and at 
 the sides of a pediment. The parts of a pediment are so similar to other parts of 
 an order already described, that they may be drawn without further description. 
 
 The learner, desirous of exercising himself on the preceding information, may 
 take as examples for practice the elevation of the principal front of the Athenaeum 
 Clubhouse, — Part I. in this Publication — drawing it to a larger scale; also the 
 design for a verandah, with its different parts, represented in two Plates in the 
 same Part, and other engravings interspersed throughout the Work. 
 
 The style of architecture commonly termed Gothic, the most fascinating to 
 the eye of the painter, has so many varieties in the exquisitely beautiful arrange- 
 ment of its parts, that it is quite impossible, in our small compass, to give examples 
 that would convey even a limited idea of their beauties. The varieties of this 
 style of architecture are arranged chronologically, and consist mainly in the form 
 of the arches. In the earlier specimens the arches are semicircular ; elevations of 
 them can then be drawn by means of the compasses : from the semicircular, a 
 change took place to the pointed arch, and the variation in the forms of these 
 constitutes an essential point in the varieties of the pointed style. In the next 
 portion of our subject, when treating of drawing arches in perspective, it will be 
 necessary to shew how these arches are described in elevation drawing, before 
 they can be put in perspective ; we therefore defer this description till treating 
 of arches in the following division, and will proceed at once to the leading prin- 
 ciples of the second division of the subject on outline. Perspective. 
 
 CHAPTER V. 
 
 ON OUTLINE. 
 Perspective. 
 
 Before commencing the branch of drawing, termed perspective, it is 
 necessary to comprehend what the term means; to understand well, before 
 commencing operations, what is required to be done. Considerable difficulty is 
 frequently experienced in making a young student in drawing thoroughly com- 
 
PERSPECTIVE. 
 
 37 
 
 prehend what is meant by perspective * The positive form of the objects that 
 daily and hourly come under our observation, is so strongly impressed on the 
 mind, that it is not easy to reconcile ourselves to the idea, that in a correct 
 representation of these objects in drawing, when they present themselves to the 
 eye at an angle, the form in outline is never the same as the real form ; never- 
 theless such is the fact.f In the perspective delineation of a square, which in 
 its geometrical description has its opposite sides equal and all its angles right 
 angles, the form completely changes ; the sides are no longer equal, nor are 
 the angles right angles; it assumes the form either of a trapezium or <^ ^ 
 
 trapezoid. The cube, which we know to have all its six sides 
 OO Q regular squares, assumes the form of an irregular hexagon. The circle 
 
 instead of being represented by a curve line, equidistant in all its points 
 from the centre,assumes the form of an ellipse, flatter or broader according to its 
 ciz:? position above or below the eye of the spectator, or the angle at which it 
 is viewed. 
 
 Let us endeavour, by examples, to explain to the student these apparent 
 changes in the form of objects in their perspective delineation. 
 
 In the annexed woodcut the reader is to suppose 
 himself looking out of a window at the objects visible 
 from it; and it must be obvious to any one, even of the 
 most ordinary capacity, that the representation here 
 given is essentially correct ; that although the opening 
 through which the view is seen, the space occupied by 
 the window, is an area of but a few feet, an extensive view 
 is visible to the spectator through it, embracing a distance 
 perhaps of many miles. It will be observed, that ac- 
 cording to the distance of objects from the station of the 
 spectator, — objects that are in reality of the same mag- 
 nitude, — they vary in their apparent size ; that the nearer 
 any of these objects are to the spectator, the larger 
 they appear, and the greater the distance so does their size apparently 
 
 * An elementary treatise on this subject, published by Mr. Weale, and addressed to amateurs and juvenile 
 students in drawing, gives a very clear insight into the use and application of the rules of perspective ; and 
 would be found a useful introduction for any desirous of making acquaintance with this indispensable branch 
 of drawing. The work is entitled " A Practical Treatise on Perspective, simplified for the use of Juvenile 
 Students and Amateurs," by G. Pyne. 1846. 
 
 f The sphere is an exception ; viewed from any point, its form is invariably the same. 
 
38 
 
 PRACTICAL RULES ON DRAWING. 
 
 diminish. The flower-pot on the window-ledge, containing the plant, is in 
 reality but a few inches in height, yet from its being so close to the spectator, it 
 is represented higher than the figure of a man mounting a ladder by the side of 
 the house opposite, though we know that man to be at least ten times the height 
 of the flower-pot. Again, we see in the cut that a single pane of glass in the 
 open window, which is quite close to the spectator, appears so large in propor- 
 tion to the opposite building as would allow the whole corner of it to be seen 
 through this single pane; and that an entire window of this building, though at 
 so short a distance, occupies but a small space in the representation of the one 
 single pane of glass so close to the eye. The plant in the flower-pot in the 
 engraving is represented much larger than the tree overhanging the wall, though 
 taking the average height of the figure to be between five and six feet, the tree 
 must be really near twenty feet in height, though the plant is not so many inches ; 
 and a very large object, the barn on the right, still farther distant from the 
 spectator, does not appear half the height of the flowerpot and geranium ; the 
 distant mountains, yet farther oflT, though probably of considerable altitude, 
 are apparently not much higher than the thickness of the bottom rail of the 
 window. 
 
 There are, in addition to the apparent decrease in the size of objects as they 
 recede from the eye, other circumstances to be observed in perspective represen- 
 tations. In a careful examination of nature, or the objects intended to be 
 represented, there is not only an obvious variation in their apparent size, but also 
 constant variation in their form, and according as the distance is increased, the 
 sharpness of the outline gradually decreases, arising from a less determined 
 light and shadow ; and a diminution of the intensity of their colour is also very 
 perceptible. Perspective drawing is the art of representing objects as they 
 appear to the eye. Linear perspective shews how to represent the various 
 apparent changes in the size and form of objects, according to their distance, and 
 the angle at which they are seen, by outline ; the rules for drawing linear perspec- 
 tive are unerring, deduced from the science of optics, and capable of mathematical 
 proof. Aerial perspective, is the art of representing the apparent distance of 
 objects, by the force or delicacy of tone and color; to use the painter's expres- 
 sion, it is the representation of atmosphere ; it depends upon taste, and careful 
 investigation of the ever changing eflfects in nature, and constitutes one of the 
 greatest excellencies of the painter. It is to linear perspective only that this 
 portion of the work applies. 
 
 The student must perfectly understand first, that in perspective delineations 
 
PERSPECTIVE. 
 
 39 
 
 all objects decrease in size according to the distance from which they are viewed 
 by the spectator ; and secondly, that as well as diminishing in size, they also 
 change their form. Let ns refer to the wood engraving at the head of the 
 chapter, and assuming the window through which the picture is seen, to be of the 
 same size as those seen in the opposite building, the one on the shadowed side 
 of the house, though represented so much smaller than that through which it is 
 seen, is considerably larger than either of those on the side of the building in 
 light, from its being nearer to the spectator; the nearest window on the light 
 side of the building will be found to be larger than the farther one, and if a row 
 of twenty windows were continued, they would gradually decrease in size as 
 the distance from the spectator increased. The figure on the ladder is represented 
 much larger than the one on the footway by the garden wall, from its position 
 being so much nearer the eye ; and if we imagine a series of figures, like a line of 
 soldiers, extending as far as the eye could reach, it must be obvious that they 
 would gradually diminish till they became mere dots. Understanding then that 
 objects represented in perspective decrease in size according to their distance 
 from the spectator, it is not difficult to comprehend that their apparent form 
 varies from their real form. The window seen on the shaded side of the house, 
 though represented mnch smaller than that through which it is seen, does not 
 change its form, and this because it is parallel to the window from which it is 
 seen ; to express this more technically, the window is said to be in a plane parallel 
 to the plane of delineation.* Not so with the windows on the light side of the 
 house ; these, which are seen at an angle with the plane of delineation, no 
 longer retain their original form of a rectangular parallelogram, but take that of a 
 trapezoid, which varies according to its position. So with the top and bottom of 
 the flower-pot and saucer, we know the form to be round or circular, or nearly so, 
 but in their perspective delineation they assume somewhat the form of the ellipse. 
 
 The great distinction between geometrical drawing and perspective drawing, 
 consists in the manner of representing the parallel lines of objects ; in the former, 
 geometrical drawing, parallel lines, if continued to any extent, would never meet; 
 whereas in the latter, perspective drawing, in the representation of parallel lines, 
 they always incline towards each other or converge, and if continued would meet 
 in a point. 
 
 Let us suppose that we have a square box (a cube) placed before us 
 in such a position as to see the front and one of the sides. Let A B C D 
 
 * This as well as some other planes will be explained in our progress. 
 
40 
 
 PRACTICAL RULES ON DRAWING. 
 
 represent the front of the box, and on the side C draw a square, C E F G, for 
 the side ; it must be quite clear that this cannot be correct, for where could we 
 possibly draw the top, if that, as well as the front and side, were visible? One 
 side of the top must join the line A of the front; another side of the top 
 must join the line F of the side. We have shewn that objects of the same 
 length appear smaller as they are removed from the spectator ; the line E must 
 necessarily then be represented shorter than the line C. The line H divides the 
 square C E F G into two equal parts ; the space therefore between H E being 
 farther off than the space C H, must be represented narrower; so that the line E 
 is represented less' than the line C, and the space C E narrower than the space 
 
 B C, as in the following diagram. Join the top and bottom of the 
 — - — r-| line C with the top and bottom of the line E, and it will give a repre- 
 ° " sentation of a cube in perspective, of which two of the sides only are 
 
 visible. 
 
 Let A B, fig. 1, Plate VIII., represent the nearer side of a window viewed at 
 an angle : now as it has been already shewn that things of the same size are 
 represented less and less according as they recede from the spectator, the side 
 of the window farthest off must be shorter than the nearest; let C D then repre- 
 sent the length of the further side of the window. It is quite clear from this, 
 that, in order to draw the top and bottom of the window by the lines from A to C 
 and B to D, these lines must incline towards each other ; and if these lines were 
 continued they would meet in a point, as at E ; the rule for finding the points 
 to which the inclined lines of a perspective drawing tend, is the first and great 
 object required, for without it we can do nothing. Let us take the front eleva- 
 tion, fig. 6, Plate I., and put it in perspective, drawing it to the same scale. We 
 must suppose ourselves standing to the right of the cottage, in such a position 
 as to see both the front and side elevations. Let A B, fig. 2, represent the line 
 8, 10, of the elevation, and C D* the line 1, 9, which being farther from the spec- 
 tator than A B, must necessarily be represented shorter; join A C and B D, and 
 continue the lines till they meet at the point E ; this point is called the vanishing 
 point, and to this j)oint all the parallel horizontal lines in the front elevation, 
 fig. 6, Plate I., must be drawn in the perspective representation. Through this 
 point, at right angles with the line A B, draw a line across the picture, and 
 through the point B draw another line parallel to this last; that passing through 
 
 * The distance from C D to A B, and the height of C D, are assumed ; the proper mode of finding the 
 distance and height of C D will be explained as we proceed. 
 
PERSPECTIVE. 
 
 41 
 
 the point E is called the horizontal line, that passing through the line B the 
 ground line. 
 
 The point F on A B represents the real height of the window from the ground, 
 taken from the elevation, fig. 6, Plate I., and from F to the point G, the height 
 of the window ; but as the window is farther off than the angle of the cottage 
 A B, the near side of it must be less than FG, and the further side less than the 
 near side. If a line be drawn from the point G to the vanishing point E, and 
 another line from the point F to the same vanishing point, it will intersect the line 
 C D at the points H J, dividing the line C D into three parts, of the same pro- 
 portions as the three parts of the line A B,* shewing the height the window 
 would appear on the line C D, between the points H J. If any point on the line 
 G J be taken, and from it a perpendicular line drawn to touch the line F H, it 
 will give the length of the line F G at that particular point; we have therefore 
 first to find at what point the nearest top corner of the first window will come 
 on the line G J. From the point A, parallel to the horizontal line, draw a line 
 the length of the front elevation A to K, and mark on it the distances of the 
 windows from the sides of the cottage at 1 and 2, and the width of the windows 
 at 3 and 4. The line A K, with the divisions on it, shews the geometrical pro- 
 portions of the width of the parts of the elevation ; the line A C is the perspective 
 representation of the line A K, and we require a point by which we can obtain 
 on A C the relative positions of the points 1, 2, 3, 4, on A K, as we have the 
 relative positions of the points GF on the line C D. From the point K, through 
 the point C, draw a line to touch the horizontal line at the point L; this will be 
 the point required, or point of distance ; from the point 1 to the point L draw a 
 line intersecting the line A C at 5 ; from the point 2 draw a line to L, intersecting 
 the line A C at 6; from the points 4 and 3 also draw lines to the point L, 
 intersecting the line A C at 7 and 8 ; these points, 5, 6, 7, 8, are the perspective 
 positions on the line A C of the points 1, 3, 4, 2, on the line A K. Between the 
 lines G J and F H, perpendicularly under the several points 5, 6, 7, and 8, draw 
 lines parallel to the sides of the cottage (A B and C D) which will give the 
 representation of the apparent size and form of the windows in the elevation (fig. 
 6, PI. I.), seen in perspective. 
 
 * To shew that the divisions on the line C D bear the same proportion to it as those on A B, a line is 
 drawn from the point a, which divides the line A B into two equal parts, to the vanishing- point, cutting the 
 line C D at 5, which will be found to divide the line C D also into two equal parts. 
 
 (On Drawing.) 
 
42 
 
 PRACTICAL RULES ON DRAWING. 
 
 This mode of finding the point for ascertaining perspective distances is very satisfactory, and may 
 be relied on for accuracy ; it is particularly serviceable in drawing from nature, as it may be used with 
 equal advantage for a part of a line as for the whole. For the satisfaction of the student as to the accu- 
 racy of this plan, let him refer to fig. 3, Plate VIII., which shews the perspective positions of the 
 points 1, 3, 4, 2, at 5, 6, 7, 8, on A C, as found in fig. 2. G M (fig. 3) is also a perspective repre- 
 sentation of the line A K. If the geometrical line were drawn from the point G to J, with its points 
 of division 9, 10, 11, 12, G J will bear the same relative proportion to it that A C bears to A K; thus 
 if a line be drawn from the point J through the point M, the point L, where it touches the horizontal 
 line, will be the point of distance for finding the positions of the points 9, 10, 11, 12, on the line G M ; 
 as is seen by the intersections 13, 14, 15, 16, on this line by those drawn from the several points 
 9, 10, 11, 12, to the point of distance L; and the student will perceive that the several points 13, 14, 
 15, 16, lie perpendicularly under the points 5, 6, 7, 8 ; the point 13 under 5, 14 under 6, &c. The 
 lines A Cj G M, and B D, are each of them the perspective representation of the same length of line 
 (A K), and the perspective positions of the sides of the windows may be set on either; so that had 
 they been required on the line B D, the geometrical line must have been drawn from B to 0, and the 
 point of distance found by drawing a line from the point 0 through D to the hoiizontal line. 
 Lines di-awn from the points of division on B 0 to the point of distance L, will give intersections on 
 the line B D perpendicularly under those on G M and A C. 
 
 This rule is particularly useful for dividing a perspective line into any nmnber of equal parts. 
 Suppose it is required to divide the line A C, fig. 4, into five parts ; from the point A draw a horizontal 
 line, and set off" on it five equal parts from A to B ; from B through C draw a line to the horizontal 
 line at D, which is the point of distance, and to it from the points 1, 2, 3, 4, di-aw lines ; the points of 
 intersection on A C will give the required distances j but it is of no moment to what length the 
 geometrical line is drawn, if the proportion of its parts is the same. Let A E represent the geometrical 
 line, and a, b, c, d, the points of division ; find the point of distance by drawing a line from E through 
 C to F ; if lines be dravra from each point of division a, b, c, d, to F, the intersections on A C will be 
 in the same points as those di'awn from the line A B to the point D. 
 
