m 
 
 LIBRARY OF 
 
 HENRY C. FALI — 
 
 and KATHARINE A. FALL 
 
 Plumber l^ 8 
 
 D^<? 0/ Purchase ^A^nS. £0 #3 
 
 C05^ N 
 
 ■ .'V^;v. ■
 
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 DESCRIPTIVE GEOMETRY, 
 
 AS APPLIED TO 
 
 THE DRAWING OF 
 
 FORTIFICATION AND STEREOTOMY. 
 
 FOR THE USE OJ" 
 
 THE CADETS OF TEE U. S. MILITARY ACADEMY. 
 
 BT 
 
 D. H. MAHAN, LL.L\, 
 
 PE0FE8S0R OF FOETTFIOATION, CIVIL BNGIKEEBING, *0., VNITED STATES illLITlET AOAI>*m 
 
 NEW EDITION, "WITH ADDITIONS. 
 
 NEW YORK: 
 
 JOHN WILEY & SONS, 
 
 15 ASTOR PLACE. 
 
 1883.
 
 ■Entered, according to Act of Congress, in the yeai 1884, by 
 
 D. H. MAHAN, 
 
 la the Clerk'* Office of the District Court of the United States for the Southern DUtrloJ 
 
 of New York. 
 
 Trow's 
 Printing and Hookrinding Co., 
 
 PRJN ll-.KS A , 1'liRS, 
 
 205-213 East 12th St., 
 
 NEW YUKK.
 
 PREFACE. 
 
 Tile subjects of the following pages have been 
 taught orally at the Militaiy Academy for many years ; 
 but, for the saving of time, and the convenience of 
 the pupils, it has been thought best to clothe them 
 in a printed dress ; and as, in this form, the volume 
 might be found useful in other schools, as an appli- 
 cation of descriptive geometry to practical questions, 
 it was also thought well to have it published.
 
 ONE PLANE DESCRIPTIVE GEOMETRY 
 
 AS APPLIED TO 
 
 FORTIFICATION DRAWING. 
 
 1. The method now in general use, among military en- 
 gineers, for delineating the plans of permanent fortifications, 
 is similar to the one which had been previously employed 
 for representing the natural surface of ground in topograph- 
 ical and hydrographical maps ; and which consists in projec- 
 ting, on a horizontal plane at any assumed level, the bounding 
 lines of the surfaces and also the horizontal lines cut from 
 them by equidistant horizontal planes, the distances of these 
 lines from the assumed plane being expressed numerically 
 in terms of some linear measure, as a yard, a foot, &c. 
 
 2. Plane of Reference or Comparison. The assum- 
 ed horizontal plane upon which the lines are projected is 
 termed the plane of comparison or plane of reference, as it 
 is the one to which the distances of all the lines from it are 
 referred, and as it serves to compare these distances with 
 each other and also to determine the relative positions of 
 the lines. 
 
 3. References. The numbers which express. the dis 
 tances of points and lines from the plane of comparison are 
 termed references. The unit in which these distances are 
 expressed is usually the linear foot and its decimal divisions. 
 
 As the position assumed for the plane of comparison is 
 arbitrary, it may be taken either above or below every point 
 of the surfaces to be projected. In the French military ser- 
 vice it is usually taken above, in our own below the surfaces. 
 The latter seems the more natural and is also more conveni- 
 ent, as vertical distances are more habitually estimated from 
 below upwards than in the contrary direction. Each of 
 these methods has the advantage of requiring but one kind 
 of symbol to be used, viz : the numerals expressing the ref
 
 2 ONE PLANT- DESCRIPTIVE GEOMETBT. 
 
 erenccs; whereas, if the plane of comparison were so taken 
 that some of the points or lines projected should lie on one 
 
 of it and some on the other, it would be then necessary 
 i i use, in connection with the references, the algebraic sym- 
 bols plus or minus to designate the points above the plane 
 from those below it. 
 
 As the distances of all points are estimated from the 
 plane of comparison, the reference of any point or line of 
 this plane will therefore be zero, (0.0); that of any point 
 above it is usually expressed in feel : decimal parts of a foot 
 being used whenever the reference is not an entire number. 
 When the reference is a whole number it is written with one 
 decimal place, thus (25.0); and when a brok< n number with 
 at least two decimal places, thus (3.70), (15.63). In writing 
 the reference the mark used to designate the linear unit is 
 omitted, in order that the numbers expressing references 
 may not be mistaken for those which may he put upon the 
 drawing to express the horizontal distances between points. 
 
 The references of horizontal lines are written along and 
 upon the projections of these lines. All other references 
 are written as nearly as practicable parallel to the bottom bor- 
 der of the drawing, for the convenience of reading them 
 without having to shift the position of the sheet on which 
 the drawing is made. 
 
 This method of representing the projections of objects 
 on one plane alone has given rise to a very useful modifica- 
 tion of the one of orthogonal projections on two planes, and 
 has been denominated one mane descriptive geometry} the 
 plane of comparison being the sole plane of projection ; and 
 the references taking the place of the usual projections on a 
 vertical plane. By this modification the number of lines to 
 be drawn is less; the graphical constructions simplified; 
 and the relations of the parts is more readily seized upon, 
 as the eye is confined to the examination of one set of pro- 
 jections alone. • 
 
 But the chief advantage of it consists in its application 
 to the delineation of objects, like works of permanent forti- 
 fication, where, from the great disparity of the horizontal 
 extent covered and the vertical dimensions of the parts, a 
 drawing, made to a scale which would give the horiz< 
 distances with accuracy, could not in most cases render the 
 vertical dimensions with any approach to the same de 
 of accuracy; or, if made to a scale which would admit of 
 the vertical dimensions being accurately determined, would 
 require an area of drawing surface, to render the horizontal 
 dimensions to the same scale, which would exceed the con-
 
 ONE PLANE DESCRIPTIVE GEOMETRY. 6 
 
 venient limits of practice. Taking for example an ordinary 
 scale used for drawing the plans of permanent fortifications 
 of one inch to fifty feet, or the scale g-^, the details of all 
 the bounding surfaces can be determined with accuracy to 
 within the fractional part of a foot, whereas a vertical pro- 
 jection to the same scale would be altogether too small for 
 the same purposes. 
 
 4. Point and Right Line. To designate the position 
 of a point, PI. 1, Pig. 1, the projection of the point-and its 
 reference are enclosed within a bracket, thus (28.50). This 
 expresses that the vertical distance of the point from the 
 plane of reference is 28 feet and fifty-hundredths of a foot. 
 The position of a right line oblique to the plane of reference 
 is designated by the projection of the line, and the references 
 of any two of its points. Thus in Fig. 1 the points a and 
 b, upon the projection of the right line, with their respective 
 references (25.15) and (28.50), determine the position of the 
 line with respect to the plane of reference. 
 
 "When the line is horizontal, or parallel to the plane of 
 reference, its projection, with the reference of one of its 
 points, will be sufficient to designate it, and fix its position 
 with respect to the plane of reference. Thus in Fig. 1 the 
 reference (25.15), written upon the projection of the line, 
 expresses that the line is horizontal, and 25.15 feet from the 
 plane of reference. 
 
 5. For the convenience of numerical calculation, the po- 
 sition of a line, with respect to the plane of reference, is 
 often expressed in terms of the natural tangent of the angle 
 it makes with this plane; but as this angle is the same as 
 that between the line and its projection, its natural tangent 
 can be expressed by the difference of level between any two 
 points of the line, divided by the horizontal distance between 
 the points. Now, as the difference of level between any 
 two points of the line is the same as the difference of the 
 references of the points, and the horizontal distance between 
 them is the same as the horizontal projection of the portion 
 of the line between the same points, it follows, that the nat- 
 ural tangent of the angle which the line makes with the 
 plane of reference is found by dmiding the difference of the 
 references of the points by the distance in horizontal projec- 
 iton between them. 
 
 The vulgar fraction which expresses this tangent is term- 
 ed the inclination, or declivity of the line. Thus the frac- 
 tion i would express that the horizontal distance between 
 any two points is six times the vertical distance, or difference 
 of their references ; the fraction §, that the vertical distaiice
 
 4 ONE PLANE DESCRIPTIVE GEOMETRY. 
 
 bttween any two points is two-thirds the horizontal distance ; 
 the denominator of the fraction, in all cases, representing 
 the number of parts in horizontal projection, and the nume- 
 rator the corresponding number of parts in vein Meal distance, 
 
 When the position of a line is designated in this way, it ie 
 said to be a line whose inclination or declivity is one-sixth, 
 two-thirds, ten on one, &c, or simply, a line of one-s'uth, &c. 
 
 6. Having the declivity of a line, the difference of refer- 
 ence of any two of its points, the projections of which are 
 given, will be found by multiplying the horizontal distance 
 between them by the fraction which expresses this declivity ; 
 in like manner the horizontal distance of any two points 
 will be obtained by dividing the difference of their references 
 by this fraction. 
 
 To obtain therefore the reference of a point of a line, 
 having its projection, the horizontal distance between it and 
 that of some other known point of the line must be deter- 
 mined from the scale of the drawing by which the horizontal 
 distances are measured ; this distance expressed in numbers, 
 being multiplied by the fraction which expresses the declivity 
 of the line, will give the difference of reference of the two 
 points ; the required reference of the point will be found by 
 subtracting this product from the reference of the known 
 
 f)oint, if it is higher than the one sought, or adding if it ie 
 ower. Thus let (25.15) be the reference of a known point 
 higher than the one sought; the horizontal distance between 
 the points being 35.75 feet, and the inclination of the line 
 T V ; then 35.75 x T \ = 3.575 will be the difference of refer- 
 ence of the points, and 25.15 — 3.575 = 21.575, the required 
 reference. The converse of this shows that the horizontal 
 distance between two points on this line whose difference of 
 reference is 3.575 will be 3.575 -h T V= 35.75 feet. 
 
 7. When the projection of a line is divided into equal 
 parts, each of which corresponds to' a unit in vertical dis- 
 tance, and the references of the points of division are written, 
 it is termed the scale of declivity of the line. In constructing 
 the scale of declivity of a line, the entire references are alone 
 put down ; one of the divisions of the equal parts being sub- 
 divided into tenths, or hundredths if necessary, so as to give 
 the fractional parts of the references corresponding to an\ 
 fractional part of an entire division. 
 
 8. The true length of any portion of an oblique line be- 
 tween two given points is evidently the hypothenuse of a 
 right angle triangle of which the other two sides are the dif 
 ference of reference ot the points, and their horizontal dis- 
 tance.
 
 ONE PLANE DESCRIPTIVE GEOMETRY. 5 
 
 9. Plane. The position of a plane oblique to the plane 
 of reference may be determined either by the projections and 
 references of three of its points; by the projections and de- 
 clivity of two lines in it oblique to the plane of reference ; 
 or by the projection of two or more horizontal lines of the 
 plane with their references. 
 
 The more usual method of representing a plane is by the 
 projection^ on the plane of reference of the horizontal lines 
 determined by intersecting it by equidistant horizontal planes. 
 These projections are termed horizontals of the plane, those 
 usually being taken the references of which are entire numbers. 
 
 10. If in a given plane a line be drawn perpendicular to 
 any horizontal line in it, the projection of this line on the 
 plane of reference will be also perpendicular to the projec- 
 tions of the horizontals. The angle of this line with the 
 plane of reference is evidently the same as that of the given 
 plane with it, and is greater than the angle between any 
 other line drawn in the plane and the plane of reference. 
 This line is, on this account, termed the line of greatest de- 
 clivity of the plane. 
 
 11. If the scale of declivity of the line of greatest de- 
 clivity be constructed, it will alone serve to fix the position 
 of the plane to which it belongs, and to determine the refer- 
 ence of any point of the plane of which the projection is 
 given. For since the horizontals are perpendicular to the 
 scale of declivity, the point where the horizontal drawn 
 through the given projection of a point in the plane cuts 
 this line will determine upon the scale the reference of the 
 horizontal, and therefore that of the point. 
 
 12. The inclination or declivity of a plane with the plane 
 of reference may be expressed in the same way as the incli- 
 nation of its line of greatest declivity. Thus a plane of one- 
 fourth; a plane of twenty on one; a plane of two-thirds, 
 express that the natural tangents of the angle between the 
 planes and the plane of reference are respectively represent- 
 ed by the fractions i, - 2 T % and f. 
 
 13. The horizontal distance between any two horizontal 
 lines in a plane, the angle of which is given, can be found 
 in the same way as the horizontal distance between two 
 points of a line, the inclination of which is given, Art. 7, 
 by dividing the difference of the reference of the two hori- 
 zontal lines by th$ fraction representing the declivity of the 
 plane ; in like manner the difference of references of any 
 two horizontal lines will be obtained by multiplying their 
 horizontal distance by the same fraction. 
 
 14. To distinguish the scale of declivity, P\. 1, Fvg. 2,
 
 6 ONE PLANE DESCRIPTIVE GEOMETRY. 
 
 from any other line of a plane, it is always represented by 
 two fine parallel lines, drawn near each other, and crossed 
 at the points of division, where the references are written, 
 by short liues which are portions of the corresponding hori- 
 zontals. 
 
 With the foregoing elements the usual problems of the 
 right line and plane can be readily solved. 
 
 15. Problems of the Right Line and Plaile. 
 Prob. 1, PI. 1, Fig. 3. Ha/ovng the prqjt ct/ions and refer 
 
 end 8 of two lines that intersect, tofnd tin a/ngh between t)<> m. 
 
 Let at) be the projection of one of the lines, the refer- 
 ences of two of its points (10.30) and (4.90) being given 
 cd the projection of the other line, (10.30), and (5.0) being 
 the references of two of its points; (10.30) being the point 
 of intersection of the two lines. 
 
 Find on each of the lines, Art. 7, a point having the 
 name reference (7.0). The line joining these tAvo points 
 will be horizontal, and projected into its true length ; taking 
 this line as the base of a triangle of which the other two sides 
 are respectively the true lengths of the portions of the two 
 given lines projected between (10.30) and (T.0), Art. 7, the 
 angle at the vertex will be the one required. 
 
 16. Prob. 2, Pig. 4. Through a point to draw a Una 
 parallel to « given line. 
 
 Let c (7.50) be the projection of the point ; ab that of 
 the given line of which the two points (7.0) and (9.0) are 
 known. 
 
 Through o drawing cd parallel to ab, this will be the 
 projection of the required line ; and as its declivity is the 
 same as that of the given line, it will be only necessary to 
 set off from c towards d, the same distance as between (7.0) 
 and (9.0), to obtain a point (9.50) as far above (7.50) as (9.0) 
 is above (7.0). 
 
 17. Prob. 3, Fig. 5. Through a point in a plane to draw 
 a line in the plane with a given inclination. 
 
 Let cd be the scale of declivity of the given plane, and 
 a (5.50) the given point ; and suppose, for example, that the 
 declivity of the plane is \ and that of the required line is T V. 
 
 Draw the horizontal of the plane (5.50) which ] 
 through the point, and any other horizontal, as (7.0). The 
 projection of the required line will pass through a, and the 
 portion of it between the two horizontals will be equal, Art. 
 f>, to the difference of their references, or 1.5 ft. divided by 
 the fraction which represents the inclination of the required 
 line. Describing, therefore, from a, an arc, with this dis- 
 tance ac or 1.5 -i- T \ = 15 ft. as a radius, and joining the
 
 ONE PLANE DESCRIPTIVE- GEOMETRY. 7 
 
 point b, where it cuts the horizontal (7.0), with #, this will 
 be the projection of the required line. 
 
 18. Prob. 4, PI. 1, Fig. 6. Saving three points of a 
 plane, to construct its horizontals and scale of declivity. 
 
 Let a (12.0), b (15.25), and c (15.50), be the projections of 
 the three points. Join a with the other two, and construct 
 the scales of declivity of the lines of junction, Art. 6. The 
 lines joining the same references on these two scales will be 
 horizontals of the required plane. Its scale of declivity is 
 constructed by drawing two parallel lines perpendicular to 
 the horizontals, and writing the references of the points 
 where they intersect the horizontals. 
 
 19. Prob. 5, PI. 1, Pig. 1. To find the horizontals of 
 a plane passed through a given line and parallel to another 
 line. 
 
 Let ab and cdhe the projections of the two lines. From 
 a point (10.0) on cd draw a line df Prob. 2, parallel to ab; 
 and by Prob. 4 find the horizontals of the plane of df and 
 cd ; these will be the required horizontals. 
 
 20. Prob. 6, PI. 1, Fig. 8. To find the horizontals of a 
 plans the declivity of which is given, and which passes 
 through a given line. 
 
 Let bd be the scale of declivity of the given line, and 
 suppose, for example, the declivity of the line to be T 'j and 
 that of the required plane to be }. 
 
 Since the horizontals of the plane must pass through the 
 points of the line having the like references, and as the dis- 
 tance in projection between any two of them, Art. 13, will 
 be equal to the difference of their references divided by the 
 fraction giving the declivity of the plane, it follows that to 
 find the one drawn through b (14.0), for example, it will be 
 simply necessary to describe from any other point, as a 
 (12.0), an arc of a circle, with a radius of 12 ft., equal to 
 the quotient just mentioned, and to draw a tangent to this 
 arc from b. If any other horizontal, as (16.0), is required, 
 which would not intersect the projection of the given line 
 within the limits of the drawing; any two points, as (12.0) 
 and (14.0), for example, may be taken as centres, and two 
 arcs be described from them, with radii of 12 and 24 ft., 
 calculated as above, and a line be drawn tangent to the 
 arc; this tangent will be the required horizontal. 
 
 21. Prob. 7, PL 1, Fig. 9. Hiving either the horizontals 
 or the scales of declivity of two planes, to find their intersec- 
 tion. 
 
 ' Join the points ah where any two horizontals, as (12.0) 
 and (14.0), in one plane intersect the corresponding horizon-
 
 8 ONE PLANE DESCRIPTIVE GEOMETRY. 
 
 tals of the other, and the line so drawn will be the projection 
 of the required intersection. 
 
 22. When the horizontals of the two planes are parallel, 
 or when they are so nearly parallel that their points of in- 
 tersection cannot be accurately found, the following method 
 may be taken : Draw any two parallel lines as cd, cd', PI. 
 1, Fig. 10 ; these may be considered as the horizontals of an 
 arbitrary plane, and having the same references, (12.0) and 
 (14.0), as the two corresponding horizontals in each of the 
 given planes. The intersections of the horizontals of the 
 arbitrary plane with those of the given planes will determine 
 two lines, mn } m'n' , which, being the projections of the in- 
 tersections of the given planes with the arbitrary plane, 
 will, by their intersection o, determine the projection of a 
 point common to the three planes, and therefore a point of 
 the projection of the intersection of the two given planes. 
 Assuming any other two parallels ab, a 'I', as the horizontals 
 of another arbitrary plane ; finding in like manner the point 
 o and joining o and o' by a line, this will be the required 
 projection. 
 
