m LIBRARY OF HENRY C. FALI — and KATHARINE A. FALL Plumber l^ 8 D^*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 " (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 , 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 : 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, 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', "i O.Rmnfft. Fiff.l r r Pl.l. O JtoLnSK PI. 2 H*fi%4if 1-H — ,^_j — i J-N ffi ; i ♦itimm/ nm»ninm.nj-: - * 7 lfM!imn,7"'?.''!"ffi'K ...3 p llll«ilHIIIIII>i/ri.iiiiiiiii.i,iin.i,,r.,..ii,.,1 RanM V. ■•--.. \ \ \\ \ \ i 1 |C' ! D'i 1 r PI.3. O.Hanm, l.>. Fiji. Fig. 5. ~~"T~^» C' D r a O.San r 2T; [li .-fj'S\ ^-' V ^i^JjXL ---J.Bt-T** CLTiJfc. * <■_ $ e ! ^ /i . _J J? !-- ft] u 7H~" "7 - r* f*> t K «J > 4' >"' i i -AUr ,^i />-/?^^^ A / <&■= / ! \ \ i ! PI -p. kU^' %3 - ■ University of California SOUTHERN REGIONAL LIBRARY ^LITY 405 Hilgard Avenue, Los Angeles ; CA90024 1388 Return this material to the library from which it was borrowed. ~T ■*•'* I IT Date: Thu, 19 Sep 91 12:47 PDT To ECL4BAT Subject: SRLF PAGING REQUEST Deliver to Shelving # UCSD CENTRAL A 000 085 639 3 Item Information Mahan, D. H. (Dennis Hart^ Descriptive geometry* as a Item ORION # : 2817y*t?7MC 1802-187 lied to Requester Inform Unit : UNK Terminal : Jser Information Name : hineline* m. Lib card : g» h ist Phone : Address : 0175-u «.« }r ~H m£i&'- 1 1 : A, • ■_ ^^B