 To return to fig. 2 : put up on the line A B, at M, the geometrical 
 height of the door, and from it draw a line to the vanishing point E, which 
 will give the perspective height of the door when the positions of its sides are 
 found ; to find these, measure on the line A K, from A, the distance of the 
 nearest side of the door from the side of the cottage at 9, and from 9 the width 
 of the door at 10, from each of which points draw a line to the point of distance 
 L, and the intersections at 11, 12, on A C, will give the perspective positions of 
 the points 9 and 10 on that line. Between the lines M E and B D draw lines 
 parallel to the sides of the cottage, perpendicularly under the points 11 and 12, 
 and the perspective outline of the door will be complete. 
 
 The line 2, 4, of the side elevation, fig. 7, PL I., being further from the spec- 
 
PERSPECTIVE. 43 
 
 tator than the Ime 1, 3, must be represented in fig. 2, PI. VIIL, like the line 
 C D, shorter than the line A B. Let N O* represent the line 2, 4, of the elevation 
 in perspective ; continue the lines A N and B O till they meet on the horizontal 
 line at the point P, which point will form the vanishing point for all the hori- 
 zontal lines of the side elevation. In order to draw the gable roof in perspective! 
 we must first find the perpendicular line passing through the centre of the paral- 
 lelogram of the elevation, fig. 7, PI. I., here represented by the lines A, N, O, B. 
 The centre of a rectangular parallelogram in perspective, is found in a precisely 
 similar way to that of a geometrical one, by the intersection of its diagonal lines ; 
 the intersection therefore of the lines A O and N B, at Q, is the centre through 
 which to draw the perpendicular line Q R. Continue the line A B to S, the 
 geometrical height of the gable, and from the point S draw a line to the vanishing 
 point P ; where this line intersects the line Q R at T, is the perspective height 
 of the gable ; draw the oblique lines T A and T N to complete the gable end of 
 the side. From the point T, the top of the gable, draw a line to the vanishing 
 point E ; this will give the perspective line of the top of the roof: there remains 
 only, to complete this figure, a line from the point C, corresponding with the line 
 A T ; this line C W is frequently drawn parallel to A T, which is incorrect. 
 There are several ways by which the point W may be found ; as simple a mode 
 as any is to draw a line from the point D to the vanishing point P, which line at 
 that end of the cottage corresponds with the line B O of the end drawn ; from 
 the point U draw a line to the vanishing point E, cutting the line D P at V ; 
 from V draw a perpendicular line cutting the line T E at W, which is the 
 point required ; join C W : the figure A T U B is the perspective represen- 
 tation of the half of the elevation, fig. 7, Plate I., and the figure C W V D is the 
 same figure in a more distant position. 
 
 The Author trusts the student has been able to follow him in the foregoing- 
 description ; he has endeavoured to render the directions as simple as possible, 
 keeping entirely to the practical part of his subject. Those who desire to acquire 
 a theoretical knowledge of Perspective must consult more elaborate works ; but 
 to understand them requires a considerable knowledge of mathematics, without 
 which they are for the most part unintelligible. Trusting then that the fore- 
 
 * The height of this line, and its distance from A B, liiie the line C D, are assumed, 
 f The gable roof in this example is represented without any projections ; the introduction of them would 
 create more intricacy than is judicious at so early a stage of our proceedings, 
 
 F 2 
 
44 
 
 PRACTICAL RULES ON DRAWING. 
 
 going is tolerably understood, we will make a few observations on the wood-cut 
 at the commencement of this Chapter, before we proceed to the next. 
 
 It is easy to imagine a sheet of glass, or other transparent medium, placed 
 between the spectator and the objects to be represented, on which the outline 
 might be traced ; suppose the woodcut to have lines drawn round it, to represent 
 this, and on the surface of the glass you trace the outline of all the objects seen 
 through it, the form of them must necessarily be such as they appear to the eye ; 
 this surface (which is in fact the picture) is called the plane of delineation, of 
 which the base line is called the ground line. The point exactly opposite the 
 eye of the spectator, is called the point of sight,* and a line drawn through this 
 point, parallel to the ground line, is called the horizontal lincf It will be 
 seen, in referring to the cut, that all lines parallel to the sides of the plane of 
 delineation, that is, all the perpendicular lines of the real objects, are also per- 
 pendicular in the representation; that the horizontal lines on the shaded side of 
 the building are all drawn parallel, and that the horizontal lines (that is, those 
 lines that are horizontal in the object itself) on the light side of the building 
 incline towards each other, and that if continued they would meet in a point. 
 Fig. 5, Plate VIII., is a rough plan of the woodcut. Let A represent the 
 position of the spectator, B C the opening of the window through which the 
 view is seen, D E that side of the building in shade, D F the side of the building 
 in light, and F G the garden wall ; H J representing the base line of the plane of 
 delineation. 
 
 It is necessary, in order to understand this jiiagram, fig. 5, that the learner should have some 
 notion of how it is that we see objects. It is by means of rays of light passing from the objects them- 
 selves to the eye, and representing therein an image of the object ; the rays always proceeding from 
 the objects in straight lines and at every possible angle. Fig. 6 is a plan shewing the positions of 
 three cubes, B, C, D, viewed from the point A ; E F representing the ground line of the plane of 
 delineation. The rays of light are sent from every point of eveiy face of the cube in every possible 
 direction. It must be evident from this figure, that none of the rays proceeding from the sides b, c, d, 
 
 * The point of sight in this representation is also the vanishing point ; and let it be remembered that the 
 point of sight is the vanishing point for all the lines of objects that are at right angles with the plane of 
 delineation. 
 
 f All lines parallel to this line, or, which is the same, to the base line of the picture, are called horizontal 
 lines ; but the line which passes through the point of sight is the horizontal line, or line representing the 
 horizon, and its height depends on the position of the eye of the spectator. Much of the pleasing effect of a 
 perspective drawing depends on the height of the horizontal line. 
 
PERSPECTIVE. 
 
 45 
 
 of the cube C in this position, could ever come to the point A, and consequently that the side a would 
 be the only side visible, and that a part only of the rays proceeding from the side a would arrive at A. 
 Where the extreme lines representing the rays from the ends of the side a in their passage to the 
 position of the spectator A, cross the plane of delineation at e f, they denote the length the line a 
 should take in the picture. In the plan of the cvibe the rays from two of the sides, a and d, woidd 
 arrive at the point A, causing both of these sides to be visible ; and the rays from the extremities 
 of the lines a and d would cross the line E F at ^, h, i; g h denoting the width of the side a on 
 the plane of delineation, and h i that of the side d. Precisely the same as with the rays from the 
 cube D proceed those from the cube B, the right side of the cube being visible instead of the left. 
 Let us suppose two other cubes placed behind the cube B, as G and H ; it is obvious that the fronts 
 of these cubes would not be visible, the cube B covering the cube G, and the cube G covering the cube 
 H; but the inner sides of them would be visible, and lines drawn from their extremities to the point 
 A would intersect the plane of delineation at the points k, I, m, n, demonstrating that as the cubes recede 
 from the spectator, the representation of them on the plane of delineation diminishes. The width of 
 the side of the nearest cube B, represented between k and /, on E F, is greater than that of the side 
 of the second cube G, represented between I and m; and this again is greater than the side of the cube 
 the farthest removed, represented between m and n. 
 
 Suppose the position of the spectator to be removed a little to the left at J, the same two sides of 
 the cube D, a and d, are visible to the eye, but by different rays, and they do not consequently preserve 
 the same relative proportion ; from the position at A, the side a is represented considerably wider than 
 the side d; from the position J, the two sides are represented much nearer the same size ; if the 
 position of the spectator were still farther removed to the left, the side d would appear the widest. 
 From the position at J, the side d of cube C becomes visible, but the representation would be extremely 
 narrow, as shewn by the intersections on the plane of delineation by the dotted lines. 
 
 To return to the plan, fig. 5 ; the lines from the points D E, the extreme 
 points of the shaded side of the building that are visible through the opening for 
 the window B C, are drawn to the point A, the position of the spectator, to shew 
 the spaces they occupy on the plane of delineation ; as also the lines representing 
 the light side of the building, with the spaces for the windows and the length of 
 the wall. It will be seen by the plan, that the line D E is parallel to the line H J 
 representing the ground line of the plane of delineation ; and as the face of the 
 building stands perpendicularly over the line D E, the face of the building must 
 be parallel to the plane of delineation. When the plane of any object is parallel 
 to the plane of delineation, the horizontal lines do not tend to a vanishing point, 
 but are parallel to the base of the plane of delineation ; consequently the repre- 
 sentations of the forms on all such planes do not change, they only vary in 
 magnitude according to their distance from the spectator ; thus the corner of the 
 building is represented a right angle, and the windows, with their subdivisions 
 
46 
 
 PRACTICAL RULES ON DRAWING. 
 
 into panes, are all rectangular, preserving their real form, but represented of a 
 smaller size. The side of the house, D F, being at an angle with the plane of 
 delineation, all its horizontal lines must tend to a vanishing point ; and standing 
 at a right angle w^ith the plane of delineation, the vanishing point will be in the 
 point of sight E ; the lines of the top and bottom of the wall also incline to the 
 same point, as the wall is in the same plane as the side of the house. All the 
 parts of the window through which the view is seen, as the sash, window ledge, 
 side walls, &c., are either parallel to the plane of delineation, or at right angles 
 with it ; all the lines representing the several parts, therefore, parallel to this 
 plane, are drawn parallel to the ground line, and all those representing the parts 
 at right angles with it, incline to the vanishing point E. The difference of form 
 in the representation of the flower pot and saucer from that of the circles denoting 
 their position in the plan, must be obvious ; but we must postpone the description 
 of representing circles in perspective to another Chapter: all that is here intended 
 is to make the reader understand that objects seen perspectively change their 
 form and size ; that the size diminishes according to the distance from the spec- 
 tator; and that the size and form that objects appear to have may be regulated 
 by drawing lines to certain fixed points. 
 
 CHAPTER VI. 
 
 Prop. 1. — To find the vanishing points for the delineation of a cube ; the 
 position of the cube B C D E, fig. 1, Plate IX., and the station A of the spectator, 
 being given, F G representing the ground line of the plane of delineation. 
 
 From the point A, parallel to the side D E of the square, draw a line to 
 intersect the line F G at H ; and on the opposite side, from the point A parallel 
 to the line E C, draw a line to intersect the line F G at J. 
 
 Construct a parallelogram, fig. 2, to represent the picture, (called the plane 
 of dfelineation,) making the base line equal to the ground line of the plane of de- 
 lineation F G, and mark on this line the points H and J ; the points perpen- 
 dicularly over these two points H and J, on the horizontal line, will be the 
 vanishing points required. We must now determine the height of the horizontal line, 
 
PERSPECTIVE. 
 
 47 
 
 which is always regulated by the height of the spectator's eye from the ground 
 plane.* Let us suppose that we are working on such a scalef that the base line 
 of the picture represents a length of twenty feet ; taking the height of the spectator's 
 eye to be five feet from the ground plane, draw a line across the picture, five feet 
 of the scale above the ground line, and parallel to it, which will represent the 
 horizontal line ; on this line, perpendicularly over the points H and J, mark the 
 points a and b. The point a will be the vanishing point to which all lines in the 
 original object (A B C D, fig. 1,) parallel to the line D E, will incline ; the point 
 h will be the vanishing point to which all the lines in the original object parallel 
 to the line C E will incline. 
 
 Prop. 2. — From the figs. 1 and 2, to complete the perspective delineation of 
 the cube. 
 
 Construct a parallelogram, similar to fig. 2, in which draw the horizontal 
 line, and mark the vanishing points a and b\ then on the plan fig. 1, from the 
 points B D C E of the square, draw lines to the point A, to represent the visual 
 rays of the extremities of each side of the square, intersecting the line F G at the 
 points 1, 2, 3, 4 ; then will the distance from 1 to 3 represent the width to draw 
 the side D E of the cube, and 2, 4, the width to draw its opposite side B C ; 3, 4, 
 denotes the width of the side C E, and 1, 2, the width of its opposite side B D. 
 Continue the line D E till it meets the line F G at K. Mark all these points on 
 the line F G, fig. 3, and you have all the points necessary to draw the perspective 
 representation required. 
 
 From each of the points 1 , 2, 3, 4, and K, on F G, fig. 2, draw up perpen- 
 dicular lines. The point K is the continuation of the line D E to the line F G 
 on the plan ; draw then a line from the point K, fig. 2, to the vanishing point a, 
 cutting the perpendicular line drawn from 3 at c, and the perpendicular line 
 drawn from 1 at t?; the line Kerf represents the line K E D of the plan in per- 
 spective, and where it passes between the perpendicular lines drawn from 
 
 * The ground plane is in fact the ground on which the objects to be delineated stand, understood to be a 
 horizontal plane. The whole space between the ground line of the picture and the horizontal line, represents 
 the ground plane, on which are placed the various objects to be represented. All surfaces parallel to the 
 ground plane are said to be in horizontal planes ; all surfaces at right angles with it are said to be in vertical 
 planes. 
 
 f It would have been equally easy to have drawn the whole of these figures to a scale giving the size of 
 the cube, its distance from the plane of delineation, and the distance of the spectator from that ; but not being 
 essential to our present object, every thing not absolutely necessary to exemplify that poi'tion of our subject 
 immediately under consideration is avoided. 
 
48 
 
 PRACTICAL RULES ON DRAWING. 
 
 3 and I, as from c to </, it represents the bottom line of the cube corresponding 
 with the line E D of the plan ; from the point c draw a line to the vanishing 
 point h ; where this crosses the perpendicular line drawn from 4 at e it repre- 
 sents the line E C of the plan ; from the point e draw a line to the vanishing 
 point a, cutting- the line d b at f\ the line e f corresponds with the line C B of 
 the plan, and the line d f corresponds with the line D B ; thus the geometrical 
 square E D B C is represented in perspective by the figure c d f e, on which to 
 construct the cube. On the line set up the height of the cube (the length of 
 either side of the plan E D B C) at ^, and from this point draw a line to the 
 vanishing point a, cutting the perpendicular lines from 3 and 1 in the points 
 h and 2 ; c h represents the height of the angle of the cube over the point E, d i 
 the angle over the point D. From each of the points h and i draw lines to the 
 vanishing point h, where the line h h intersects the perpendicular line drawn from 
 
 4 at ek represents the angle of the cube standing over the point C; from the 
 point k draw a line to the vanishing point a. Where the line k a intersects the 
 line i b at I, J' I represents the angle of the cube standing over the point B. This 
 completes the drawing of a cube in perspective viewed from the position shewn 
 in the geometrical plan fig. 1 . 
 
 By reference to the plan, it will be seen that the angle over the point E is the nearest, conse- 
 quently the line c h, representing this angle in the perspective drawing, is the longest ; the point D 
 being nearer to the spectator than the point C, the line d i, though less than h c, is longer than k e • 
 the angle over the point B being farthest removed from the spectator, / / in the perspective drawing 
 is the shortest of the four lines representing the four perpendicular angles : and the same will be 
 observed of the lines representing the horizontal angles, the line c (D E of the plan) is longer than 
 the line/e (B C of the plan), and the line i h, the nearest horizontal angle at the top, is longer than 
 the opposite angle I k; %o also the lines representing the nearest horizontal angles c e and h k are 
 longer than those representing the opposite angles d f and i I, demonstrating that objects of the same 
 size, in their perspective representation, diminish as their distance increases from the station of the 
 spectator. 
 
 In finding the points I and /, which determine the width of the two sides of the cube that arc not 
 seen, those between k e and / and d i and / 1, they were found by the intersections of the lines i b 
 and k a for the point /, and the lines d b and e a for the point /, not making use of the perpendicular 
 line drawn from the point 2. But it will be observed, that these intersections come exactly on this 
 perpendicular line ; shewing that if a perspective dravdng is executed with nicety you may arrive at 
 the same result by different processes. 
 