 When the horizontals of the two planes are parallel, one 
 point, as o, will be sufficient to determine the required pro- 
 jection, as it will be parallel to the horizontals. 
 
 23. Prob. 8, PI. 1, Fig. 11. To find where a given line 
 pierces a given plane. 
 
 Through the projections of any two points of the given 
 line, as m/, n' , having the same references, (12.0), (14.0), as 
 two horizontals of the given plane, draw two parallel lines, 
 ah, a'b ', which may be taken as the horizontals of an arbitrary 
 plane. The projection of the line of intersection, m, of 
 this plane with the given plane being determined by Prob. 
 7, the point o where it intersects the projection of the line 
 m'n' will be the projection of the required point, the refer- 
 ence of which can be found from the scale of the plane. 
 
 24. Prob. 9, PI. 1, Fig. 12. To draw from a given 
 point a perpendicidar to a given plane, and find its length. 
 
 Let a (12.0) be the projection of the given point ; and 
 let the given plane be represented by its scale of declivity. 
 
 The projection of the required perpendicular will pass 
 through a, and be parallel to the scale of declivity of the 
 given plane. The angle which it makes with the plane of 
 reference is the complement of that between this plane and 
 the given plane ; its tangent therefore will be the reciprocal 
 of the tangent of that of the given plane. 
 
 Drawing therefore through a the line ac parallel to bd, 
 and constructing its scale of declivity, Art. 7, this will be
 
 ONE PLANE DESCRIPTIVE GEOMETRY. 9 
 
 the projection of the required perpendicular. The projec- 
 tion of the point o where it pierces the given plane is found 
 by Prdb. 8, and the true length of the perpendicular by 
 Art. 8. 
 
 25. Geometrical and Irregular Surfaces. 
 
 All other surfaces may, like the plane, Art. 7, be repre- 
 sented by the projections on the plane of reference of the 
 curves or lines cut from them by equidistant horizontal 
 planes, together with the references of these curves ; as many 
 of these projections being drawn as may be requisite to de- 
 termine all the points of the surface with accuracy; and 
 their references being written in the same way as those of 
 the horizontals of a plane. 
 
 In the more simple geometrical surfaces, a single hori- 
 zontal curve, with the projection of some point or line of 
 the surface, will alone suffice. For example, the cone may be 
 represented by the projection and reference of any curve cut 
 from it by a horizontal plane, with the projection and refei 
 ence of its vertex ; a cylinder by the projection and reference 
 of a like curve, with the projection and reference of its axis, 
 or of one of its right line elements ; a sphere by the projec- 
 tion and reference of its centre and that of its great circle 
 parallel to the plane of reference. 
 
 26. This method of projection is more particularly ad- 
 vantageous in the representation of irregular surfaces which, 
 like the natural surfaces of ground, for example, are not sub- 
 mitted to any geometrical law, and in solving the various 
 problems of tangent and secant planes to surfaces of this 
 character. These surfaces can, for the most part, be alone 
 represented by the projections of the horizontal curves cut 
 from them by equidistant horizontal planes, and by suppos- 
 ing the zone of the real surface contained between any two 
 horizontal curves to be replaced by an artificial zone, sub- 
 jected to some geometrical law of generation, which shall 
 give an approximation to the real surface sufficiently accu- 
 rate for the object in view. The usual method of doing this 
 is to take two consecutive horizontal curves as the directrices 
 of the artificial surface of the zone, and to move a right line 
 bo as to continually intersect each of them,^ and be perpen- 
 dicular to the consecutive tangents to one of them, the upper 
 one being usually taken for this last condition. 
 
 If in PI. 1, Fig. 13, for example, (6.0), (7.0), &c, are the 
 orojections of the horizontals of a surface, the zone between 
 the curves (6.0) and (7.0) may be replaced by an artificial 
 surface, the position of the projection of the generatrix of 
 which, at any point of the upper curve (7.0), will be deter-
 
 10 ONE PLANE DESCRIPTIVE ^""JiOMETBY. 
 
 mined by constructing the horizontal tangent at that point 
 as 0, for example, and drawingoS perpendicular to it and 
 
 intersecting the lower curve. The position of the generatrix 
 a'b' at any other point a' is constructed in like manner. 
 
 '27. To obtain any horizontal of the artificial zone inter- 
 mediate to the two directrices, it will be only necessary to 
 construct several positions of the generatrix, and to find on 
 these the points having the same reference as the required 
 curve. The horizontal of the surface (6.50), for example, 
 will bisect the projections of the generatrix in its varioua 
 positions. 
 
 Problems of Irregular Surfaces and the Right 
 Line and Plane. 
 
 28. Proh. 10, PI. 1, Fig. 14. Through a given point in 
 a vertical plam vahich intersects a surface, to draw a tangent 
 to the curve of intersection of the plane and surface. 
 
 Let a (5.50) be the given point, and ab the trace on the 
 plane of reference of the given plane. The points where 
 this trace intersects the horizontal curves of the surface will 
 be the projections of points of the curve cut from the surface 
 by the plane. 
 
 Let any arbitrary line as ac be now drawn through a, 
 and its scale of declivity be constructed ; and let lines be 
 drawn between the points having the same references on ac 
 and on the horizontal curves where ah intersects them. These 
 lines will be the projections of horizontal lines and will gen- 
 erally make different angles with ac. The one as (T.'M, 
 which makes the smallest angle with it, towards the descend- 
 ing portion, will determine the projection o of the tangential 
 point. For, construct the scale of declivity of the line of 
 which a (5.50) is the projection of one point, and o (7.0), on 
 ab, another. Comparing now the references of the points 
 on the line, and which is assumed as the projection of the 
 required tangent, with the references of the points of the 
 curve having the same projection, it will at once be evident 
 that these two lines have only the point projected in (7.0) in 
 common, and that every other point of the right line, of 
 which adb is the projection, is exterior to the curve, and 
 therefore the line itself must be tangent to the curve at the 
 point determined as above. 
 
 29. Proh. 11, PI. 1, Fig. 15. To construct the elements 
 of a cone, with a given vertex, which shall envelope a given 
 surface. 
 
 Let (10.0), &c, be the horizontals of the given surface ; 
 and a (6.0) the projection of the vertex of the cone. 
 
 From a, draw lines ab, ab' , &c, as the horizontal traces
 
 0XE PLANE DESCRIPTIVE GEOMETRY. 11 
 
 of vertical planes whicli pass through the vertex and inter 
 sect the surface. Construct, by Prob. 10, the tangents from 
 a to the curves cut from the surface by the planes ab, &c. 
 These tangents will be the required elements. 
 
 30. Prob. 12, PI 1, Fig. 15. To find the curve of in- 
 tersection of a cone enveloping a given surface by a Jwrizon- 
 tal plane. 
 
 Let (9.0) be the reference of the given horizontal plane. 
 Having found, by Probs. 11 and 12, the elements of the cone, 
 and constructed the scale of declivity of each one ; then 
 joining the points o, o', o", having the same reference on 
 each scale as the given horizontal plane, a continuous line 
 mo"o'on will be obtained, which will be the projection of 
 the points where the elements pierce the given plane, and 
 therefore the projection of the required intersection. 
 
 31. Prob. 13, PI. 2, Fig. 1. A limited extent of surface 
 being given, and a point exterior to it, to find the limits with- 
 in which planes may be passed through this point and lie 
 above all the given surface. 
 
 Let a (8.0) be the projection of the given point ; (10.0), 
 (9.0), &c, the horizontals of the given surface, the limits of 
 which are the sector contained within the arc BDC, and 
 the two radii aB and aC. 
 
 Taking a as the vertex of a cone which shall envelope 
 the given surface, the elements of this cone can be found by 
 Probs. 11 and 12. Any plane tangent to this cone, which 
 does not intersect the surface within the given limits, will 
 satisfy the conditions of the problem. 
 
 From the position of the vertex of the cone with respect 
 to the surface, it will be seen that a horizontal plane, passed 
 through the vertex, will cut from the cone two elements 
 which will be projected in the two horizontals ab' and ah" 
 (8.0) of the cone, the first of which will be tangent to the 
 horizontal (8.0) of the surface, and the second ab" will 
 pierce the surface, where the limiting arc BDC cuts the 
 same horizontal (8.0) ; and that all the elements projected 
 within the angles Bab' and Cab" will lie below the horizon- 
 tal plane (8.0). Now, if the elements within these angles 
 be prolonged beyond the vertex, they will form two portions 
 of cones having the same elements as the portions below the 
 vertex, and it is evident that any plane passed tangent to 
 either lower portion, as b'aB, within one of these angles, 
 will leave this portion below it, and the corresponding por- 
 tion, formed by the prolonged elements, above it ; and, in 
 order that thisplane shall satisfy the conditions of the prob- 
 lem, it must also leave the portions of the cone within the
 
 12 ONE PLAITS DESCRIPTIVE GEOMETRY. 
 
 angles b'ab" , and b"aC, also below it. The same reasoning 
 applies to planes passed tangent to the portions of the cone 
 within each of the other two angles. It is therefore evident 
 that a plane, which shall satisfy the conditions imposed, 
 must leave all that portion of the cone which lies above the 
 horizontal plane (8.0) through the vertex, below it, and all 
 the prolonged portions, corresponding to the portions below 
 the plane (8.0), above it. 
 
 To find any such plane, let the cone be intersected by a 
 horizontal plane, as (9.0), by Prob. 12. This plane will cut, 
 from the portion of the cone within the angle b'ab", a curve 
 of which non' is the projection ; the two extreme points of 
 this curve, within the limits, being at the points nn', where 
 the horizontal (9.0) of the surface cuts the limiting arc; it 
 will also cut, from each of the prolonged portions, a curve, 
 the one mr, and the other m'r' j the extreme poini m of m? 
 being on the prolongation of the extreme element aC 'j that 
 m! of the other on the extreme element aB, on the other 
 side, prolonged. Having obtained these three curves, let 
 tangent lines, ms, m's ', be drawn, from the points m and m ', 
 to the curve non' . A plane passed through either of these 
 tangents and through the corresponding element of the cone 
 as or as', drawn through the tangential point, will be a tan- 
 gent plane to the cone ; and as either of these planes will 
 leave the curve non! on one side of it, and the two curves 
 mr, and m'r', on the other, it will leave all the portion of 
 the cone corresponding to the first curve below it, and the 
 portions corresponding to the other curves above it ; and 
 will therefore satisfy the required conditions. The same will 
 hold true for any tangent plane to the cone along any ele- 
 ment drawn between the points s and «'/ since the tangent 
 drawn to any point of the curve non', between the points s 
 and s', will leave this curve on one side of it, and the other 
 two, mr and m'r', on the other. 
 
 32. Prob. 14, PI. 1, Fig. 16. Hi/rough a given line to 
 pass a plane tangent to a surface. 
 
 1st. Let ab be the projection of the given line, and (10.0), 
 (9.0), &c, the horizontals of the surface. From the points 
 on the line, as (10.0), &c, draw lines tangent to the horizon- 
 tals having the same references; the tangent which makes 
 with the projection of the line the least angle towards the 
 descending portion, will, with the line, determine the requir- 
 ed plane. 
 
 For, let the tangent (10.0) be the one which makes with 
 ab the least angle; from the other points, (9.0), eve., of ab, 
 draw lines parallel to the tangent (10.0); these lines will lie
 
 ONE PLANE AND DESCRIPTIVE GEOMETRY. 13 
 
 in the plane that contains this tangent and ah, and will be 
 horizontals of this plane ; they also lie respectively in the 
 planes of the horizontals (9.0), (8.0), &c, of the surface, but, 
 since they fall exterior to these horizontals, it follows that 
 their plane also lies exterior to every horizontal curve of the 
 surface, except at the curve (10.0), and where it touches the 
 surface at the point of contact of its horizontal (10.0) with 
 this curve. 
 
 2d. When the line ah, PI. 1, Fig. 17, is horizontal, let 
 tangents be drawn to the horizontal curves and parallel to 
 ah. These tangents may be regarded as the elements of a 
 cylinder which envelops the surface, the tangent plane to 
 which will be tangent to the surface. To find the element 
 of contact of the plane and cylinder, let the cylinder and 
 given line be intersected by an arbitrary vertical plane, of 
 which od is the trace. From the point o, (6.5), where the 
 line pierces this plane, let a tangent line be drawn to the 
 curve cut from the cylinder by the plane, by Proh. 10. The 
 point of contact will determine the position of the element 
 of the cylinder along which the plane, through ah, will be 
 tangent ; since the tangent to the curve projected in od, with 
 the line ah, will determine the tangent plane to the cylinder. 
 
 3d. "WTien the line ah, PI. 1, Fig. 18, is so nearly hori- 
 zontal that tangents cannot be drawn from its points, within 
 the limits of the drawing, to the horizontal curves. Let any 
 point of the line, as o, (7.0), be taken as the vertex of a cone 
 enveloping the surface ; a plane passed through the line and 
 tangent to the cone will be tangent to the surface. 
 
 Find, by Probs. 10 and 11, the projection mm of the 
 curve cut from this cone by the horizontal plane (8.0) ; from 
 the point (8.0) of ah draw a tangent to mm. This tangent, 
 with the line ah, will determine the required plane. 
 
 33. Proh. 15, PI. 1, Fig. 19. To find approximately 
 the point where a given right line pierces a surface. 
 
 Let (8.0), (9.0), &c, be the horizontals of the surface, and 
 df the scale of declivity of the line. Through any two 
 points, as a (9.0) and c (8.0), draw two parallel lines, as am 
 and en, which may be taken as the horizontals of an arbi- 
 trary plane passed through the given line. Joining the 
 points m, n where the horizontals of the arbitrary plane in- 
 tersect the corresponding horizontals of the surface, this 
 line mn will be the approximate intersection of the plane 
 with the zone of the surface between the horizontals (8.0) 
 and (9.0), and the point o where mn intersects df will be 
 the approximate point required.
 
 14 OXE PLANE DESCRIPTIVE GEOMETRY. 
 
 34. Prdb. 16, PI. 1, Fig. 20. To >w2 Me intersection 
 of a plane and surface. 
 
 Let (10.0), (9.0), tfcc., be the horizontals of the surface, 
 ef the scale of declivity of the plane. 
 
 Draw the horizontals of the plane having the same refer- 
 ences as the horizontals of the surface, the points of inter- 
 section of the corresponding lines will be the projections of 
 points of the required intersection. 
 
 When it is desired to find a point of the curve of inter- 
 section intermediate to two horizontal curves ; if the refer- 
 ence of the required point is fixed, it will be necessary to 
 construct, Art. 27, the horizontal of the surface, and the 
 horizontal of the plane having this reference ; their intersec- 
 tion will give the projection of the required point. If the 
 reference of the required point is not fixed, draw any gene- 
 ratrix, as ao of the zone on which the required point is to be 
 found, and by Prob. 8, Fig. 11, find the projection of the 
 point, as o, where ac pierces the given plane; this will be 
 the required point. 
 
 35. Application of Preceding Problems. 
 
 The following problems will aid as illustrations of the 
 preceding subject in its application to the determination and 
 delineation of lines and surfaces. 
 
 36. Prob. 1, PI. 2, Fig. 2. The plane of site of a ivork, 
 the exterior line and scale of declivity of its terre-plein being 
 given; to construct the plane of the rampart-slope and its 
 foot; also a ramp of a given inclination along the rampart- 
 slope leading from the plane of site to the terre-plein. 
 
 Let a (74.50) and b (76.0) be the references of two points 
 on the exterior line of the terre-plein, and mn its scale of 
 declivity; let the rampart-slope be ■§, the declivity of the 
 ramp |, its width 4.30 yards ; and the plane of site be hori- 
 zontal and at the ref. (60.0). 
 
 The foot of the rampart-slope lying in the plane of site 
 will be horizontal, and will be determined, Prdb. 6, Fig. 8, 
 by finding the line of the slope at the ref. (60.0). 
 
 Having the two bounding lines of the rampart-slope, the 
 inner line cd of the ramp is constructed, by assuming a 
 point <?, on the foot of the rampart-slope, as the point of 
 departure, and determining the line of £ drawn from c on 
 the rampart-slope by Prdb. 3, Fig. 5. Having found this 
 line, which is also the line of greatest declivity of the ramp, 
 the exterior line ef of the ramp is drawn parallel to it, and 
 at a distance 4.30 yds., equal to the width assumed for the 
 ramp. The horizontals of the ramp will bo perpendicular 
 to these two lines. The foot of the ramp, ce, will be a hor-
 
 ONE PLANE DESCRIPTIVE GEOMETRY. 19 
 
 izontal line, drawn through the point of departure. The 
 top of it, df, will he determined by Prof). 7, Fig. 9, by find- 
 ing the intersection of the ramp and the terre-plein, one 
 point of which will be the point d (76.30), the intersection 
 of the inner line of the ramp and the interior line of the 
 terre-plein. 
 
 The ramp is terminated on the exterior, by passing a 
 plane through its exterior line ef having the same slope as 
 the rampart-slope. This plane will intersect the plane of 
 site in a line parallel to the foot of the rampart-slope, and 
 the terre-plein in one parallel to the exterior line of the 
 terre-plein. 
 
 37. Prob. 18, PI. 2, Fig. 3. Having given the lines of 
 the parapet of a work and the scales of declivity of the planes 
 of its interior crest and terre-plein, to determine the lines and 
 surfaces of a ba/rbette in its salient for five guns. 
 