 Prop. 3. — To find the vanishing point for the delineation of a cube, with its 
 
PERSPECTIVE. 
 
 49 
 
 sides parallel to, and at right angles with, the plane of delineation ; and to draw 
 the cube in perspective. 
 
 B D C E, fig. 3, represents the position of the square on which the cube 
 stands, and A the position of the spectator ; F G the ground line of the plane 
 of delineation. Proceed first to find the vanishing point. It must be evident 
 that in this position of the cube, the lines D E and B C being parallel to F G, 
 no line drawn from the point A parallel to D E could ever intersect the line 
 F G, as parallel lines produced to any length can never meet; the sides of the 
 square D E and B C then can have no vanishing point, but must be drawn 
 parallel to the ground line. The point J, the position of the vanishing point for 
 the sides E C and D B, is perpendicularly over the point A, and is the point also 
 denoting the point of sight ; continue the line B D to the point K, and draw the 
 visual rays from the angles of the square, as in fig. 1, to the point A. Mark all 
 the points on F G in the plan, on F G, fig. 4, the ground line of the picture, from 
 each of which draw perpendicular lines. Mark the vanishing point b, and to it 
 draw a bne from the point K; where this intersects the line drawn from 1, it 
 gives the point d, marking the position in perspective of the point D of the plan, at 
 its distance D K from the line F G : from the point d draw a horizontal line to 
 meet the line drawn from 3 at c ; dc represents the perspective length of the line 
 D E, fig. 3. From the pointy, (Kg being the geometrical height of the cube), 
 draw the line g b; the intersection of this with the line drawn from 1 at i, gives 
 the length d i of that angle of the cube over the point D of the plan ; a horizontal 
 line from i meeting the line from 3 at h, represents the nearest upper horizontal 
 angle; the line between h c, the angle standing over the point E; dike repre- 
 sents one of the planes of a cube parallel to the plane of delineation. Where the 
 lines g b and K b intersect the line drawn from 2 at the points / and f, I /repre- 
 sents the angle of the cube standing over the point B. From the points / and f 
 draw horizontal lines to meet the line from 4 at A; and e\ f e represents the line 
 BC; and / ^ the opposite upper horizontal line to ^ ; </y represents the line 
 D B ; and c e the line EC; i I and h k the upper horizontal lines of the cube at 
 right angles w ith the plane of delineation. 
 
 These two examples, figs. 2 and 4, elucidate the remarks made in the foregoing Chapter. In 
 noticing the view from the window in the woodcut, we remarked that those windows represented on 
 the shaded side of the house, though represented on a small scale, ])reserved their original rectangular 
 form, because that front of the house was in a plane parallel to the plane of delineation. By the 
 example given, figs. 3 and 4, it is shewn that this must be so, as whatever distance the horizontal lines 
 
 (On Ubawino.) p 
 
50 
 
 PRACTICAL RULES ON DRAWING. 
 
 of objects in planes parallel to the plane of delineation may be removed from the station point of the 
 spectator, they can have no vanishing point ; thus the figui-e dike is as perfect a square as the 
 original square of the cube D B C but smaller, from its distance from the ground line (K to D) ; 
 and the figure f I k e is also a true square, but being still farther off, is represented smaller. It was 
 also remarked, that where a plane was at an angle with the plane of delineation, the horizontal lines 
 on it would incline towards each other, and meet in a point ; if this angle was a right angle, the 
 vanishing point would be in the point of sight. This is shewn in figs. 3 and 4, by the lines D B 
 and C E, which being at right angles with the line F G, have their vanishing point in the point 
 of sight. 
 
 If the angle of the plane of any object with the plane of delineation, be other than a right angle, 
 then the vanishing point will be found on some other point of the horizontal line, as seen figs. 1 and 2. 
 The point of sight is a point found perpendicularly above the point A, shewn on the horizontal line, 
 fig. 2 at O ; all the sides of the cube being at an angle with the plane of delineation, and neither of 
 them at a right angle, their vanishing points come on points removed from the point of sight ; there- 
 fore we may state as fundamental rules, that 
 
 All Horizontal lines in a plane parallel to the plane of delineation, in a perspective drawing, are 
 parallel to the ground line of the picture. 
 
 All Horizontal lines in a vertical plane at right angles with the plane of delineation in a perspec- 
 tive drawing, incline towards a point in the horizontal line, opposite the eye of the spectator, called 
 the point of sight. 
 
 All Horizontal lines in a vertical plane at any angle with the plane of delineation, not a right 
 angle, in a perspective drawing, incline towards a point in the horizontal line, either to the right or 
 left of the point of sight, called the vanishing point. 
 
 CHAPTER VII. 
 
 Before proceeding farther, let us here strongly impress on the mind of the 
 student the necessity for his exercising assiduity and patience ; bearing ever in 
 mind, "that nothing is denied to well-directed labour." Let him not be dis- 
 couraged if he does not thoroughly comprehend what he has read up to this 
 point ; but if such should be the case, recommence from the beginning of this 
 part of the subject, (Perspective,) page 36, give it a patient perusal, carefully 
 drawing the different figures on a larger scale ; let him draw, line by line, each 
 figure according to description, and he will find that what at first may have 
 appeared confused and obscure, will become clear and satisfactory. After 
 
PERSPECTIVE. 
 
 51 
 
 drawing the examples given, Plate IX., let him draw a variety of cubes, changing 
 their positions with reference to the plane of delineation, till he feels himself 
 capable of drawing a cube in perspective in any position in which it may be 
 placed ; he will then have become practically acquainted with the broad leading 
 principles of perspective — for upon thes'e rules, as a groundwork, he may proceed 
 to put into perspective the whole face of a building. 
 
 Fig. 3, Plate X., represents the perspective representation of a desk, a portfolio, 
 and a rule, lying on a square table, of which fig. 1 is the plan. B C D E repre- 
 sents the surface of the table ; A the position of the spectator ; and F G the 
 ground line of the plane of delineation. 
 
 Prop. 4. — From the above plan to find the perspective positions of the objects. 
 
 Proceed as in figs. 3 and 4, Plate IX., to put the square representing the table 
 in perspective, as shewn at c d e f, fig. 4 ; — observing that the vanishing point 
 (perpendicularly over the station point A) is between the two sides of the square, 
 instead of being on one side. Having constructed this figure, (B C D E, fig. 2), 
 draw with ink the lines representing the picture F G J K, the horizontal line, 
 and the square in perspective B C D E, and rub out all the pencil lines used from 
 the plan to put the square in perspective, represented in the plate by dotted 
 lines.* Let abed, fig. 1, represent the form and position of a writing-desk 
 standing on the table. To avoid any confusion with what has been previously 
 done, let the student imagine this oblong figure to be the only figure to be repre- 
 sented in perspective, without reference to the square table. Let F G represent 
 as before the plane of delineation ; and A the station point of the spectator ; 
 F G J K the picture on which it is to be drawn. Draw the figure precisely in 
 the same manner as the last, observing that as the side a </ is in a direct line with 
 the spectator's eye, the points a and d will come on the same point on the line 
 F G, and that will be on the point representing the point of sight ; consequently 
 that side will be represented by a perpendicular line. Draw the lines abed, 
 fig. 2, in ink, and as before rub out all the pencil lines that have been used from 
 the plan, so as to be enabled to proceed with the next figure without any confu- 
 sion from the lines necessarily used for the preceding ones. The table and desk 
 
 * The multiplicity of lines perplexes, and almost alarms many students ; but it must be evident, that in 
 our examples we are compelled to leave every line that has been used to produce the various figures ; the 
 student, however, can rub out his pencil lines on the completion of each part of his drawing ; by this proceeding 
 each individual part of this figure becomes a separate problem. 
 
 G 2 
 
52 
 
 PRACTICAL RULES ON DRAWING. 
 
 are both of them represented to stand parallel to the plane of delineation, conse- 
 quently the vanishing point for the sides is found in the point of sight ; but the 
 next figure, efgh, which represents the form and position of a portfolio lying on 
 the table, is at an angle with the plane of delineation ; the vanishing points must 
 therefore be found away from the point of sight. To accomplish this, the vanish- 
 ing points L, M, of the figure efgh, must be found in the same manner as H and J, 
 those for the figure B C D E, fig. 1, Plate IX. ; and the position and form of the 
 portfolio drawn in a similar manner to the figure c d ef, fig. 2, Plate IX. Draw 
 the figure in ink, and rub out the pencil lines. The figure i k I m, representing 
 the form and position of a flat rule, also lies at an angle with the plane of deli - 
 neation, but not parallel with the last figure drawn; other vanishing points, N, P, 
 therefore must be found as in fig. 1, Plate IX. ; and proceed to draw this figure 
 precisely in the same way as the last. This will complete the plan in perspective 
 of the table, desk, portfolio, and rule, of the geometrical plan, fig. 1. 
 
 If the height of the desk, the thickness of the sides of the portfolio, and of the 
 rule, were given, the mode for completing the drawing of the figures is in every 
 respect the same as that given for the completion of the figures of the cubes, 
 figs. 2 and 4, Plate IX. Tliese additional parts are only omitted in this figure, to 
 avoid the confusion that would unavoidably arise in the number of lines requisite 
 for this purpose; but fig. 3 is a representation of the same objects from a similar 
 station, drawn to double the scale. 
 
 The student will observe in this figure, that the table and desk, being parallel to the plane of 
 delineation, have their vanishing points for the sides at right angles with it in the point of sight, as 
 would any number of objects similarly situated ; that the portfolio and rule being at an angle other 
 than a right angle with the plane of delineation, have their vanishing points away from the point of 
 sight ; and that from their lying at different angles with the plane of delineation, their vanishing 
 points also are different. If twenty portfolios were piled one upon another, so as to lie evenly with 
 the edges of one immediately over the edges of anothei", the lines of the sides would all be drawn to 
 the same vanishing points ; but if they were laid unevenly, so that the side of one portfolio vi-as at an 
 angle with the side of another, each portfolio must have its distinct vanishing points. The corner 
 point of the portfolio e, on the plan, is purposely placed exactly at the edge of the table, in order to 
 shew the accuracy with which the relative positions of the points are preserved between the perspective 
 and geometrical drawings — the same corner of the portfolio e coming exactly at the edge of the table 
 in the perspective drawing. 
 
 The foregoing subject, Plate X., has been chosen as a familiar example to 
 shew how, from a plan, a variety of objects in different planes may be put in 
 
PERSPECTIVE. 53 
 
 perspective ; but whether the objects to be represented are, as in the example, a 
 table or a book, &c., or the different parts of the ground plan of a building, the 
 mode for finding the points for the perspective representation is precisely the 
 same as for those figures given in the preceding examples. The figures hitherto 
 chosen are simple parallelograms, but the directions given for drawing them are 
 sufiicient to enable the student to put any right-lined figure into perspective. 
 
 Prop. 5 — From the plan B C D E F G, fig. 5, Plate IX., the plan of a hexagon 
 viewed from the point A, K L representing the ground line of the plane of deli- 
 neation, to draw the perspective form and position. 
 
 From each of the points B, C, D, E, F, G, draw the visual rays to the station 
 point A, intersecting the line K L at the points 1, 2, 3, 4, 5, 6. The sides B C 
 and E F being parallel to the plane of delineation, will have no vanishing point, 
 but must be represented by horizontal lines; the vanishing points for the lines 
 at an angle with the plane of delineation, are found as in the preceding examples. 
 BG and DE being parallel, will have the same vanishing point at H, and CD 
 and G F being parallel, will have the same vanishing point at J. Mark all the 
 points described on K L, fig. 5, on the ground line fig. 6, and raise perpendicular 
 lines from each point, marking the point of sight O and the vanishing points 
 H and J on the horizontal line. B C and F E being parallel to the plane of 
 delineation, their width may be determined by lines drawn to the point of sight. 
 From the point B of the plan draw a perpendicular line to the line KL at a, and 
 place the point a on the ground line, fig. 6, from which point rule a line to the 
 point of sight 0, then a B, fig. 6, will represent in perspective a B, fig. 5, of 
 the geometrical plan ; from the point B draw a horizontal line to meet the line 2 
 at C, the distance between 1 and 2 representing the perspective width of the side 
 B C. From the point B draw a line to the vanishing point H, till it meets the 
 line 6 at G, the distance between 1 and 6 being the perspective width of the side 
 B G of the hexagon ; from the point G draw a line to the vanishing point J, till 
 it meets the line 5 at the point F, from 5 to 6 representing the perspective width 
 of the side G F. On the opposite side draw, in like manner, the lines C D and 
 D E, and join the points E and F; the line uniting them will be found exactly 
 parallel to the line B C, and the whole figure will be the representation of the 
 hexagon, fig. 5, in perspective. 
 
 Here it will be well to remark that this figure is an excellent example to shew that the same 
 
54 PRACTICAL RULES ON DRAWING. 
 
 result may be arrived at, from whatever point we may commence. Had we commenced from tlie 
 point B to di-aw the line B G instead of the line B C, G B in the geometrical drawing must have been 
 continued tiU it met the line K L at 6, and the point b marked on the ground line fig. 6, from which 
 point a line drawn to the vanishing point H would have intersected the line 1 in the same point B as 
 the line drawn from the point a to the point of sight 0. So with the point F. The dotted line B F, 
 fig. 6, represents the dotted Hue B F, fig. 5, and the line « B F, fig. 6, intersects the line 5 in the 
 same point as the line di-awn from the point G to the vanishing point J. If it had been requisite to 
 commence the perspective figure with the line G F, the line G F of fig. 5 should have been continued 
 to meet the line K L at the point c placed on the ground line fig. 6, and a line ruled from it to the 
 vanishing point J, as shewn by the dotted line c G which cuts the lines 6 and 5 in the same points 
 G and F that were found by the previous method. 
 
 To construct, on such a plan as fig. 6, a solid figure, we should proceed in the 
 same way as described for constructing the cubes ; construct a square or oblong 
 on the line B C, and from the upper line draw the hexagon in the same way 
 exactly as that fi^om the lower line. Fig. 7 represents a solid hexagonal figure 
 constructed on the foregoing plan. 
 
 If the plane of delineation were such as is shewn by the dotted lines N P, 
 every side of the hexagon being at an angle (not a right angle) with the plane of 
 delineation, would have its vanishing point away from the point of sight ; the 
 opposite sides of a hexagon being parallel, three vanishing points would be 
 required. 
 
 Any other rectilinear figure, either a triangle or polygon, may be drawn from 
 these directions ; the student is recommended to draw according to the foregoing 
 description a pentagon, heptagon, and nonagon. 
 
 CHAPTER VIII. 
 
 Trusting that the matter in the foregoing Chapters has been sufficiently 
 explicit to enable the student to comprehend the drawing of the various figures 
 of which it is descriptive, we will proceed to put into perspective some figures 
 
PERSPECTIVE. 
 
 55 
 
 of common objects; premising that what we are about to perform depends 
 entirely on the rules previously given, but that in executing the figures, those 
 lines only will be used which are absolutely requisite for the object in hand. 
 
 Prop. 6. — From fig. 1, the plan, and fig. 2, Plate XI., the elevation of a square 
 slab standing on two steps, viewed from the point A, to draw the perspective 
 representation, B C marking the plane of delineation. 
 