 Let ab be the scale of declivity of the plane of the interior 
 crest, which, as the terre-plein is parallel to the plane of 
 the interior crest and 8 feet below it estimated vertically, 
 will also serve as the scale of declivity of the terre-plein, by 
 subtracting 8 feet from the references of the former to obtain 
 the corresponding references of the latter. Having con- 
 structed a pancoupe of 4 yds. in the salient, find the intersec- 
 tion of the top surface of the barbette, which is horizontal 
 and assumed on the drawing at the reference (82.75) with 
 the planes of the interior slope, this intersection will deter- 
 mine the foot of the genouillere of the barbette. From this 
 last line at the pancoupe set back along the capital a distance 
 of 8 yds., and from the extremity of this line draw a per- 
 pendicular to the interior crest of each face. The pentag- 
 onal figure thus marked out will be the space for the gun in 
 the salient. From the foot of each of the perpendiculars 
 set off along the faces distances of 12 yds. for the lengths 
 along the interior crests to be occupied by two guns on each 
 side of the salient. Setting back from the extremities of 
 these two last distances perpendiculars to the interior 
 crest of 8 yds. and drawing lines through the extremities of 
 these perpendiculars parallel to the interior crests, they with 
 the two perpendiculars will mark out the exterior bounding 
 lines of the barbette. By passing planes of } or 45° through 
 these exterior lines, and finding by Prob. 7, Fig. 9, their in- 
 tersections with the terre-plein, these lines will be the foot 
 of the barbette slopes. A ramp having a slope \ leads from 
 .the terre-plein to the top of the barbette ; the width of this 
 ramp is 3.30 yds., its interior line in projection being on the 
 prolongation of the foot of the banquette slope. The ramp
 
 16 ONE PLANE DESCRIPTIVE GEOMETRY. 
 
 is terminated by side slopes of }, the intersections of which 
 with the terre-plein and the slopes of the harbette and ban* 
 quette are found by Prob. 7, Fig. 9. The foot of the ramp 
 or its intersection with the terre-plein is also found by the 
 same problem. 
 
 As the top surface of the barbette is horizontal, it may 
 De necessary in some cases to make the interior crest along 
 the barbette tlso horizontal, in which case the superior slope 
 of the parapet along the barbette being higher than the 
 rest of it, the two planes will be connected by a plane of 
 45°, as at C. 
 
 38. Prob. 19, PI. 2, Fig. 4. To determine the bounding 
 surfaces of a ramp leading up an irregular surface and so 
 placed that its axis or centre line shall nearly coincide with 
 the irregular surface. 
 
 Let (8.0), (9.0), &c, be the horizontal curves of the sur- 
 face, and let a (8.0) be the point of departure or foot of the 
 ramp. Assuming the declivity of the ramp ±, for example, 
 from a, with a radius of 9 units, describe an arc, and join 
 by a right bine the point b where it cuts the horizontal (9.0) 
 with the point. Repeat this construction from b to c on the 
 horizontal (10.0) ; and so on to the top e or point of arrival. 
 The broken line a-b-c-d-e will be the projection of the axis. 
 But, to avoid the angular changes of direction, the straight 
 portions of the axis may be connected at the angular points, 
 by setting off from b, for example, the equal distances ba ', 
 be', and connecting these points by an arc of a circle tangent 
 to the straight portions. The same construction being re- 
 peated at the other angular points, the broken line will be 
 replaced by the sinuous line aa'c', tfcc, as the axis. Having 
 determined the axis, the exterior and interior lines of the 
 top surface are drawn parallel to the axis, and at a distance 
 from it equal to half the assumed width of the ramp. 
 
 From the position of the axis the exterior half of the 
 ramp will be in embankment and the interior in excavation. 
 To determine the side slopes of the embankment pass planes 
 through the straight portions of the exterior edge of the 
 ramp, and find by Prob. 6, PI. 1, Fig. 8, the horizontals of 
 these planes, and by Prob. 16, Fig. 20, the intersections of 
 these planes with the irregular surface. The plane surfaces of 
 the side slopes thus determined are connected by curved sur- 
 faces which pass through the curved lines of the exterior edge. 
 These surfaces may be determined as follows : Take, for ex- 
 ample, the point n at the foot of the plane side slope A where 
 it cuts the radius through a' prolonged, of the arc a'c' ; and 
 the point o on the radius through c' where it cuts the foot of
 
 OXE PLANE DESCRIPTIVE GEOMETRY". 17 
 
 the plane side slope B. The lines of which nv anil ou aro 
 the projections will evidently have the same inclination, and 
 they may be assumed as the lines of junction of the plane 
 slopes A and B and the curved side slope x. This curved 
 side slope may then be generated by the motion of a right 
 line which has the top line of which vu is the projection for 
 its directrix, whilst in its motion it makes a constant angle 
 with the plane of comparison, and its projections are con- 
 stantly normal to the arc vu. From the construction com- 
 prising these conditions the foot no of the curved portion x 
 of the side slope is determined. The same constructions are 
 repeated to obtain the portions G of the plane, and y, s of 
 the curved side slopes, with the line m-n-o-p-q^r-8 the foot of 
 these slopes. 
 
 The side slopes of the part in excavation A', B' , C and 
 x\ y' with the line m'-n'-o'-jp'-g'-r' are determined by like 
 constructions. 
 
 The portions of the top surfaces of the ramps bounded 
 by the arcs of circles are helicoidal surfaces, of which the 
 axis is the directrix and the plane of comparison the plane 
 director. 
 
 The curved surface side slopes are also evidently helicoi- 
 dal surfaces, the directrices of which are the curved lines 
 above mentioned, and the vertical lines through the centres 
 of the arcs which are the projections of those curved lines. 
 
 Remarks. In the figure the declivity of the side slopes 
 of the embankment is one-half the excavation. The decliv- 
 ities of the curved portions of the top are greater than those 
 of the plane surfaces, the difference depending on the angle 
 between the straight portions of the axis. 
 
 39. Observations on the best mode of executing 
 Drawings. 
 
 Accuracy. The first requisite in all drawings is minute 
 accuracy, both in the geometrical constructions, and in writ- 
 ing down all letters and numbers which serve either as ref- 
 erences, or to give dimensions. To attain this, so far as 
 regards the geometrical part, judgment is to be exercised in 
 the selection of the means for establishing;; on the drawing 
 the positions of the various points which are either given or 
 to be found ; as one method although in theory as correct as 
 some other may not, in practice, be found to yield as satis- 
 factory results. The following remarks will serve to illus- 
 trate this point : 
 
 1st. In setting off from, a scale of equal parts several dis- 
 tances, along a line, whether equal or unequal, the most ac- 
 curate method is to commence by first setting off the entire 
 2
 
 18 ONE PLANE DESCRIPTIVE GEOMETRY. 
 
 distance, and then the several parts ; talcing care to verify, 
 from the scale, the aggregate of the several partial distances ; 
 thus in the example PI. 2, Fig. 5, where the aggregate of 
 all the partial distances is 60.33 feet, commence by setting 
 off the entire distance 60.33 feet ; next 50.33, which is the 
 sum of the two distances 20' and 30'.33, then verify the re 
 maining 10' by the scale. 
 
 2d. When a distance to be set off is so small that it can- 
 not be laid down with accuracy by the points of the dividers, 
 the following method may be employed : set back, from the 
 point from which the required distance is to be set off, any 
 arbitrary distance, then set forward, from this last point, a 
 distance equal to the sum of this arbitrary distance and the 
 one required ; thus in PL 2, Fig. 6, where 2' is to be set off" 
 from a towards c, set back from a say 30' to b, then from b 
 32' toe. 
 
 3d. To set off a point at a given perpendicular distance 
 from a Une, it will mostly be found more speedy, and more 
 accurate, to take off from the scale the given distance, in 
 the dividers, and, setting one point on the paper, bring the 
 other so that the arc described by it, with the given distance 
 as a radius, shall be tangent to the line, than to employ the 
 usual method of first erecting a perpendicular to the line 
 and then setting off the required point along the perpendic- 
 ular; thus in Pi. 2, Fig. 7, wishing to set off c at 20' from 
 ab, take 20' in the dividers, and, by the eye, find where one 
 point must be placed so that the other describing an arc will 
 touch ab. This method will be found convenient in drawing 
 a parallel' to a line at a given distance from it by setting off 
 another point in the same way, and drawing through the 
 two the required parallel. 
 
 4th. In setting off several points for the purpose of draw- 
 ing several parallels to a given line, as, for example, the par- 
 allel lines which bound the planes of a parapet, it will be 
 found most speedy and accurate to draw first upon a slip of 
 smooth thin paper two lines perpendicular to each other, 
 then marking on one of the lines the respective given dis- 
 tances of the parallels from the other, and cutting the paper 
 close to the line along which the given points are marked 
 off, so that the strip when laid upon the drawing so as to 
 have one of its lines to coincide with that to which the par- 
 allels are to be drawn, their distances from it can be pricked 
 off by a sharp pointed pencil, or in any other way. In PI. 
 2, Fig. 8, ab is the line of the drawing ; A the strip of paper, 
 fc, fd, fe, &c, the distances at which the parallels are to be 
 drawn from ah, marked off on the edge of A perpendicular
 
 ONE PLANE DESCRIPTIVE GEOMETRY. 10 
 
 to the line f, which line when A is laid on the dn-tving, 
 should coincide with ah. If the line ah is somewhat long, it 
 will be better to set oft* these points near each of its extrem- 
 ities and join them by lines, than to draw the parallels in the 
 usual way by aid of the ruler and triangle. 
 
 5th. When a point is to he constructed hy means of t/i6 
 intersection of two lines arbitrarily chosen, such a position 
 should be assumed for the arbitrary lines that they shall not 
 form a very acute angle at their point of intersection, as in 
 that case this point might not be so distinct to the eye as to 
 be marked with accuracy. For example, in erecting a per- 
 pendicular to a line at a given point, and in like problems, 
 in which points are found by the intersections of arcs of cir- 
 cles, it will be best and most convenient to take for the radii 
 of the arcs the distance between their centres, as the angle 
 between the tangents to the arcs at their point of intersection 
 will then be 60°, which is a sufficient angle to give accurately 
 the point where the lines cross. In cases like Figs. 10, 11, 
 Arts. 22, 23, the arbitrary lines ah, a'h\ (fee, should be so 
 chosen as to intersect the horizontals nearly at right angles, 
 and so, also, that the resulting lines, by which the points o, 
 o' , are determined, shall not intersect in too acute an angle. 
 
 In all such cases of determining points, and where a point 
 :'s pricked into the paper, it will be found well to designate 
 the point thus O, by a small circle drawn around it with the 
 lead pencil, in order that the eye may see it with more dis- 
 tinctness. 
 
 6th. In determining a portion of a line hy the construc- 
 tion of two arbitrary points, the points should be so chosen 
 that the portion required may fall between them and not 
 beyond them. In Pi. 1, Fig. 10, for example, if the requir- 
 ed portion of the line of intersection of the planes extended 
 on either side, beyond o, or o', or beyond both, the lines ah, 
 cd, &c, should be so chosen as to bring o and o', as far apart, 
 at least, as the length of the required portion of the line 
 which they serve to determine. 
 
 7th. No means of verifying the accuracy of the construc- 
 tion of points, or lines, should be omitted. In Pi. 1, Fig. 9, 
 for example, other corresponding horizontals should be drawn, 
 and, if the line of intersection determined by the two points 
 first found is correct, their points of intersection also will 
 fall upon it. In PI. 1, Figs. 9, 10, the scale of declivity of 
 the line of intersection being determined, the references of 
 the points, where it intersects the scales of declivity of the 
 planes, should be the same as the same points on the scales, 
 if the line has been accurately determined. A- general and
 
 20 ONE PLANE DESCRIPTIVE GEOMETRY. 
 
 minute verification of all the parts of the drawing, should be 
 made before any portion of it is put in ink. 
 
 Neatness. This is a not unimportant element in the at 
 tainment of accuracy in drawing. A few minutiae, when 
 attended to, will subserve this end. 
 
 That part of the paper on which the draughtsman is not 
 working should be kept covered with clean paper, pasted on 
 the edge of the board, so as to fold over the drawing, and 
 the parts which are finished should be similarly protected. 
 
 Before commencing the daily work the paper should be 
 carefully dusted, and the scales, rules and triangles be care- 
 fully wiped with a clean dry rag. 
 
 As few lines of construction as possible should be drawn in 
 pencil ; and only that part of each which may be strictly ne- 
 cessary to determine the point sought. As, for example, where 
 a point is to be found by the intersection of two arcs of cir- 
 cles ; when the position of the point can be approximately 
 judged of by the eye, only a portion of one arc, which will 
 embrace the point, may be drawn, and the point where the 
 second arc would intersect the first be marked without 
 describing the arc. In PI. 1, Fig. 10, instead of drawing 
 the entire lines ah, cd, &c, it would be simply necessary to 
 mark the points only where they cut the horizontals ; and, 
 in like manner, the points o and o' might be marked without 
 drawing the entire lines. 
 
 ]STo more of any line of the drawing should be made in 
 pencil than what is to remain permanently in ink. The ob- 
 ject of these precautions is to keep the paper from becoming 
 covered with dirt and the lines from being defaced by the 
 wear of the paper. 
 
 Inking. In inking the lines the following directions will 
 be found useful : 
 
 Efface carefully all pencil lines that are not to be inked; 
 and those parts of the permanent lines which are not to re- 
 main, before commencing to ink. 
 
 When right lines are tangent to curves, put in ink the 
 curve before the right line ; draw all arcs of equal radii at 
 once, one after the other ; if several arcs are to be described 
 from the same centre, it will be well to put a thin bit of quill 
 over the point for the end of the dividers to rest on, to avoid 
 making a large hole in the drawing. 
 
 If the drawing is not to be colored with the brush, all 
 the lines of one color should be put in before commencing 
 on those of another. 
 
 If one of the bounding lines of a surface is to be made 
 heavier than the others, its breadth should be taken from
 
 TlaM'l. 
 
 Tig. 14. 
 
 ; '°oj 
 
 ■-■----- — - -(9.0) 
 
 ! *• ^ '" - 
 
 "
 
 ONE PLANE DESCRIPTIVE OEOMETBY. 21 
 
 the surface they limit and not be added to it ; aid when the 
 heavy line forms the boundary of two surfaces, its breadth 
 must be taken from the one of greatest declivity. 
 
 Coloring. "When the drawing is to be colored, all lines 
 that are not to be black may be put in first with black, mak- 
 ing them very faint, so that they may receive their appro- 
 priate colors after the drawing is otherwise completed. 
 
 ]STo heavy line should be put in until the work with the 
 brush is completed. 
 
 When all the lines are in, the drawing should be thor- 
 oughly cleaned with stale bread-crumb ; and then have sev- 
 eral pitchers of water dashed over it, the board being placed 
 in an inclined position to allow the water, colored by the ink 
 lines, to escape rapidly, and not to discolor the paper. 
 
 In using the brush, whether for flat tints, or graded, the 
 requisite depth of tint should be reached by a number of 
 faint tints laid over each other ; this is especially necessary 
 in laying tints of blacks, browns, and reds. 
 
 To obtain an even flat, or graded tint, on dry paper re- 
 quires considerable skill. The best plan for this is, first to 
 wet with a large brush, or clean rag, the surface on which 
 the tint is to be laid, then, with a slightly moist rag, clear 
 the surface of water, and before the paper has time to dry 
 to lay on the tint. With this precaution, the heaviest tints 
 of Chinese ink, the most difficult of all to manage on dry 
 paper, can be neatly laid down. 
 
 Titles, &c. The lettering and numbering of a drawing 
 should be in ordinary printed character ; this is particularly 
 requisite in the numbering, to avoid misapprehensions which 
 often arise from individual peculiarities in writing numbers. 
 
 As has been already remarked, references are written in 
 black, within brackets which, when practicable, embrace the 
 point referred to. When not practicable, a small dotted line 
 may lead from the point to the reference ; thus, 0. . .(25.50) ; 
 but to distinguish references from other numbers the desig- 
 nation of the unit is omitted. 
 
 All horizontal distances between points are written upon 
 a dotted line drawn between the points, with an arrow-head 
 at each end ; where several partial distances in a right line 
 are marked, it will be also well to mark the total distance : 
 the latter may be written above or beneath the former, PI. 
 2, Fig. 5. ^ 
 
 In writing horizontal distances, the usual designation of 
 the unit is always written thus, y for yards, ' for feet, (tec. 
 All the numbers must be expressed in the same unit ; the 
 fractional parts being in decimals.
 
 22 ONE PLANE DESCRIPTIVE GEOMETRY. 
 
 References and horizontal distances cannot be too miu h 
 multiplied, in order to avoid misapprehensions, and the re- 
 sults of errors of construction, as well as to save the time 
 that would be taken in applying dividers to the drawing to 
 find from the scale affixed to it the dimensions of any part. 
 
 Scale. A scale very accurately constructed should bo 
 affixed to the drawing before it is cut from the board ; so 
 that the shrinkage of the paper, which is about 5^5 ma y 
 affect all the parts equally, and the scale thus be made to 
 correspond to the real lengths of the lines on the drawing. 
 The scale should be divided according to the decimal system, 
 as being most convenient for counting off. 
 
 The first division of the scale should furnish the units 
 and also their decimal parts, if the scale bears that propor- 
 tion to the true dimensions of the object represented which 
 will admit of these divisions. This first division is numbered 
 from right to left, PI. 2, Fig. 9, the zero point being on the 
 right, the 10 point on the left ; the succeeding divisions, to 
 50 inclusive, should each be equal to the first division, con- 
 taining ten units each. The remaining divisions may con- 
 tain fifty units each. It will be seen that any number of 
 tens, units, or fractional parts of a nnit can thus be readily 
 taken off from the scale by the dividers. The scale should 
 be long enough to give the dimensions of the longest line on 
 the drawing. 
 
 The proportion which the scale bears to the true dimen- 
 sions of the object should be written above the scale ; thus, 
 Scale one inch to ten yards, or 3 }q. And the designation 
 of the unit of the drawing should be annexed to the last 
 division on the scale, as yds. for yards, ft. for feet, etc. 
 
 Note. — For more detailed directions on the mechanical 
 or instrumental methods of geometrical drawing, see Mahan'a 
 Industrial Drawing.
 
 Hale 2.
 
 STONE CUTTING. ^3 
 
 STONE CUTTING. 
 
 The object of tins Article is to explain the geometrica 
 methods of representing the more usual and elementary 
 combinations of blocks of stone -in walls and arches, by 
 means of their projections ; and from these and the data of 
 the problem to deduce the true dimensions of the bounding 
 surfaces and lines of each part. 
 
 Walls bounded by Plane Surfaces. In walls of cut 
 stone a the blocks are usually separated by horizontal and 
 vertical joints; the latter being in vertical planes perpendic- 
 ular to the face of the walls, and which are termed plane* 
 of right section, to distinguish them from other vertical 
 planes of section. When the face of the wall is inclined to 
 the horizon, its slope, or batir, is usually expressed by the 
 ratio of the base of the slope to the perpendicular, measured 
 in the plane of right section ; or the slope is said to be so 
 many base to so many perpendicular. In the right section 
 A' It' CD', (PI. A, Fig. 1,) for example, the inclination of 
 the face A'C to the base of the wall A B', is measured by 
 dividing the perpendicular C E' from C upon the line A'B\ 
 by the distance A E' between the point A' and the foot of 
 the perpendicular. The quotient ^', thus obtained, is ev- 
 idently the natural tangent of the angle C'A'E' ; and the 
 most convenient method of representing the batir is by a 
 fraction ; the numerator expressing the number of units in 
 the perpendicular, and the denominator the corresponding 
 number of units in the base ; thus a batir of f expresses a 
 slope of six perpendicular to one base ; a batir of f , one of 
 three perpendicular to two base, &c. 
 