 This figure, for the purpose of shewing the student how any object may be 
 represented in perspective from measurement, is drawn to a scale of one-eighth of 
 an inch to a foot in the plan and elevation ; the perspective drawing, for the advan- 
 tage of more clearly shewing the various points and lines, is drawn to a scale of a 
 quarter of an inch to a foot. Let the student first, as in figs. 1 and 2, Plate IX., 
 find the vanishing points for the sides a b and c d at E,* and for the sides a c and 
 ^ at D ; then draw the visual rays of the four angles of the outer square to the 
 station point, and continue the line b a to the line B C ; mark all the points of 
 intersection on the ground line of the plane of delineation 1 23 4 5, and the 
 positions of the vanishing points E D, on the ground line of the picture, fig. 3, 
 and as in the preceding figures draw the perspective square a 6c f/. The only 
 use of the point d in this figure, is for drawing in perspective the diagonal lines 
 a d and h c, which will be found of great value. From the point 5 set up a per- 
 pendicular line to serve as a line of projection for the measurement of the heights 
 of the steps, and on it mark the height of the first stepf at 6, from which draw a 
 line to the vanishing point E ; where it crosses the lines 1 and 2 at e and f, it 
 gives the perspective drawing of one side of the lowest step, corresponding with 
 the side ab ef oi fig. 2. From the point e draw a line towards the vanishing 
 point D till it meets the line 3 at s, which will complete the drawing of the other 
 side of the lower step. Continue the lines 7 and 8 of the plan to the line ab, 
 and from their points of contact draw visual rays to the station point, and carry 
 the points of intersection, 9, 10, to the ground line of the picture; from the point 
 9 draw a perpendicular line intersecting ab 2X g, and ef ^tli\ then aegli will 
 represent in perspective the square ae g h oi the elevation. From the point g 
 
 * This vanishing- point is found nearly at the extremity of our Plate ; consequently the vanishing point 
 for our perspective drawing must be some distance beyond our limits. The lines however are all drawn towards 
 the letter E, and if continued would be found to meet in a point at its correct position on the horizontal line. 
 
 f Bear in mind that throughout this figure the measurements for the perspective representation are double 
 the measurements of the plan and elevation ; the student, however, as he will not be cramped for room, can 
 draw the plan, elevation, and perspective drawing to the same scale. 
 
56 
 
 PRACTICAL RULES ON DRAWING. 
 
 draw a line to the vanishing point D ; this will give i, a point on the diagonal 
 a d corresponding with the point i of the plan, and I a point on the diagonal h c 
 corresponding with the point / of the plan. From the point i draw a line to the 
 vanishing point E; this will give k, a point on the diagonal b c corresponding 
 with the point k of the plan. 
 
 Observe here that the width from a e to (/ h is found by means of the visual ray drawn from ff to 
 A and its point of intersection on B C carried to the ground line of the picture ; \, 9, representing the 
 perspective width of a ^ of the plan ; each of the corresponding points of the corners of the inner 
 square could have been found iu the same manner, but the process here taken is equally correct 
 and far more simple. If the point marked m in the plan had been required, the intersections of 
 two lines drawn from k and I to the vanishing points D and E would give this point, and would fall 
 on the diagonal line a d; this point however is not required in this i-epresentation. 
 
 From each of the points i, k, and I, draw a perpendicular line ; from the point 
 h a line drawn to the vanishing point D where it crosses the perpendicular lines 
 drawn from i and will represent the bottom line n o of the second step ; from 
 the point n a line drawn to the vanishing point E at its intersection with the per- 
 pendicular line drawn from k, will represent the bottom line of the second step on 
 the other side. On the perpendicular line drawn from 5, set up the height of the 
 top of the second step at 11, and from it rule a line to the vanishing point E; 
 where this line crosses the perpendicular line drawn from the point 9 at* p, it 
 gives the point for determining the height of the second step in perspective ; from 
 p draw a line to the vanishing point D ; where this intersects the perpendicular 
 line drawn from i at q, it denotes the nearest top corner of the square; where it 
 intersects the perpendicular drawn from I at r, it denotes the farthest top corner 
 on the right; from the point q a line must be drawn to the vanishing point E ; 
 where this line intersects the perpendicular drawn from k at t, it denotes the 
 further top corner on the left. 
 
 Proceed to draw the square slab on the top exactly in the same manner as 
 the slab forming the second step. From the point 10 draw up a perpendicular 
 line ; from its point of intersection on a ft draw a line to the vanishing point D ; 
 this will give the points on the diagonal lines over which stand the angles of the 
 right side of the square ; from the point of intersection on the diagonal a d draw 
 a line to the vanishing point E; this will give a point on the diagonal line be 
 over which stands the angle to the left ; from each of these points on the diagonal 
 lines draw a perpendicular line. From the point in which the line drawn from 
 1 1 (the geometrical height of the bottom of the slab) to the vanishing point E, 
 
PERSPECTIVE. 
 
 57 
 
 intersects the perpendicular line drawn from 10 atw, a line must be drawn to the 
 vanishing- point D ; where this line intersects the perpendicular lines drawn to 
 represent the right hand angles at v and w, it denotes the lower right-hand line of 
 the slab ; from the point of intersection with the nearest perpendicular at v, a line 
 must be drawn to the vanishing point E; where it meets the perpendicular line 
 drawn to represent the left-hand angle of the slab, it represents the bottom line 
 on the left. To get the height of the top of the slab in perspective, set the geo- 
 metrical height on the perpendicular line of projection 5 6 II, at 12, and from 
 this draw a line to the vanishing point E ; where this line intersects the perpen- 
 dicular line drawn from 10 at x, it denotes the point from which to draw the 
 upper lines of the slab ; draw from x a line to the vanishing point D, and its 
 intersections with the lines denoting the upright angles of the square at i/ and z 
 will complete the right-hand side of the slab ; from the nearest point of inter- 
 section, 1/, draw a line to the vanishing point E, which, where it meets 
 the upright line of the left angle, will complete the drawing of the left side of the 
 slab. From the corner / draw a line to the vanishing point D, to meet t,he 
 angle of the second step ; from the point t to the same vanishing point, draw a 
 line to meet the angle of the upper slab. From the point s draw a line to the 
 vanishing point E, to meet the angle of the step above, and from the point 
 to the same vanishing point a line to meet the angle of the upper slab ; from the 
 points z and its opposite angle draw lines to the respective vanishing points E 
 and D, intersecting at the point F, and the perspective figure is completed. 
 
 In the examples hitherto made use of, the objects have been represented below 
 the eye of the spectator; let us take an example where part of the object is below 
 and part above the eye. The mode of drawing such a figure is the same as before 
 described ; but it will be seen, in our progress, that where any projection occurs in 
 any object above the eye, it hides a part of the object above it. 
 
 Prop. 7. — From the given plan and elevation of an obelisk, with the station 
 point and plane of delineation, to draw the perspective representation. 
 
 The plan, fig. 1, Plate XII., is figured to correspond with the elevation, fig. 2 ; 
 1 , 2, representing the square of the base ; 3, 4, the square of the coping ; 5, 6, the 
 square of the plinth, &c. ; 11, 12, the inclined lines of the pyramid ; and 13, the 
 apex. 
 
 Proceed as in the last figure to put the plan into perspective, getting all the 
 
 ^On Drawing.^ H 
 
58 
 
 PRACTICAL RULES ON DRAWING. 
 
 points in it from 1 to 13. Set up the line A B as a line of projection for the per- 
 spective heights of the various parts. From the point A set up the height of the 
 base at b, and draw the base as in the foregoing example.* 
 
 Note here, that to get the heights of the parts of the elevation in the perspective representation, 
 it is required to find them first on perpendicular lines drawn from the line 1 to 14, by means of lines 
 drawn from the geometrical measurements on A B, and carried from the points of intersection by 
 horizontal lines to the perpendiculars drawn from the plan ; thus to find the point of the apex at the top 
 of the pyramid in the elevation, which stands perpendicularly over the centre of the squares at 1 3, both 
 in the plan and perspective representation, a perpendicular line must be drawn from the point 13 ; 
 somewhere on this line must come the point/. From the point 13 draw a horizontal line to meet the 
 line 1 14; and from the point of intersection di-aw up another perpendicular line; from the point / 
 on A B (the geometrical height of the apex) di'aw a line to the vanishing point ; where this intersects 
 the last perpendicular drawn from the line 1 14, it denotes the perspective height of the apex ; and a 
 horizontal line drawn from it, intersecting the perpendicular drawn from 13 at/, is the perspective 
 position of the apex of the obelisk. 
 
 From the points 5 and 6 draw perpendicular lines to represent the sides 5, Q, 
 of the elevation ; continue the line 6 5, to 15, from which point draw a perpen- 
 dicular line ; from c on A B draw a line to the vanishing point ; where this inter- 
 sects the line drawn from 15, is the perspective height of ac; draw from this 
 point a horizontal line ; and w here it passes between the perpendicular lines 
 drawn from 5 and 6 it represents the line c of the elevation ; how to draw the 
 base line of this, shewing where it stands on the bottom slab, has been shewn 
 in the preceding figure. From the points 3 and 4 raise perpendicular lines to 
 represent the sides 3, 4, of the projection in the elevation ; continue the line 4, 3, to 
 the line 1, 14, at 16, and find the perspective representation of the parallelogram of 
 the elevation from it (16) and the points c and cl on A B as before directed, in this 
 and the preceding problems, for similar figures. From the points 7, 8, draw per- 
 pendicidar lines to find the perspective positions of the points 7, 8, of the elevation, 
 from which the base of the shaft springs ; continue the line 8, 7, to the line 1, 14, 
 from the point of meeting draw a perpendicular to meet the line drawn from d to 
 the vanishing point, from whence draw a horizontal line, cutting the lines drawn 
 from 7 and 8 at 17 and 18, which are the points required. Here we see that the 
 
 * One side of the object being parallel to the plane of delineation, and the figure being rectangular, the 
 vanishing point is in the point of sight. 
 
PERSPECTIVE. 
 
 59 
 
 line d of the elevation, in the perspective representation, hides the base of the 
 shaft ; nevertheless it is necessary these ])oints should be found, as without them 
 we should not be able to draw the perspective representation of the lines 7, 9; 
 8, 10. Draw perpendicular lines from the points 9, 10, and continue the line 10, 9 
 to the line 1, 14; from its i)oint of contact draw a perpendicular line to meet a 
 line drawn from e to the vanishing- point, and from the [)oint of intersection draw 
 a horizontal line, cutting the perpendiculars from 9 and 10 in the points 19, 20 ; 
 join 17 and 19, and 18 and 20, and we have the figure 7, 8, 9, 10, of the elevation 
 in perspective. We have already shewn how to find the apex f \n perspective; 
 join 19 f and 20 f, and we have the whole figure of the elevation in perspective. 
 
 Let us here remark, that the plan of this figure is represented parallel to the plane of delineation, 
 and that the individual parts, as a 6 1 2j 5 6 i and 3 4 c e? of the elevation, retain in the per- 
 spective representation their original form, because they are parallel to the plane of delineation ; but 
 the figure 7 8 9 10, and the figure 9 10/, are neither of them parallel to the plane of delineation, 
 and consequently do not retain their original form; the perspective figui-e 17 18 19 20, being a 
 difierent form to the elevation 7 8 9 10, which it represents ; also the perspective figure 19 20 /, 
 is different to the figure 9 10/ of the elevation. 
 
 The rules for drawing the right-hand side of this figure have been so fully 
 explained in the preceding figures, that it is needless to recapitulate them ; the 
 heights of all the further corners are regulated by lines drawn from the near 
 corners to the vanishing point; the width of every part by lines drawn from the 
 points of intersection on the diagonal line. 
 
 The whole of the foregoing figure, which may appear somewhat intricate to the young student, 
 may be clearly understood by the diagrams, figs. 4, 5, 6, PL XI. Fig. 4 and fig. 5 represent the plan 
 and elevation of a double cross, each forming a similar figure. Put the plan in perspective similar to A, 
 fig. 6, and the elevation similar to B, fig. 6. From each of the points 1 to 12, (A, fig. 6,) draw a 
 perpendicular line, and from the points a to m, (B, fig. 6,) draw horizontal lines, crossing the perpen- 
 dicular lines di-awn from A, fig. 6. The intersections of these lines will give all the points required 
 for drawing this figm-e in perspective. The horizontal lines from g h, crossing the perpendicular lines 
 from 1, 2, will give the nearest square of the cross 13, 14, 15, 16. The horizontal lines from i k, 
 crossing the perpendicular lines from 10, 11, will give the square behind it, 17, 18, 19, 20; if the 
 points 15 and 19, 16 and 20, and 14 and 18, are joined, it will complete the perspective of a solid 
 cube, the nearest of the seven cubic solids forming the figure ; and if these lines were continued on, 
 they would meet in a point, the vanishing point used for putting the plan and elevation in perspective ; 
 the same will be found in every cube of which this figure is composed. So wath the last figure ; if we 
 had put the ground plan in perspective, and by the side of it the elevation, perpendicular lines drawn 
 
 H 2 
 
60 
 
 PRACTICAL RULES ON DRAWING. 
 
 from all the points of the former, and horizontal lines drawn from all the points of the latter, would 
 have given, by their intersections, all the points requisite for the perspective representation. This is 
 really the process by which the preceding figiu*e has been drawn, as may be seen by the dotted lines 
 representing one half of the figui-e ; haK of the figure only being required to draw the perspective 
 representation. 
 
 CHAPTER IX. 
 
 Having given thus much information, sufficient to enable the student to draw 
 almost any rectilinear figure in perspective, we will now proceed to give direc- 
 tions for drawing curves. The only means by which curvilinear figures can be 
 represented in perspective is by intersecting the curves by straight lines, putting 
 those straight lines in perspective, and thus finding the positions of their inter- 
 secting points in perspective, through which the curves must be drawn. 
 
 In looking at any round object, as a plate, waiter, &c., unless we were to 
 stoop over it, or that it were set upright, so as to bring the centre imme- 
 diately opposite the eye, it would not appear to us a circle. Let us imagine a 
 circular card, dark on the upper side and white beneath, with a perpendicular 
 string or wire, from the ceiling to the floor, passing through its centre. If this 
 card were slid up the string, above the eye of the spectator, he would lose sight 
 of the black side, and the white would become visible ; if it were slid on the 
 string so as to come exactly even with the eye, neither side would be visible, but 
 the thickness only of the card would be seen. The student can readily compre- 
 hend that in the gradual progress of the card from the ground to a position even 
 with the eye, it must be constantly changing its form ; the diameter one way 
 always remaining the same, but the diameter the other way constantly decreasing 
 till it becomes perfectly invisible. Fig. 1 represents this ; the card at the position 
 «, the lowest, shews the broadest figure and the dark side ; at the position b, 
 higher up, the dark side is still visible, but much narrower ; at c, being exactly 
 even with the eye, neither side is visible; at d, above the eye, the white under 
 side becomes visible. To put a circle in perspective, we must first strike a 
 circle with the compasses, and construct about it a square, as in fig. 2, Plate XIII., 
 
PERSPECTIVE. 
 
 61 
 
 draw the diameters 1, 5, and 3, 7, and the diagonal lines intersecting the curve of 
 the circle in the points 2, 4, 6, 8 ; and draw perpendicular lines through the points 
 4, 6, and 2, 8, to meet the opposite sides of the square. 
 
 Prop. 8. — From the diagram, fig. 2, to put the circle in perspective. 
 
 From the preceding examples the student ought to be able to put this recti- 
 linear figure in perspective without assistance ; but the Author, from long expe- 
 rience in teaching, being aware that the most trifling variation will sometimes 
 confuse the pupil, will proceed with the directions for this figure ; giving the 
 student to understand that the directions for drawing it will serve for any figure 
 representing a circle or part of a circle in perspective. 
 
 In giving instructions for drawing this figure, it is unnecessary to go over all 
 the ground we have trodden before ; let us hope that this square could be put in* 
 perspective from any given station-point ; we will therefore consider the figure 
 A B C D, fig. 3, to be drawn so as to represent the square A B C D, fig. 2, in 
 perspective. In the perspective square draw the diagonal lines A C and B D ; 
 carry the points £ 3 and F, fig. 2, to the ground line, fig. 3, and draw from each 
 point a line to the vanishing point. Where the line drawn from E intersects the 
 diagonal A C, it gives a point corresponding with the point 2 of fig. 2 ; where it 
 intersects the diagonal B D, it gives a point corresponding with the point 8, fig. 2. 
 Where the line drawn from F intersects the diagonal B D, it gives a point cor- 
 responding with the point 4, fig. 2 ; where it intersects the diagonal A C, it gives 
 a point corresponding with the point 6, fig. 2. The line drawn from 3 to the 
 vanishing point, where it intersects the line C D, gives a point corresponding 
 with the point 7, fig. 2 ; a horizontal line drawn through the centre of the per- 
 spective square (shewn by the intersection of the diagonal lines) will give a point 
 on the line A D corresponding with the point 1, fig. 2, and on B C a point cor- 
 responding with the point 5, fig. 2. Thus we have the perspective positions of 
 all the points 1,2, 3, 4, 5, 6, 7, and 8 ; through which points if a curve line be 
 drawn, it gives the representation of a circle in perspective, seen in a horizontal 
 position. 
 