 Prob. 1. Having given the right section of a wall, to 
 construct the projections of its hounding lines, and the edges 
 of the horizontal and vertical joints. 
 
 Let A'B'D'C, (PI. A, Fig. 1,) be the right section ; the 
 base A'B' and the top CD' being horizontal ; the face A'C 
 having a batir — = f ; and the back B'D' vertical. 
 
 Draw a lino A A, , to represent the foot of the face in pl?n. 
 Parallel to A A, draw CO, , and at a distance from it eq/ial x o 
 
 • See Mahau's Civil Engineering, Art. 351.
 
 24 STONE CUTTING. 
 
 A ' E\ the distance of the point A' from the foot of the per 
 pendicular drawn from C in the plane of right section ; CC 
 will be the top line of the face in plan. Parallel to CC I , and 
 at a distance CD , the breadth of the wall at top, draw 
 BB t , which will be the projection of the hack. 
 
 At any convenient distance from BB ' draw A" A'" par- 
 allel to it, to represent the foot of the wall in elevation; the 
 top C"D" will be drawn parallel to A" A'", and at the 
 height BD' of the top above the base. 
 
 To draw the projections of the horizontal edges, suppose 
 the wall divided into four equal courses by horizontal joints. 
 As the batir of the wall is f , the base of the slope of each 
 of these equal courses will be one-sixth of its height; if 
 •lines therefore are drawn parallel to the foot AA / , and at a 
 distance from each other equal to $ the height of each course, 
 these lines will be the projections in plan of the horizontal 
 edges, as shown on the right of the plan. As the projections 
 of the vertical edges are contained in planes of right section, 
 they will be drawn perpendicular to the horizontal ones, 
 and breaking joints with them ; as represented on the same 
 portion of the plan. 
 
 The horizontal edges in elevation will be drawn parallel 
 to A" A'" , and at a distance from each other equal to the 
 height of a course. The vertical edges will be drawn per- 
 pendicular to these last, and corresponding to their projec- 
 tions in plan. 
 
 Remark. If the projections of any horizontal line of 
 the face, at a given height above the foot of the wall, are 
 required, as mn and m"n', for example, it is evident that 
 m"n' will be drawn in elevation at the given height above 
 the foot A" A'", and that mn will be parallel to AA / in plan, 
 and at a distance from it equal to } the height of m"n'. 
 
 JProb. 2. Having given the batir of the faces of two 
 walls that intersect, the foot of each being in the same hori- 
 zontal plane, to draw the projections of the line of intersec- 
 tion of the faces. 
 
 Let Aa, {Fig. 1,) be the foot of one wall in plan, and the 
 batir of its face f ; ah the foot of the other, and the batir of 
 its face \. 
 
 It is evident that the point a will be one point of the 
 intersection in plan. If now a horizontal line be drawn in 
 each face at the same altitude, they will intersect and give a 
 second point of the intersection of the faces. Assuming any 
 altitude for the horizontal line on one face, its projection 
 mn, in plan, will be parallel to Aa, and at } of the assumed 
 altitude from it. In like manner, the projection no of the
 
 6T0XE CUTTING. 25 
 
 corresponding line on the other face having the batir f will 
 be parallel to ah, and at £ the assumed altitude from it. 
 Drawing, therefore, a line through the points a and n, this 
 will be the intersection in plan. 
 
 To obtain the intersection in elevation, draw the foot of 
 the wall A" a' in elevation, also the line m"n' at the assumed 
 altitude. The point a will be projected in a', and the point 
 n in n '; and the line a'n' , drawn through them, will be the 
 intersection in elevation. 
 
 Proh. 3. To construct the projections of the hounding 
 Lines and edges of the joints of a huttress against the inclin- 
 ed face of a given wall ; the hase of the huttress heing given, 
 and heing in the same horizontal plane as the hase of the 
 wall; the faces of the huttress, its end, and the top to have 
 given slopes. 
 
 Along the foot AA t set off the breadth of the buttress 
 ad at its base, and construct the two sides ah, cd; and the 
 end he of the base. Let the batir of the face of the wall 
 foe \ ; that of the two faces and the end of the buttress f ; 
 and that of its top }. 
 
 By Prob. 2 construct the projections, in plan and eleva- 
 tion, of the intersections ae, a'e , and dh, d'h', of the faces 
 of the buttress and wall ; and those hf hf, and eg, c'g', of 
 the end and faces. 
 
 To construct the projections of the top surface of the 
 huttress. 
 
 Suppose that the top line of the buttress eh, e'h', where 
 it joins the wall, is of the same altitude as the top of the 
 wall ; and that the top surface from this line outwards from 
 the wall has the given slope j. Now, if a line as y'zL 
 be drawn parallel to e'h' the top line, and at any assumed 
 distance below it, this line may be regarded as the projection, 
 of a horizontal line in the top surface of the buttress ; and 
 its corresponding projection in plan will be a line yz parallel 
 to eh, and at six times the distance from it that y'z' is below 
 e'h'. Having drawn these two lines of indefinite length, 
 construct the projections of the horizontal in the face of the 
 buttress which is at the same distance below the top. The 
 projection in elevation will be a continuation of the same 
 line y'z', and in plan its projection will be parallel to the 
 foot ah, and at a distance from it equal to \ its altitude 
 *bove it. Drawing an indefinite line xy parallel to ah at 
 this distance from it, the point y, where it intersects the line 
 yz, will be a point of the projection in plan of the intersec- 
 tion of the top surface and face of the buttress. The point 
 e. is another point; joining e and y by a line and prolonging
 
 26 STONE CUTTING. 
 
 it unt.l it intersects the projection of the intersection of the 
 face and end at f, this line ef will be the projection of one 
 of the top surface in plan. The other side hzg will be 
 found by a like construction. The points / and g being 
 joined will be the projection of the end of the top surface. 
 
 To obtain the corresponding lines in elevation; the points 
 ?/ and z are projected into //' and z' \ the points e', y 1 , and h', 
 z\ are joined by lines which are prolonged to meet the lines 
 which are the projections of the exterior edges of the hut- 
 tress at/' and g', which correspond to /and g in plan, and 
 the points /' and g' are joined ; e'f'g'ti is the elevation of 
 the top surface. 
 
 The projections of the edges of the vertical joints in plan 
 will be perpendicular respectively to the lines be and cd, and 
 breaking joints as shown on the right portion of the plan. 
 The projections of these lines in elevation will be found by 
 projecting their extremities into the corresponding projec- 
 tions of the horizontal edges in elevation, as shown on the 
 elevation of the face and a portion of the end, on the right. 
 
 Prob. 3, Case 2 (PL B, Figs. 1, 2, 3). Having given the 
 cross section of a brook, or other smaU natural water wag, 
 over which a full centre archedstone culvert is to be thrown, 
 to support an embankment of a roadway, of a given height 
 above the natural surface of the ground, to construct the 
 bounding lines of a wing wall with plane bounding sur- 
 faces. 
 
 Figs. 1, 2 are the elevation and plan, or the vertical and 
 horizontal projections of the parts ; P Q being the ground 
 line. Fig. 3 is a section and elevation, through the axis of 
 the arch, on the vertical plane of which 11 S is the ground 
 line. M, M' are the slopes of the embankment, jST, N' the 
 bottom of the brook. 
 
 Let C B Z', Fig. l,be the cross section of the side bank 
 of the water way, and of the adjacent level ground; the 
 centre of the semicircle of the full centre arch, taken on the 
 level C B of the natural surface; L I the level of the top of 
 the embankment; 1/ E', Fig. 2, the foot of the embankment 
 
 The wing wall and arch {Figs. 1, 2), are supported upon 
 a general substructure, the height of which is A A'; the 
 plan of that portion of which, supporting the wing wall, is 
 shown by A' B' C. The faces of this substructure being 
 vertical, and projecting a distance, represented by Z A, be- 
 yond the springing line of the arch, the foot of the wing wall, 
 and the foot of the embankment; the point B' being taken 
 on the crest B' K of the side bank. 
 
 Through the line Z D, Z' D', the foot of the wing wall,
 
 STONE CUTTING. 27 
 
 the plane of its face is passed; the top point of which X, X' 
 is found by drawing a line m n parallel to Z' D', and at a 
 distance from it equal to the base of the batir corresponding 
 to the height of the embankment L I above D Z, and taking 
 its intersection X, X' with the top line of the head of the 
 arch. Joining X Z, it will be the elevation of the intersec- 
 tion of the face of the wing wall with the end of the arch. 
 
 The top surface of the wing wall X." T>" (Fig. 8), receives 
 the same slope as the side slope of the embankment M, M', 
 and is here taken to coincide with it. The wing wall is ter- 
 minated at the end by a vertical plane D" F" parallel to the 
 head wall of the arch. The thickness I X, FX' of the wing 
 wall at top is assumed. 
 
 Joining the points X' D' and X D, the interior edge of 
 the top line of the wing wall is found. Drawing I H and 
 I' IF respectively parallel to these, the exterior lines are 
 found. 
 
 The lower end of the wing wall is terminated by what is 
 termed a newel stone, which serves, in this case, as a buttress. 
 The height of this stone D" F" is arbitrary, as is also its 
 slope F" G" on top. Assuming these, the intersection of 
 the vertical plane, terminating the wing wall with its face, 
 will be the lines F' D', F D, parallel respectively to X' Z' 
 and X Z. The lines F' G', F G, and H' E', H E, which also 
 are parallel, will be found by Prob. 3. 
 
 The vertical joints of the face of the wing wall are per- 
 pendicular to its face. Drawing the line X' Y' perpendicular 
 to D' Z', and its corresponding projections X Y, X'' Y" on 
 Figs. 1, 3, the directions of the edges of the vertical joints, 
 as x' y\ x y, x" y", will be parallel, on their respective Figs., 
 to these lines. 
 
 The top of the wing wall, instead of a coping, is formed 
 with elbow joints uniting with the horizontal joints. The 
 portion of the joint forming the elbow is perpendicular to 
 the top surface. Drawing then a line Z" W" (Fig. 3), per- 
 pendicular to D" X", and its corresponding projections Z W, 
 Z' W on Figs. 1, 2, these will be the directions z w, z' w', 
 z" w", of the elbows. The depth of the elbow is arbitrarily 
 assumed, by drawing a line n" q" on Fig. 3 parallel to 
 D" X".
 
 28 STONE CUTTING. 
 
 CYLINDRICAL 
 
 AND OTHER 
 
 A.H O H E S. 
 
 To facilitate the geometrical operations for determining 
 the bounding surfaces and lines of the voussoirs of arches, a 
 few preliminary problems and theorems, on which these 
 operations are based, will first be explained. 
 
 Prob. 4, {PI. A, Fig. 2.) Saving given a semi cylinder, 
 the right section of which is a semicircle, and its axis and 
 two bounding elements being horizontal, to construct the pro- 
 jections of the intersection of the cylinder by a plane inclined 
 to its axis and having a given inclination to the horizontal 
 plane containing the axis / also, the projection of the inter- 
 section of this semi cylinder with another semi cylinder with 
 a semicircle also for its right section, the axis and bounding 
 elements of this last being in the same horizontal plane as 
 those of the first ; and then to develop the portion of the 
 first semi cylinder which lies between the given plane and 
 the other cylinder. 
 
 Let a'c'b' be the right section of the given cylinder, and 
 o' its centre ; the line a'V being horizontal. Let a' A and 
 b'B be the horizontal projections of its bounding elements, 
 and o'G that of its axis. Let ab be the trace of the given 
 inclined plane on the horizontal plane of the bounding ele- 
 ments ; AB one of the bounding elements of the other cylin- 
 der, and LM its axis. The quadrant AL' the half of the 
 right section of this cylinder ; L the centre of this quadrant. 
 ° 1st. Taking any two elements of the given cylinder, at 
 the same height, as x'x^ and y f y t , above a'b', they will be 
 projected in plan parallel to the axis o'C\ and will be drawn
 
 STONE CUTTING. 29 
 
 indefinitely through the points a? 2 and y v The given ineli ned 
 plane will cut these elements at the same height x'x„ and if 
 the projection xy of a horizontal line in this plane, at the 
 height x'x„ be drawn, the points x and y, where it cuts the 
 two elements of the cylinder, will be two points of the re- 
 quired projection in plan. To construct this line xy, let the 
 given inclination of the plane be \ ; the projection of this 
 horizontal line, which is at the height x'x t above the foot ab 
 of the plane, will be {Prob. 2) parallel to ab, and at a distance 
 from it equal to \ of x'x t ; drawing therefore xy parallel to 
 ab, and at this distance, it will be the required projection in 
 plan. The points x and y thus found will be two points of 
 the projection in plan required. In the same way any num- 
 ber of points can be found, and the curve axeyb, traced 
 through them, will be the required projection in plan. 
 
 The construction just explained, although very simple, 
 may be abridged as follows : Through a' draw a'f' perpen- 
 dicular to a'b' ; prolong y'x' to the left, and set off from m'", 
 where it cuts a'f', the distance m'"x'" equal to mx, as before 
 found. Through a'x'" draw the indefinite line a'e. Now, 
 to construct the projection of any other point in plan, as c 
 on the element at the height d ; through d draAV a line par- 
 allel to a'b', take the part o'"d" intercepted between a'f and 
 a'e' and set it off from o, where the projection of the ele- 
 ment through d cuts ab, to c along the projection of the ele- 
 ment; c will be the required point. This is evident from 
 the relations which the heights and horizontal distances con- 
 sidered bear to each other. 
 
 2d. To find the projection in plan of the intersection of 
 the cylinders. Draw AD perpendicular to AL. If a dis- 
 tance Ar" equal x'x 3 is set off on this line, and a parallel to 
 AL be drawn through r", the point u" where it cutis the 
 quadrant will give the point on it through which the element 
 of the second cylinder, at the height x'x i of the two elements 
 at x' and y', is drawn. Through u" drawing an indefinite line 
 parallel to AB, the bounding element of the second cylin- 
 der, it will be the projection in plan of the element at the 
 height Ar"=x'x i ; and the points w and v, where it cuts the 
 two projections of the elements of the first cylinder at the 
 same height, will be two points of the required projection. 
 In the same way other points would be found, by construct- 
 ing the projections in plan of corresponding elements on 
 the two cylinders. 
 
 This operation, like the former, may be also abridged aa 
 follows : Through V draw a perpendicular b'd to a'b'. With 
 a radius equal to AL describe a quadrant tangent to I'd' at
 
 30 stoxe crrrrs-G. 
 
 b'. Now, if a'y' be prolonged to the right, it is evident that 
 the distance r'"u'", intercepted between bd and the quad- 
 rant, is equal to r"vl\ or to rw. In like manner, the point 
 G in plan is obtained by setting off from t, on the element 
 -4_Z?and along oC, the distance W=HC. The curve AmOoB^ 
 drawn through the points thus determined, will be the pro- 
 jection in plan of the intersection of the two cylinders. 
 
 3d. To make the development of the portion of the cyl- 
 inder which lies between the two intersections thus deter- 
 mined, it will be necessary to obtain the distances of the 
 points of these two intersections from a curve of right sec- 
 tion ; since the tangent to this curve at any point being per- 
 pendicular to the element of the cylinder at the same point, 
 the curve when developed will also be perpendicular to the 
 elements when developed, and will therefore develop into a 
 right line. 
 
 To determine the relative positions of these curves in 
 development ; first develop the curve of right section a'c'b', 
 on the line a'b' prolonged to the right, by setting off the dis- 
 tances b'y", b'c", &c, to a", equal respectively to the lengths 
 of the arcs b'y', b'c', &c, to a'. Through the points y", c" , 
 dec, draw lines perpendicular to b'a" ; these will be the de- 
 veloped elements of the cylinder through y', c, &c Set 
 off along these lines the distances y"v / , c"O t , <kc, respectively 
 equal to the distances y„v, o'C, &c, and through the points 
 jd, v„ C„ w x , A x , draw a curve. This is the developed inter- 
 section of the two cylinders. Make the same constructions, 
 on the same developed elements, with respect to the distances 
 of the points y, c, x, a, from a'b' ; the curve &?,«„ drawn 
 through these points, will be the developed intersection of 
 the oblique inclined plane and the cylinder. 
 
 In like manner, if the given cylinder were cut by a plane 
 perpendicular to the horizontal plane and oblique to its axis, 
 of which ab, for example, is the trace, the developed curve 
 of its intersection would be obtained by setting off along the 
 developed elements the distances of the points a, m, o, cVrc, 
 from a'b', and through the points thus determined drawing 
 a curve bo x a im 
 
 Rema/rk. The curve of right section in development 
 serves only as a fixed line from which the relative positions 
 of other points, with respect to it and to each other, can be 
 determined; since it develops into a right line, and the 
 elements in development are perpendicular to it. The posi- 
 tion of this curve may be therefore fixed arbitrarily, as maj 
 be found most convenient for the purposes of the drawing.
 
 STONE CUTTING. 31 
 
 Prob. 5, (PI A, Fig. 4). Let the semicircle A'C'B' be 
 tLe curve of intersection of a given semi cylinder by a ver- 
 tical plane, the diameter AB 1 being horizontal, and suppose 
 the bounding elements through A' and B' to be limited at 
 a horizontal plane at the distance A A below A'B', and to 
 pierce this plane at the points E and D, on a line parallel 
 to AB and at a given distance from it ; these elements being 
 oblique both to the vertical plane and to the horizontal plane, 
 and therefore not projected on either into their true lengths, 
 
 Let DM be the axis, and ED a bounding element of 
 another semi cylinder, the right section of -which is a semi- 
 circle; L2£ and ED being also in the given horizontal 
 plane. 
 
 It is proposed, with these data, to find the lengths of the 
 elements of the obit 'que 'cylinder intercepted between the ver- 
 tical plane of ABC' and the horizontal cylinder; the curve 
 of right section of the oblique semi cylinder at any assumed, 
 point; and the development of the portion of it which lies 
 between- the given vertical plane and the horizontal semi cyl- 
 inder. 
 