 If it were required to make a vertical representation of a circle in perspective, 
 the process would be the same. Set up perpendicularly over the point A the line 
 A from which draw the perspective square A 5 c D ; draw, as in the former 
 figure, the diagonals A c and b D ; mark on the line A b the divisions e 3 f, rule 
 a line from each to the vanishing point, and draw a perpendicular line through 
 
62 
 
 PRACTICAL RULES ON DRAWING. 
 
 the perspective centre of the square ; the intersections of these lines will give in 
 this figure all the points corresponding with the points 1, 2, 3, 4, 5, 6, 7, 8, of the 
 preceding, through which the curve may be drawn to represent a circle in per- 
 spective seen in a vertical position. 
 
 Prop. 9. — To draw the perspective representation of the shaft of a column, 
 the superior and inferior diameters of which are equal. 
 
 It is unnecessary in these problems, which are given merely as examples for 
 drawing the representations of certain forms in perspective, to give a plan for 
 each individual figure ; it would be a waste of space and time. Any one of the 
 plans in the preceding figures representing a square, seen from a stated position, 
 may be supposed to contain a circle, and from it the student can make his 
 drawing according to rule. In the following figures it is assumed that all the 
 preliminary steps are gone through, and the points from which the representation 
 of the square was drawn had been previously found. 
 
 Let A B, fig. 4, represent the diameter of the circle ; from the point 3 describe 
 the half circle,* from which, as ift the preceding figures, find the points E and F. 
 From these points, A, E, 3, F, B, construct a square, with the intersecting lines, 
 marking the points to correspond with the last figure. At the proposed height 
 of the shaft draw a horizontal line, and mark on it perpendicularly over the 
 corresponding points on the ground line A, E, 3, F, B. From the points A and B 
 draw lines to the vanishing point O- Draw up perpendicular lines from the 
 points D and C, and join their points of intersection with the lines A Q, and B O, 
 at C D. This upper figure, A B C D, represents a square in perspective, the 
 sides and angles of which are perpendicularly over the sides and angles of the 
 lower square A B C D. Draw the diagonal lines in the upper square, lines 
 from the points E 3 F to the vanishing point, and a horizontal line through the 
 centre, and you will have points of intersection corresponding with the points 
 1, 2, 3, 4, 5, 6, 7, 8, of the lower figure; the point 1 of the upper figure being 
 perpendicularly over the point 1 of the lower figure, 2 over 2, 3 over 3, &c. &c. ; 
 draw the curve to represent the circle in perspective through the respective points 
 in each figure, and join the side extremities of the top and bottom curves ; this 
 will complete the perspective representation that was required. 
 
 In the directions for drawing the orders of architecture, it was shewn that the 
 
 * It is needless to draw more lines than are necessary ; the half circle being quite sufficient for the 
 purpose required. 
 
PERSPECTIVE. 
 
 63 
 
 shafts of columns taper from the base upwards ; we must therefore shew how 
 such a column is to be represented in perspective. 
 
 Prop. 10. — To draw the perspective representation of the shaft of a column, 
 the superior diameter of which is less than the inferior diameter. 
 
 Let A B, fig. 5, represent the inferior diameter of the shaft ; draw the per- 
 spective square A B C D, in which construct a similar figure to the last drawn, 
 and describe the curve ; at the proposed height of the shaft draw a horizontal 
 line, and set on it the points A B perpendicularly over the points A B on the 
 ground line ; from the points A and B draw lines to the vanishing point, and, as 
 in the last figure, draw the upper perspective square A B C D, and within it the 
 diagonal lines A C, B D. Let a b, on the ground line, represent the superior 
 diameter of the shaft; from the centre c draw the geometrical figure, for which, 
 observe, the same diagonals serve ; find the points d and e, and mark between 
 A and B on the upper line the several points a, d, c, e, b, perpendicularly over the 
 points a, d, c. e, b on the ground line. From each of these points draw a line to 
 the vanishing point ; then f g h i will represent the square in perspective, in 
 which the curve is to be drawn. Draw a horizontal line through the centre, and 
 you will have intersecting points in the square corresponding with the intersecting 
 points in the preceding figures, through which draw the curve ; join the extre- 
 mities of the upper and lower curves, and the figure required is completed. 
 
 Prop. 11. — To draw the perspective representation of a bridge with two 
 semicircular arches, of which fig. 6 is the elevation on a small scale. 
 
 First construct a parallelogram in perspective to represent abed, fig. 6, as 
 in fig. 7, which is drawn to double the scale of the elevation ; and here we will 
 make use of a rule by which we shall find the width of the piers and arches, and 
 the points through which to draw the curves, by a shorter process than finding 
 them from a plan. Let us suppose the perspective parallelogram to have been 
 drawn from a regular plan ; measure off on the ground line to the right of the 
 point a the geometrical width of the piers and arches from a to e (double the 
 length of the elevation, fig. 6) ; from e through c rule a line till it meets the hori- 
 zontal line at f ; this point will give the perspective distances on the line a c of 
 any geometrical distance measured on the line a e* as correctly as if they were 
 drawn from the plan. Mark then in the spaces for the arches on a e, the points 
 (1, 2, 3, 4, 5) required to draw the perspective curve, and carry these points to 
 
 * See Plate VIIL, and Chap. I, on Perspective. 
 
64 
 
 PRACTICAL RULES ON DRAWING. 
 
 the line a c by means of intersecting lines drawn to the point of distance / ; from 
 each of these points draw a perpendicular line to meet a line from g, the geome- 
 trical height of the arches ; from the middle point on a e, to each of the corners 
 on the line drawn from g, draw a straight line, and you will have the inverted 
 rectilinear figure below the line a e set upright in perspective in its correct 
 situation on the line a c. We now require to draw the thickness of the arches, 
 and to facilitate the student we have shewn the mode for accomplishing this in a 
 fresh figure. 
 
 Fig. 8 represents on a larger scale the parallelogram of the bridge abed, 
 and the first arch, with all the lines erased that were used for drawing it ; the 
 points 1, 2, 3, 4, 5, only being left, as they are required for what follows ; first 
 from a and b draw lines to the vanishing point P, and mark the width of the 
 bridge in perspective ;* from the point g draw a line to the vanishing point P, 
 cutting i h at k ; then from the points k h draw lines to the vanishing point ©f . 
 From each of the points 1, 2, 3, 4, 5, draw a line to the vanishing point P, inter- 
 secting the line k q at the points 6, 7, 8, 9, 10, from each of which points draw a 
 perpendicular line to the line h O ; join 12, 6, and 12, 10, and you will have a 
 rectilinear figure for drawing the inner arch corresponding in all its intersecting 
 points with that from which the outer arch was drawn ; draw the curve through 
 the points of intersection, and from the point I a line to the vanishing point P, 
 cutting the perpendicular line from 10 at 11. Proceed with the second arch in 
 the same manner, and the representation will be completed. 
 
 It is necessary to shew the whole of eveiy process by which a drawing is to be put in perspective, 
 as from being acquainted with different ways of producing the same result, one may be serviceable 
 for one purpose, another for some other ; so various are the ways of arriving at the same end, that it 
 is not uncommon to see two draughtsmen, working at the same subject, producing fac-simile drawings, 
 and yet finding almost all the points by different processes. That which is accomplished by the fewest 
 possible lines, is always best. 
 
 A very simple mode of drawing a series of arches is shewn in figs. 9 and 10. Fig. 9 is part of 
 the elevation of an archway (all that is essential for our pui'pose). The student will hei-e observe that 
 not a line is thrown away ; a quadrant of a circle only is enclosed in a square, with only one diagonal 
 di'awn to get the point of intersection for the curve, and this is carried horizontally to the line ab at c 
 the height of the piers, from where the arches spring, is also carried horizontally to the line a b at d; 
 
 * This is taken here at will, and chosen of a trifling width, the better to understand the problem, 
 f This vanishing point, from want of space, is out of the picture ; nevertheless the student will understand 
 that all lines said to be drawn to it would meet in the same point if continued. 
 
PERSPECTIVE. 
 
 65 
 
 cany these points c d to the liae a b, fig. 10. Upon the line b e mark the geometrical width of the 
 piers and arches, with the centre of each arch as at 1, 2, 3 ; 1, 2, 3, &c. : and carry all these points as 
 before directed to the line 6 © at 4, 5, 6 ; 4, 5, 6, &c. : from all the points 4 and 6 draw perpen- 
 diculars between the lines a 0 and h 0 ; draw the lines d Q and c O > draw a perpendicular line from 
 each of the points 5 to the line</0 at 7, 7, &c. ; join all the points 4, 7, and 6, 7, through which lines 
 the intersections of the line d Q will give points for drawing the curves for fifty arches if required. 
 
 Fig. 11, A, represents the common method of drawing a simple pointed arch 
 geometrically, by the intersection of two semicircles; fig. 11, B, represents the 
 same in perspective. This however would be but a clumsy mode of proceeding. 
 Fig. 12, A, is a pointed arch drawn similar to fig. 11, A, enclosed in a rectan- 
 gular figure, with points of intersection taken as in the semicircular figures, figs. 
 7 and 9. Fig. 12, B, is the same figure represented in perspective. 
 
 Where arches are represented on a large scale, it is necessary to find more 
 points of intersection for drawing the curves. Fig. 13, A, is the geometrical 
 figure of a pointed arch, having three points of intersection, through which to 
 draw the curve; fig. 1.3, B, is the perspective representation of this figure. Any 
 additional number of intersecting points the draughtsman may require to enable 
 him to describe the curve, may be made in the geometrical drawing. 
 
 The student will easily comprehend, from the foregoing examples, that any 
 curve may be put in perspective by enclosing it in a rectilinear figure, getting 
 points of intersection in certain parts of the curve, and then finding their per- 
 spective positions. To exercise himself in drawing curvilinear figures in per- 
 spective, let him refer to the plate on Conic Sections (Plate VII., Elementary, A, 
 Part IV.) Let him first take the figure of the ellipse; draw the parallelogram in 
 perspective; then the transverse and conjugate axes ; divide the transverse axis 
 into 10 perspective divisions, and each of the sides of the parallelogram parallel 
 to the conjugate axis, also into 10 perspective divisions ; then draw the lines 
 similar to those in the geometrical figure, and they will give points of intersection 
 through which a curve drawn will represent the ellipse in perspective. So with 
 the figures of the parabola and hyperbola; find the perspective positions of all 
 the points on the geometrical rectangular figures, then draw^ lines from the re- 
 spective points to get the perspective positions of the intersections, through which 
 points a curve drawn will represent the figure in perspective. By the foregoing- 
 directions the student should be able to draw in perspective the whole of the 
 figures of mouldings given in Plate II. on Drawing. 
 
 In a similar manner to that used for drawing the regular figures represented 
 
 (On Uhawi\o.) f 
 
66 
 
 PRACTICAL RULES ON DRAWING. 
 
 in Plate XII., may be drawn in perspective the representation of ornaments. 
 To find all the points of intersection for the curve lines in such figures as those 
 of figs. 4, 5, 6, Plate VII., would be an endless labour; but leading points must 
 be found, so as to regulate the height and width of the principal features, at cer- 
 tain distances, and the intermediate parts drawn by the eye. This requires 
 practice; but the student will find, by diligence, that his eye will gradually 
 become accustomed to perspective form and proportion, and in his progress he 
 will discover that muchfewer points are required to execute a perspective drawing, 
 than he may be led to imagine at the commencement. To make the rules 
 intelligible, we are compelled at the commencement to go through the whole 
 process, secundum artem ; but as we proceed, all extraneous matter is left out; 
 witness the simphcity with which a long series of arches may be drawn by the 
 rules in the example given, figs. 9 and 10, Plate XIII. 
 
 The rules and examples given in the preceding pages are sufficient to enable 
 an attentive student to put in perspective the whole of a building ; still the com- 
 plication of parts may render such an attempt a little puzzling ; the Author may 
 probably in a Supplementary Part recur to this subject, and from the foregoing 
 matter, (referring to the different problems as he proceeds in the several parts,) 
 from a regular plan, and front and side elevations of some familiar structure, shew 
 how to put the whole in perspective. 
 
 CHAPTER X. 
 
 ON LIGHT AND SHADE. 
 
 It is almost superfluous to attempt an explanation of what is meant by light 
 and shade. Every one must be aware that when the sun shines on any object, it 
 is said to be in light; that if this object is interposed between the stm and 
 another object it casts upon it what is termed a shadow; that those parts of an 
 object that do not face the sun are said to be in shade. Painters make use of 
 the terms light, shade, and middle tint, to express the different gradations ; middle 
 
LIGHT AND SHADE. 
 
 67 
 
 tint, however, may be either light or shadow. An object may be in light, yet that 
 light so subdued as to be quite as dark as some of the parts in shade. This 
 materially depends on the direct or oblique manner in which the sun's rays are 
 received on the object, as also on the local quality of the object to be represented. 
 It may readily be conceived that a newly white-washed building receiving the 
 direct rays of the sun, must appear infinitely brighter than a dusky brick or dark- 
 painted building. If any object stand at an angle with an object in bright light 
 it becomes partially lighted by it ; this is called reflected light.* It is usual to 
 represent those shadows cast by one object on another, darker than the object 
 casting the shadow ; and in ordinary drawing it is best to follow this rule, though 
 it may not always be correct, for an object of a dark material, casting its shadow 
 on any very light surface, may appear darker than the shadow it casts ; the shadow 
 of a dingy brick chimney thrown on a white plaster building would be lighter 
 than the chimney casting the shade. Rules have been given, attempting to regu- 
 late the quantities of light and shade in a picture. It has been said that a good 
 picture should consist of one portion of positive dark, one of positive light, and 
 six of middle tint; and numerous examples have been brought forward to prove 
 that some of the finest productions of art derive their excellence from such a dis- 
 tribution ; but it is impossible to lay down any fixed rules for the quantities of light 
 and shade; a judicious arrangement of light and shadow constitutes one of the 
 great excellencies of painting ; but it is impossible to make a picture from a 
 recipe. Middle tint is most essential to the good effect of a picture, and should 
 occupy the principal portion of the subject; it gives value to the light and to the 
 darker shades ; but the quantity must depend on the nature of the subject, the 
 hour of the day intended to be represented, and above all, on the judgment and 
 feeling of the artist. Mr. West, the president of the Royal Academy, in speak- 
 ing on this subject some half century back, said that the principles of light and 
 shadow might be exemplified by a bunch of grapes. Whether or no the idea 
 originated with this venerable painter is of little importance ; the choice of sub- 
 ject for illustrating in a simple manner the beauty of light and shade is most 
 happy ; the endless variety of shades cast by the irregular form and disposition 
 of the grapes — the little points of dark in the angular depths between the grapes 
 
 * If any one place an object in the sun against a dark curtain, a bust for instance, he will perceive that 
 side of the object on which the sun shines to be in bright light, and the side next the curtain in deep shade ; if 
 a sheet of white paper be placed between the object and the curtain, the side in shade becomes instantly lighted 
 up by the rays reflected from the white paper, so as to shew every part distinctly. 
 
68 PRACTICAL RULES ON DRAWING. 
 
 — shew admirably how, by a judicious arrangement of light and shade, reflected 
 light and cast shadows, a representation of the most simple object may be made 
 agreeable to the eye. 
 