 1st. The simplest method of finding the true lengths of 
 the elements between the vertical plane and horizontal cyl- 
 inder will be to construct their projections on another vertical 
 plane parallel to them. Let BN be the trace of such a plane 
 on the horizontal plane containing the axis LM of the hori- 
 zontal semi cylinder, and Be' its trace on the given vertical 
 plane. This assumed plane cuts from the horizontal semi 
 cylinder an ellipse of which DN is evidently the semi trans- 
 verse axis, and the radius of the semi cylinder, which is 
 equal to the distance between the axis and the bounding 
 element ED, is the semi conjugate ; setting off this distance 
 from N to N\ on a perpendicular to DN, and describing 
 the quadrant of an ellipse DN' , on these lines as semi axes, 
 it will be the half of the curve cut from the semi cylinder, 
 and will be its position when the plane in which it lies is 
 revolved around its trace BN to coincide with the given 
 horizontal plane. In this revolved position of the plane, the 
 line Be', in which it cuts the given vertical plane, will be 
 found in BO" perpendicular to BN. As this plane contains 
 the bounding element of the oblique semi cylinder projected 
 in BD, this element will be found in DB ' when revolved, 
 the height BB" being equal to BB', the height above AB 
 in which the element pierces the given vertical plane. 
 
 jSTow, if any other vertical plane be passed parallel to 
 the one assumed, as that of which OG / , and CC', parallel 
 respectively to BN and Be' are the traces, it will cut from
 
 32 STOXE CUTTING. 
 
 the horizontal semi sylinder an ellipse, equal to the one 
 already described; and from the given vertical plane the 
 line CC ; and. as it contains also the element projected in 
 CC\, by revolving this plane also, like the last, tracing the 
 quadrant of the ellipse cut out, the line CC\ and the element 
 in their revolved positions; the portion of this element be- 
 tween the vertical plane and the horizontal cylinder may 
 thus be determined. In like manner, the corresponding 
 lengths of any other elements, as those which are projected 
 in mm / , and nn / , might be found. But as these successive 
 operations would be long, a more simple and expeditious 
 method is resorted to, as follows: As the elements of the 
 horizontal semi cylinder are parallel to the given vertical 
 plane, if all the points on this plane and cylinder are pro- 
 jected on the assumed vertical plane of which BS and Be' 
 are the traces, by a system of lines oblique to this plane and 
 parallel to the elements of the horizontal cylinder, it is evi- 
 dent that all the ellipses cut from the horizontal cylinder 
 will be projected into the one cut from it by the assumed 
 vertical plane; that all the lines, as CC, mm'. . ■•-■'. &c, cut 
 from the given vertical plane, will be projected in Be' ; and, 
 in the revolved position of the assumed vertical plane, will 
 be found in BC" ; whilst the portions of elements of the 
 oblique semi cylinder, which lie between the horizontal semi 
 cylinder and the given vertical plane, will be projected on 
 the assumed vertical plane in their true lengths, and, in its 
 revolved position, will be found parallel to B'D, and drawn 
 through points n", C", &c, at the same height above B, on 
 the line BC", as the corresponding points ml, < , n', are 
 above AB. Drawing these parallels, the portions n'"m , 
 CO'", &c, between the line BC" and the curve 1)X', will 
 be the lengths required. 
 
 As there are two elements on the oblique semi cylinder, 
 one on each side of the highest one, projected in CC, , as 
 those projected in mm, , and nn t , which are of tlte same alti- 
 tude, the lines B"D, n'"m", Arc, will be respectively the 
 revolved positions of the projections of the corresponding 
 pairs of these elements. 
 
 Rema/rk. By using the system of oblique projecting 
 . instead of the usual mode of perpendicular ones, 
 the relative positions of the lines projected are not changed, 
 since these lines, being all parallel to the assumed vertical 
 plane, will be projected on it in their true lengths, whether 
 the projecting lines be oblique, or perpendicular to this 
 plane. By the system of perpendicular projections, a sepa- 
 rate construction, like the first, would have been requisite to
 
 STONE CUTTIXG. 33 
 
 determine each element ; whereas by the oblique one used 
 the construction of one ellipse DN n and of the line BC" are 
 alone sufficient. 
 
 2d. The curve of right section must lie in some plane 
 perpendicular to the elements of the oblique cylinder. To 
 tix such a plane, draw a line XY perpendicular to the pro 
 jections of the elements on the horizontal plane ; and from 
 Y, where it cuts BD, another line YZ perpendicular to the 
 projections of the elements on the assumed vertical plane. 
 These two lines may be taken as the traces of a plane on 
 these two planes ; and, as these traces are perpendicular to 
 the projections of the elements on the two planes, the plane 
 itself will be perpendicular to the elements, and will cut 
 from the cylinder a right section. 
 
 To construct this curve of right section, it will be neces- 
 sary to find, in the first place, on the assumed vertical plane, 
 the projections of the points in which the elements of the 
 oblique semi cylinder pierce the plane of right section. To 
 do this, it is evident that the vertical planes which contain 
 these elements, as the one, for example, of which CC\ and 
 CO' are the traces, cut the plane of right section in lines 
 parallel to YZ, its trace on the assumed vertical plane. To 
 find the projection on this plane of the line cut out by the 
 vertical plane of which CC[ and CO' are the traces, it is 
 plain that the point z, where the horizontal traces XY and 
 CC l intersect, will be one point of the required hue. This 
 point, being in the assumed horizontal plane, will be pro- 
 jected into the line BX, the ground line of the assumed 
 vertical plane, at z', by drawing a line through z parallel to 
 AB, according to the method of oblique projections adopted. 
 If from z' a line be drawn parallel to YZ, this line z'z" will 
 be the required projection of the line cut from the plane of 
 right section by the vertical plane which contains the ele- 
 ment projected in GC. The point z", where this line cuts 
 C"C", the projection of this element on the assumed verti- 
 cal plane, will be the projection on this plane of one point 
 of the curve of right section. In like manner, projecting 
 the points X, x, y, &c, into the ground line BX at X, x\ 
 y , &c, and, from these last points, drawing parallels to YZ, 
 the points X", x", y", &c, in which they cut the correspond- 
 ing elements in projection, will give other points ; and the 
 curve X'x"z"y" Y will be the projection of the curve of 
 right section required. 
 
 Remark. Since the elements are projected on the as- 
 sumed vertical plane into their true lengths, it is evident 
 that taking any point of this projection of the curve of
 
 34 . 8T0XE CUTTING* 
 
 right section, as x" for example, the distance x"m" will be 
 the true distance of the point of which x" is the projection 
 from the horizontal semi cylinder, as measured on the ele- 
 ment projected in n/"m," j and x"n'" will be its true distance 
 from the vertical plane containing the semicircle A ' C' ' B '. 
 In like manner, the true distances of other points of the 
 curve of right section from this plane and from the horizon- 
 tal semi cylinder measured along the elements may be found 
 from the projection of this curve. 
 
 Having found the projection of the curve of right sec- 
 tion, the curve itself can be found by revolving the plane 
 of right section upon the horizontal plane around its trace 
 XY. The distance zz^ of the point projected in z" from 
 XY \$> evidently equal to z'z", since this line is the pro- 
 jection, in its true length, of the one in the plane of right 
 section drawn through the point z in the vertical plane 
 containing the element projected in CG\ In like manner, 
 the distances XX ^ xx„ yy n , being set off from XY, along 
 the perpendiculars to it through these points, and equal 
 respectively to the distances XX\ x'x", &c, the curve 
 X^z^y^ Y l will be the required one. The line X t Y l which 
 corresponds to X" Y in projection will be the diameter of 
 this curve. 
 
 3d. Having the curve of right section in its true length, 
 as well as the elements of the oblique cylinder, and the 
 points where they cut this curve, it will be easy to make 
 the required developments which, for convenience, will be 
 done on the assumed vertical plane in its revolved position. 
 To do this set off on the line JTZ, from the point Y, the 
 distances Y'z 3 , Yx s , YX^ respectively equal to the arcs 
 Y x y % z„ Y&, &c, of the curve of right section. The right 
 line YX t will be the development of this curve. Through 
 the points z 3 , x 3 , &c, thus set off, draw perpendiculars to 
 YX 2 , these will be the indefinite lengths of the developed 
 elements drawn through the points z 3 , x 3 , &c. From these 
 last points set off z 3 C 3 , z 3 C i respectively equal to z"C" and 
 z"C" j in like manner, set off the distances x 3 n 3 and x 3 m % 
 respectively equal to x"n" and x"m'", &c. The curves 
 B"C 3 n 3 A n and DC 4 m 3 E /} will be the developments respec- 
 tively of the semicircle A!G'B\ in which the oblique semi 
 cylinder cuts the given vertical plane through AB, and of 
 the curve in which it intersects the horizontal semi cylinder. 
 The developed portion of the oblique semi cylinder which 
 lies between these curves and the developed positions B"I) 
 and A X E X of its bounding elements will be the one required,
 
 STONE CUTTING. 35 
 
 4th. To obtain the horizontal projection of the curve in 
 which the semi cylinders intersect. Project the points C" , 
 m", &c, into I?.zv, by perpendiculars ; from the foot, as C„ 
 of each perpendicular draw an oblique projecting line 
 parallel to AB; the points, as C[, n„ ra„ &c, in which these 
 intersect the projections of the corresponding elements, will 
 be points in the required projection ; and the curve Dn x C;m x E 
 will be the one required. 
 
 Theorem 1, {PI. A, Fig, 3.) If the lines AB and DE, 
 which bisect each other and are horizontal, are the transverse 
 axes of two semi ellipses, t/ie planes of the two curves being 
 vertical, and having the same semi conjugate axis, projected in 
 the point C, then will the lines which join the points of the 
 curves at the same height above the horizontal plane of the 
 transverse axes be parallel to each other. 
 
 Let A'C'B' be the projection of the semi ellipse, having 
 AB for its transverse axis, on a plane parallel to itself; and 
 let the other curve be revolved about the common semi 
 conjugate into the plane of the first and be projected into 
 D'U'E'. Drawing any line, as m'n', parallel to AB' , it 
 will cut the two curves at the points ml, n', and o',p', at 
 the same height above the horizontal plane ; the first two 
 being projected horizontally in m and n ; the second in o, 
 and p x in the revolved position of the second curve, and in 
 o and p in its original position. Joining the points o and 
 m, also p and n, then will these lines be parallel to each 
 other, and to the lines AD and EB which join the lowest 
 points of the two curves. For, from the properties of two 
 ellipses having a common conjugate axis, the corresponding 
 ordinates of the curves to this axis will be proportional to 
 their semi transverse axes ; that is, 
 
 en 1 :cp' :: C"B' : C"E'-, 
 or, On: Up :: CB : CE; 
 by substituting the equal lines in horizontal projection. But 
 when this last proportion obtains, the lines pn and EB are 
 parallel. In like manner, mo may be shown to be parallel 
 to AB, and consequently to EB. The same holds true for 
 the lines mp and on, with respect to AE and DB. • 
 
 It follows from this that two semi ellipses, having the 
 above conditions, will be the curves of intersection of two 
 genii cylinders, the axes of which lie in the horizontal plane 
 of the transverse axes of the curves, and are parallel respec- 
 tively to the lines joining the extremities of these axes. 
 The converse of this proposition is also evidently true, viz. : 
 if the axes of two elliptical or circular semi cylinders lie in 
 the same horizontal plane and intersect, and the highest elo*
 
 36 STONE CUTTING. 
 
 ment of each is at the same height abo-\ t this plane, then 
 will their curves of intersection be plane curves, and be pro- 
 jected on the horizontal plane in the two right lines which 
 join the- opposite points of intersection of the lowest ele- 
 ments. 
 
 Theor. 2, {PL A. Fig. 5.) Let A'CB' be the vertical 
 projection of a send ellipse situated in a vertical plane, of 
 whtch AD, parallel to A7B', is the horizontal trace} and let 
 tin point O, on theperp< ndicular OC to A15 which bisects it, 
 he the horizontal projection of a vertical Vine through O, and 
 let the semi ellipse and this vertical be taken as the directrices 
 of a surface, generated by moving a line parallel to the hor- 
 izontal plane, and m each of its successi/vt j i sitions touching 
 the vertical at O and the semi ellipse y then will any section 
 of this surface by a plane, as ad, parallel to the plane of 
 the ellipse-, be also an ellipse of wh/ich the line ah intercepted 
 between OA and OB, will be one axis, and the line OC equal 
 to the other semi axis. And a tangent line drawn to this 
 ellipse at any point, as the one projected in n, n', urill pierce 
 the horizontal plane in a line OD, draicnfrom the point O 
 to thepoint J), where a tangent to the directing ellipse at the 
 point projected in m, m', at the same height as n', also pierces 
 the horizontal plane. 
 
 1st. Through the point projected in n, n' , draw the pro- 
 jections On i. and o'm' of an element of the surface, and 
 project the line ab into a'b '. As the lines BO and bo are 
 parallel, there obtains 
 
 730 : bo :: mO : no; 
 
 hut BO=B'0'; bo=b'0',tmdmO=m'o'; no=n'o'; 
 therefore, BO' : I/O' :: m'o' : n'o', 
 which shows, from the properties of ellipses having a com- 
 mon axis, that the curve b'n'C is an ellipse. 
 
 2d. Let tangents be drawn to the two ellipses at the cor- 
 responding points m' and n', at the same height above A'B'. 
 These tangents will intersect the common axis at the same 
 point E' , and the other axis at points I)' and d' such, that 
 
 01/ : O'd'v.m'o : n'o'. 
 Projecting the points I)' and d' into the respective planes 
 of the two ellipses at D and d, and observing that OD' = 
 OB} Od'=od/ m'o' =m ( ' ; an.dn'o'=noj there obtains 
 
 OB : Cm :: cd : <n; 
 that is, the line joining the points I> and d passes through 
 the point 0. 
 
 Remark. This surface is a right conoid of which the 
 horizontal plane is the plane director.
 
 BIOXE CUTTING. 37 
 
 Prok. 0. To construct a tangent plane and a normal, 
 line to the conoid at any point, as n, n'. 
 
 Draw the projections of the element Om, o'm! of the 
 surface at the given point. Find where the tangent to the 
 directing ellipse at the point m, m pierces the horizontal 
 plane. Join this point JJ with 0. Through n draw a par- 
 allel to CD, and where it cuts OP, at d, draw a line XJ? 
 parallel to Om. This is the horizontal trace of the tangent 
 plane at n, n' . For the tangent line at the given point to 
 the ellipse projected in ah pierces the horizontal plane at d, 
 this is therefore one point of the horizontal trace required, 
 and as the element of the surface is contained in the tangent 
 plane and is horizontal the trace XYwill be parallel to Om.. 
 
 The line zv, drawn through n perpendicular to XY, is 
 the indefinite projection of the normal to the surface at 
 n, n'. 
 
 Prob. 1, {PI. A, Fig. 6.) To draw a tangent plane and 
 a normal line to a helicoidal surface at a given point on the 
 surface. 
 
 Suppose XYZ, the involute of any given curve xyz, to 
 be the base of a vertical cylinder; and let the line Zy' , tan- 
 gent to this curve at Z, be the horizontal trace of a plane 
 tangent to the cylinder along the element projected in Z; 
 and in this tangent plane, revolved on the horizontal plane, 
 let any inclined line Zy" be drawn through the point Z. If 
 the tangent plane be now returned to its vertical position, 
 and be wrapped around the vertical cylinder, the inclined 
 line Zy" will form a helix on the cylinder, and the points 
 y", m", &c, in their position on the cylinder, will be pro- 
 jected into its base, at the points Y, m, <fec. ; such that the 
 arcs ZY, Z/n, will be equal to the distances Zy', Zm' , &c, 
 from Z to the projection of these points in the tangent 
 plane into its horizontal trace. If a right line be moved 
 along this helix so that in all its positions it shall be parallel 
 to the horizontal plane, and be projected on this plane in 
 lines as Zz t , Yy, normal to the curve XYZ, this line will 
 generate a helicoidal surface, the elements of which will be 
 normal to the cylinder of which XYZ is the base, and tan- 
 gent to the one of which pyz 1 is the base. 
 
 Let the point projected in Y, and which is at the height 
 y'y" above the horizontal plane, be the one at which it is 
 required to construct a tangent plane, and a normal line to 
 the surface. As the helix makes a constant angle, equal to 
 the one Zy"y'', with the elements of the cylinder, at the 
 points where they intersect ; and as a tangent to the helix 
 at any point n akes the same angle as the helix dues at this
 
 38 9T0XE CUTTING. 
 
 point with the element of the cylinder ; it ifl evident that 
 the tangent to the helix, at the point projected in Y, will 
 pierce the horizontal plane, at a point z, on the tangent 
 drawn to the curve at Y, at the same distance from this 
 point as the distance Zy' — ZY. This point s will therefore 
 be one point of the horizontal trace of the required tangent 
 plane. But, as the tangent plane contains the horizontal 
 element of the surface at the given point, its trace on the 
 horizontal plane will he parallel to this element. Drawing 
 a line AB through z parallel to Yy, this will he the trace 
 of the required plane. The normal will evidently he pro- 
 jected in Yz, as this line is perpendicular to AB the requir- 
 ed trace. 
 
 To find the time length of this normal let CD, parallel 
 to Yz, he the trace of a vertical plane. The element through 
 the given point will pierce this plane at a height YY above 
 CD equal to y'y ". The normal line will be projected on 
 this plane in its true length, and will pass through the point 
 Y. But, as the vertical plane through CD is also perpen- 
 dicular to the tangent plane at the given point, the line 
 Yz' will be the projection as well as the trace of the tangent 
 plane on this vertical plane. The line YS drawn perpen- 
 dicular to Yz' will therefore be the indefinite projection of 
 the required normal. 
 
 Remark. The traces of all tangent planes to the surface, 
 at points on the element considered, will evidently be par- 
 allel to Yy, the projection of this element; and the trace 
 of any one may be readily found by means of the point z. 
 .For, through y, where the projection of the element is tan- 
 gent to the curve ocyz, through the point z draw an indefinite 
 line yy^j then if at any point on the projection of the ele- 
 ment lya line be drawn parallel to Yz, the projection of 
 the tangent at Y, the point where this parallel cuts yy l will 
 be a point in the horizontal trace of the tangent plane at 
 the assumed point. This may be readily proved as follows: 
 Let the tangent projected in Yz, and the element of the 
 cylinder projected in y be taken as the directrices of a warp- 
 ed surface of which the horizontal plane is the plane director 
 This warped surface will be tangent to the helicoidal surface 
 throughout the entire element projected in Yy; for they 
 have the same plane director; a common tangent plane at 
 the point projected in Yj and also one, which is vertical, 
 at the point projected in y. Now, as the warped surlace in 
 question is a hyperbolic paraboloid, any vertical plane par- 
 allel to Yz will intersect it in a right line which will cut the 
 element projected in Yy, and pierce the horizontal plane ir
 
 STONE CUTTING. 39 
 
 the line yy, wliere the trace of the assumed vertical plane 
 cuts it. This point will therefore be a point in the trace ol 
 a tangent plane to both surfaces at the point where the line 
 cut from the hyperbolic paraboloid intersects the element 
 common to the two surfaces, and along which they aro 
 tangent. 
 
 Prob. 7, PI. 1. To construct the projections and true 
 dimensions of the bounding lines and surfaces of the vous 
 soirs of a horizontal full centre arch. 
 
 This problem comprises several cases, according to the 
 character and the positions of the surfaces which form the 
 ends or heads of the arch. 
 
 Case 1st. The arch being terminated at one end by a ver- 
 tical plane oblique to its axis and at the other by a vertical 
 plane perpendicular to the axis. 
 
 Let the semicircle {Fig. 1) of which B ' C is the di- 
 ameter, be the right section of the soffit of the arch, and 
 having divided t it into any odd number of equal parts, as 
 five for example, draw through the points of division F\ 
 F\ &c, radii which prolong to the semicircle described on 
 A! D as a diameter. These radii will be directions of the 
 joints in the right section, A' ' B' being their common thick- 
 ness. Through the points A', I\ G\ &c, drawing vertical 
 and horizontal lines, they will be the exterior bounding lines 
 of the voussoirs in right section ; the line K'K" which 
 bounds the top of the keystone being assumed at pleasure. 
 
 Let AD {Fig. 2) be the trace on the horizontal plane 
 containing the axis and lowest elements of the arch of the 
 vertical plane oblique to the axis which forms the front end 
 of the arch, and ad the trace of the one perpendicular to 
 the axis that bounds the back end. 
 
 As the edges of any voussoir, as the one of which Q'M' 
 N'O'P' {Fig. 1) is the right section, are all parallel and 
 horizontal, they will be projected into their true lengths in 
 plan between the two traces of the end planes. The one 
 corresponding to M' will be projected in plan {Fig. 2) in 
 Mpn l ; that corresponding to N' in N 1 n l , &c. 
 
 As all the joints, except the two lowest, are oblique to 
 the horizontal plane, their horizontal edges alone are pro- 
 jected into their true dimensions in plan. The two lowest, 
 corresponding to A'B' and CD, are projected into AabB, 
 and CcdD respectively. To find the true dimensions of the 
 others, and the development of the soffit, develop in the first 
 place the right section of the soffit, by setting oft' the dis- 
 tances B'F", E'F\ &c, {Fig. 5) respectively equal to 
 the arcs B ' E' \ &c {Fig. 1) ; and then, by Fig. 2, PL A t
 
 40 8TONE CUTTING. 
 
 find the points B, E t , M n &&, of the developed curve 
 in which the sollit is intersected by the oblique plane of 
 front end; also the right line a, e, , m t , &c, which is 
 the development of the right section in which it is cut bv 
 the plane of the back. The surface bounded by the devel- 
 oped curves and the two lowest elements of the semi cylin 
 der is the developed soffit. 
 
 To find any oblique joint in its true dimensions, as the 
 one projected in plan between the lines Mjm, and N ,n, , set 
 off from M', (1'i'j. 5) to the right along ML" a distance 
 equal to M 'N' {Fig. 1), the breadth of the joint. From 
 the point N' {Fig. 5) draw a line perpendicular to JB < . 
 and from N' set off N'N t equal to n'N J {.Fig. 2). Join the 
 points M,N I ; the trapezoidal figure 3f / ]Y / n / m / is the requir- 
 ed joint. In like manner the true dimensions of all the 
 other joints are found as shown on Fig. 5. 
 
 Case 2d. The arch being terminated at one end by a plane 
 oblique to its axis and to the horizontal plana of its two low- 
 est elements and at the other by a vertical plane perpendicu- 
 lar to the axis. 
 
 Let AD {Fig. 2) be the horizontal trace of the oblique 
 plane, and let the angle which it makes with the horizontal 
 plane be f . Drawing any line vw parallel to AD {Prob. 1) 
 as the projection in plan of a horizontal line of this plane, 
 and from any point as A a perpendicular Ax to vw, the 
 height of the point x above the horizontal plane will be 
 three times the distance Ax. Setting off from A' {Fig. 3) 
 the length Ax to x', at a?' erecting the perpendicular x'x"= 
 ZA'x' and joining A', x", the angle x'A'x" will be the angle 
 between the oblique and horizontal planes. Taking now 
 the distance Ay {Fig. 2) in which the line vw cuts the line 
 Aa, parallel to the axis, and setting it off from A' to y' 
 {Fig. 3), erecting a perpendicular y'y"=x'x", and joining 
 A'y", the angle yA'y" will be the one between the oblique 
 and horizontal ] Janes measured in the vertical plane parallel 
 to the axis of which Aa {Fig. 2) is the trace. 
 
 Having drawn the line A'y", and the vertical at A', 
 the projections in plan of the points in which the oblique 
 terminating plane cuts the bounding horizontal lines of the 
 voussoirs corresponding to the points F', 1 ", F', &c. {Fig. 1) 
 in right section are readily determined by applying the con- 
 structions in Prob. 4. The points in plan corresponding to 
 E' and /' {Fig. 1), for example, Mill be found by drawing 
 lines through E' , I parallel to A'y', taking the lengths 
 E"E"\ A!' A'" and -dting them ofl from F) and /, [Fig. 
 2) from the tra :e A]> to F and /, along the projections in
 
 STONE CUTTING. 41 
 
 plan of the corresponding edges of the vcussoirs, The 
 curve BEF, &c, thus obtained, and the pentagonal figures 
 EllIGF, &c.j are the projections in plan of the olltique 
 sections of the soffit and voussoirs of the arch. 
 
 Remark. As a verification of the accuracy of the con- 
 structions, the lines IE, GF, &c, prolonged should pass 
 through the point L where the axis cuts the trace AD; and 
 the lines A t l, JIG, &c, should be parallel to AD, as the 
 top surfaces of the voussoirs are horizontal planes. 
 
 Case 3d. The arch being terminated by an oblique plane, 
 as in either of the preceding cases, and at the other extremity 
 by a semi circular cylinder, its axis and two bounding ele- 
 ments being in the same plane as the corresponding lines of 
 the arch and perpendicular to them. 
 
 Let ad {Fig. 2) be one bounding element of the given 
 semi cylinder. Having set off the radius of this cylinder 
 from D, {Fig. 4) on the line A'D prolonged, and described 
 a portion of the semicircle tangent to the vertical DN" , 
 draw through the points 31, W, &c, {Fig. 1) parallel lines 
 to A'D. By Prob. 4 find the projection in plan of bef, &c, 
 of the curve of intersection of the soffit of the arch and the 
 semi cylinder; also the projections of the points i, h, g, &c, 
 in which the horizontal edges of the voussoirs intersect the 
 semi cylinder; the pentagonal figures efghi, &c, will be the 
 projections in plan of the lines in which the surfaces of the 
 voussoirs intersect the semi cylinder. Any point in plan, as 
 n, is found by setting off a length n t n from ad, along the 
 projection of the edge corresponding to N' , equal to the dis- 
 tance N"N'" {Fig. 4). 
 
 Remark. The lines ei,fg, &c, are portions of ellipses, 
 which prolonged pass through the point /, in which the axis 
 of the arch cuts the bounding element ad. The lines gh, 
 no, &c, are right lines, being the projections of the inter- 
 section of the horizontal surfaces of the voussoirs with the 
 semi cylinder. The lines hi, op, &c, are the projections of 
 arcs of circles in which the side vertical planes of the vous- 
 soirs corresponding to H'l', OP', intersect the semi cyl- 
 inder. 
 
 The true dimensions of the joints in either of the two 
 last cases are found by setting off' from the line B'C {Fig. 
 5) the lengths along the perpendiculars, at the points E', I, 
 &c, which correspond to the distances respectively of the 
 points E, /and e, i, {Case Zd, Fig. 2) from A'D. 
 
 Remark. As a verification of and aid to accuracy of 
 construction, let a line LI {Fig. 5) be drawn parallel to the 
 edge M 'm and at a distance fi ^m it equal to I'M' {Fig. 1)
 
 42 STOXE CUTTING. 
 
 the radius of the rig-lit section; the right lines MJV and 
 M X t prolonged should intersect the line LI at the same 
 point L, such that the length LI {Fig. 5) shall be the same 
 as IX {Fig. 2). In like manner the curve .m/ m prolonged 
 should intersect the same line LI at the point I which cor- 
 responds to the one I {Fig. 2). Similar constructions of 
 verification will he found on Fig. 5 {PU. 3 and 4). 
 
 From an examination of Fig. 2 it will he seen, that the 
 projection in plan of the voussoir corresponding to M'JV'O', 
 &G., {Fig. 1) in right section, will in Case 3d be hounded on 
 one end by the figure MJTO, &c., on the other by the one 
 mno. &e. ; and by the parallel hues which join the corres- 
 ponding points Mm, ISn, 6zc. 
 
 Application. Having found the principal dimensions of 
 the voussoirs, let it be required to cut from a single block of 
 stone of the form of a rectangular parallelopipedon the 
 voussoir corresponding to M'N'X)\ &c {Fig. 1) in Case 3d. 
 Drawing a line S'T through Q' perpendicular to 0'P\ and 
 prolonging O'N' to R' on the line drawn through M' par- 
 allel to 0'P\ the rectangle R'T will evidently be the di- 
 mensions of the end of the block within which the voussoir 
 in right section can be inscribed. The dimensions of the 
 length of the block will evidently be determined by drawing 
 through the point Q {Fig. 2) a line B ' Q parallel to ro. 
 Having inscribed on the end of a block of the form and 
 dimensions thus found, the figure in right section, the block 
 would be prepared by cutting away those portions, as 
 M.S' Q\ &c, which are exterior to the figure. This being 
 done, the points corresponding to M, iT, 0, &c, and m, n, 
 o, &c, {Fig. 2) can be set off on the corresponding edges, 
 and the two ends of the voussoir, the one terminated by 
 the oblique plane, the other by the semi cylinder, be obtain- 
 ed. In cutting away the portions of the block to form the 
 curved surfaces of the soffit and of the end of the voussoir, 
 a model cut from a thin board, by shaping it en the lack to 
 the form of the arc M'Q' {Fig. 1), and a like model cut to 
 the form of the arc DN"\ would be requisite as a guide to 
 the workman, to be applied, from time to time, in a direc- 
 tion perpendicular to the elements of the cylinders, until it 
 is found that the models coincide accurately at all points 
 with the prepared surfaces. 
 
 Models also of the true forms of the joints, determined 
 in Fig. 5, may be cut from thin pasteboard, or any like ma- 
 terial, and be used to verify the work. These last would 
 evidently not be requisite to guide the workman in setting 
 off his points where he works from a block of the above
 
 STONE CUTTING. 43 
 
 form. But, in cases where a block of irregular shape has 
 to be taken, they may be found the most convenient for set- 
 ting off the points to determine the form of these joints on 
 the stone. 
 
 Prob. 8, PI. 2. To construct the projections and true 
 dimensions of the voussoirs in the groined and cloistered 
 arches. 
 
 In each of these cases the soffits of the arches are formed 
 by the intersections of two semi cylinders, the axes of which 
 are in the same horizontal plane, and their top elements at 
 the same height above this plane. From these conditions, 
 the curves of intersection of the soffits (Theor. 1) will be 
 plane curves, and will be projected in plan in the diagonal 
 lines which join the intersections of the lowest elements of 
 the semi cylinders. 
 
 In the cases selected to illustrate this problem (PI. 2), 
 the curve of right section of one of the semi cylinders is a 
 semicircle {Fig. 1), that of the other (Fig. 2) a semi ellipse, 
 each having the same rise L K '• and their axes are taken 
 perpendicular to each other. The joints in each arch cor- 
 responding to F'G', &c, are normal to the soffit, or surfaces 
 of their respective cylinders ; the upper and lower edge3 of 
 the corresponding joints in each arch being in the same 
 horizontal plane, as well as the top surfaces, as G'H' (Figs. 
 1, 2), of the voussoirs. 
 
 Remark. Fig. 1 is the right section of the semi circular 
 arch ; Fig. 2 that of the semi elliptical arch ; Fig. 3 above 
 the line ah is a portion of the plan of the groined arch, the 
 soffit of the serai circular arch being projected within the 
 angle BKG and the corresponding soffit of the semi ellipti- 
 cal arch, only the half of which is shown in plan, being pro- 
 jected within the angles aKB and bKC. Fig. 5 on the 
 right represents two of the joints of a groin stone belonging 
 to the semi circular arch with the development of the por- 
 tion of the soffit between them ; that on the left the true 
 dimensions of the corresponding parts of the same stone 
 which forms a part of the other arch. 
 
 Fig. 4 below the line ab is a portion of the plan of the 
 cloistered arch, the soffit of the semi elliptical portion being 
 projected within the angles B I KG I : that of the semi circu- 
 lar portion being projected within the angles B 2 KC / and 
 B / F0 2 . Fig. 6 on the right represents the two joints of a 
 groin stone which forms a part of the semi circular arch 
 with the portion of the soffit between them ; that on the left 
 the corresponding parts of the same stone of the other arch,
 
 44: STONE CUTTING. 
 
 Groined Arch. Having constructed (Fig. 1) the right 
 section of the semi circular arch as in Prob. 7, assume Ji ( " 
 {Fig. 2) as the transverse axis of tlie ellipse of right section 
 of the other arch, and placing it at any convenient position, 
 on the left, perpendicular to the direction B'( " (Fig. 1), set 
 off the semi conjugate L'K' equal to the radius of the semi- 
 circle, and describe the semi ellipse. Find on the semi 
 ellipse, as shown by the lines A' /> \ &c, on the left of Fig. 
 1, the points J.\ /•', &c. 5 at the same height above B'C, as 
 the corresponding points of the semicircle. ( ionstruct tan- 
 gents to the semi ellipse at these points, and at the same 
 normals for the directions of the joints. Find on these 
 normals the points T, G', etc., at the same heights as the 
 like points in Fig. 1. Through the points /', 6r, (Fig. 2) 
 draw the vertical and horizontal lines I'll' and G'H', for 
 the bounding lines of the voussoirs. Having completed, in 
 this way, the right section (Fig. 2), draw, in plan (Fig. 3), 
 the projection of the axes, and the bottom elements of the 
 arches, corresponding to the points B\ C (Figs. 1, 2). 
 Drawing the diagonal lines BB, and CC\ between the points 
 where these elements intersect in plan, they will be (Theor. 
 1) the projections of the ellipses in which the two semi cyl- 
 inders intersect, and which form the edges of the groins. 
 
 To find the projections in plan of the voussoir of the groin 
 which corresponds to the one M'N'O', &c. (Fig. 1), and 
 F'G'R', &c. (Fig. 2), draw Mm (Fig. 3), the projection of 
 the lower edge of the joint corresponding to 31' (Fig. 1), 
 and Jlm i the corresponding projection for the point F' (Fig. 
 2) ; and in like manner the lines Nn and Nn t , the projec- 
 tions of the upper edges. Joining M and N gives the pro- 
 jection of the intersection of the planes of the two joints. 
 Find in like manner Qq and Qq t • Pjp and 1'jr , the projec- 
 tions of the edges corresponding to the joints Q'P' and 
 EI'. Joining Q and P gives the projection of the inter- 
 section of the planes of the two last joints. Having found 
 the projections in plan of the edges of the joints and their 
 intersections, the voussoir is terminated on the semi circular 
 arch by a joint of right section mp, taken at any suitable 
 distance from the point P j and on the semi elliptical arch 
 by a like joint iMJ> r The required voussoir in plan will be 
 the figure Mrn/pPjp^ih t . The part above the line MG be- 
 longing to the semi circular arch ; that to the right of it to 
 the other. 
 
 To find the true dimensions of the joints of this voussoir 
 and of the portion of thesofnt which belongs to the semi < ir- 
 cular arch, draw a line mp (Fig. 5) to correspond to the one
 
 STONE CUTTING. 4:5 
 
 mp {Fig. 3) of the plane of right section by which the vous- 
 soir is terminated. On this line set off mn=M'JV' {Fig. 1) 
 the breadth of the upper joint; nq=M'Q' the length of 
 the arc between the joints; and qp=QP' the breadth of 
 the lower joint. Through these points draw perpendiculara 
 to the line, and set off on them nN=mM {Fig. 3) ; mM= 
 nN ;qQ=qQ, &n&pP=pP. Join the points 31, J\ T , and 
 Q, P, by right lines ; the points JV, Q, by a curve line, an 
 intermediate point of which can be found by constructing 
 in plan the element yx of the soffit which corresponds to 
 the middle point x' {Fig. 1) of the arc 3FQ' and setting 
 this line off on Fig. 5 from y, the middle point of nq to x. The 
 Fig. 5 will give the joints required in their true dimensions, 
 also the developed portion of the soffit between them, and 
 which is bounded at one end by the curve of right section 
 corresponding to mq, and at the other by the portion of the 
 ellipse of the groin corresponding to MQ {Fig. 3). 
 
 In like manner the true dimensions of the joints, and 
 the portion of the soffit between them, which belong to the 
 same stone, and form the portion of the elliptical arch ter- 
 minated by the joint of right section m i p i {Fig. 3) may be 
 found, as shown {Fig. 5) on the left ; by setting off the dis- 
 tances p q, , q / m l , m l n l , respectively equal to E'l', F'F' 
 and F'G' {Fig. 2), and on the perpendiculars to p l n l , through 
 the points j?, , q t , &c, setting off the distances p,P, q,Q, &c, 
 respectively equal to pP &c, in plan {Fig. 3). 
 
 The portion of the keystone which forms the top of 
 the groins at the point K {Fig. 3), is limited on the semi 
 circular arch by the joint of right section G t JV t j and by the 
 one Hh on the semi elliptical cylinder, with a corresponding 
 one on the right of K. 
 
 The joints of right section of the different courses, as 
 Trip and GN t , are arranged to break joint. 
 
 Cloistered Arch. The constructions for determining 
 the projections, &c, of the joints and their true dimensions, 
 are precisely the same in this case as in the preceding. 
 
 In Fig. 4c aapj) are the exterior lines in plan, and 
 BfiBfi^ the interior lines of the top of the walls, or the 
 imposts of the arches ; the semi circular arch springing from 
 the lines B n G and GB, and the semi elliptical form G t B t . 
 The lines G t K and B t K qxq the projections of the half of 
 each groin. 
 