 In landscape painting, the morning and evening are almost always chosen for 
 studying effects for pictures; a finer effect of light and shade being produced from 
 the inclination of the sun's rays at these times; in subjects of buildings the light 
 and shade is better when the sun is higher, as the full advantage of the depth of 
 the shadows is produced by the various projections. 
 
 Light and shade is a most valuable adjunct to elevation drawing ; it enables 
 the draughtsman to define what is uncertain in mere outline, and to shew the 
 projections of the various parts without the necessity of a section; for instance, 
 fig. 1, Plate XIV., is a representation of a double cross, similar to figs. 3 and 4, 
 Plate XI. ; but from this simple outline it is impossible to say whether it is intended 
 to represent an elevation or a plan. Fig. 2 is the same figure, but shaded ; here 
 we see at once that it is a solid figure, with the centre square projecting from the 
 other parts, by the shadows thrown under and at the side of it; also that it is a 
 standing figure, from the shadow from its base. The common mode of shading 
 mechanical drawings is to make the width of the shadow equal to the ex- 
 tent of the projection; drawing the shadows at an angle of 45°; thus by 
 measuring the depth of the shadow below the centre square, it will be found 
 that as it is exactly equal to the width of the square, the projecting figure that 
 casts the shadow is a cube. Fig. .3 is the simple outline of a parallelogram, but 
 without the aid of the shadows it is not possible to know precisely what it is in- 
 tended to represent ; but the same figure, with a line of shade under the top line 
 and on one of its sides within the figure, shews at once that it represents a recess, 
 as fig. 4 ; and again, a similar figure, with a line of shade below the bottom line 
 and on one of the sides without the figure, clearly shews that it represents a pro- 
 jection. Supposing the height of the figure to represent six feet, if the width of 
 the shadow is made one-twelfth the height of the figure it will shew that the pro- 
 jection is six inches. Fig. (i, a rectangular figure with a semicircular top, might 
 be supposed to represent various things, but the slightest indication of shadow, 
 as in figs. 7 and 8, shews immediately the positive figure intended to be repre- 
 sented. The following six figures shew how much is to be conveyed by light and 
 shade. Each of the figs. 9 to 14 in outline is the mere form of a circle ; the diflferent 
 manner of shading them shews each to be a different figure ; fig. 10 representing 
 a sphere; fig. 11 a cylindrical recess; fig. 12 a cylindrical projection; fig. 13 
 
LIGHT AND SHADE. 
 
 69 
 
 represents a circular recess forming a portion of a hollow sphere ; and fig. 14 a 
 circular projection, forming part of a sphere. Light and shade gives to the same 
 outline a distinct difference of figure, and hence becomes a most valuable assis- 
 tant to elevation drawing ; as we have shewn that by outline and light and 
 shade all the proportions of height, width, and projection of the various parts of 
 an object may be defined. 
 
 CHAPTER XI. 
 
 But few directions are necessary for the mere purpose of enabling the student 
 to wash in the forms of his shadows ; he must be provided with a hair pencil, 
 and a cake of color — either Indian ink, Sepia, or Cologne earth, will answer the 
 purpose. Flatness and equality of strength are most essential ; a quantity of the 
 color should be rubbed in a saucer, and diluted to the required strength, which 
 will insure uniformity of tone ; the pencil should be kept as full of color as pos- 
 sible, and where it is requisite, as is constantly the case, to leave one part whilst 
 washing the color round the outline of another, the edge of the part left should 
 be quite floating ; this will admit of returning to the part before the edges are 
 dry, and obviate any appearance of joining the washes. Great care must be 
 taken to work the shadows clean up to the outline, touching the lines, but not 
 going beyond them ; any deviation from this produces a most unpleasing effect. 
 The full depth of the shadows should not be laid on at once, but produced by 
 repeated washes, and the student, in his earliest attempts, will find it of advantage 
 to wash in the first tint very faint ; by so doing he will find the color flow more 
 freely, and any trifling error may be more easily remedied. If a dark and light 
 shadow touch one another, fill in both in the first tints, with the same wash, and 
 darken the deepest shade afterwards ; this obviates any hard lines w hich would 
 arise were the shades washed in separately. 
 
 Let us go back to our first lesson in elevation drawing, fig. 6, PI. I., and see 
 how we are to proceed to put in the light and shade. In describing the drawing 
 of this figure, it was stated that the roof projected in front of the cottage one foot ; 
 we must therefore draw a faint horizontal line, at the distance of one foot of the 
 
70 
 
 PRACTICAL RULES ON DRAWING. 
 
 \ 
 
 i 
 
 scale, below the lowest line of the roof, and carefully wash in the shadow be- 
 tween the two lines. Suppose the recesses of the windows to be six inches, a 
 faint line must be drawn six inches of the scale below the upper line of each 
 window, and as the shadows fall from left to right (the usual mode of represent- 
 ing them in elevation drawings) a faint line must be drawn six inches from the 
 left side of each window, and the spaces filled in with color. Suppose the re- 
 cess of the door to be nine inches, a line must be drawn in like manner nine 
 inches below the upper line, and on the left side of the door, and the space filled 
 in with color; this is shewn in fig. 15, Plate XIV. In the side elevation, fig. 7, 
 Plate I., the sloping sides of the roof project one foot beyond the wall, as shewn 
 at 22, 23, fig. 6, Plate I., a faint line must therefore be drawn one foot from the 
 sloping side on the left, and the space filled in with color as shewn in fig. 16, 
 Plate XIV. Thus the two figures 15 and 16 shew by outline the height and width 
 of the several parts, and by the shadows the extent of the recesses and pro- 
 jections, from which a perspective drawing may be made without further direc- 
 tions. In shading elevations, as cast shadows only are represented, they may be 
 made of one uniform tint ; but in perspective drawings, when the thicknesses of 
 the parts are represented, the cast shadows should be darker than the side in 
 shade. 
 
 In shading perspective drawings, it must be observed that the shadows 
 thrown from the projecting parts follow the same rules as the parts themselves, 
 as shewn in fig. 17, the perspective representation of figs. 15 and 16; the depth 
 of the line of the shadow from the roof must be measured on the line of projec- 
 tion, and a line ruled from it to the vanishing point, which will give the gradual 
 diminution of the width of the shadow as it recedes from the spectator. To find 
 the shadows in the recesses of the windows and the door, those thrown down- 
 wards must be measured on the line of projection for finding the perspective 
 heights, and ruled to the vanishing point ; those thrown sideways must be mea- 
 sured on the line of projection for finding the width of the parts, and their per- 
 spective width found by means of the point of distance(figs.3, 4, Plate VIII.) That 
 portion of the perspective drawing representing the elevation, fig. 16, would be 
 entirely in shade, but the projecting roof would receive the shadow of the side of 
 the house, and is consequently represented darker than the side of the building 
 that casts the shadow. 
 
 To enter into the rules necessary for thoroughly comprehending the pro- 
 jection of shadows, would be overstepping what was proposed in this work ; Ave 
 
LIGHT AND SHADE. 
 
 71 
 
 must therefore be content with making the shadows equal in width with the ex- 
 tent of the projection; but as the forms of the shadows are not the same as the 
 objects that cast them, we will point out the mode by which the forms are to be 
 ascertained. Let A, fig. 18, be the position of a rod projecting- at right angles 
 from a wall ; B C the plan of the rod, shewing its length ; B D the ground line. 
 From the point A, at an angle of 45°, draw downwards the line A a, and from the 
 point C at an angle of 45° draw upwards the line C c. The line A a shews the 
 direction the shadow of the rod would take on the wall ; the line C c, where it 
 cuts the line B D at E, shews at what distance the shadow of the rod should 
 terminate. Draw a perpendicular line from E, cutting the line A a at F, then 
 A F would be the shadow of a rod projecting at right angles from the point A of 
 the length B.C. Let A b represent the height of a plane projecting at right angles 
 from the wall of which BC is the length of the projection ; from the point b draw a 
 line at an angle of 45° to meet the perpendicular E F, then the figure X¥ db w ould 
 represent the form of the shadow thrown from such a projecting plane. Such 
 would be the shadow of a shutter opened at right angles. It mnst then be evident 
 that to get the correct form of the shadows we must have a plan of the projec- 
 tions. Let fig. 19 represent part of the elevation of an entablature, with the plan 
 of the projection of its parts; B represents the extreme projection of the upper 
 part, C the position of the plane on which the shadow is thrown. From the 
 point A draw a line downwards at an angle of 45°, as A a,- from the point B 
 draw a line upwards at an angle of 45°, till it meets the line C at 6; from b draw 
 a perpendicular line to meet the line A a, this will giv^ the depth of the shadow 
 on the plane over the dentils. This plane projects beyond the dentils the dis- 
 tance from C to E. From each of the points D and c draw lines at an angle of 
 45° towards each other, and from the point of intersection of the line drawn from 
 c with the line E raise a perpendicular ; where this intersects the line from D it 
 gives the depth of the shadows on the dentils. Continue the lines from D and c. 
 Where the line from c meets the line F draw a perpendicular to meet the line 
 drawn from D; this gives the depth of the shadow thrown by the projection from 
 C to F. From each of the points G and H draw lines downwards at an angle of 
 45°, and from each of the points J and K draw lines upwards at the same angle ; 
 from the points where the lines drawn from J and K intersect the line F draw 
 up perpendicular lines to meet those drawn from G and H, which will give the 
 form of the shadow of the dentil ; others can be drawn in the same manner or by 
 measurement. To avoid confusion we will give the depth of the shadow of the 
 
73 PRACTICAL RULES ON DRAWING. 
 
 entablature, and the form of the shadow of the capital in a separate figure. Fig. 20 
 is a similar figure to the last with the addition of the pilaster and capital, both in the 
 elevation and plan. From the points A and 1 draw down the lines A a* and 1,2; 
 from the point B draw up the line Bb; raise a perpendicular from the point b cutting 
 A a at a, from which draw a horizontal line cutting 1, 2, at 2, then the figure 
 A 1, 2, a would be the form of the shadow of the entablature without the mould- 
 ings. From the points C, D, E, F, draw the lines C c, D c?, E e, and F From 
 the points G and H draw the lines G ^ and H h; from the point o- draw a per- 
 pendicular, cutting C c and D <^ in the points c and d; from the point h draw a 
 perpendicular cutting the lines E e and F Jin the points e and J] From the point 
 c draw a horizontal line cutting E e at 3, and from the point f a horizontal line 
 cutting the continued line D at 4, then c, 3, e,/, 4, d, is the form of the shadow 
 of the capital of the pilaster. From the points J K draw the lines J j and K k, 
 and from the points j and k draw perpendicular lines for the width of the shadow 
 of the pilaster, which will complete the outline of the form of the shadow thrown 
 between the pilasters. The student may here see that the depths of the shadows 
 of the several parts are exactly equal to the projections that cast them, by com- 
 paring the shadow with the plan of the projections, 5 6 being equal to 7 10; 5 e 
 to 7 11; Ecto 7 8, &c. 
 
 Such are the leading rules for shading mechanical drawings ; in working from 
 Nature it is not necessary to be so extremely nice ; few artists when so doing pro- 
 ject their shadows by rule — they usually wash them in on the spot as they see 
 them, and practice enables* them to do so with great accuracy. Much beauty of 
 light and shade is frequently produced by incidental shadows ; these we con- 
 stantly meet with in studying from nature ; a passing cloud, trees, buildings, any 
 thing in fact, though not represented in the picture, may cast a shadow, the in- 
 troduction of which frequently produces a most happy effect; incidental lights 
 are equally advantageous. The most agreeable pictures are those produced by 
 artists who by long assiduity have become so habituated to the rules of per- 
 spective, and light and shadow, that when drawing from Nature, they trust to 
 their eye and knowledge without the necessity of having recourse to the ruler 
 and compasses ; this, however, can only be the result of long and patient study. 
 " How much liberty may be taken to break through rules, and as the Poet ex- 
 
 * It is unnecessary to repeat every time "at angle of 45"," all the inclined lines are to be drawn at this 
 angle, both upwards and downwards. 
 
ON COLOR. 
 
 73 
 
 " presses it, To snatch a grace beyond the rules of art, may be a subsequent con- 
 " sideration when the pupils become masters themselves. It is then, when their 
 " genius has received its utmost improvement, that rules may possibly be dis- 
 " pensed with. But let us not destroy the scaffold, until we have raised the 
 " building'."* 
 
 CHAPTER XII. 
 ON COLOR. 
 
 We have now arrived at a portion of our subject, to lay down rules for which, 
 in the present advanced state of water color painting, is attended with consider- 
 able difficulty. In the preceding parts, elevation and perspective drawing both 
 purely mechanical, every line can be accomplished with the ruler and compasses 
 by established rules ; so, to a certain extent, is it with light and shade ; there are 
 fixed rules for finding the forms of shadows, and a little practice will soon enable 
 the student to wash in his tints with flatness and precision ; but color depends 
 principally on feeling, and though much may be done by pointing out the most 
 judicious selection of colors, and those required for mixing the different tints 
 best adapted to the representation of various objects at their respective distances 
 in a picture, still to most persons the aid of a master is very requisite at the com- 
 mencement. 
 
 Water color painting, which is universally made use of for all ordinary pur- 
 poses, from the rapid improvement it has undergone during the last few years, is 
 almost anew art; half a century back and this art was but little understood ; the 
 best works of that period were hardly superior to colored engravings. Drawings 
 of that day were, in fact, made on the same principle as colored prints ; the sha- 
 dows were laid in with grey, and the local color of the various parts washed 
 over light and shadow together; the shading certainly was varied, a bluish grey 
 being used for the distant parts, warm grey for the foreground, and a blending of 
 
 * Sir Joshua Reynolds' Discourses on Painting at the Royal Academy, 
 
 On Drawino.) K 
 
74 
 
 PRACTICAL RULES ON DRAWING. 
 
 the two for the middle distance. This style of drawing could be most happily 
 imitated by aquatint engraving, printed in two colors, and the impressions tinted 
 from the original drawings. At the period of which we are speaking, the painters 
 in water colors did not aim at much power, and the proper term applied to 
 their works was tinted drawings; the term paintings, which is applied to the 
 works of art now produced by this material, is far more correct than that of 
 drawings, for they are most unquestionably paintings, and of a high quality, ri- 
 valling in many respects the finest works in oil. 
 
 It is to be lamented that the artists employed by architects, to execute draw- 
 ings from their designs, for the most part, have not kept pace with the improve- 
 ment displayed in other branches of water color painting. It is an ordinary 
 occurrence for an artist to receive from an architect's office, a drawing of some 
 building, to which he has to add the sky and landscape, and too frequently the 
 difficulty is not confined to his having so to arrange his part as to distract the 
 eye from the formality of the design, but the building with which he is for- 
 bidden to interfere, is so monotonous and so meagre in colour, that he is fettered 
 in his endeavours to produce an agreeable picture, by being compelled to make 
 his landscape fitting with the building; in artistical language, to have the whole 
 picture in keeping; were the artist to work the landscape to his own feeling, the 
 architectural part would appear like a phantom. The formality of architectural 
 subjects is assigned as a reason for the want of interest often to be observed in 
 their representation. It would be idle to pretend that a drawing of an entirely 
 new building, is capable of juoducing so pleasing an effect as one representing 
 an ancient edifice ; the crumbling quality of old material whatever it may be, 
 the discoloration produced by time, and the diversity of line arising from dila- 
 pidation, forming together what is called the picturesque, afford a wider scope 
 for the exercise of the artist's powers ; still the representation of a good design, 
 though new, with a judicious foreground and background, and what is most 
 essential in architectural subjects, a well arranged sky, both as to form and 
 color, may be made to produce a most agreeable whole ; witness the works of 
 Mackenzie, Nash, &c., &c. There are many, arguing from the works of artists 
 who unmercifully sacrifice accuracy of drawing and truth of light and shadow 
 for what they term breadth of effect, contend that correctness in these matters 
 is a secondary consideration ; such however is a gross error, for inaccuracy of 
 drawing is as offensive to a correct eye, as a false note in music is painful to a 
 correct ear. Among the early works of the great Turner, who stands pre-eminent 
 
ON COLOR. 75 
 
 as a painter in water colours, are to be found many drawings of architectural 
 subjects ; these though treated with the greatest simplicity, from the refinement of 
 his taste even at this early period of his career, have never been surpassed ; he 
 then attempted to represent nature under her simple effects, but since, he has 
 produced the most marvellous works of art, and it is not affirming too much to 
 say, that those of his latter productions, even with the most extravagant effects, 
 are still faithful representations of the extraordinary appearances in which nature 
 sometimes displays herself. It ought not to be matter for surprise that such 
 works are not generally admired, for they are not generally understood ; it 
 requires a refined judgment to fully appreciate such works as, emanating from 
 the most highly cultivated intellect, are addressed to the mind. The great point 
 to keep in view, in making a picture, is to draw the line between a too great 
 subservience to rule, and being led away too far from it, by following the steps 
 of those who produce those catching effects, fascinating at the first view, at the 
 expense of truth. 
 