 By drawing in plan the edges of the joints corresponding 
 to P'Q' and M'JY' {Figs. 1, 2), and joining the points P, 
 Q, and 3f, N {Fig. 4), the intersection of these joints are 
 obtained in plan. The groin stone which corresponds to
 
 46 8T0XE CUTTING. 
 
 these joints is limited by a plane of right section nip (Fig. 4) 
 taken at pleasure on the semi circular arch, and a like <>ne 
 /// ji on the semi elliptical. All the parts of this groin stone 
 -will therefore be projected within the figure iwpPv m JJ '. 
 
 To construct the joints and the portion of the soffit in 
 their true dimensions which belong to this stone, commence 
 {Fig. 6) by setting off on a right line the distances nm, ma, 
 and qp, respectively equal to S'J/', 21 Q , and QP' (Ftg. 
 1), and which correspond t-o the joint of right section mm 
 {Fig. 4). Draw through the points n, m, &c, thus set oft - , 
 perpendiculars to np, and along these perpendiculars set off 
 the distances nN, m2L. qQ, and pP, respectively equal to 
 the same lines on Fig. 4. Join the points NM and QP by 
 right lines; and 21 Q by a curved line, an intermediate point, 
 of which corresponding to x {Fig. 1) is found by setting off 
 from y t the middle of mq the distance y i x i equal yx t {Fig. 4). 
 
 In the same way the corresponding portions of the groin 
 stone belonging to the joint of riijht section m / p / on the 
 semi elliptical arch are found from Figs. 2 and 4. 
 
 The top groin stone at K {Fig. 4) which forms a portion 
 of the two arches is represented as a single stone. 
 
 The joints of right section in the different courses of 
 voussoirs are arranged as shown in plan to break joint. 
 
 Application. Having determined the projections in plan 
 of the edges of the joints of a groin stone with the true 
 dimensions of the joints, and the portion of the soffit of 
 each arch belonging to it, their uses in shaping the stone 
 from, the solid block will be easily understood. Taking, for 
 example, the groin stone of the groined arch, the right sec- 
 tions of which are given in Figs. 1 and 2; the plan in Fig. 
 3; and the true dimensions of the joints, &c, in Fig. 5, it 
 will be readily seen that, supposing a block from which the 
 stone is to be shaped to be a rectangular parallelopid, its 
 thickness must be such that the right section 21'JV'O', &c, 
 {Fig. 1) can be inscribed within the rectangle of the end 
 that corresponds to the joint of right section vp {Fig. 3) of 
 the semi circular arch, and the figure F'G'JB. , Arc, be in- 
 scribed within the end corresponding to the joint m / p l of 
 the semi elliptical arch; and the length of the block must 
 be equal to rnM, and its breadth to Mm . 
 
 •Having inscribed upon the ends of the block the two 
 figures of cross section, the portions of the solid exterior to 
 them are gradually worked off until the dressed surfaces 
 coincide with Fig. 5, which may be ascertained either by 
 measurement, or by the actual application of these figured 
 cut from some thin flexible material.
 
 STONE CUTTING. 47 
 
 Remark. A careful examination of the lines of tho 
 figures will show the geometrical methods for determining 
 the tangents to the points on Fig. 2 which correspond to 
 Fig. 1. 
 
 ■ Prob. 9, Pis. 3, 4. To construct tJte projections and true 
 dimensions of the voussoirs of the rampant cylindrical arch. 
 
 This problem, which comprises two cases, is a variation 
 of Prob. 7. The arch in this, as in Prob. 7, being termi- 
 nated at one end by a vertical plane, and at the other by a 
 horizontal semi cylinder having its axis and elements paral- 
 lel to the vertical plane of the end ; the elements of the soffit 
 and the edges of the voussoirs of the arch being oblique both 
 to the vertical plane and to the horizontal plane of the plan. 
 
 Case 1, PL 3. Construct (Fig. 1) the semicircle B'T'C, 
 to represent the oblique section of the arch by the vertical 
 plane of the end. Let ad (Fig. 2), assumed at any conve- 
 nient distance from A'D', be the projection in plan of the 
 lowest element of the semi cylinder of the other end, which 
 with the axis is taken in a horizontal plane at the distance 
 a" 'a" (Fig. 3) below the line A'D' ; and let the lowest ele- 
 ments of the arch drawn from the points A'D' CD', and its 
 axis from F , (Fig. 1) be taken to intersect the line ad, and 
 to lie in vertical planes perpendicular both to the vertical 
 plane of the end and to the horizontal plane. The edges 
 of the voussoirs as A'a, B'b, (fee, in plan will be perpendic- 
 ular to A'D'. 
 
 To obtain the edges of the voussoirs in their true dimen 
 sions, it will be necessary to find their projections, as in Prob. 
 5, on a plane parallel to them. 
 
 Let the vertical plane which contains the edge projected 
 in A'a be taken for this purpose, and suppose it revolved 
 around the line A ' K' 1 ' , its trace on the vertical plane of the 
 end, to coincide with this plane. In this new position of 
 tl>e side vertical plane the edge projected in A a will be ob- 
 tained on it by setting off A'a'"=A'a / erecting at a'" a 
 perpendicular = a'"a", and joining A'a'". The edges 
 drawn from B', C , D', and the axis of the arch will all 
 evidently be projected into the same line A'a". 
 
 The projections of the other edges on this plane will 
 evidently be parallel to A'a' (Fig. 3), and their positions 
 will be found by drawing through the points E', I', <fec, 
 (Fig. 1) lines parallel to A'D', and from the points E ", &c, 
 in which they cut A'K'" drawing parallels to A'a" (Fig. 3). 
 
 As a' is the revolved position of the point in which the 
 vertical side plane cuts the lowest element of the semi cyl- 
 inder of the end, and as the axis is in the same horizontal
 
 48 STOKE CUTTING. 
 
 plane as this element, by setting off on the horizontal through 
 a", and to the left of a", the radius of the semi cylinder 
 and from the point thus set off describing an &rca"X', it 
 will be the one cut from the semi cylinder by the vertical 
 side plane. The lines, as H"II'", E"E '" intercepted between 
 the arc a"X' and the line AX", are the projections in their 
 true dimensions of the required edges of the voussoirs. 
 
 To obtain a right section of the arch, for the purpose of 
 constructing the joints of the voussoirs in their true dimen 
 sions, and the development of the soffit of the arch ; from the 
 point A' draw the line A' Y' perpendicular to the projec- 
 tions of the edges on the vertical side plane ; this line may 
 be regarded as the trace of a plane of right section on the 
 side vertical plane, and the line A'D' as its trace on the 
 horizontal plane through A'D'. If the plane of right sec- 
 tion thus fixed be revolved around A'D' until it becomes 
 horizontal, the right section contained in it will be deter- 
 mined in its true dimensions. The points as <?', f, &c, 
 {Fig. 4) will be fouud, after this revolution, by setting oft' 
 from the line A'D' along the projections of the elements in 
 plan corresponding to the points E ', F', &c, distances equal 
 to A'e', A'f', &c, {Fig. 3) measured from the point A' along 
 the line A' Y'. The curve B'e'f'm'q'C {Fig. 4) thus deter- 
 mined is the curve of right section of the soffit, and the fig- 
 ure 7n'n'o'j?'q' the right section of the voussoir corresponding 
 to M'X'O'P'Q' {Fig. 1) of the end. 
 
 Having obtained the right section, the development of 
 the soffit and the joints in their true dimensions are found, 
 as in Prdbs. 5 and T, as follows : Havin^drawn a line B' C 
 {Fig. 5) set off along it the distances B'e\ e'f',f'm', &c, 
 respectively equal to the arcs Be', e'f &c, on the curve of 
 right section {Fig. 4). The right line B'C will be the de- 
 velopment of the curve. Through the points B', e',f, &c. 
 draw perpendiculars to B'C which will be the elements of 
 the soffit in development, which are the lower edges of the 
 joints and correspond to the points B\ E\ &c, {Fig. 1), 
 As the true distances of the extremities of these edges from 
 the plane of right section are given on Fig. 3, and are the 
 distances e'e", e'E'" for the edge projected in its true length 
 between the line A'E'" and the curve d'X'j by setting off 
 e'e" and e'E'", from e' to E', and e' to E'" {Fig. 5), the 
 points E' and E'" will be respectively points of the devel- 
 opment of the curves in which the soffit intersects the verti- 
 cal plane of the one end of the arch and the horizontal 
 c} T linder that terminates the other. In like manner the 
 points F', M\ &c, of the developed curve B'T'C are
 
 8T0NE CUTTING. 49 
 
 found, and those as h", E'", &c, of the other end. To ob- 
 tain the true dimensions of any joint, as the one correspond- 
 ing to EI' {Fig. 1), set off the distance e'i', {Fig. 5) equal 
 to the breadth of the joint in right section, which is e'i 1 
 {Fig. 4). Through i' draw a perpendicular to B'C, and set 
 off from it the distance i'l', i'A'", respectively equal to 
 a' A", a' A'" {Fig. 3). Join I', E' , by a right line, and 
 A"E'" by a curved line, the figure A'"I'E'E'" will be the 
 required joint in true dimensions. The other joints are 
 found in like manner. 
 
 To find the dimensions of a block of the form of a rec- 
 tangular parallelopiped from which one of the voussoirs, as 
 the one corresponding to M'N' 0' P' Q' could be cut, it will 
 be observed that the edges of this voussoir are projected in 
 their true dimensions on Fig. 3, between the line H" H'" , 
 which corresponds to the points N' , 0' {Fig. 1), and the line 
 E"E'", which corresponds to Q'; drawing therefore from 
 H" a perpendicular to E"E'" prolonged, and from E" one 
 to H" H f " prolonged, the rectangle thus formed will be the 
 true dimensions of one side of the block. As m'n'o'p'q' 
 {Fig. 4) is the cross section of the same voussoir, the breadth 
 of the rectangle of the end of the block must be equal to 
 m'p\ in order that the figure m'n'o' , &c, can be inscribed 
 within it. The manner of setting off the different lines on 
 the block, with the view of dressing it into shape, will be 
 readily seen from what has already been stated on this point 
 in the preceding problems. 
 
 Remarks. From the preceding constructions, the joints 
 and the development of the soffit for any other plane end 
 passing through the line A ' D' , and having the line A'Z', for 
 example, for its trace on the vertical side plane, can be 
 readily found, by setting off, on Fig. 5, from the line B'C, 
 the distances between the corresponding points on the lines 
 A' Y' and A'Z', as h'h", for example, in the same way as 
 for those between A' Y' and A ' K" ' , and through the points 
 e",f", &c, drawing the developed curve of intersection of 
 the soffit and assumed plane, and constructing the corres- 
 ponding joints as e"i"A'"E"'. 
 
 Prdb. 9, Case 2d, PI. 4. This case is a variation of the 
 preceding one, the axis and elements of the soffit of the 
 arch being oblique both to the horizontal plane, and to the 
 vertical plane which terminates the arch at one end, but 
 situated in vertical planes oblique to the vertical plane of 
 the end. The position of the semi cylinder which terminates 
 the other end is the same in all respects as in the preceding 
 case.
 
 50 STONE CUTTING. 
 
 Fig. 1 represents the end of the arch in the vertical 
 plane. The curve of the soffit B'F'M ' C in this plane is a 
 semicircle. The arch for the mere illustration of this case 
 consists of only three voussoirs. 
 
 Fig. 2 represents the projections of the elements in plan ; 
 the axis of the arch and the lowest elements of its soffit 
 intersect the lowest element of the semi cylinder, which is 
 projected in the line ad, parallel to AD', and lies in a hor- 
 izontal plane at the distance JD'D" below AD', at the 
 points b, I and c. 
 
 Fig. 3 represents the projections of the edges of the 
 voussoirs on the vertical side plane, parallel to them, of 
 which D'T and D' V are the traces on the horizontal plane 
 of the plan, and the vertical plane of the end. The system 
 of projecting lines in this case is the same as the one usee 
 in Prob. 5. 
 
 Fig. 4 represents the revolved position, on the horizontal 
 plane, of the right section of the arch, contained in the plane 
 of right section of which XY, perpendicular to the projec- 
 tion of the axis L'l is the horizontal trace, and TV, perpen- 
 dicular to D"d, the projection of the axis on the side vertical 
 plane, is the trace on this plane. 
 
 Fig. 5 represents the joints and the development of the 
 soffit in their true dimensions. 
 
 Having constructed Figs. 1 and 2, find, by Prob. 5, the 
 projections of the edges of the joints on the vertical side 
 plane of which D'T is, the horizontal trace, assuming, in the 
 first place, the line D"d, as the projection of the axis and 
 lowest elements on this plane, and parallel to which all the 
 other projections of the edges are drawn ; the one corres- 
 ponding to the point M' (Fig. 1), for example, is found by 
 setting off from D" {Fig. 3) on the revolved position 
 D'K"' of the trace of the side vertical plane with that of 
 the end, the distance D'M" equal to MM' {Fig. 1), and 
 drawing M"M'" parallel to D"d. 
 
 Representing, by the line drawn through T parallel to 
 ad, the axis of the semi cylinder, the ellipse cut from the 
 semi cylinder of the end will have dT, and TS' equal to 
 the radius of the semi cylinder, for its semi axes. Having 
 described the quadrant dS' of the ellipse in its revolved po- 
 sition, the projections of the edges of the joints intercepted 
 between it and the line D'K'" will be the true lengths of 
 the edges. 
 
 To obtain the right section, take XY perpendicular to 
 the axis L'l in plan, and Yz' perpendicular to its projection 
 ty'd on the vertical side plane, as the traces of the plane
 
 STONE CUTTING. 51 
 
 of right section. Having found {Fig. 4, PI. A,) the projec- 
 tions c\ m t , n', &c. {Fig. 3) of the points in which the ele- 
 ments of the soffit and the edges of the joints pierce the 
 plane of right section, next construct from these {Fig. 4) the 
 right section as revolved on the horizontal plane. 
 
 To construct the soffit and joints in their true dimensions, 
 draw {Fig. 5) a line a t D' {Fig. 4, PL A) and set off on this 
 from b, to c t the length of the curve of right section b.c / 
 {Fig. 4). Drawing through the points b / , j t , ra, , c, , per- 
 pendiculars to afi' , set off on them above and below a t D' 
 distances m M' m if'", respectively equal to m 3f" and 
 m,M '" {Fig. 3). The curves B'F'M. ' C, and bM' h c, drawn 
 through the points thus obtained, will be the developments 
 of the intersection of the soffit with the end plane and semi 
 cylinder. To construct the joint of which M'M'" is one 
 edge, set off m / n / on a.E' equal to m l n l {Fig. 4), the width 
 of the joint in cross section ; through n / draw a parallel to 
 M' IF", and set off on it n t N' , n^tf"' ', respectively equal 
 to n'N" and n'JV'" {Fig. 3). Joining M' N' by a right 
 line, and M'"W" by a curve line, the figure obtained is the 
 required joint. The others are found in like manner. 
 
 Remark. A comparison of the lines on Fig. 5 with 
 those on Fig. 3 will point out the manner of constructing 
 an intermediate point as v of the curve M" ' N'" . 
 
 Prob. 10, PI. 5. To construct the projections and true 
 dimensions of the voussoirs of the hemispherical dome. 
 
 Let the semicircle {Fig. 1) BL"C be the vertical section 
 of the soffit of the dome, and suppose it divided into seven 
 equal parts at the points E\ F r , &c. Drawing radii through 
 the points F', &c, set off upon them the equal distances 
 E'l', F'G', &c, and complete, as in the preceding cases, 
 the figures JEP'1I"G'F', &c, to represent the sections of 
 the voussoirs. If Fig. 1 be supposed to be revolved about 
 the vertical radius PL" as an axis, any section of a vous- 
 soir, as EIH"G'F r would generate the entire voussoir of 
 the dome comprised between the horizontal circles on the 
 6offit projected in the lines E'Q' and F'M'. The lines 
 EI', F'G', in this revolution generate the joints between 
 the voussoir in question and the two in contact with it, 
 which joints are portions of a cone of which the centre of 
 the dome is the vertex, and LL' the axis. The line F El' 
 will generate a cylindrical surface having the same axis, and 
 the line H'G' a plane. 
 
 Having in this manner determined the bounding surfaces 
 of each course of voussoirs, the course is divided into blocks 
 of suitable dimensions, by joints of right section, formed by
 
 52 STONE (TUTTING. 
 
 intersecting the course by vertical planes through the axis 
 L'L". If IE, and I I F I {Fig. 2), for example, be taken aa 
 the projections in plan of two joints of right section, the 
 figure ll F ,E \ will be the projection in plan of a block 01 
 voussoir of the course in question. The figure E'I'I'F" 
 {Fig. 1) is the projection of the lower conical joint of this 
 voussoir; F G'G"F" that of the upper conical joint; and 
 E'FF"E" that of the portion of the soffit. The joints 
 of right section of the adjacent courses break joints, aa 
 shown at E'E', V"v', &c, on the curves V'L", Y"L', 
 &c. {Fig. 1), which are the projections of the circles cut 
 from the soffit by the joints of right section. 
 
 Application. Having determined the projections of the 
 bounding lines and surfaces of a voussoir, their true dimen- 
 sions can be easily determined, and from them the size of a 
 block from which the voussoir can be cut. Taking, for ex- 
 ample, the voussoir projected in plan {Fig. 2) in II \F \E t , 
 from an inspection of the projections {Figs. 1, 2) it is obvious 
 that the dimensions ,of a block from which it can be cut 
 must be such that the figure IIF I E I {Fig. 2) can be inscribed 
 within its base, and its thickness be equal to the vertical 
 height between the lines H'G" and E'E" {Fig. 1), the tot a" 
 depth of the voussoir. Having selected a block of the suit- 
 able dimensions, the different lines and surfaces of the vous- 
 soir can be obtained in their true dimensions from its projec- 
 tions {Figs. 1, 2, 3), and marked out on the sides and bases 
 of the block. 
 
 It will be seen that the end of this voussoir, which forms 
 a portion of the soffit, is comprised between the two merid- 
 ian planes IE, and I / F / {Fig. 2) and the upper and lower 
 conical joints. The points F', F", E\ E", {Fig. 1) are 
 therefore the projections of the four angular points of the 
 voussoir, on the soffit, and lie upon the circumference of a 
 small circle of the dome passing through the points of which 
 these are the projections on Fig. 1. This small circle can 
 be readily constructed {Fig. 4), since the lengths of the 
 chords joining the two upper points FF", and the two 
 lower E'E" {Fig. 1), are given in their true dimensions EF t 
 and EF {Fig. 2); and the diagonals projected in EF t and 
 E,F{Fig. 2) can be readily obtained in their true dimensions. 
 Having set out the small circle determined from these ele- 
 ments {Fig. 4), it will limit the portion of the soffit on the 
 end of the voussoir, and will serve as a guide to the work- 
 man in working it out. 
 