 CHAPTER XIII. 
 
 The student, supposing him to have acquired a tolerable freedom in the 
 use of the pencil by working his drawings in one color — that is, that he has 
 acquired a facility in drawing outlines, and putting in the light and shadow, from 
 drawings, prints, and from nature, which it has been our aim to assist him in 
 effecting — may proceed with color. The first necessary preliminary is to provide 
 himself with the requisite materials, and to do this, without the assistance of some 
 person conversant with them, he would find himself a little puzzled. Every 
 Artists'-colorman has a collection of from fifty to sixty colors ; many of these, 
 for general purposes, are worse than useless ; some are expressly for flower- 
 painting, others for miniatures, &c. ; a collection of twelve colours is ample for 
 all the ordinary parts of an architectural or landscape drawing. Artists differ a 
 little in the choice of colors, but the following list embraces sufficient for all 
 
 K 2 
 
76 
 
 PRACTICAL RULES ON DRAWING. 
 
 general purposes : — 1. Indigo ; 2. French Blue ;* 3. Gamboge ; 4. Yellow Ochre; 
 5. Roman Ochre ; 6. Raw Sienna ; 7. Burnt Sienna ; 8. Venetian Red ; 9. Crimson 
 Lake ; 10. Cologne Earth; 11. Vandyke Brown; 12. Lamp Black. Sometimes 
 the introduction of an extra bright bit of color on a figure, piece of drapery, or 
 some local object, gives great zest to a drawing; for this purpose, colors brighter 
 than any in the foregoing list are required : as vermilion for an extra bright 
 red; Naples yellow, or chrome, for an extra bright yellow, &c.; but these colors 
 should not be used for putting in the general effect. In addition to the colors, 
 the student must provide himself with three or four hair pencils and a large flat 
 brush ; skies and large masses are always best washed in with a large pencil, it 
 enables the artist to work with more freedom ; the under-touches of small parts, 
 such as the deepest shades in ornamental work, require a fine-pointed pencil. 
 The improvement in the manufacture of the materials used in water-color paint- 
 ing, has kept pace with the improvement in their use, and the student, before he 
 commences, should be careful in his choice. There are numerous advertisers of 
 cheap water-colors, but as the best quality is not very expensive, it is advisable 
 to go at once to a respectable dealer^ and procure the requisite articles. A good 
 deal depends on the choice of paper, but of this article there is so much variety, 
 both in size and quality, that it is difficult to specify any particular sort ; stout 
 paper (not cardboard) is preferable to thin paper, and it should not be over 
 smooth ; imperial drawing paper (not hotpressed) answers well for ordinary 
 drawings. Let us then suppose the student to have completed his outline, and 
 ready to commence coloring. In drawing architectural outlines, the hands, the 
 rulers, and the Indian rubber are so constantly passing over the paper, that it 
 becomes frequently both greasy and dirty ; the whole surface of the paper there- 
 fore should be washed with water by means of a flat brush or a sponge; the latter 
 is best if it can be used with sufficient delicacy, so as to avoid obliterating the out- 
 line ; should the paper be very greasy, a little gall added to the water will be 
 found of advantage. Before the paper becomes thoroughly dry from this ablu- 
 tion, the first large masses of color should be washed on ; the colour runs more 
 
 * This is a color prepared to imitate an expensive one called ultramarine, and is superior to it in respect 
 of its working with more freedom. 
 
 f During the last few years, the increase of manufacturers and dealers in all things appertaining to this 
 art has been immense, and there are doubtless many highly respectable houses besides the undermentioned ; 
 from any of these, however, the quality of the goods may be relied on : — Newman, Soho Square ; Ackerman, 
 Strand and Regent Street ; Rowney, and Windsor and Newton, Rathbone Place ; Roberson, Long Acre ; 
 and Smith, Tichborne Street. 
 
ON COLOR. 
 
 77 
 
 freely, and it enables theartist to take his pencil from one part to another, without 
 the edges of the part left getting dry before he can return to it. 
 
 It is usual to commence by laying the color for the sky. For architectural 
 subjects, grey skies have generally the best effect; the building forming the prin- 
 cipal feature, should be thrown out as much as possible by the secondary parts, 
 and buildings being generally represented of a warmish yellow, approaching to 
 orange, are best relieved by grey. Architectural subjects, consisting of a multi- 
 plicity of straight lines, should be relieved by a diversity of form in the sky ; and 
 a cloudy sky, the clouds having a rotundity of form, is best suited to this class of 
 subjects; a small quantity of pure blue may be introduced with good effect, but it 
 should be towards the upper part, and made to appear as if seen through openings 
 in the clouds. The student must carefully avoid, in sketching the forms of 
 clouds, following the sloping lines of the architecture, but rather endeavour to 
 bring them in an opposite direction, and, as much as possible, bring the dark 
 clouds against the light side of the building, and relieve the dark side, if admis- 
 sible, by light ones; the opposition produced by attending judiciously to these 
 remarks, forms in a picture what is termed effect. The forms of the clouds being 
 sketched, a grey, made by a mixture of French blue and Venetian red, with a very 
 small addition of lake, should be washed evenly nearly all over the sky, leaving 
 only the bright lights of the clouds ; this tint will form the lightest shade of the 
 clouds, and should be allowed to dry before the next tint is laid on. The several 
 shades of the clouds must then be washed in, one over the other, till the full 
 depth of the darkest cloud is laid in ; where the clouds are darkest, a little indigo 
 may be added to the grey above. The tints of the clouds must be varied a little, 
 some cooler and some warmer ;* a little pure French blue may now be laid on to 
 represent patches of the blue sky seen through openings in the clouds."}" 
 
 In proceeding with the building, the ordinary process is to commence by 
 putting in the shadows ; as a general color for these, Venetian red and indigo 
 may be used ; if the local color of the building be yellow, a little lake added to 
 
 * Blue is a cold color, red and yellow warm ; the tones of color are said to be cool or warm as either of 
 these colors preponderates. Thus a little lake added to a wash of blue in a sky, not sufficient to make it pur- 
 ple, is called a warm blue. 
 
 f The student, as he proceeds, will acquire the power, in the first process of laj'ing in the clouds, to vary 
 the tints according to his feeling, making in his first wash one part darker than another, and also changing the 
 tone of color — making one portion a little cooler, another a little warmer ; he will also, by practice, be able 
 to put in a second wash of color before the first dries ; but at the commencement, unless he allow one wash to 
 dry before beginning another, he will be sure to find himself in trouble. 
 
78 PRACTICAL RULES ON DRAWING. 
 
 the red and blue will prevent any greenish appearance in the shadow, after the 
 local color is put on. Lamp black and lake form a good mixture for putting in 
 the cast shadows, but care must be taken that the latter color is not too violent. 
 In an attentive observance of nature, it will be seen that few surfaces present one 
 even color ; almost every object, on close inspection, has a decided variety in its 
 tones ; some a little brighter, some a little more grey. These varieties of tone in 
 nature, may in the representation be a little exaggerated ; one simple wash of 
 color over the face of a building has a tame effect, and gives to a drawing the 
 appearance of a colored print. Supposing the edifice to be of stone, the gene- 
 ral tone of which is yellowish, inclining to orange, mix several tints in separate 
 saucers : — 1. Venetian red and yellow ochre, the red predominating; 2. Yellow 
 ochre purej 3. The two just mentioned equally mixed; these will give you orange, 
 yellow, and pale orange ; 4. The grey made with lamp black and lake, very 
 pale. Fill the pencil with one of the warm tints and commence at the top 
 corner of the building in light, nearest the shaded side, and continue washing 
 over the whole face of the building, changing the tints from time to time from 
 the different saucers, and running one tint into the other ; this will produce a 
 pleasing variety of tones ; and occasionally leave a sharp thin streak of light to 
 represent the divisions between the stones, taking care that those representing the 
 horizontal divisions follow the perspective directions the lines should take. If 
 the building be of brick, both the light and shade must have more strength of color. 
 For the shadows, Venetian red, indigo, and burnt sienna may be used ; occasionally 
 a small portion of lake may be added. For the lights, — 1. Venetian red and yellow 
 ochre; 2. Burnt sienna and lake; 3. A mixture of the two ; 4. The same grey as 
 the last. These tints should be laid in similar to the four above-named for re- 
 presenting stone, and small spaces left between the color, here and there, to in- 
 dicate the mortar between the bricks. Roofs of tile may be colored similar to 
 brick, but, being usually of a darker tone, burnt sienna should] be used more freely 
 than yellow ochre. Slates may be represented with a mixture of indigo, Venetian 
 red, lake, and a small quantity of yellow ochre. Windows, from their reflect- 
 ing any objects within their range, may be represented in a variety of ways; 
 sometimes they appear so intensely bright as to dazzle the eye, at times they re- 
 flect the deepest blue, at others dark bottle-green; the most usual mode of repre- 
 senting them is by a moderately dark blue grey, with occasional reflections of 
 light; dark reflections in the windows, on the light side of the building and vice 
 versa, produce the best relief. 
 
ON COLOR. 
 
 79 
 
 Much judgment is required in the arrangement of the landscape, so as to 
 divert the eye from the rigid forms unavoidable in architectural representations. 
 Branches of trees thrown out so as occasionally to project beyond an angle of the 
 building to break a long line, and by a little light and playful foliage take oft" 
 from the weight of the structure, have great value, but care must be taken, in so 
 doing, that no part is hidden essential to the development of the architect's design. 
 The foreground is where the artist has the best opportunity of- exercising his 
 judgment and imagination; he is here unfettered by rules in managing his light 
 and shade, and by various manceuvres he may introduce form and color so as to 
 give considerable effect to the principal subject; the sweep of a road or path, 
 the introduction of circular or oval beds of flowers, shrubs, &c., afford ample op- 
 portunity for giving interest and diversity of form; and the introduction of such 
 objects as a garden roller, watering pot, &c. &c., insignificant as they may be 
 looked upon, produce, by a judicious treatment, both a useful and pleasing effect. 
 It is impossible at present to enter into the minutiae of coloring landscape ; the 
 variety of tints in foliage is endless, embracing all the gradations from the fresh 
 green of a spring morning, to the almost red tone of an autumn evening. The 
 color of an object depends entirely on the effect under which it is seen; for the 
 same object seen on a cool grey morning, which may then appear of a cold grey 
 or white, will, under an effect from a setting sun, appear of an intense orange. 
 There are few who do not know that the mixture of blue and yellow produces 
 green ; but as there are several blues and many yellows, the variety of greens to 
 be made from them is interminable. Gamboge, indigo, and burnt sienna are the 
 ordinary colors used for producing green ; indigo and gamboge produce a fresh 
 green ; the more the blue predominates the cooler is the green. Burnt sienna 
 added to the above produces warm green, and the more predominant the sienna 
 the warmer will be the tone. Raw sienna and indigo form a good color for the 
 lights on foliage; Roman ochre* and indigo is useful for dark trees and shrubs, 
 as is yellow ochre and indigo for sober greens ; where great depths are required, 
 lamp black and gamboge may be used, and a trifling quantity of burnt sienna will 
 give it warmth. Great care must be taken by the young artist to avoid making his 
 pictures too green ; to prevent this, it is best to lay in all the shadows of the foli- 
 age with Venetian red and indigo, with a slight addition of lake ; in this grey the 
 
 * Where force of color is required, Roman oclire may be used as a substitute for yellow ochre, or com- 
 bined with it. 
 
80 PRACTICAL RULES ON DRAWING. 
 
 red must preponderate much more than in that used for the shadows of buildings ; 
 as green, which will be washed over it, always neutralises red, so red will always 
 be found to neutralise green. The observation above, cautioning the student 
 against making the color of foliage too generally local, will apply to almost all 
 objects to be delineated. One of the great excellencies in a picture, is the ap- 
 proximation only, in its general tones, to the local color of the real matter; every 
 thing being viewed from some distance, the tones necessarily appear subdued, 
 and partake more or less of grey, and the student will find more diflSculty in 
 producing a grey sober picture, than a bright gaudy one. We must now, at all 
 events for a season, take leave of the reader ; we have already exceeded the limit 
 allotted to this portion of the work, but trust, in its progress, to be allowed an 
 additional space to introduce on each part of our subject a little supplementary 
 matter with some additional Plates. 
 
PRACTICAL.-ON CONSTRUCTIVE ART. 
 
 THE GIRARD COLLEGE FOR ORPHANS. 
 
 This establishment is situated in the North Western environs of Philadelphia, 
 on a tract of land containing forty- five acres. Its founder, Stephen Girard, was 
 born in Bordeaux, on the 20th of May, 1750; and died in Philadelphia on the 
 26th of December, 1831. He commenced the world in extreme poverty, from 
 which he emerged with steady progress, until he attained an unrivalled eminence 
 both as a Shipping Merchant and a Banker: — by these pursuits he amassed 
 an immense fortune, nearly the whole of which he bequeathed to benevo- 
 lent objects. 
 
 For the endowment of the College he appropriated two millions of dollars ; 
 and further provided, that if the interest of what may be left of that sum, after 
 the buildings are constructed and furnished, should prove inadequate to the 
 support and education of all the " poor white male orphans" who may apply 
 for admission ; then such further sum shall be taken from the final residuary 
 fund, (consisting of several millions of dollars), as may be necessary to construct 
 as many additional buildings as may be required to accommodate and educate 
 all the additional pupils who apply for admission. 
 
 The buildings provided for by the will, as a nucleus to the establishment, are 
 " a permanent College, with suitable out-buildings, sufficiently spacious for the 
 residence and accommodation of at least three hundred scholars, with the 
 requisite teachers and other persons necessary in such an institution." — In 
 reference to the College, Mr. Girard directs that, "it shall be at least one 
 hundred and ten feet front, east and west, and one hundred and sixty feet, 
 north and south;" that "it shall be three stories in height, each story at least 
 fifteen feet high in the clear from the floor to the cornice ;" and that " in each 
 
 DIV. B. A 
 
2 
 
 ON CONSTRUCTIVE ART. 
 
 story there shall be four rooms ; each room not less than fifty feet square in the 
 clear ; the four rooms on each floor to occupy the whole space east and west on 
 such floor or story, and the middle of the building north and south, so that in 
 the north of the building, and in the south thereof, there may remain a space of 
 equal dimensions, for an entry or hall in each for stairs and landings ;" " the 
 steps of the stairs to be made of smooth white marble, with plain square edges ;" 
 " the outside walls to be faced with slabs or blocks, of marble or granite, at least 
 two feet thick ;" and " the floors and landings, as well as the roof, to be covered 
 with marble slabs." 
 
 The designs for the establishment were made by Thomas U. Walter, Architect, 
 of Philadelphia, and are now being executed under his directions. The accom- 
 panying plates exhibit the details of the plan of the main building with the four 
 out-buildings, and the following diagram shows their relative position. 
 
 120 
 
 120 
 
 The centre building covers a space of 184 by 243 feet ; it consists of an 
 octostyle, peripteral superstructure composed in the Corinthian order of Grecian 
 Architecture, resting on a base of twelve steps extending around the whole 
 edifice ; — a pyramidal appearance is thus given to the substructure, and a means 
 of approach afforded to the porticoes on every side. The dimensions of the 
 stylobate are 159 by 217 feet, and the cell or body of the building measures 111 
 by 169 feet. The whole height from the ground to the top of the pediment is 
 above 97 feet. 
 
 The columns are thirty-four in number ; the diameter of the shaft at the top 
 of the base is six feet, and at the bottom of the capital five feet ; the height of 
 the capital is eight feet five inches, and its width from the extreme corners of the 
 abacus, nine feet ; the whole height of the column, including capital and base, is 
 fifty-five feet. 
 
THE GIRARD COLLEGE. 
 
 3 
 
 The entablature is sixteen feet six inches high, and the greatest projection of 
 the cornice from the face of the frieze is four feet nine inches. 
 
 The capitals of the columns are proportioned from those of the monument of 
 Lysicrates, at Athens ; they are divided in height into four courses ; the first 
 embraces the water-leaf, and consists of a single stone of sixteen inches in 
 thickness; the second course is also composed of a single stone, the height of 
 which is two feet ten inches ; the annular row of acanthus leaves occupies the 
 whole of this course ; — the third division embraces the volutes and cauliculi ; 
 this course, which is likewise two feet ten inches in height, is composed of two 
 pieces, having the vertical joint between the cauliculi, on two opposite faces ; — 
 the fourth, or upper course, being the abacus, is one foot five inches in height. 
 
 The doors of entrance are in the centre of the north and south fronts ; they 
 are each sixteen feet wide in the clear by thirty-two feet high ; their outside finish 
 consists of antepagmenta of two feet seven inches wide, the supercilium of which 
 is surmounted with a frieze and cornice ; — the cornice is supported by rich 
 consoles of six-and-a-half feet in height, and the cymatium is ornamented with 
 sculptured honeysuckles. 
 
 The exterior of the whole structure is composed of fine white marble obtained 
 from quarries in Pennsylvania and Massachusetts, The columns are composed 
 of frustra measuring from three to six feet in height ; the bases each consist of a 
 single piece, the diameter of which is upwards of nine feet ; the capitals are all 
 wrought on the ground, and present an admirable specimen of architectural 
 carving: — the whole cost of each column, including its capital and base, is 
 thirteen thousand dollars. 
 
 The vestibules in the first story, and the lobbies in the second and third 
 stories, are all vaulted from marble entablatures supported by antcB and columns, 
 the shafts of which are each composed of a single stone. 
 
 There are sixteen columns and as many antae in each story ; those in the 
 first story are Ionic from the temple on the river llissus, and those in the second 
 and third are modifications of the Corinthian. 
 
 The building is three stories in height, measuring twenty-five feet from floor 
 to floor; it contains four rooms on every floor, each of which is fifty feet square 
 in the clear. The first and second stories are vaulted with groin arches, and the 
 third with pendentive domes falling below the plane of the roof, and lighted with 
 skylights of 16 feet in diameter. 
 
 The roof is composed of marble slabs of four feet by four feet six inches, 
 
4 ON CONSTRUCTIVE ART. 
 
 supported on nine inch walls; the joints are covered with saddle tiles, and the 
 slabs are so formed as to render the whole water-tight without the use of cement. 
 
 It being altogether of recent date, Anglo-American architecture has, instead 
 of passing through, passed over that phase of the art which was produced by 
 European medi(Bvalism'' It was but natural, perhaps, that a people compelled 
 to adopt a style ready shaped out for them, should take up that which had been 
 proclaimed by the universal consent of all civilized nations to be the most perfect 
 the world had ever beheld, — the one upon which the art had been anchored ever 
 since the so-called Revival of the arts, but which, though professedly imitated, 
 had been only to a certain extent, and had been corrupted in modern times by 
 so many innovations, and by so many elements altogether foreign to its constitu- 
 tion having been mixed up with it, as to have lost its original character. In the 
 idea of adopting and re-establishing Grecian architecture in all its purity, there 
 was a temptation not to be resisted. Admiration for it was secured beforehand 
 by the dazzling halo of its classical fame ; added to which it possessed another 
 strong recommendation, namely, its extreme simplicity. It required, or seemed 
 to require, nothing more than an acquaintance with the Grecian orders, and 
 fidelity of imitation in copying them ; for which last purpose numerous archi- 
 tectural publications supplied the requisite studies and authorities. European 
 artists and archaeologists had carefully delineated and explained the remaining 
 monuments of Greece ; but, availing themselves of such preliminary labours, the 
 architects of America aimed at nothing less than the exemplifying in their own 
 structures the practical restoration of Grecian architecture and Grecian taste. 
 That style accordingly began to be employed by them indiscriminately for pur- 
 poses the most opposite — not merely for public edifices of monumental character, 
 but for domestic buildings. Modern country residences were put into the cos- 
 tume of Grecian villas, or rather were made to look at the first glance like 
 Grecian temples, by the mere addition of Greek porticos or colonnades to the 
 house itself ; a mistake that has been committed here at home. 
 
 It was not perceived that in the fancied simplicity so attained, there was, 
 instead of the unity almost indispensable for simplicity, a marked duality, — that 
 of two opposite systems, the antique and the modern, not artistically blended 
 together so as to compose a third system out of the elements of the other two, 
 but left to conflict together in harsh contrast, the one by the side of the other. 
 It was not perceived — does not even seem to have been so much as suspected — 
 that the very simplicity of Grecian architecture, the quality for which it is so 
 
THE GIRARD COLLEGE. 5 
 
 loudly extolled, admirable as it is in itself, more or less unfits it for actual prac- 
 tice in modern times. Architecture must necessarily comply with the necessities 
 of building, with the circumstances and habits of the age and the people who 
 employ it. To endeavour to accommodate buildings to a style which never con- 
 templated, and therefore never provided for, what is demanded by the present 
 state of civilization, is a most perverse course — one totally opposite to that by which 
 all styles have naturally grown up, and been moulded according to the actual exi- 
 gencies of their own period. Excellent as it was for Greece itself, Grecian archi- 
 tecture is by far too limited to be at all suitable as a general system at the present 
 day. While it affords nothing but colunmiation, (which in modern buildings, so 
 far from being at all needed, is at the best only embellishment,) it so obstinately 
 rejects all besides, that the more we endeavour to adhere strictly to the imperfect 
 models it holds out to us — imperfect with regard to what we really require — the 
 greater are the incongruities we fall into, and the more evident do we make our 
 misapprehension of the style we profess to admire. The most correct imitation 
 can produce nothing more than the semblance of a Greek temple ; and provided 
 the nature of the structure itself be such that its exterior can readily be made to 
 assume such form, the imitation may be perfect. Such is the case with the 
 Madeleine at Paris, and the Walhalla near Regensburg (Ratisbon), in Bavaria ; 
 the former of which is externally the model of an octostyle Corinthian, the other 
 of an octostyle Doric temple, it being an avowed imitation, or even a facsimile, 
 of the Athenian Parthenon, with the exception of its sculpture. In those 
 instances — merely exceptional ones — there is no obvious inconsistency ; because 
 windows, that is, windows in the walls, could be dispensed with, the interior 
 being lighted entirely from the roof ; besides which, the interior is what the exte- 
 rior bespeaks it to be, namely, a single large room, answering to the cella of a 
 temple, and occupying the entire height of the structure from floor to roof. In 
 each case the plan itself, and the constitution of the building, are so exceedingly 
 simple, that there is nothing which directly militates against the model taken for 
 the occasion. Yet those two edifices are, as has already been observed, extreme 
 and exceptional instances, which only serve to prove how altogether unfit is Greek 
 architecture for general purposes, — and Greek temple architecture most of all. 
 When the building is divided within into stories, by floors, and consequently all 
 but the uppermost story must be lighted by windows in the walls, a character 
 totally different from that of the classical temple style must inevitably take 
 
6 ON CONSTRUCTIVE ART. 
 
 place; wherefore, to adhere strictly to the latter, becomes impossible, and as the 
 attempt must prove abortive, it is one that ought not to be made. 
 
 After these remarks, it cannot be supposed that the Girard College is here 
 offered as an example to be followed, but it is one that holds out an instructive 
 lesson, and is one from which much may be gathered. Still it is certainly of 
 interest as a monumental work of modern times, executed in the most costly and 
 durable materials, without regard to expense ; and serves to make evident both 
 what can and what cannot be accomplished by strict adherence to Greek autho- 
 rity, as far as such precedent goes. The order itself is upon a scale that may 
 well be termed colossal in comparison with that of the largest Greek temples, 
 and being applied strictly in accordance with the arrangement of a classical 
 peristylar edifice, it affords a magnificent and impressive representation of 
 Grecian architecture. But, though both on that account, and as a monumental 
 fabric constructed of and covered with marble, the main building of the Girard 
 College is entitled to admiration, and may fairly be classed among the most 
 remarkable works of modern times, it is by no means perfectly satisfactory as a 
 production of art. While originality, either of conception or design, is disclaimed 
 for it altogether, the necessity for windows has prevented that consistent imita- 
 tion of the antique which excuses the want of originality, and renders mere 
 imitation a merit. 
 
 As a practical style, Greek architecture has now had a tolerably fair trial, and 
 has, after many experiments, been found so inadequate to modern purposes, even 
 under the most favourable circumstances, that the affectation of it is beginning to be 
 laid aside, and architectural taste has now veered about from ultra-Greek to ultra- 
 Italian on the one hand, and the mediaeval styles on the other. The more per- 
 fect a style is in relation to the particular purposes it was intended for, and the 
 particular conditions it was expressly adapted to, the less suitable is it for general 
 application, or for buildings that are at all differently constituted from those edifices 
 which are monuments of such style. Unfortunately, while Greek architecture 
 demands columniation, as essential to its characteristic physiognomy, it rejects 
 fenestration, as being contrary, or even destructive to, that physiognomy ; yet 
 though columns may be applied for the nonce, windows cannot be dispensed 
 with at pleasure. Had the main building of the Girard College been a single 
 spacious and lofty apartment, intended to serve as the College Hall, and accord- 
 ingly admitted of being lighted from the roof, its temple character could have 
 
THE GIRARD COLLEGE. 
 
 7 
 
 been consistently kept up, and the exterior and interior of the structure would 
 then have corresponded with each other ; the former indicating", and the latter 
 being, a single room analogous to the cella of an ancient peripteral temple. At 
 present the degree of resemblance obtained by scrupulous regard to Greek pre- 
 cedent, as far as such precedent could be followed, amounts to no more than very 
 imperfect imitation. Now, in art, whereas free imitation is not only excusable 
 but laudable, imperfect imitation, occasioned by the impossibility of adhering to 
 the model which is avowedly intended to be copied, inevitably partakes more or 
 less of mere mimicry. As to the American architects, it is no wonder that they 
 should have fallen into a mistake which both the doctrine and the practice of 
 their European brethren in modern times had prepared for them by cherishing 
 error all along. No doubt architecture requires to be governed by immutable 
 principles ; but obstinate fixity of forms and stagnation of ideas are not only 
 distinct from, but subversive of, all genuine principles, both of composition and 
 construction. To use an Americanism, Architecture has, in consequence of such 
 unhappy error, been brought to " a complete fix," — to a dead stand — without 
 any alternative than that of oscillating between former styles, be they Greek, 
 Gothic, or any other. Every step is backwards, without any effort being made 
 to take a single one forward. We are well aware that architecture cannot be 
 made to advance by sudden leaps and jumps ; the progress to a style accom- 
 modated to our actual social wants and habits, must of necessity be a gradual 
 one, and not only gradual but slow ; but unless a beginning be made somewhere 
 — unless the first step be taken — we must for ever remain just where we now 
 are, and be content to be the literal copyists of those who have gone before us. 
 As an art, architecture has, it seems, run its course ; it can accomplish nothing 
 more than it already has done; — such is the opinion which is, we do not say 
 openly avowed, but betrayed, by present practice, and by that complaisant 
 criticism which bestows on works of mere reproduction the encomiums that 
 should be reserved for originality — for genuine artistic conception. 
 
THE FIVE EJNGRAVINGS. 
 
 Plate I. — Half plan of Foundations — Main Building. 
 Half plan of first Story. 
 
 Plate H. — Half plans Western Out-buildings ; 
 Students' Buildings. 
 Half plans of Eastern Out-buildings ; 
 Professors' Dwellings. 
 
 Plate HI. — Elevation of Front Main Building. 
 
 Plate IV. — Half of side elevation of Main Building. 
 
 Plate V, — Longitudinal Section through half of the Main Building. 
 
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 lA.SUurj fivm JKitcJven. 
 N. Uppej- pait of Kitchen 
 
 Flan oT tlw ihowuiFLeoi: 
 
 R< fcrc?jce tp FloJt of 
 First TIjDor. 
 
 i\.Libra7'ii _ 47 d . 29. o 
 ^ . S ecretarifs Rooni 
 C. Open Area. 
 JJ. IVater Closets 
 K.Bilhard Room . 
 
 Uppa-part of Stazrease. 
 ^y. Jhawbig Jtoom. 
 
 Scale ,'f II" I ' 
 
 /"Ian- 
 
 ,,, F/a?/ i^f Fw First Floo?-. 
 
 ^J-eer 
 
 S.W.H-aiiaXL.dtl. 
 
 John Weale.Mai/ lJS-17. 
 
Jtilm Weal^. June 2.2847. 
 
I'r.i./i.;i/ H. 
 
 Jrt/n/r.'^ Ciiih //ciisr 
 
 22 e/ ff .1 I 1' :i -I g 
 
 Scale ^^ illl lllllllll I I I I I 
 
 Joh?i nV.zU.Jufjfi 1.1847. 
 
L.'iuloii Ji'lni WcuU,.o:i. ffu//i HvlUrr:i ,mn. 
 
I'ractifaL B. 
 
/Ill /III //'(■/// >■ . 
 
 I'riirhnll IS . 
 
 Miin/i/.'.fs el' //i/i/hn/'s. 
 J'imiili// 1 
 
 Sirfl.Pt'eKf JJiii/Sf 
 
 Pi irr lliinJ< ii.( 
 
 J„/,i, // ;■„/,■. . /une.J. /S4 7, 
 
fa 
 
 < 
 
\ 
 
W INDH on CA STLE 
 
 Jf/////ivi ni Mrmso't Mndow Fnu/u:<,- tj/id Sashes i/i //i^ J'rimte ^pa/tr?imti\ 
 
 A //fan ,^<le/t. 
 
S T L E 
 
 ^krtj^ Male. J047. 
 
Pid rinmc; a/ul sas/us are of TPamso) 
 
 Tfie jae// li,irs a/itf rel'aUs (ue fif soM CippL-r 
 
 '<->fi//,r ill/- 
 
 X 
 
 A SrcLss La/lvs and Jcrafs f/uxi t//e /w/i>w 
 
 m,a/ ie r<7noyed tah; ott the nvnt B 
 
 yet at t/ie rferp/ifs. 
 
 fas/i bars and rd>atas- of solid Copper. 
 
 /- Brass CoUars and Siren's. 
 
 De^zmed iv Sir J- m^iztiiae, & made Ifessr^ Jmu-traju, £■ SmffA.Am^ 
 
 Sidf size 
 
 ■Win ^^ea/e. JS-/7 
 
 • ■ u/,,,/,r'i,. m/,, 
 
rr.„-/i,;,/ , /S. 
 
 Loiuicn John IV,„U..lrcJnt,;-lurnl /.,/,niri:S^'Mi'//' H.'llwrii 1,14/ . 
 
 If . I B,m,- 
 
ECCLESIASTICAL WOOD WORK. 
 
 %\JRy DEL & SC. 
 
 THE ROOF OF THE NAVE.S. MA RY.WESTONZOYLAND. SOMERSETSHIRE. 
 
\ 
 
GETTY CENTER LIBRARY 
 
 3 3125 00070 7725