 Fig. 3 gives the true dimensions of the side of the vous- 
 soir in the meridian plane IE, {Fig. 2).
 
 STONE CUTTING. 53 
 
 Prdb. 11, PI 6. To construct the projections and trite 
 Ibnensions of the voussoirs of the gate-recess. 
 
 The line BG {Fig. 2) represents the trace of the vertical 
 face or front of a wall ; ad that of the back, also vertical. 
 Through this wall an arched gate-way is to be so constructed, 
 that the gate, composed of two leaves, may be placed mid- 
 way between the face and back, as at AD, and when open 
 the leaves shall be thrown back, taking respectively the 
 positions A t a, D,d, by revolving around the vertical axes 
 projected in A, and D / . In this way, the gate occupies a 
 recess within the wall, from which circumstance the problem 
 is named. 
 
 Let B', F', 0', (Pig. 1) be the curve of right section of 
 the right arch, BB I C, G (Fig. 2) its plan. Let the top of 
 the gate when closed be a semicircular cylinder, AD' (Fig. 
 1) being its diameter, and the rectangle A A D D (Fig. 2) 
 its plan ; the gate when closed shutting against the plane 
 surface ring projected (Fig. 1) between the two semicircles; 
 and Fig. 2 in the line AD. 
 
 The problem to be solved consists in so arranging the 
 surface of the recess under which the leaves swing in being 
 opened or closed : 1st, that it shall offer no obstruction to 
 the play of the leaves ; 2d, that it shall be one of easy geo- 
 metrical construction ; 3d, that it shall present a pleasing 
 architectural effect. 
 
 The lines A,a, D,d, (Fig. 2) being the traces of the ver- 
 tical side planes of the recess against which the leaves rest 
 when closed, these planes are each terminated at top by an 
 arc of a circle assumed at pleasure, but of greater radius 
 than A'L' (Fig. 1). To construct this arbitrary arc (Fig. 3), 
 revolve the side plane D t d around the axis projected in D, 
 (Fig. 2), and D'D' (Fig. 3), parallel to the face of the wall, 
 into the position D / d / . Assume dl" the highest point of 
 the arbitrary arc in the revolved position, at the same height 
 as Til (Fig. 1) is above X', and construct an arc passing 
 through D'dl" and tangent to D'D , and let this be taken 
 for the required arc. Supposing the side plane revolved 
 back to its primitive position, Del'' will be the projection 
 of the arc ; and d'd" that of the vertical edge of the back 
 and side planes. 
 
 Let this arbitrary arc, the semicircle projected in A'lc 'D 
 (Fig. 1) and A t D t (Fig. 2), and the axis of the arch pro- 
 jected in L' (Fig. 1), LI (Fig. 2), be taken as three direc- 
 trices of a warped surface, to form a portion of the upper 
 surface 3f the recess. The projections of the extreme posi-
 
 54 STONE CUTTING. 
 
 ticii of the element of this surface will be d"x'L' {Fig. 1\ 
 day {Fig. 2). A like surface covers the opposite side. 
 
 To form the top, the two warped surfaces determined are 
 connected by a third, which must be tangent to each of them 
 along the extreme element d"L', a"L' of each, so that the 
 three surfaces may appear as a continuous surface, and thus 
 satisfy the 3d condition. To satisfy this condition, let 
 the axis of the arch and the semicircle, which are two of 
 the directrices of the two first warped surfaces, be taken 
 as two of the directrices of the third. This will give two 
 tangent planes common to the surfaces along each of the 
 elements d'F, a"L '. Construct now a tangent plane to 
 the warped surface found at the point d", by drawing a tan- 
 gent to the curve projected in D'd" at this point, and through 
 this tangent and the element projected in d" L passing a 
 plane. . The element pierces the vertical plane of which 
 A D t is the trace at x, a?'/ the tangent to the curve D \d" at 
 d" intersects the vertical line D'D" at D"; joining then 
 D , x', it will be the projection of the trace of the tangent 
 plane on the vertical plane AD, j the projection of its trace 
 on the vertical plane ad is v'd"w', parallel to x'D". Draw- 
 ing an arc of a circle passing through a" and d" and tangent 
 to v'w', if it is taken as the third directrix of the second 
 surface, the two surfaces will be tangent, as they have a 
 third common tangent plane at d". 
 
 The 2d condition Is satisfied by taking warped surfaces 
 to form the soffit, or top surface. 
 
 Having constructed the warped surfaces, with these arbi- 
 trary conditions, it will be necessary to ascertain whether thej 
 satisfy the 1st condition. To do this, it will be observed 
 that the top of the leaf describes, in its revolution, a surface, 
 and which, to satisfy this condition, should not intersect the 
 warped surfaces within the side plane, as Aa, for example. 
 This intersection will be found by the usual methods for 
 finding the intersections of two given surfaces. The line 
 r's' {Fig. 1), for example, may be assumed as the vertical 
 trace of a horizontal plane intersecting the two surf; 
 This plane will cut from the surface described by the top of 
 the leaf an arc of a circle, projected in sr {Fig. 2) and from 
 the warped surfaces an arc 8% t rj and as these intersect at ?', 
 without A lt a, the surfaces do not interfere along this hori- 
 zontal plane. The same construction would be made for 
 other points. 
 
 The hounding lines of the top surface being found, the 
 arch is divided off into five equal parts, as B'A , ivc. The 
 planes of the voussoir joints pass through the axis of the
 
 STONE CUTTING. 55 
 
 arch and extend to the points /', 67', &c, arbitrarily chosen 
 and from these last points the voussoir joints are vertical. 
 
 To determine the true dimensions of the joints M'X 
 and Q P {Fig. 1); let each of them be revolved around 
 their lower horizontal edges, projected in M', Q ', parallel to 
 the horizontal plane; taking JUf'JST', this is done by draw- 
 ing through M ' a line parallel to L'D' and setting off along 
 it, from 21'. distances equal to G'F'=M'N\ F'i', F h', 
 and, to simplify the construction, drawing through the 
 points thus set off, lines parallel to L'M' to intersect L'D'; 
 from these last points drawing lines parallel to L'l the points 
 LfoF'i'h'G'I {Fig. 4) will be found, which joined will give 
 the figure and true dimensions of the joint through M'N'. 
 In a like manner the figure Fk [ E 'if 'g ' G ' 1 and true dimen- 
 sions of the joint through Q P' are obtained. 
 
 Application. To cut the voussoir out of a block of the 
 form of a rectangular parallelopiped, its dimensions must 
 be such that the figure N'jI'QP'O' {Fig. 1) can be inscrib- 
 ed within the end, and its length be equal to IG' {Fig. 4). 
 Having set out the bounding lines of the different surfaces 
 from Fig. 1, the plane and cylindrical portions will be first 
 cut off; next the portion of the warped sarfaces, by first 
 working down to the positions of several of the intermediate 
 elements, determined from the drawing, and then finishing 
 off by the eye the portions of the surface between these ele- 
 ments. 
 
 Prob. 12, PI. 1. To construct the projections and true 
 dimensions of the steps of the geometrical stairioay. 
 
 Let ABGD {Fig. 1) be the polygonal base of a vertical 
 wall, along which a flight of stone steps is to be built. Let 
 XYZ, xyz, be two curves having the relation of involute 
 and evolute to each other ; the one XYZ being the base of 
 a vertical cylinder, the surface of which limits the ends of 
 the steps, and which is termed the well of the stairs. Let 
 X \ Y,Z / be another curve parallel to the one XYZ, and at 
 the distance from it that persons going up or down the stairs 
 would naturally take; and where, on this account, the top 
 of each step, or the tread should have a uniform breadth. 
 This tread added to the height, or rise, being usually assumed 
 at twenty-two inches, as a convenient distance for each step. 
 
 The problem consists : 1st, in arranging the tread and 
 rise with these conditions ; 2d, in making the under sur- 
 face of the stairway a helicoidal surface ; 3d, in arranging 
 the joints between the steps so as to be plane surfaces, and 
 normal to the helicoidal surface at the middle p< nt; 4th, in
 
 56 STOXE CUTTING. 
 
 determining from these conditions the form and dimensions 
 of each step. 
 
 Having set off the equal arcs XI, 1-2, 2-3, &c. (Fig. 
 1) along the curve X t l t Z t , equal to the assumed tread, 
 draw, through these points, lines tangent to the curve an/z, 
 and prolong them to the line bed parallel to BCD / the 
 quadrilaterals thus formed, between X. YZ and bed, will he 
 the true dimensions of the top surface of each step. ]\i id- 
 way between the equal arc, as at Y, , Z l} cvjc, draw lines 
 also tangent to soys, and let these be assumed as the projec- 
 tions of "the edges of the joint3 along the helicoidal surface ; 
 and, to fix their position, let the edge of each joint be taken 
 at the distance of half the rise below the top of the step. 
 The points thus determined will lie on a helix, which at 
 each of these points is half a rise below the top of the step, 
 and the inclination of the tangent to which at any point 
 will be the rise or height of each step divided by the uniform 
 tread. 
 
 Having fixed the position of the helix, the helicoidal 
 surface is generated by moving a right line along it so as to 
 be parallel to the horizontal plane, and, in all of its posi 
 tions, be projected normal to the curve XYZ. 
 
 Let y t , the middle point of the lower edge Ym, be taken 
 as the point at which a normal plane is to be passed to the 
 helicoidal surface for the joint in question. This plane is 
 determined as shown on Fig. 2, by Prob. 7 {PI. A, Fig. 6), 
 and in like manner the one at z t on Zo as shown in Fig. 3. 
 
 Having determined these planes, their intersections with 
 the tops of the steps will give the lines Xn, Op, parallel to 
 Ym, Zo, which are the top edges of the joints. 
 
 "With the data now determined, the form and dimensions 
 of the ends of the step to which these two joints belong 
 can be determined. The larger end of the step is contained 
 in a vertical plane of which mq is the trace on the horizontal 
 plane. Draw a line B C parallel to be and at any assumed 
 distance from it; this may be taken as the revolved position 
 of the top line of the ena, on the horizontal plane. From 
 the points m, n, o, p, and q draw perpendiculars to mq', 
 set off on these the distance q"q' for the rise of the step ; 
 m"m' equal to half a rise; o' half a rise below p' ; join the 
 points thus set off. The figure m'n'q"qp'o' is the one requir- 
 ed. To find that of the other end (Fig. 5), the portion of the 
 cylinder of the well YM is developed, and the correspond 
 ing points set off on it; the figure Y'N'M"2£'0'Z' is the 
 we requi-ed.
 
 STONE CUTTING. 57 
 
 Application. To cut the stone, a block must be taken 
 upon the top of which the figure YmqM {Fig. 1) can be 
 set off" and on the large end Fig. 4. Having dressed off 
 the plane and cylindrical surfaces, the portion of the 
 warped surface can be dressed off as in the preceding prob- 
 lem. 
 
 Prdb. 12, PI. 8. To construct the projections and true 
 dimensions of the voussoirs of the groined annular and 
 radiant arch. 
 
 The soffit of this arch is formed of the surfaces of an an- 
 nular and radiant arch, the intersections of which form the 
 curves of the groin. 
 
 To find the horizontal projections of these curves, let (Fig. 
 2, PI. 8) B' C' be the diameter, and ~K" the centre of a semi- 
 circle, contained in a vertical plane passed through the ver- 
 tical line projected at L horizontally. If the semicircle be 
 revolved around the vertical L it will generate the surface of 
 an annular arch. 
 
 From the point L (Fig. 1) in the horizontal plane of the 
 diameter B' 0' of the semicircle let two lines L B' and L c 
 be drawn, making any convenient angle with each other, and 
 let the chord of the arc B' c, included between them, de- 
 scribed by the point B' of the semicircle, in its revolution, 
 be taken as the transverse axis of a semi-ellipse, contained in a 
 vertical plane of which B' c is the trace; the conjugate semi- 
 axis of this ellipse being equal to the radius of the semicircle. 
 Then if a right line be so moved that it shall, in all of its 
 positions, be parallel to the horizontal plane of the semi- 
 transverse axis of the ellipse, intersect the vertical through 
 L, and rest on the curve of the semi-ellipse, it will generate 
 the soffit of the radiant arch. 
 
 The horizontal projections B' L, C, and C L c, which are 
 those of the groins, will be obtained, by finding the inter- 
 sections of the projections of the corresponding elements cut 
 from the two soffits by horizontal planes. The point L, 
 being that of the highest point; and the points B ; C Cj c 
 being those of the lowest points of the groins. 
 
 If now the semicircle be divided into five equal parts, and 
 the right sections of the voussoirs of a cylindrical arch be 
 drawn (Fig. 2) ; the joints E' I', F' G', &c, of this right sec- 
 tion, in the revolution of the vertical plane containing them, 
 will describe zones of conical surfaces, the vertices of which 
 will be on the vertical through L, where these joints pro- 
 longed intersect it. In like manner, the lines I' H', G' G", 
 &c, will describe cylindrical surfaces, having the same verti- 
 cal for their common axis; and the horizontal line6, as G' H',
 
 58 STONE CUTTING. 
 
 &c, will describe zones of circles. Thus completing the 
 bounding surfaces of the voussoirs belonging to the annular 
 arch. 
 
 As the soffit of the radiant arch is a right conoid, having 
 the horizontal plane, containing the semitransverse axis of 
 the directing semi-ellipse, for its plane director, and the ver- 
 ticil through L as its right line directrix, its joints, along the 
 rectilinear elements corresponding to the horizontal circles 
 described by the points Q', M', F', E', of the semicircle, to 
 be normal throughout to the soffit, would require to be 
 warped surfaces. To avoid the inconvenience of constructing 
 these, a plane surface joint is substituted for each stone in- 
 stead of the other; and this is so taken, that it shall be nor- 
 mal to the soffit at the middle point of that portion of the 
 right line element of the soffit which belongs to the joint. 
 Taking for example the element projected in L E, e 4 , and as- 
 suming that the groin voussoir is bounded, on the radiant 
 portion of the arch, by the lines F,/,; by/, I, which is the 
 projection of the line cut from the soffit by a vertical cylin- 
 der ; by the projection F, E, of the groin curve, and by the 
 line E' ft,; then F, / will be the lower edge of the radiant 
 joint, corresponding to the lower edge of the annular joint, 
 described by the point F'; and E, <? 2 will be the lower edge 
 of the joint below, corresponding to the one E' I'. 
 
 Having the lower edge E, e s of the joint, the other bound- 
 ing lines of it are found by constructing a normal plane to 
 the soffit containing the right line element through E„ at the 
 middle point e of E, e„ and finding the intersections g s I of 
 this plane with the vertical cylinder f I that limits the 
 voussoir; with the conical zone described by E' F, and which 
 is projected in E, z ; and finally with the horizontal plane 
 which passes through F (Fig. 2), and which will be projected 
 through I parallel to E, e v 
 
 In like manner the plane joint projected in F, f g x Gr can 
 be found. 
 
 The portion of the groin stone, belonging to the annular 
 arch, is limited by a vertical plane passed through the points 
 L, F„, H; the line F, E, of the groin; the upper and lower 
 conical joints ; the cylindrical surface projected in z H ; aud 
 the horizontal plane"th rough G' H' (Fig. 2). 
 
 Note. — To construct the normal planes of the plane joints 
 of the radiant arch see Prob. 6. 
 
 All the horizontal lines of the surfaces bounding the groin 
 voussoir in question, are projected in their true dimensions 
 n Fig. 1. 
 
 Having found the horizontal projections of all the lines
 
 STONE CUTTING. 59 
 
 bounding the groin stone considered, the true dimen- 
 sions of all the developable surfaces by which it is bounded 
 can be found by methods already used in the preceding 
 Probs. 
 
 Take, for example, the plane joint of the radiant arch, 
 projected in E, <? 2 I z. Having first determined the tangent 
 plane to the middle point e % of the lower edge of the joint, 
 and its trace t, t, on the horizontal plane of the springing 
 lines, by Prob. 6, let this last plane be intersected by a ver- 
 tical plane X Y perpendicular to the lower edge E e 2 (Fig. 1), 
 and let it then be transferred, parallel to itself, to X Y (Fig. 
 3). The plane X Y will cut from the tangent plane a line 
 which, in the revolved position of the plane X Y (Fig. 3), 
 will be projected in t, E' ; E" E' (Fig. 3) being equal to E" 
 E' (Fig. 2). From E' (Fig. 3) drawing a line perpendicular 
 to E' 4 it will be a normal to the soffit of the radiant arch at 
 e 2 (Fig. 1), and where this normal intersects at I' a line par- 
 allel to E'' t„ and at the same height above it as I' (Fig. 2) 
 is above E'^the line E' I' (Fig. 3) will be the true width of 
 the plane joint considered. Now, revolving this plane joint 
 around the line E' I' to coincide with the vertical plane X Y 
 (Fig. 1), the points e v E : (Fig. 1), for example, will fall in a 
 perpendicular to E' I' (Fig. 3) as far from it, at e" and E", 
 as they are in horizontal projection from X Y (Fig. 1). In 
 like manner, the points i' and. 1'" (Fig. 3) are found ; and 
 e" E" V" i will be the true dimensions of the plane joint. 
 E" V" wall be the intersection of the plane radiant joint with 
 the corresponding conical joint of the annular arch ; and e' 
 i' the intersection of the same joint with the cylindrical joint 
 f x I of the radiant arch. 
 
 Fig. 4, showing the true dimensions of the upper plane 
 joint, is constructed by a like series of operations. 
 
 Fig. 5 is the development of the cylindrical joint of the 
 radiant arch projected in/", I; and of the cylindrical surface 
 of the groin stone of the annular arch projected in z H. 
 
 The projections of the conical joints of the annular arch 
 are easily found, by developing the cones to which they be- 
 long. 
 
 Fig. 6 is the development of the end surface of the exterior 
 voussnir of the radiant arch which joins the groin voussoir 
 considered. 
 
 These arches rest, as in the cylindrical groined arch, on 
 pillars. The tops of these pillars, on a level with the spring- 
 ing lines of the arches, are shown in the trapezoids B B' a' a, 
 CC'b' b, &c. (Fig. 1.) 
 
 The dimensions of the block for the groin stone in ques-
 
 60 
 
 STONE CUTTING. 
 
 tion area 1 ' given in the projections, sections, &c, of Figs. 1, 
 2, 3, 4, 6, and the developments of the conical joints. With 
 „hese elements, the bounding lines can be marked out on the 
 block, and the voussoir be worked off, by methods similar to 
 those pointed out in the two preceding problems.
 
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